Proceedings of the 15th Annual Congress

of the Association for Mathematics Education

of South Africa (AMESA)

“Mathematical Knowledge for Teaching”

29 June – 3 July 2009 University of the Free State

Bloemfontein

Editors: JH Meyer & A van Biljon

VOLUME 1

Copyright © reserved Association for Mathematics Education of South Africa (AMESA) P.O. Box 54, Wits, 2050, Johannesburg 15th Annual AMESA National Congress, 29 June – 3 July 2009, Bloemfontein, Free State. Volume 1 All rights reserved. No production, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be produced, copied or transmitted, except with written permission or in accordance with the Copyright Act (1956) (as amended). Any person who does any authorised act in relation to this publication may be liable for criminal prosecution and civil claim for damages. First published: July 2009 Published by AMESA ISBN: 978-0-620-44225-1 Printed by: BYTES DOCUMENT SOLUTIONS (Xerox Authorised Distributer)

Foreword “Mathematical Knowledge for Teaching” is, of course, non-negotiable in any reputable education system. How can you teach mathematics without the necessary knowledge? More important questions are: What kind of mathematical perspective does your knowledge create in your mind? Is your knowledge sufficient enough to react sensibly when you have to deal with the inquisitive minds amongst your learners? Are you prepared to broaden your knowledge on a continual basis, until the day you retire (or die!), or do you just stagnate to the narrow avenues of syllabus-knowledge, sufficient for your learners to obtain a pass mark in the examination? Most importantly: How does your knowledge inspire you to inspire your learners? It is hoped that this conference will, in the least, create fresh viewpoints on and an awakening of mathematical knowledge - that knowledge exists not only to pass on, but to stimulate the creation of further knowledge. There are a total of 82 papers to be presented at this congress: 14 long papers, 9 short papers, 21 “How I Teach” papers, 37 workshops (pre-conference workshops included) and one poster. In addition, there will be five plenary sessions, two panel discussions, a keynote address and several interest group discussions. There will also be presentations by exhibitors in the maths market and activity centre. We are also pleased to see a number of presentations on mathematics literacy. This certainly adds to the broadening of the spectrum of mathematical knowledge. Another encouraging fact is the substantial number of presentations by teachers from the maths4stats project, funded by Stats SA. It is delightful to see an increasing interest in involvement in our mathematics education community. It is hoped that this trend will continue. The reviewing process certainly had its hiccups. Some of the reviewers had to be reminded several times to send feedback. Some of the authors offered the same kind of difficulty – to send the final versions of their manuscripts in time. Each of the long and the short papers was sent to three reviewers. As a general rule, papers with at least two recommendations for acceptance were selected. Many authors were requested to make modifications to their papers, as requested by the reviewers, before they could be accepted for publication in the proceedings and be included in the congress programme. Each of the “How I teach” papers, workshops and posters was reviewed by at least one person and most of them were accepted (after some rework). We trust that the general standard improved due to this process. Let me express my sincere thanks to all those reviewers and authors who reacted swiftly and who took the deadlines seriously. Johan Meyer Academic Programme Director

Acknowledgement Each paper submitted to the congress was sent to n reviewers, where }3,2,1{∈n , for blind reviewing. We hope that the contributors found the comments and suggestions useful and we trust that this process helped to improve the quality of the papers. Many thanks to our reviewer corps who reviewed the papers in a constructive and helpful spirit: Hennie Boshoff Lorraine Botha Laurie Butgereit Michael de Villiers Gawie du Toit Stephan du Toit Johann Engelbrecht Faaiz Gierdien Nico Govender Belinda Huntley Paul Laridon Caroline Long Solomon Mabena Themba Mthethwa Vimolan Mudaly Nirendran Naidoo Hercules Nieuwoudt Marc North Alwyn Olivier Craig Pournara Marc Schafer Gerrit Stols Anelize van Biljon Nelis Vermeulen

Contents Long Papers Author Title Page

De Villiers, Michael From the Fermat point to the De Villiers points of a triangle

1

Du Toit, Stephan The use of metacognitive strategies in the teaching and learning of mathematics

9

Essien, Anthony A. From context to concept? An analysis of the introduction of the equal sign in three grade 1 textbooks 

22

Gierdien, Faaiz & Olivier, Alwyn

When pre-service teachers learn to function using spreadsheet-based algebraic approaches

33

Jaffer, Shaheeda Breaking up and making up: a feature of school mathematics pedagogy

45

Long, Caroline From whole numbers to real numbers: applying Rasch measurement to investigate conceptual complexity in Key Concepts

57

Mabizela, Mdumiseni G. Learners fail mathematics: an argumentative essay on contributing factors

69

MacKay, Roger Remarks on the inefficient use of time in the teaching and learning of Mathematics in five secondary schools

79

Matoti, Sheila & Junquira, Karen

Assessing the academic behavioural confidence (abc) of first-year students at the Central University of Technology, Free State

86

Mthethwa, Themba M. An analysis of Mathematical Literacy curriculum documents: cohesions, deviations and worries

103

Pournara, Craig Two approaches to learning the Mathematics of annuities

114

Roberts, Anthea Impact of language on the constitution of Mathematics in pedagogic contexts: a case drawn from a research and development project

124

Siyepu, Sibawu W. The zone of proximal development in the learning of differential calculus

136

Southwood, Sue Plenty of Pythagoras proofs 146

Short Papers Author

Title Page

Cameron, Bridget Reflections of South African Teachers on teaching Math in the USA

151

Du Toit, Gawie Effective learning of algebra at school 155

Faleye, Sunday & Mogari, David

The effect of habitual use of calculators on the arithmetic proficiency of first year university students

162

Faleye, Sunday & Mogari, David

A reflection on the teaching of Fluid Mechanics in some South African universities

173

Jaca, Prince S. An explanatory framework – ‘Speedometer’ 179

Mabotja, Tlou R. Learners thinking and reasoning about the concepts area and perimeter of two-dimensional shapes

185

Miranda, Helena Mathematics Teacher Professional Development: A reflection

196

Nyaumwe, Lovemore J. Primary school teachers’ mathematical content knowledge on division of proper fractions: Some theoretical illustrations

204

Van Biljon, Anelize Quartiles and percentiles: which formula? 210

Plenaries

Author

Title Page

Adler, Jill Mathematics for teaching matters 217

Meyer, Johan Thinking outside the box 234

Setati, Mamokgethi & Duma , Bheki

When language is transparent: supporting Mathematics learning multilingual contexts

235

Stylianides, Andreas J. Towards a more comprehensive “knowledge package” for teaching proof

242

1

From the Fermat point to the De Villiers points of a triangle Michael de Villiers

University of KwaZulu-Natal profmd@mweb.co.za

In August 2008, I accidentally found out to my great surprise that two special points of a

triangle have been named the De Villiers points after me at the WolframMathWorld

(Weisstein, no date) and that they are also referenced as Points 1127 and 1128 at the

Encyclopedia of Triangle Centers (Kimberling, no date). But it was also immediately

humbling (and bemusing) to note that there are more than 3500 special points known in

relation to the simple, elementary triangle, so these are only two amongst thousands!

Be that as it may, the purpose of this paper is to provide a brief background

leading up to my discovery of these points, and the proofs involved, which should be

accessible and informative for talented mathematics learners and their teachers.

Let’s start by considering the following Sketchpad

investigation from De Villiers (2003b).

Airport Problem

Suppose an airport is planned to service three cities of more or less equal size. The planners decide to locate the airport so that the sum of the distances to the three cities is a minimum. Where should the airport be located?

Solution

Rotate triangle ADC by - 60° around point C to get

triangle A'D'C. From the rotation, it follows that CD

= CD', and since angle D'CD measures 60°, it

follows that triangle DCD' is equilateral. Since AD =

A'D' from the rotation, we now have AD + CD + BD

A B

C

D

DC = 2.006 cm

DB = 1.663 cm

DA = 2.653 cm

DC + DB + DA = 6.321 cm

D

C

BA

A ’

D ’

2

= A'D' + D'D + DB. But the path from A' to B (e.g. A'D'

+ D'D + DB) will be a minimum when it is straight, in

which case, angle A'D'C = 120°, and therefore angle

ADC = 120°. From symmetry it follows that the other

two angles around D will also be equal to 120°. Thus,

the solution of the problem is to place the airport where

these three angles around D all equal 120°. It is now not

hard to see that D can be located simply by constructing

equilateral triangles A'AC, B'BA and C'CB on the sides

of triangle ABC (see the diagram at right) and constructing the straight lines A'B, B'C and

C'A to meet at D.

Figure 1: Fermat and Torricelli

Historical Notes

The point D is usually called the inner Fermat point1 of a triangle after Pierre de Fermat

who first posed the problem in the 1600s of finding a point inside an acute triangle so that

the sum of the distances to the vertices is a minimum. However, more correctly, it should

probably be called the Fermat-Torricelli point as the Italian mathematician and scientist

Evangelista Torricelli was the first to solve the problem and propose constructing

equilateral triangles on the sides to locate the optimal point. Of some cultural-historical

interest is that the Italian and French mathematical communities are apparently still

arguing about who the point should be named after! The transformation proof given

above was more recently invented in 1929 by the German mathematician J. Hoffman.

The centroid of a triangle 1 The outer Fermat point is obtained by constructing the equilateral triangles inwardly, then similarly drawing the concurrent lines A’B, B’C and C’A.

A B

C

D

D ’

A ’

C ’

D ’

B ’

D ’

3

The following fundamental geometry result still appears in some high school geometry texts, but unfortunately mostly without proof: “The three medians of any triangle are concurrent at the centroid.” Let’s consider a non-traditional proof based on areas, but that will give us further insight leading to an interesting, and important generalization.

C

B

A

ED

G

F Figure 1

Proof 1. Let AE and CD be medians intersecting at point G as shown in Figure 1. Join B

with G and extend to F on AC. We now have to show that F is the midpoint of AC. (In other words, that BF is also a median and therefore that all three meet in the same point G.)

2. If we denote the area of a triangle by the following notation, area

!

"ABC

!

"(ABC), we have:

!

(BAF)

(BFC)=

12h1AF

12h1FC

=AF

FC and

!

(GAF)

(GFC)=

12h2AF

12h2FC

=AF

FC.

Therefore:

!

AF

FC=(BAF)

(BFC)=(GAF)

(GFC)=(BAF) " (GAF)

(BFC) " (GFC)=(BAG)

(BCG) ... dividendo.

Similarly, we find:

!

CE

EB=(ACG)

(BAG) and

!

BD

DA=(BCG)

(ACG).

3. But it is given that BE = EC and BD = DA. Therefore, (BCG) = (ACG) and (ACG) = (BAG) which implies (BAG) = (BCG). But the areas of these two triangles are proportional to AF and FC as shown by the second equation. Thus,

!

AFFC = 1

implies AF = FC and completes the proof.

Looking back: Ceva’s Theorem Now look back carefully at the proof. Only consider the product of the three ratios

!

AFBC ,

!

CDDB and

!

BEEA expressed in terms of areas in Step 2. What do you notice about this

4

product, and whether in deriving these three ratios, the properties that E and D are midpoints were used at all? What can we therefore conclude from this? Note that:

!

AF

FC"CD

DB"BE

EA=(BAG)

(BCG)#(ACG)

(BAG)#(BCG)

(ACG)=1. More over, the properties that E and D

are midpoints were not used at all in this derivation! Therefore we can immediately generalize, e.g. if in any triangle, line segments AD, BF and CE are concurrent (with D, F

and E respectively on sides BC, AC and AB) then

!

AF

FC"CD

DB"BE

EA=1. The converse of

this result is also true, and can be proved by using proof by contradiction. Pedagogically, the above example beautifully illustrates the discovery function of proof as mentioned in De Villiers (2003a), whereby sometimes proving a result and reflecting on the proof carefully, can lead to a further generalization. This interesting, major result is called Ceva's Theorem after an Italian mathematician

named Giovanni Ceva (1648-1734) who published his theorem in 1678 and proving it by

considering centers of gravity and the law of moments. In his honour the line segments

AE, BF and CD joining the vertices of a triangle to any given points on the opposite

sides, are called cevians. (Note that apart from the medians, the altitudes and angle

bisectors of a triangle can be considered as cevians if extended to meet the opposite

sides). The converse of Ceva’s theorem is a powerful theorem for proving various

concurrencies of lines, and all learners preparing for the 3rd round of the South African

Mathematics Olympiad should know it.

Generalizing the Fermat-Torricelli point

The Fermat-Torricelli point can be generalized further by congruent, similar isosceles or

similar triangles on the sides, but all are special cases of the following unifying

generalization from De Villiers (1995):

"If triangles DBA, ECB and FAC are constructed outwardly (or inwardly) on the sides of any ∆ABC so that

!

"DAB ="CAF ,

!

"DBA ="CBE and

!

"ECB ="ACF then DC, EA and FB are concurrent."

5

In order to prove this result we will use the following lemma, which is stated here without proof.

Figure 2

Lemma Triangle ABC is given. Extend AB and AC to D and E respectively so that DE//BC. Choose any point Y on BC and extend AY to X on DE (see Figure 2). Then BY/YC = DX/XE.

Proof of the Fermat generalization Assume that the lines we want to prove concurrent intersect BC, CA and AB respectively at X, Y and Z. Extend AB to G and AC to H so that GEH//BC (see Figure 3). Label BE, EC, CF, FA, AD and DB respectively as

!

s1,s2,s3,s4,s5 and

!

s6. Then

!

"BGE ="ABC and

!

"BEG = b.

Figure 3

According to the sine rule:

A

B C

D EX

Y

6

!

GE

sin "GBE( )=

s1

sin "ABC( )

GE

sin b +"ABC( )=

s1

sin "ABC( )

GE =s1sin b +"ABC( )sin "ABC( )

Similarly we obtain

!

EH =s2sin c +"ACB( )sin "ACB( )

.

According to the preceding Lemma therefore

!

BX

XC=GE

EH=s1sin b +"ABC( )sin "ABC( )

#sin "ACB( )

s2sin c +"ACB( )

.

In the same way we have

!

CY

YA=s3sin c +"ACB( )sin "ACB( )

#sin "CAB( )

s4sin a +"CAB( )

AZ

ZB=s5sin a +"CAB( )sin "CAB( )

#sin "ABC( )

s6sin b +"ABC( )

Therefore,

!

BX

XC"CY

YA"AZ

ZB=s1

s2

"s3

s4

"s5

s6

... (3)

Applying the sine rule to triangles ECB, FAC and DBA we obtain

!

s1

s2

=sin c( )sin b( )

;s3

s4

=sin a( )sin c( )

;s5

s6

=sin b( )sin a( )

By substitution into (3) therefore

!

BX

XC"CY

YA"AZ

ZB=1 so that AX, BY and CZ are concurrent

according to the converse of Ceva's theorem. But then EA, FB and DC are also concurrent.

The generalization is not new, and the earliest proof I’m aware of is from 1936 by N. Alliston in The Mathematical Snack Bar by W. Hoffer, pp. 13-14. Of practical relevance is that the Fermat-Torricelli generalization can be used to solve a ‘weighted’ airport problem, for example, when the populations in the three cities are of different size. I was

7

also a few months ago contacted by a mathematical biologist from the USA who was looking at its application in the branching of larger arteries and veins in the human body into smaller and smaller ones.

The De Villiers points of a triangle On the basis of an often-observed (but not generally true) duality between circumcentres and incentres, I conjectured in De Villiers (1996) that the following might be true from a similar result for circumcentres (Kosnita’s theorem), namely:

The lines joining the vertices A, B, and C of a given triangle ABC with the incentres of the triangles BCO, CAO, and ABO (O is the incentre of

!

"ABC), respectively, are concurrent (in what is now called the inner De Villiers point).

Investigation on the dynamic geometry program Sketchpad quickly confirmed that the conjecture was indeed true. (For an interactive sketch online, see De Villiers, 2009). Using the aforementioned generalization of the Fermat-Torricelli point, it was now very easy to prove this result.

Figure 4

Proof As shown in Figure 4 we have that

!

"DAB = 1

4"A ="CAF ,

!

"DBA = 1

4"B ="CBE and

!

"ECB = 1

4"C ="ACF , and from the Fermat-Torricelli generalization it therefore

follows that DC, EA and FB are concurrent. The outer De Villiers point is obtained when the excircles are constructed as shown in Figure 5, in which case the lines joining the vertices A, B, and C of a given triangle ABC

CB

A

O

F

E

D

8

with the incentres of the triangles BCI1, CAI2, and ABI3 (Ii are the excentres of

!

"ABC), are concurrent. The proof follows similarly from the Fermat-Torricelli generalization.

Figure 5

Concluding comment

Unfortunately so far I’ve been unable to find any additional, interesting properties of the

De Villiers points, but hope that I or someone else may do so in the not too distant future.

References

De Villiers, M. (1995). A generalization of the Fermat-Torricelli point. The

Mathematical Gazette, July, pp. 374-378.

De Villiers, M. (1996). A dual to Kosnita’s theorem. Mathematics & Informatics

Quarterly, 6(3), Sept, pp. 169-171.

De Villiers, M. (2003a). The role of proof in Sketchpad. Rethinking Proof with

Sketchpad. Emeryville: Key Curriculum Press, pp. 5-10.

De Villiers, M. (2003b). Airport Problem. Rethinking Proof with Sketchpad.

Emeryville: Key Curriculum Press, pp. 115-118.

De Villiers, M. (2009). http://math.kennesaw.edu/~mdevilli/devillierspoints.html

Kimberling, C. (no date). Encyclopedia of Triangle Centers. Available online:

http://faculty.evansville.edu/ck6/encyclopedia/ETC.html

Weisstein, E. W. (no date). De Villiers Points. Available online:

http://mathworld.wolfram.com/deVilliersPoints.html

9

THE USE OF METACOGNITIVE STRATEGIES IN THE TEACHING AND LEARNING OF MATHEMATICS

Stephan du Toit

University of the Free State The broad aim of this study was to investigate the use of metacognitive strategies by grade 11 mathematics learners and grade 11 mathematics teachers. Two objectives were stated: To investigate which metacognitive strategies grade 11 mathematics learners and mathematics teachers can employ to enhance metacognition among learners; and to investigate the extent to which grade 11 mathematics learners and mathematics teachers use metacognitive strategies . Questionnaires were used to obtain quantitative data about the use of metacognitive strategies by learners and teachers. The findings indicate that planning strategy and evaluating the way of thinking and acting were used to the greatest extent by both the teachers and the learners. Journal-keeping and thinking aloud were used to the least extent by teachers and learners.

Introduction and background The purpose of teaching mathematics is to empower learners to “make sense of society” (NDE, 2003: 9). Various stakeholders in society, for example parents, employers and tertiary institutions, exert pressure on mathematics education because mathematical competence “contributes to personal, social, scientific and economic development” (NDE, 2003: 9).

South African learners do not perform very well in mathematics. The aim of the Department of Education was for 50 000 learners to pass mathematics with more than 50% in the 2008 NSC (NDoE, 2008: 12; Naude, 2007: 17). This aim was achieved, a total of 63 038 learners scored above 50% in the 2008 National Senior Sertificate (NCS) mathematics examination. When the total numbers of learners are considered that wrote mathematics, a more distressing picture emerges. A total of 270 097 learners wrote mathematics in 2008, therefore only 23,34% of those learners achieved more than 50% in the examination (NDE, 2008: 10, 12). On international level, an even worse scenario emerges. South Africa’s grade 8 learners scored the lowest of 46 countries with a score of 264 in the 2003 Trends in Mathematics and Science Study (TIMSS), 11 points lower than in 1999 (TIMSS, 2003: 5, 7). South Africa did not participate in the 2007 TIMSS (TIMSS, 2008: 2).

How could learners’ mathematical competence and performance be improved? Campione (1987: 136) observes that knowledge about a domain, specific procedures for operating in that domain, and general task-independent regulatory processes are three prerequisites for effective performance within some domain. De Corte adds affective components as another prerequisite (1996: 34-36) by stating that expert performance in a given domain necessitates the integrated acquirement of the following four categories of aptitude, namely a structured, accessible domain-specific knowledge base; heuristic methods; affective components; and metacognition.

10

Defining metacognition Papaleontiou-Louca states that, in the field of cognitive developmental research, metacognition has become a foremost topic since 1973 (Papaleontiou-Louca, 2003: 9). In this regard, Schoenfeld (1992: 9), describes “metacognition” as a term that was coined in the 1970s and only occasionally appearing in the literature of the early 1980s, but appearing with growing frequency through the decade, becoming (with problem-solving) probably the most clichéd and least understood buzz words of the 1980s.

Definitions of metacognition vary, Schoenfeld (1992: 2, 38, 39), for example, asserts that “metacognition has multiple and almost disjoint meanings (for example, knowledge about one’s thought processes, self-regulation during problem-solving) which make it difficult to use as a concept”. Hacker (1998: 11) states that there is general agreement that the definition of metacognition should at least include the following aspects, namely knowledge of one’s knowledge; the conscious monitoring and regulating of one’s knowledge; and cognitive and affective states. Metacognition is the knowledge and beliefs about cognition, in addition to the skills and strategies enabling the self-regulation of cognitive processes (De Corte, 1996: 35, 36), while Papaleontiou–Louca (2003:12) defines metacognition as “…all processes about cognition, such as sensing something about one’s own thinking, thinking about one’s thinking and responding to one’s own thinking by monitoring and regulating it”. These various definitions of metacognition have in common the emphasis on the knowledge of cognition and the monitoring and regulation of cognitive processes. The summaries of the different aspects facets of metacognition by Hacker (1998: 11) and Schoenfeld (1992: 38, 39) contain an additional reference to the awareness and regulating of one’s affective state.

Metacognition and academic performance According to Schraw (1998: 114), performance is improved by metacognitive regulation as learners utilize resources and existing strategies better. The claim that cognitive monitoring enhances learning is supported by Paris and Winograd (1990: 15) where they argue that “students can enhance their learning by becoming aware of their own thinking as they read, write and solve problems at school”.

A study conducted by Camahalan (2006: 194) found that students’ academic achievement is more likely to improve when they are given the chance to self-regulate and explicitly taught metacognitive learning strategies. Metacognitive strategies are one category of metacognition; the other three categories, according to Flavell (1979: 906) are metacognitive knowledge, metacognitive experience, and metacognitive goals. Butler and Winne (1995: 245) assert that there is agreement among theoreticians that the most effective learners are self-regulating. Boekaerts and Simons (1995: 85) view self-regulation as synonymous to metacognitive strategies.

Metacognitive strategies Metacognitive strategies refer to the conscious monitoring of one’s cognitive strategies to achieve specific goals, for example when learners ask themselves questions about the work and then observe how well they answer these questions (Flavell, 1981: 273). Boekaerts and Simons (1995: 91) view metacognitive strategies as the decisions learners make before, during and after the process of learning. There are various metacognitive strategies aimed at developing learners’ metacognition (Costa, 1984: 59-61; Blakey & Spence, 1990: 2, 3).

11

Planning strategy At the start of a learning activity, teachers should make learners aware of strategies, rules and steps in problem-solving. Time restrictions, goals and ground rules connected to the learning activity should be made explicit and internalised by the learners. Consequently, learners will keep them in mind during the learning activity and assess their performance against them. During the learning activity, teachers can encourage learners to share their progress, their cognitive procedures and their views of their conduct. As a result, learners will become aware of their own behaviour and teachers will be able to identify problem areas in the learners’ thinking (Costa, 1984: 59). When learning is planned by someone else, it is difficult for learners to become self-directed (Blakey & Spence, 1990: 3).

Generating questions Ratner (1991: 32) views the questioning of given information and assumptions as a vital aspect of intelligence. Learners should pose questions for themselves before and during the reading of learning material. Learners will pause regularly to determine whether they understand the concept; if they can link it with prior knowledge; if other examples can be given; and if they can relate the main concept to other concepts. Here Muijs and Reynolds (2005: 63) argue that the connection of prior knowledge and new concepts should take place during the lesson and not only when a new concept is introduced. This integration of prior knowledge and new concepts enables the learner to understand the unified and interconnected nature of knowledge, while also facilitating profound understanding of subject matter (Ornstein & Hunkins, 1998: 240). Integration adheres to the second of the principles for quality mathematics education (NCTMP), stated by the National Council of Teachers of Mathematics (NCTM), (NCTM, 2000: 2), namely a coherent curriculum in which students’ mathematical concepts are linked and build on one another. In support, Blakey and Spence (1990: 2) state that learners should ask themselves what they know and what they do not know at the beginning of a research activity. As the research activity progresses, their initial statements about their knowledge of the research activity will be verified, clarified and expanded.

Choosing consciously Teachers should guide learners to explore the results of their choices before and during the decision process. Therefore, learners will be able to recognize underlying relationships between their decisions, their actions and the results of their decisions. Non-judgemental feedback to learners about the consequences of their actions and choices promotes self-awareness (Costa, 1984: 60), and it enables the learners to learn from their mistakes, thereby supporting the fourth principle of the NCTMP of “learning… understanding, actively building new knowledge from experience…” (NCTM, 2000: 2).

Setting and pursuing goals Artzt and Armour-Thomas (1998: 9) define goals as “expectations about the intellectual, social and emotional outcomes for students as a consequence of their classroom experiences”; these goals support the first principle of the NCTMP of high expectations and support for learner. Learners who are self-regulating strive to attain a self-formulated goal. Self-regulated behaviour can be adapted with changing circumstances (Diaz, Neal & Amaya-Williams, 1990: 130).

Evaluating the way of thinking and acting Metacognition can be enhanced if teachers guide learners to evaluate the learning activity according to at least two sets of criteria (Costa, 1984: 60). Initially, evaluative criteria could be jointly developed with the

12

learners to support them in assessing their own thinking. As an example, learners could be asked to assess the learning activity by stating helpful and hindering aspects and their likes and dislikes of the learning activity. Accordingly, learners keep the criteria in mind when classifying their opinions about the learning activity and they motivate the reasons for those opinions (Costa, 1984: 60). Guided self-evaluation can be introduced by checklists focusing on thinking processes and self-evaluation will increasingly be applied more independently (Blakey & Spence, 1990: 3).

Identifying the difficulty Costa (1984: 60) advises teachers to discourage the use of phrases like “I can’t”; “I am too slow to…”; or “I don’t know how to…”. Rather, learners should identify the resources, skills and information required to attain the learning outcome. As a result, learners are assisted to distinguish between their current knowledge and the knowledge they need. They also have more resolve in seeking the right strategy for solving the problem.

Paraphrasing, elaborating and reflecting learners’ ideas Teachers should support learners to restate, translate, compare and paraphrase other learners’ ideas. Consequently, learners will be better listeners to other learners’ thinking and also to their own thinking (Costa, 1984: 61). The teacher can, for example, ask: “What you are explaining to us is…”; “I understand that you are suggesting the following…”.

Carpenter and Lehrer (1999: 22) assert that the ability to articulate one’s ideas requires profound understanding of significant aspects and concepts. They view the ability to reflect as a prerequisite for articulation in that articulation requires the identification of the essence and critical elements of an activity.

Clarifying learners’ terminology Learners regularly use vague terminology when making value judgements, for example “The question is not fair” or “The question is too difficult”. Teachers should elucidate these value judgments, for example “Why is the question not fair?” or “Why is the question too difficult?” (Costa, 1984: 61).

Problem-solving activities In problem-solving, existing knowledge is applied to an unfamiliar situation to gain new knowledge (Killen, 2000: 129). Problem-solving activities are ideal opportunities to enhance metacognitive strategies, as good problem-solvers are generally self-aware thinkers. Learners with superior metacognitive abilities are better problem-solvers. The ability to analyze their problem-solving strategies and reflect on their thinking reveals the learners’ metacognitive skills (Blakey & Spence, 1990: 2; Panaoura, Philippou & Christou, 2001: 3).

After the problem-solving process, teachers should encourage learners to clarify their course of action, instead of merely correcting the learner (Costa, 1984: 61). Goos and Galbraith (1996: 231) state that non-cognitive aspects, like learners debilitating beliefs about the nature of mathematics and about themselves, could have a positive or negative effect on cognitive and metacognitive processes involved in problem-solving.

When the whole class works on a problem, the teacher, instead of steering the learners to the answer, helps the learners to take full advantage of those aspects that they have produced. During this process of guiding the learners, the teacher will ask questions like: “Are you all convinced that you understand the problem?”;

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and “Which of the suggestions to solve the problem should we attempt first, and why?” After the class has worked on the problem for about five minutes, the teacher could ask them whether the process is going well, and if not, to reassess the strategy. If the class decides to reject that strategy, the teacher could ask whether anything helpful could be recovered from their effort. When a solution is reached, the teacher reviews the whole problem-solving process and indicates where the class went wrong initially. The teachers also lead the class in finding alternative solutions to the problem (Schoenfeld, 1987: 202). In this regard, Muijs and Reynolds (2005: 64) list reflection as one of the elements of constructivist teaching strategies. They describe reflection, a key learning moment, as the comparing of solutions between learners. They also regard reflection as the process learners engage in when they think about problem-solving strategies and their effectiveness.

Schoenfeld (1987: 202) considers whole class problem-solving as promoting self-regulation, because the teacher’s role as a moderator compels the learners to focus on control decisions made by themselves, and not by the teacher. Another aspect of whole class problem-solving that Schoenfeld (1987: 202) discusses is the opportunity it affords to pose problems that evoke beliefs about mathematics. An example is mentioned of the belief that problems can be solved relatively quickly if the subject matter is well understood. To challenge this belief, a problem is assigned that would probably take the class a few days, or even weeks, to solve.

Schoenfeld’s (1987: 206) aim with small group problem-solving is to provide the learners with a range of problem-solving strategies (heuristics), and then to train them to use those strategies effectively. When learners are only taught about heuristics and then have to work on problems at home, the teacher is not present in the midst of problem-solving when his input could have promoted the use of self-regulation skills, for example, the teacher informs the learners that they are going to be asked the following three questions whenever they work on a problem: “What exactly are you doing?”; “Why are you doing it?”; and “How does it help you?” Gradually, it becomes a matter of practice for the learners to start asking the questions themselves, thereby improving their problem-solving skills and operation on a metacognitive level.

Thinking aloud Teachers should promote the habit of thinking aloud when learners solve problems (Costa, 1984: 61). Talking about their thinking will help learners to identify their thinking skills (Blakey & Spence, 1990: 2).

Muijs and Reynolds (2005:64) use the term “articulation” to describe learners’ expression of their thoughts and ideas. They recommend that learners should discuss complex tasks and present their ideas to fellow learners. They furthermore suggest that group work could be very effective in promoting articulation. In this regard, Blakey and Spence (1990: 2) mention paired problem solving, where one learner describes his thinking processes while his partner helps him to clarify his thinking by listening and asking questions.

A main aspect of Vygotsky’s developmental theory is that children start using language not to only communicate, but also to regulate their activities by guiding, planning and monitoring (Diaz et al., 1990: 135). Three consequences for self-regulation through the use of language can be identified. First, children organize and restructure their perceptions in terms of their goals. Second, children’s actions are less impulsive as they allow them to act reflectively according to their goals. Finally, language not only enables children to regulate their way of perceiving stimuli, but also to regulate their behaviour (Diaz et al., 1990: 135, 136). Camp, Blom, Hebert and van Doornick, (1977: 160) developed a program called Think Aloud to improve self-control. Children are taught to use the following four questions when solving problems: “What is my problem?”; “How can I do it?”; Am I using my plan?”; and “How did I do?”

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Journal-keeping Keeping a personal diary throughout a learning experience facilitates the creation and expression of thoughts and actions. Learners make notes of ambiguities, inconsistencies, mistakes, insights, and ways to correct their mistakes. Preliminary insights can be compared with changes in those insights as more information is gathered or obtained through feedback from assessment, thereby supporting the fifth principle of the NCTMP, namely, that assessment should support the learning of mathematics (Costa, 1984: 61; Blakey & Spence, 1990: 3; NCTM, 2000: 2).

Cooperative learning Cooperative learning creates the opportunity for learners to work together in small groups to enhance learning. It entails more than group work, as group work is considered as a modification of whole-class discussion. In cooperative learning, the teacher gives indirect guidance as the group works together to achieve specific learning outcomes (Killen, 2000: 73). Cooperative learning may promote awareness of learners’ personal thinking and of others’ thinking. When learners act as “tutors”, the process of planning what they are going to teach lead to independent learning and clarifying the learning material (Blakey & Spence, 1990: 2).

Modelling The NCTM lists effective teaching as a third principle of the NCTMP (NCTM: 2000: 2). Modelling occurs when teachers demonstrate the processes involved in performing a difficult task, or when teachers tell the learners about their thinking and the motivation for selecting certain strategies when solving problems (Muijs & Reynolds, 2005: 63). Modelling and discussion enhance learners’ thinking and talking about their own thinking (Blakey & Spence, 1990: 2). Schoenfeld (1987: 200) refers to the importance for teachers of not always presenting the finished, neat presentation of the answers on the board, but to sometimes model the problems and working through the problem step by step. Consequently, the processes yielding the correct answer (for example false starts, recoveries from false starts and interesting insights) are exposed and the chief purpose of the modelling approach is achieved, namely the centring of learners’ awareness on metacognitive behaviours.

Costa (1984: 61) suggests that modelling could be the most effective strategy used to enhance metacognition among learners because they learn best by imitating adults. Teachers will, by thinking aloud throughout planning and problem-solving activities, demonstrate their thinking processes. Teachers, therefore, have a great responsibility because “ a fair proportion of the learning problems in mathematics are actually taught to the children…” (Moodley, 1992: 8). Van der Walt and Maree (2007: 235) found that mathematics teachers employed question-posing strategies and think-aloud models, but that they did not sufficiently promote the implementation and practice of these strategies among learners.

Aspects that denote teachers’ modelling behaviour include explaining their planning, goals and objectives to the learners and motivating their actions; acknowledging their temporary inability to answer a question, but developing pathways for finding the answer; making human mistakes but demonstrating how to correct those mistakes; requesting comments and assessment of their actions; acting in accordance with an explicitly stated value system; the ability to explain what their strengths and weaknesses are; and expressing an understanding and valuing of learners’ ideas and feelings (Costa, 1984: 61). Regarding the expression of understanding and the valuing of learners’ ideas and feelings, Muijs and Reynolds (2005: 65) state that flexibility, an element of the constructivist teaching strategies, is the process whereby learners partly guide the progress of the lesson as teachers interact with learners.

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Vygotsky’s developmental theory proposes that the development of self-regulation originates and is enhanced by the teacher-learner social interactions (Diaz et al., 1990: 128). Diaz et al. (1990: 139) identify three characteristics of teacher-learner interactions that promote self-regulation, namely the use of reasoning and supplying reasons for commands; the gradual withdrawal of teacher control; and the combination of the previous two aspects in an atmosphere of emotional warmth and affective nurturance. De Abreu, Bishop and Pompeu (1997: 235) also stress the importance of affect in arguing that, although learners experience mathematics cognitively and affectively, they only have the opportunity to express the cognitive aspect.

Aims of the study The broader aim of the study was to investigate the use of metacognitive strategies by grade 11 mathematics learners and grade 11 mathematics teachers in the teaching and learning of mathematics in the Motheo district. The following research questions were formulated:

Which metacognitive strategies can grade 11 mathematics learners and their mathematics teachers employ to enhance learners’ metacognition?

To which extent do grade 11 mathematics learners use the identified metacognitive strategies?

To which extent do grade 11 mathematics use and encourage learners to use the identified metacognitive strategies?

Research Design

Form of inquiry Information gathered from a literature study provided an answer to the first research question, and survey research as a form of inquiry was used to collect the quantitative data required to answer the second and third research questions. A learner and a teacher questionnaire, based on the literature study, were constructed to determine the extent of the use of the metacognitive strategies in the teaching and learning of mathematics.

Questionnaire The learner questionnaire and the teacher questionnaire comprised 37 and 47 questions respectively that were based on the use of the metacognitive strategies. The learner questionnaire determined the extent to which learners use the metacognitive strategies, except modelling, in the learning of mathematics. The teacher questionnaire investigated the extent to which teachers use the metacognitive strategies in the teaching of mathematics, and encourage the use of the metacognitive strategies in the learning of mathematics. In both questionnaires, respondents could choose any of the following options on a Likert-scale: almost never, sometimes, usually, almost always. Table 1 reflects the correspondence between questionnaire items and the metacognitive strategies.

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Metacognitive strategy

Learner questionnaire item(s)

Teacher questionnaire item(s)

Planning strategy

1, 2

1, 2

Generating questions 3, 4, 5 3, 4, 5 Choosing consciously 6, 18 6, 18 Setting and pursuing goals 7, 27 7, 27 Evaluating the way of thinking and acting

8, 9, 10, 11, 28, 29, 30 8, 9, 10, 11, 28, 29, 30

Identifying the difficulty 12, 13, 14, 15, 16 12, 13, 14, 15, 16 Paraphrasing, elaborating and reflecting learners’ ideas

17 17

Clarifying terminology 15, 16 15, 16 Problem-solving activities 19, 31, 32, 33, 34, 35, 36, 37 19, 41, 42, 43, 44, 45, 46, 47 Thinking aloud 20 20 Journal-keeping 21, 22, 23 21, 22, 23 Cooperative learning 24, 25, 26 24, 25, 26, Modelling 27, 28, 29, 30, 31, 32, 33, 34, 35,

36, 37, 38, 39,40

Table 1 Correspondence between questionnaire items and metacognitive strategies

Reliability of the questionnaire The Cronbach Alpha procedure is regarded as the most suitable type of reliability for survey research where items are not scored right or wrong and where each item could have different answers (McMillan & Schumacher, 2001: 246, 247). The reliability scores of the learner questionnaire and teacher questionnaire were 0.88 and 0.95 respectively, indicating a high reliability on both questionnaires.

Piloting of the questionnaire Four grade 11 learners were asked to complete the pilot learner questionnaire and to note any ambiguous or vague questions. Two current grade 11 teachers and five former mathematics teachers completed the learner and the teacher pilot questionnaires.

Sampling This study focused on grade 11 mathematics teachers and grade 11 mathematics learners in the Motheo district. Of the five districts in the Free State, the Motheo district was the leading district regarding the pass percentage in the Senior Certificate Examination of 2006 (FsDoE, 2007a: 4).

In the mathematics Higher Grade (HG) Senior Certificate Examination of 2006 in the Motheo district, averages of the top 10 schools for the examination were between 69.75% and 60.13% (FSDoE, 2007b: s.n.) Only those schools with more than 20 learners who had written the Senior Certificate Examination in mathematics HG (five schools) were selected for the study. Table 2 contains information about the position of the five selected schools according to the 2006 Senior Certificate Examination (mathematics HG) results; the number of learners who wrote the examination; and each school’s average.

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School Position in district according to mathematics HG results

Number of learners who wrote mathematics HG

Average percentage obtained in mathematics HG

1 2 25 68.96 2 4 80 67.46 3 6 97 65.46 4 7 26 63.24 5

8 77 62.81

Table 2 Achievement in the 2006 Senior Certificate Examination (mathematics HG) In total, 394 learners and their teachers from 16 classes in five schools participated in the study. Thirteen teachers participated in this study; three teachers had two classes each. The respondents numbered 83% of the total number of learners (respondents and non-respondents).

Limitations of the study Since it was not one of the aims of the study to generalize the findings to the whole population, the findings have limited value. The following aspects are considered as limitations of this study: the use of two metacognitive strategies was determined by only one item on the teacher and learner questionnaire. Therefore, the reliability of those subscales could not be determined (see Table); and two items on the teacher and learner questionnaire were used to obtain information about the use of more than one metacognitive strategy (see Table 3.1).

Findings and discussion One of the research questions was to determine the extent to which each metacognitive strategy is used by teachers and learners. The precise means (X) and standard deviations (SD) of the extent to which teachers and learners use metacognitive strategies are indicated in Table 3.

Teachers Learners Strategy X SD X SD

Planning strategy 4.00 0.00 3.07 0.64 Generating questions 3.47 0.47 2.52 0.70 Choosing consciously 3.38 0.53 2.83 0.66 Setting and pursuing goals 3.41 0.56 2.44 0.69 Evaluating the way of thinking and acting 3.55 0.35 3.15 0.48 Identifying the difficulty 3.28 0.63 2.83 0.50 Paraphrasing, elaborating and reflecting learners’ ideas 3.16 0.83 2.53 0.97 Clarifying terminology 2.91 1.01 2.61 0.72 Problem-solving activities 3.24 0.52 2.68 0.53 Thinking aloud 2.73 1.15 2.28 1.02 Journal-keeping 2.78 0.97 2.38 0.79 Cooperative learning 3.23 0.70 2.24 0.61 Modelling 3.22 0.48 n/a Metacognitive total 3.28 0.43 2.72 0.38

Table 3 The extent to which metacognitive strategies are used by teachers and learners

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The metacognitive strategies that were used most often by the teachers were planning strategy (4.00); evaluating the way of thinking and acting (3.55); and setting and pursuing goals (3.41). The learners employed evaluating the way of thinking and acting (3.15); planning strategy (3.07); choosing consciously (2.83); and identifying the difficulty (2.83) to the greatest extent. Planning strategy and evaluating the way of thinking and acting were used to the greatest extent by both the teachers and the learners. This could indicate that teachers and learners were well organized and aware of their strengths and weaknesses in mathematics.

Metacognitive strategies used to the least extent by the teachers were thinking aloud (2.73); encouraging journal-keeping (2.78); and clarifying terminology (2.91). The learners employed cooperative learning (2.24); thinking aloud (2.28); and journal-keeping (2.38) to the least extent. Thinking aloud and journal-keeping were used to the least extent by both the teachers and the learners. This could imply that the keeping of a reflective journal is not encouraged among learners and that learners are not keeping a written record of mistakes they tend to make and insights they gain. When considering that learners use evaluating the way of thinking and acting to the greatest extent, it seems that learners can identify their strengths, weaknesses, mistakes and successes in mathematics, but they do not keep a written record of this self-knowledge. The fact that the learners used cooperative learning, which requires the articulation of one’s ideas, to the least extent, could explain why thinking aloud was used to the second least extent, as learners would be more inclined to verbally express their thoughts in a group setting than individually.

Each mean score for the extent to which a specific metacognitive strategy was used, was higher among the teachers. The teachers’ mean score (3.28) for the extent to which all the metacognitive strategies were used falls in the category “usually” to “almost always” on the 4-point Likert-scale, whereas the learners’ mean score (2.72) falls in the category “sometimes” to “usually”. Teachers used metacognitive strategies to a greater extent than the learners, as the teachers’ metacognitive total of 3.28, as compared to the learners’ metacognitive total of 2.72, indicates.

Conclusion and recommendations From the researcher’s experience of teaching, many mathematics learners do not like mathematics because they regard it as too difficult, and they cannot see the relevance of mathematics for their everyday lives or future lives. Learners also regularly enquire about effective study methods in mathematics. The use of metacognitive strategies could address these concerns as teachers, by valuing learners’ ideas and feelings (modelling), could assist in improving learners’ attitudes towards mathematics. Learner self-regulation could also be improved by the keeping of a reflective journal. By assigning real-life problems (problem solving activities), teachers have the opportunity to show the relevance of mathematics in learners’ everyday lives and future lives. The metacognitive strategies identified in this study could serve as a guide in ensuring effective teaching and assisting learners to study and learn mathematics effectively.

It is recommended that teachers and learners are assisted with the implementation of all the identified metacognitive strategies in the teaching and learning of mathematics, especially those that were used to the least extent by teachers and learners. Modelling and problem solving activities in a cooperative learning context are also regarded as focus areas. Further research could investigate the following aspects: the factors that play a role in the extent to which specific metacognitive strategies are used by teachers and learners; the reasons why certain metacognitive strategies are used to a greater or lesser extent by both the teachers and the learners; and the influence of the teacher-learner ratio, teaching experience, teaching qualifications, and allocated time of teaching on the use of the metacognitive strategies by the teachers.

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The relation between learner age, home language, language of instruction, gender, and race on the use of the metacognitive strategies by learners could be further researched.

In a speech delivered by Naledi Pandor, the Minister of Education (Pandor, 2008:1, 2), she stated that the government intends to launch an intensive teacher support programme for the improvement of teaching and learning. As the link between better academic performance and the use of the metacognitive strategies has been established by previous research, the researchers believe that teacher support programmes must include training in the use of the metacognitive strategies to ensure better teaching and learning of mathematics.

BIBLIOGRAPHY

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FROM CONTEXT TO CONCEPT? AN ANALYSIS OF THE INTRODUCTION OF THE EQUAL SIGN IN THREE GRADE 1

TEXTBOOKS

Anthony Anietie Essien

Marang Wits Centre for Maths and Science Education, University of the Witwatersrand

anthony.essien@wits.ac.za

This article is an attempt to analyse how the concept of the equal sign is introduced to learners in Grade One textbooks. In doing this, three grade 1 textbooks (learner’s book and their accompanying teacher’s guide) were analysed in the light the concepts of esoteric knowledge and realistic knowledge, and in the light of Bernstein’s concepts of recognition and realisation rules – concepts which focus on how context play a role in learners’ epistemological access to mathematics (by different groups). Analysis reveals that while both textbooks promote esoteric and realistic mathematical knowledge, both are structured in such a way that they do not enable learners to possess neither the realisations rules nor the recognition rules as far as the equal sign is concerned. The author makes an argument for the introduction of the equal sign first (using appropriate pictorial representations and artefacts) before the introduction of the plus and minus signs.

INTRODUCTION

That the equal sign is one of the most used, if not the most used, notation in mathematics is unequivocal. In virtually every branch of mathematics from geometry to algebra, the equal sign is a tool - a relational symbol - without which the learners’ mathematical explanation or solution would be meaningless. This article seeks to analyse three Grade One textbooks (both Learner’s book and Teacher’s guide) – Maths for all 1 (SDU, 2003), Classroom Mathematics Grade 1 (Jenkins et al, 2003) and Successful Numeracy Grade 1 (Chantler et al, 2008) in order to mark out the forms of mathematical knowledge they privilege and make available to both teachers and learners as far as the introduction of the equal sign is concerned. It must be, therefore, noted that the present analysis is not aimed at the comparative analysis of the three textbooks.

Grade One textbooks only were chosen because it is at this phase that the equal sign is introduced to learners. The three textbooks were chosen for a number of reasons: first, because they are books that are commonly used by learners and especially educators in Grade One; second, the books are written for the Outcomes Based-Education (OBE) – the framework for Revised National Curriculum Statement; third, the textbooks reflect the manner in which teachers generally introduce the concept of the equal sign as revealed by research (see for example Essien & Setati (2006), Kieran (1981, 1992) and Behr et al (1980)); finally, the three textbooks each have an accompanying teacher’s guide. This is critical to the present

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analysis because, one presupposition could be that Learner books are usually written with the learners in mind, while the guidelines for the educator and didactical underpinnings are provided in the educators’ guide. As such, it is quite logical that both learner book/activity book and the educator’s guide became the subject of analysis.

It must be noted that there is a dearth of Grade 1 learner’s books. In my visits to many primary schools in the Johannesburg area, I observed that what is available and used in most schools are rather the learner’s workbooks/activity book (rather than learner’s book). In the three textbooks under consideration, one (Successful Numeracy Grade 1) is a learner workbook and the other two are learner books.

In doing my analysis of how the concept of the equal sign is introduced in the three textbooks, I first make an argument on what counts as mathematical knowledge in the bid to explore what form of mathematical knowledge is promoted by the textbooks. I then draw on the concepts of realisation and recognition rules and use them as analytic tools in my exploration of the concept of the equal sign in the two texts. Doing this involved an analysis of how context is used by the textbooks to introduce the concept of the equal sign. The overall argument in this article is that the textbooks are structured and organised in such a way that enables and blends esoteric and realistic forms of mathematical knowledge in learners. I also argue that in the introduction of the equal sign, the textbooks are deficient in enabling learners possess recognition and realisation rules that would enable them (learners) access the concept of the equal sign.

THEORETICAL ORIENTATION

Bernstein (1996: 31) defines recognition rules as that which enables learners to “recognise the speciality of the context they are in”. In order words, recognition rules enable learners to correctly interpret the demands of the context thereby enabling them to respond to the task appropriately. Bernstein (1996) gives an example of a seminar which consists of participants from different disciplines and practices. He argues that without recognition rules in such a setup which orientates participants to the speciality of a particular context under consideration, contextually legitimate communication would be impossible. Bernstein argues that the unequal distribution of recognition rules might be well responsible for the silences of learners from the marginal class in the classroom.

Possession of recognition rules which allow learners distinguish context is not enough prerequisite for legitimate communication (Bernstein, 1996). Realisation rules need to complement recognition rules. By realisation rules is meant that which enables the production of “legitimate text” (Bernstein, 1996: 32). Hence, while recognition rules “regulate what meanings are relevant, realisation rules regulate how the meanings are to be put together to create the legitimate text” (Bernstein, 1996: 32). Put differently and in the context of mathematics, while recognition rules enable learners to identify the context under which the mathematics is studied, realisation rules enables learners to access the mathematics from the context thereby producing the legitimate text.

From the above definitions, it can be deduced that it is impossible to talk about realisation and recognition rules except in the light of what Cooper & Dunne (2000) refer to as “esoteric” mathematical knowledge and especially “realistic” mathematical knowledge. While realistic mathematics items “embed mathematical operations within contexts containing people and/or non-mathematical everyday objects” (Cooper & Dunne, 2000: 117), esoteric mathematics items do not. In other words, if an esoteric or context-free mathematics problem is “15 divided by 3”, a realistic problem designed to apply the same mathematical problem would be “if 3 learners need to share 15 oranges, how many would each get?”

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Research has shown that while attempts to incorporate the everyday into mathematics in texts and in the teaching and learning of mathematics is important (for example, in making the mathematics more meaningful to learners), such application or use of real life context to access mathematical knowledge is problematic in different ways. When context is used in a text, for example, it becomes difficult to separate the task from the context (Elwood, 1998). The presumption here is that the context offers a unique meaning for all users of the text regardless the sex and the social class. Research by Boaler (2000), Bernstein (1996), Saljo & Wyndhamn (1993), to mention but a few, have proved that this assumption is erroneous and misleading. Their findings have shown that Boys, for example, respond differently to girls in contextualised test items and that the middle-class learners reason more in the abstract compared to their working class counterparts.

At any Grade One level, when teaching (or learning) the concepts of the minus sign, addition sign and the equal sign, it goes without say that some form of context is always involved in the pedagogic process (and/or in the textbook used by learners). This is visible either through the use of physical objects or through pictorial representations or both, in the textbooks or in the pedagogic process. That being the case and in the light of the problems of using context discussed above, the question that remains to be answered is whether the nature of the context used in the three textbooks privilege one social class (and/or sex) of learners over the others. Analysing the texts based on realisation and recognition rules present an effective way of accomplishing this.

CRITICAL ANALYSIS OF THE EQUAL SIGN IN THREE FOUNDATION PHASE TEXTBOOKS

Introduction of the equal sign in the three textbooks

Maths for all is a mathematics textbooks series used widely in South African schools. Written in the light of Outcomes-Based Education, Math for all attempts to integrate the activities used in the book to other learning areas, and to learners’ daily activities in the home, school, etc. Activities used in the textbook are also such that encourage learners to work in a range of ways – talking, writing, singing, listening to stories, playing games, drawing, collecting, sorting, etc. The learner book, at the bottom of each page, provides the assessment focus and instructs teachers as to what they should ask learners to do.

In Maths for all 1 the introduction of the equal sign is preceded by the introduction of the counting (number) system using various diagrams and strategies. The first appearance of the equal sign occurs with the introduction of addition. Learners are asked to put one counter next to another and to say how many counters there are altogether. The teacher is instructed to place an item on the desk, and then place another one next to it and ask the learners how many items are there. The idea is to show, for example, that ‘one and one makes two’ and to show the learners that this is written as 1 + 1 = 2 (see page 12). Pictorial representations of items to be added, with the placeholders after the equal sign like the one shown below are also used in the introduction of the plus sign to learners (see page 17 for the actual image in the textbook):

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FIG. 1

The equal sign is also used in the introduction of the minus sign, using representation items such as insects, eggs, counters, fruits, etc. After contrasting addition and subtraction, the placeholder is used with the equal sign in an exercise which the learners are to do using a number line (see page 29).

In the educator’s guide for Maths for all 1, there is a detailed description of how the educator should introduce the addition sign from the use of counters and other objects (balls, trees, etc) to demonstrate the combination process. The same process is used to introduce the subtraction symbol. The educator’s guide stresses the importance of understanding the value of a number and the correct use of the language of operations (such as add, plus, subtract, take away, etc). Nowhere in the educator’s guide are there any explicit instructions on how the educator ought to introduce the equal sign. In fact, it is taken for granted that the learners would automatically know what the equal sign means when placed between objects or when used with a placeholder. Even when the commutative property of addition is introduced later in the chapter (through placeholders) there is no mention of the equal sign as signifying equivalence relations. The learner is left to believe that equal sign means ‘makes’ as in ‘one plus one makes two’ used in the introduction of addition.

Classroom Mathematics Grade 1 is also a mathematics textbook series that adheres to the principles of Outcomes-Based Education inasmuch as it advocates integration between Learning Areas and uses real life contexts in the introduction of several mathematics concepts.

The first appearance of the equal sign in Classroom Mathematics Grade 1 also occurs during the introduction of addition after the notions of “more” and “less” have been introduced to learners. Learners are given the exercises below to complete.

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FIG. 2 (pages 41 & 43, Courtesy of Heinemann Publishers)

In the first exercise involving robots, learners are supposed to colour in the correct answers with red, orange or green. I argue that even though there is context involved in the exercise, what the learner attempt to do first, is the addition of the numbers, and only after finding the answers do they (learners) apply it to the real life context. In the second exercise, the learners physically see the objects (dots) and how the addition of the objects is written mathematically. The emphasis here, according to the educator’s guide, is to teach learners how to write number sentences. Learners are then given many drilling questions with placeholders and diagrams to reinforce the concept of addition.

The same process is used to introduce the subtraction sign symbol. Learners are given a set of questions, first without diagrams, and then with diagrams and asked to write the correct number in the placeholders (See pg 65). There is therefore, in both the introduction of addition and subtraction, a movement from concept to context.

Like Maths for all 1, nowhere in the Classroom mathematics learner’s book are there any explicit pictorial diagrams aimed at enabling learners see and understand the significance of the equal sign in the addition or subtraction process. In the educator’s guide, however, educators are urged to show learners that ‘is equal to’ means ‘the same as’. The educator’s guide also urges educators to draw pictures of, say, 1 + 1 = 2 and to explain this to the learners (pg 38). There is also an elaborate explanation and activities around subtraction and its introduction in the educator’s guide, but nothing on the equal sign at this stage.

In Successful Numeracy Grade 1, the equal sign appear first in an activity where learners are asked to count how many objects are in the picture, and how many are left after crossing out some of the items. In so doing, the book introduces the concept of minus. After this, just like Maths for all, pictorial diagrams and the accompanying mathematical sentences are used to introduce the addition sign (using placeholders). Like Classroom Mathematics, while there are several activities showing learners and educators what to do/how to introduce the minus and plus symbols, the learner’s workbook of Successful Numeracy has no activities aimed at explicitly showing learners the significance of the equal sign. The educator’s guide, however, urges educators to introduce the equal sign, without explicitly saying how educators are to accomplish this (pg 39). The educators guide later encourages educators to ask learners to match pictures of shape with their real objects like: a rectangle = a table or a chalkboard; a square = a window plane or a chair seat. As I will explain later, placing the equal sign between objects only lead to misconceptions about the significance of the equal sign.

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How the textbooks reflect the intentions of the RNCS

The RNCS does not foreground the importance of the equal sign in grade 1 school mathematics. Neither does the mathematics curriculum for the foundation phase emphasis the importance of teaching the equal sign. The notion of the equal sign is, however, embedded in learning outcome 1, which requires of learners to be able to “recognise, describe and represent numbers and their relationships, and to count…” (DoE, 2002: 6) and to be able to order and compare collections of objects using the words ‘more’, ‘less’ and ‘equal’ (DoE, 2002: 14). The Curriculum also expects learners to know how to perform calculations using the appropriate symbols to solve problems involving addition and subtraction, etc. The Principles and Standards for School Mathematics, unlike the RNCS, makes a particular allusion to the equal sign symbol by referring to it as “an important algebraic concept that students must encounter and begin to understand in the lower grades (NCTM, 2000: 94). The Principles also notes that the common learners’ understanding of the equal sign at this stage should be more accurate than the limited understanding of the equal sign as signifying ‘the answer is coming’. Learners need to understand that the equal sign “indicates a relationship – that the quantities on each side are equivalent” (NCTM, 2000: 94).

The three learner’s books analysed above, in sympathy to the RNCS, and in varying degrees, do not in themselves foreground the importance of the equal sign as they ought to do. The learner books and the curriculum seem to take for granted the fact that the equal sign, (unlike the plus sign or minus sign), needs to be highlighted explicitly in texts for learners to visualise and conceptualise.

Another feature of the three textbooks analysed above is that of integration across other concepts – a feature of RNCS. All textbooks integrated the concepts of addition, subtraction and the equal sign (even though more implicitly in some than in others as far as the equal sign is concerned). And real life contexts are used to further explain the meaning of the addition and subtraction signs. The NCS proposes that learning be relevant and connected to real-life situations. To this end, all three textbooks are apt in using examples from real life context familiar to learners to introduce the concept of the equal sign. The use of balls, counters, plates, fruits, tables, weight systems, etc, rather than unfamiliar objects (dinosaurs for example) represent an attempt to use objects that are more or less related to grade 1 learners’ everyday experience.

Forms of knowledge privileged in the three textbooks

Taylor’s (1999) fundamental criticism of Curriculum 2005 (C2005) deals with the assumption C2005 makes - that the aim of integration across learning areas “will be a profound transferability of knowledge in real life” (DoE, 1997 in Taylor, 1999: 118). Taylor’s critique has therefore to do with the predominance of everyday knowledge (profane) in the C2005 to the detriment of formal school knowledge (sacred). His point was that such mixing of the sacred and the profane would most unlikely lead to profound systematic conceptual development. Formal knowledge, in such a situation, runs the risk of being “submerged in an unorganised confusion of contrived realism” (Taylor, 1999: 121). The three textbooks under consideration did not fall prey to the mistaken notion that working with everyday knowledge is sufficient and can be translated into formal (sacred) knowledge. Both esoteric and realistic forms of knowledge are promoted in the textbooks. The introduction of concepts (including the equal sign to an extent), the hands-on activities and the exercises in both textbooks constitute a fine blend of both realistic and esoteric mathematics items. The textbooks privilege, therefore, both esoteric and realistic mathematical knowledge, that is, both everyday and formal school knowledge.

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Practical implications of using the textbooks

In two of the three textbooks, we see a movement from the everyday (context) to the formal mathematical knowledge (concept) in the “introduction” of the concept of the equal sign. To what extend does starting with the context serve as a tool to the understanding of the concept of the equal sign? Does it matter if the movement is from formal mathematical knowledge to everyday mathematical knowledge as is the case with Classroom Mathematics? To respond to this question, I turn to Piaget.

The preoperational stage (roughly from age two to age six or seven) is the second stage of the Piagetian genetic epistemology. Most learners in grade one fall into this category. At this stage, according to Piaget, the child learns to manipulate his environment symbolically through inner representations, or thoughts, about the external world. Also, during this stage, the child learns to represent objects by words and to manipulate the words mentally, just as he/she earlier manipulated the physical objects themselves. That said, it makes more sense to argue that moving from the context – from the pictorial representations – to the concept is more likely to promote mathematical understanding of the concept than otherwise.

But due to the fact that, in the textbooks, the equal sign is introduced (implicitly) together with the addition and subtraction symbols, learners are more likely to identify the context of addition and subtraction under which the mathematics is studied (recognition rule) than the context under which the equal sign is used. They (learners) are also more likely to access the concepts of addition and subtraction from the context (realisation rule) than they would access the concept of the equal sign. When we look at the diagrams above from the textbooks, it is easy to see why learners would see/understand the concept of addition and subtraction than they would understand the concept of the equal sign. From the pictorial representations in the learner books, it is impossible or difficult for learners to access the concept of the equal sign from the everyday context that is used in the three textbooks (realisation rule). From the illustrative examples above of the introduction of the concept of the equal sign, and the other exercises in the textbooks under consideration, can middle class learners, even with their ability to reason in the abstract, know that the equal sign mathematically denotes an equivalent relationship - a relational symbol to compare two or more quantities? Is a cursory explanation of the equal sign as “the same as” or “makes” enough to firmly entrench the meaning of the equal sign in learners? Research done in this field and my practical experience with learners all point to the contrary. As far as realisation and recognition rules are concerned therefore, all groups are disadvantaged in the introduction of the concept of the equal sign in all three textbooks. Learners, because they are introduced to the equal sign in the way the textbooks do, would always tend to conceive of the equal sign as simply a tool for writing the answer. From the diagrams, learners may well see that the equal sign as used in the text, plays a role in the right- and left-hand relation of the question. But it will be difficult for learners to know that this relation is that of quantitative sameness between objects on the left-hand-side and objects on the right-hand-side. The temptation for learners would be to see the use of the equal sign in the texts as a demand to find the answer to what is on the left (or right) or to do something (compute) to what is on the left (or right). What is the practical implication of this understanding of the equal sign?

Kieran (1981), Stacey & Macgregor (1997), Falkner et al (1999) and Carpenter et al (2003) hold that one of the major stumbling blocks in the learning of algebra is a limited understanding of the equal sign because virtually all manipulations in algebraic equations requires a grasp of the equal sign as signifying a relation. A learner with a do-something or find-the-answer understanding of the equal sign may have difficulties understanding a simple algebraic equation like a + b = ? in later years of study. This is so because s/he is confronted with a situation where there is nothing to be done (unlike in the equation, 4 + 5 = ?). Or worst still, how would s/he understand the equation/relation, π = 3,14159…? (Freudenthal,

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1983)? Only the understanding of the equal sign as denoting a relationship (and not as a “do-something symbol”) would suffice in the understanding of such an equation. The same applies to an equation such as 4x + 4 = 4x + 1. Only the understanding of the equal sign as denoting a relation of quantitative sameness (and in later years, as denoting identity) would enable a learner to understand that it makes sense to subtract 4, for example, from both sides. Hence, the neglect in the explanation of the significance of the concept of the equal sign in the textbooks has far reaching implications for the teaching and learning of mathematics in the learners later years. It is against this backdrop that, in the next section, I deal with recommendations for improvement of the chapters analysed in this essay in future mathematics textbooks.

RECOMMENDATIONS

As has been argued above using Piaget, Grade 1 learners are in the preoperational stage and therefore need symbols and representations to mediate mathematical understanding. To this effect, future Grade One mathematics textbooks should be encouraged to use pictorial representation of object familiar to the context of learners of that age bracket to introduce the concept of the equal sign. Care must be taken, however, to avoid using the equal sign between two objects as this does not represent a relationship of equality between numbers and therefore, does not focus on the significance of the equal sign. Also, the use of realistic items must be balanced with the use of esoteric items and must be such that any group (social class or sex) is not privileged over another. The text must be organised in such a way that there is a balance in the promotion of realistic mathematical knowledge and esoteric mathematics knowledge. All three textbooks analysed in this article presented a fine blend of both realistic and esoteric mathematics knowledge.

In the textbooks too, a common feature in was that the equal sign is introduced alongside the addition and subtraction signs with elaborate and articulated ways of introducing the plus and minus signs. In all three textbooks, there were elaborate pictorial representations to illustrate the meaning of the plus and minus signs in the learner’s book. There is none of such to illustrate the significance of the equal sign. There are, in fact, no activities targeting the introduction of the equal sign. Even in two of the three books where there is an explicit mention of the equal sign, the Teacher’s Guides only say, “teach the equal sign”, or get learners to use the equal sign in number sentences. The equal sign does not, thus, enjoy equal status as the plus and minus signs. Yet, the equal sign is as important as these other two signs.

The minus and plus signs, it can be argued, are an invitation to do something since there cannot be an addition or subtraction sign in a question (in the foundation phase) that does not require the learners to compute. This is probably why the learners also take the equal sign as a command to do something since all three signs are introduced simultaneously, hence the need for textbooks to emphasis on the correct use and understanding of the concept of the equal sign and to also exhort teachers to teach it (the equal sign) explicitly. How can texts begin to use placeholders if learners have no clear definition and understanding of the equal sign?

I propose that of the three signs, the equal sign be introduced first. One way of doing this is to introduce the equal sign after the learners has been taught the constructs of “less” and “more” which precede the concepts of plus and minus both in the Curriculum and in the textbooks under consideration. The learners, would thus, understand when a quantity is ‘more’, ‘less’ or equal. After this, the teacher/textbook needs to continue explaining the role of the equal sign while introducing the addition sign. Equal sign should not be taught formally only when dealing with addition, but also reinforces when the concept of subtraction is introduced, and further reinforced with the teaching of the constructs of ‘heavy’ and ‘light’ which are the

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next constructs the Curriculum and in the textbooks I analysed. The scale balance such as the one below can be used, starting with when the constructs of ‘more than’ and ‘less than’ are introduced to learners and latter too, when the constructs of ‘heavier than’ and ‘lighter than’ are introduced to learners.

Not only should the educator’s guide contain instruction on how to teach the equal sign, the learner’s book must also have activities around the role or significance of the equal sign. As Falkner et al (1999: 233) argues, “teachers [and I would add textbooks] should…be concerned about children’s conceptions of equality as soon as symbols for representing number operations are introduced”. Learners must not only be taught how to use the equal sign, but more importantly, the significance of the equal sign in a mathematical sentence.

CONCLUSION

Recognition and realisation rules are both important if learners are to identify the context under which the mathematics item is embedded and access the mathematics by producing legitimate text (Bernstein, 1996). Maths for all 1, Classroom Mathematics Grade 1 and Successful Numeracy Grade 1 all use contexts learners can relate to to introduce the concept of the equal sign in such a way that privileges both the everyday knowledge of learners and formal school knowledge. In this essay, I have argued that as far as realisation rules and recognition rules are concerned, the three textbooks are deficient. I have also proposed one way of introducing the equal sign to learners – that it should be introduced when the learners are being taught the constructs of ‘more’ and less’, and before the introduction of the addition and minus symbols. The use of pictorial representations to introduce the equal sign is a good idea. But such representations must be accompanied by explanations of what the equal sign signify to avoid misinterpretations by learners.

References:

Behr, M., Erlwanger, S. & Nichols E. (1980). How Children view the Equals Sign. Mathematics Teaching, 92, 13-15.

Bernstein, B. (1996). Pedagogy, symbolic control and Identity: theory, research, critique. London: Taylor and Francis.

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Boaler, J. (1997). Experiencing School Mathematics: teaching styles, sex and setting. Open University press. Buckingham

Carpenter, T.; M. Franke; L. Levi (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth: Heinemann.

Chantler, E., Hoffmann C., Stephanou, L. (2008). Successful Numeracy Grade 1 (Workbook). Cape Town, Oxford

Cooper, B. & Dunne, M. (2000). Assessing Children’s Mathematical Knowledge: Social class, sex and problem-solving. Open University press. Buckingham.

Department of Education (2002). Revised National curriculum Statement Grade R-9 (Schools), Overview. Pretoria.

Elwood, J. (1998). The use of context in examination and assessment items: A source of inequality in assessment outcomes. British Journal of Curriculum & assessment, 8, 31-38.

Essien, A. & Setati, M. (2006). Revisiting the equal sign: Some Grade 8 and 9 learners’ interpretations. African journal of research in Science, Mathematics and Technology Education, 10(1), 47-58.

Falkner, K., Levi, L. & Carpenter, T. (1999). Children’s Understanding of Equality: A foundation for Algebra. Teaching Children Mathematics, 6, 232-236.

Freudenthal, H. (1983). Didactical Phenomenology of Mathematical structures. Boston: D. Reidel Publishing Company.

Jenkins, T. Buthelezi, P. Greig S., Dhlamini, M., Lubombo A., Mogorosi, P., Ponte, J. (2003). Classroom Mathematics Grade 1 (Learners’ Book). Sandton, Heinemann.

Jenkins, T. Buthelezi, P. Greig S., Dhlamini, M., Lubombo A., Ponte, J. (2003). Classroom Mathematics Grade 1 (Educator’s Guide). Sandton, Heinemann

Kieran, C. (1981). Concepts associated with the equality symbol. Educational studies in Mathematics, 12, 317-326.

Kieran, C. (1992). The learning and teaching of school Algebra. In, Grouws, D. (ed). Handbook of research on Mathematics Teaching and learning. New York: Macmillan Publishing Company.

NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

Saljo, R. & Wyndhamn, F. (1993). Solving everyday problems in the formal setting: An empirical study of the school as context for thought. In Chiaklin, S. & Lave, J. (eds) Understanding practice: Perspectives on activity and context. New York: Cambridge University press.

Schools Development Unit of the University of Cape Town, (2003). Maths for all 1 (Learner’s Book). Manzini: Macmillan

Schools Development Unit of the University of Cape Town, (2003). Maths for all 1(Teacher’s Book). Manzini: Macmillan

Smith, P., Chantler, E., Hoffmann C., Stephanou, L. (2008). Successful Numeracy Grade 1 (Teacher’s Book). Cape Town, Oxford

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Stacey K. & Macgregor M. (1997). Building foundations for Algebra. Mathematics Teaching in the Middle School, 2, 252-260.

Taylor, N. (1999). Curriculum 2005: Finding a balance between school and everyday knowledge. In, Taylor, N. & Vinjevold, P. (eds). Getting Learning Right. Joint Education Trust. Department of National Education

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When pre-service teachers learn to function using spreadsheet-based algebraic approaches

Faaiz Gierdien & Alwyn Olivier

Research Unit for Mathematics Education, University of Stellenbosch (RUMEUS) faaiz@sun.ac.za & aio@sun.ac.za

In this report we present a conceptualisation of our work with pre-service mathematics teachers, wherein we exploit spreadsheets as a particular example of Information Communication Technology through which to reform school algebra.We start by describing the working space between spreadsheets and school algebra (and vice versa), and what it means to make spreadsheets a ‘mathematical instrument.’ These we connect with two of the ‘central tasks’ in pre-service teacher preparation, namely, building a ‘beginning repertoire’ and ‘subject matter or mathematical knowledge for teaching’ with respect to spreadsheets-based algebraic approaches. An analysis of one student’s work for a spreadsheet task called ‘Factoring’ shows the extent of interactions between her algebraic knowledge and that of spreadsheets. These have implications for ‘mathematical knowledge for teaching.’ INTRODUCTION The introduction of Information Communications Technology (ICT) in the South African school curriculum has implications for the reform of school mathematics, and concomitantly, for the pre-service preparation of teachers, at both the primary school and/or high school grade levels.

There is a growing body of literature on the use of different types of ICT, such as hand-held graphing calculators, Geometer’s Sketch Pad, and spreadsheets in mathematics education. For example, Mbekwa and Julie (2003) investigated what they call pre-service teachers’ ‘mathematical engagement’ with the TI-92 graphing calculator, while Berger (2007) did a study on undergraduate students’ uses of Computer Algebra Systems as a tool for learning mathematics in a mathematics department.

In South Africa, we know very little about the understanding of (and consequently experience with) spreadsheets, on the part of pre-service primary and high school teachers. The decision to focus our research on spreadsheets is motivated by the fact that almost all computers in schools have spreadsheets in the form of Excel. Spreadsheets are the most accessible ICT to mathematics teachers and students in schools, and therefore worthwhile in terms of a research focus for pre-service teacher education students. It should be noted that the use of ICT in mathematics classrooms, both at high school and primary school levels, is advocated in South African curriculum policy documents (Revised National Curriculum Statement Grades R-9 (Schools) Mathematics (RNCS) (DoE, 2002) through the use of phrases such as ‘appropriate,’ ‘available’ and ‘electronic and other technology,’ which makes spreadsheets a commendable candidate. The conditions under which practicing teachers use spreadsheets in their mathematics classrooms could be the focus of another project, however, we are pursuing our research on spreadsheets independently of current policy statements.

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In South African pre-service preparation, or ‘initial teacher education’, (Kruss, 2008) there are opportunities and challenges associated with ‘preparing’ students to become familiar with spreadsheets, and to exploit this potential in relation to the reform of school mathematics. We agree with Feiman-Nemser (2001) that ‘preparation’ is a ‘treacherous’ idea when it comes to education, because we can never match the exact circumstances that our students will encounter beyond the pre-service phase; Dewey (1938). The envisaged role for students is one in which they will be able to become reasonably conversant with the capabilities and limitations of spreadsheets, in relation to areas the school mathematics curriculum, such as algebra. They should be able to transform their ‘pencil and paper’ knowledge of algebra with regard to the syntax peculiar to spreadsheets, in order to more effectively perform the tasks we assign them. Put differently, when pre-service teachers use spreadsheets, they have to learn an algebra-like notation in order to make the spreadsheets a mathematical or algebraic instrument. This does not happen without difficulty. The problem then becomes how and why one should study this phenomenon, which brings us to the following research question, namely, what problems are likely to be encountered when pre-service teachers learn to function using spreadsheet-based algebraic approaches?

Theoretical framework To answer the research question, we draw on a variety of literature such as school algebra, spreadsheets, instrumentation issues and teacher learning in pre-service preparation. This literature will be reviewed and integrated in such a way as to enable us to better analyse student work.

Figure 1 shows the conceptual space in which our students (pre-service teachers) find themselves when they perform their tasks using spreadsheets. To work effectively with spreadsheets, they need to have conceptual understanding of the multiple representations of algebraic objects in terms of pencil and paper versions. Furthermore they need to understand the demands and capabilities of spreadsheets, such as the peculiar syntax required, to use them effectively.

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Figure 1: From spreadsheets to algebra, and vice versa Figure 1 is central in the construction of our theoretical framework, and shows the dialectical interaction of the two ‘knowledge components’ (Mishra & Koehler, 2006; 2007), namely technological knowledge (spreadsheet knowledge), and content knowledge (algebraic knowledge). The student (or user) brings her pencil & paper algebraic knowledge to the spreadsheet-based task, which is used to appropriate the spreadsheet syntax, so that the latter reflects the former. This is how the reader needs to view Figure 1. We choose the phrase ‘dialectically interact’ to emphasise how the student’s understanding and knowledge of the interaction between these two components can progress and evolve over time.

Spreadsheets play a key role in understanding and exploring functions, or functional relationships, as objects of study when it comes to school algebra. For example, there are several studies on spreadsheets in relation to the subjects’ knowledge of algebra, as an algebraic and an arithmetic environment representative of functions or expressions. A classic study involving school children exists, in which Sutherland and Rojano (1993) demonstrate how a judicious use of spreadsheets can enhance algebraic understanding. In terms of Figure 1 this study shows interplay between algebra and spreadsheets. Haspekian’s (2005) research demonstrates how spreadsheet potentialities have a hybrid status of being ‘an arithmetico-algebraic tool.’ This implies that arithmetical or computational processes performed through spreadsheets have an algebraic angle to them, because of a possible overlap and similarity in syntax, i.e. via the use of symbols. Bills, Ainley and Wilson (2006) explored the interplay between ‘modes of algebraic communication’, which goes from spreadsheet syntax to standard notation, i.e. pencil and paper notation. Their use of ‘interplay’, based on children in a secondary school, is what we attempt to capture in

STUDENTS

ALGEBRA

SPREADSHEET

ALGEBRA

SPREADSHEET

STUDENTS

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Figure 1. The same group (Ainley, Bills and Wilson, 2005) researched what they call the ‘construction of meaning for algebra’ or ‘purposeful algebra’ using spreadsheets. They argue that algebra and its objects attain ‘meaning’ in spreadsheet-based computer environments. A spreadsheet can thus be considered as an algebraic environment that ‘mediates’ between users’ knowledge of arithmetic and algebra. Dettori, Garuti and Lemut’s research (2001) shows how middle-grade schoolchildren used spreadsheets to solve problems that require ‘modelling processes’ for problem resolutions via spreadsheet syntax. They make the point that spreadsheets are ‘function oriented’, in that formulas involve concepts of variable and function.

To function effectively, the pre-service teachers need to make spreadsheets a mathematical ‘instrument’ through specific processes, in order to understand the objects as they appear both in pencil and paper versions, and in terms of spreadsheet syntax. The objects, whether numbers or letters in cells, can take on ‘mathematical’ or algebraic meaning - such as representing functions - because a spreadsheet is a ‘function oriented’ environment (Dettori, Garuti & Lemut, 2001). The artefact or technical tool (Vérillon & Rabardel, 1995), in this case the spreadsheet programme, does not, initially, have an instrumental value. This we distinguish from the instrument it can become, through a process called ‘instrumental genesis’ (Artigue, 2002; Vérillon & Rabardel, 1995).

This instrumental genesis works in two directions, hence we refer to ‘dialectical interaction.’ Firstly, it is directed towards the artefact or spreadsheet programme, which, when loaded progressively, and utilised specifically, has the potential to transform into performing the required task, as in the case of our students. This is called the ‘instrumentalisation’ of the artefact, (i.e. the spreadsheet programme). Secondly, instrumental genesis is directed towards the subject or user, leading to the development of schemes of instrumented action that progressively take shape as techniques, which permit an effective response to given tasks. The latter direction is properly called instrumentation (Artigue, 2002). Figure 2 is an illustration of how we see an ‘instrument’ coming into being.

Artefact + subject (user) = instrumental genesis instrument

Figure 2: A simplified version of understanding an instrument Two general observations about ‘techniques’, as in ‘instrumented techniques’ in a spreadsheet environment, need to be made. These are adapted from Artigue’s (2002) work in Computer Algebra Systems, and Haspekian’s (2003; 2005) work in spreadsheets and instrumentation processes in general.

Firstly:

Techniques have pragmatic value, which permits them to produce results.

In other words, there are technical or instrumental processes that the user will have to learn and undertake, which have ‘productive potential’ i.e. they are considered efficient to do the task (Artigue, 2002, p. 248). This can be considered the ‘technical’ part of the mathematics involved in the task (Haspekian, 2003).

Secondly:

Techniques have epistemic value.

They thus partially comprise an understanding of the mathematical objects that live on the screen, in relation to their possible pencil-and-paper counterparts, which arise when the pre-service teacher attempts to do the tasks. They can therefore become a source of new questions, such as how one converts particular mathematical objects into spreadsheet syntax. As we have argued earlier, these techniques require

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conceptual understanding of algebraic knowledge, in that the user must grasp the epistemic aspects, such as the process/object duality of the mathematical objects.

What we observe from the spreadsheet research reported here is an overlap with Sfard and Linchevski’s (1994) research on the process-object duality, in the case of school algebra. They expound eloquently as to how algebraic expressions and functions can be thought of as computational processes, and as objects. They discuss the example of 3(x + 5) + 1 as an expression or process, which can also be considered a function, [f(x) = 3(x + 5) + 1] or an object. For instance, a computational or arithmetic process such as a ‘modelling process’, or a function when performed through spreadsheet syntax, takes on an algebra-like notation, which can be considered an object, and necessarily involves functional relationships between variables.

To do their tasks effectively, the pre-service teachers should know the process-object duality and the accompanying numerical, tabular and graphical representations, or a multiple representation of this duality. More importantly, irrespective of the mathematical object with which they have worked, they need to be aware of ‘cognitive linking’ between the multiple representations of the object, and how particular ‘syntactic manipulations’ of its symbols have corresponding numerical, tabular and graphic effects (Yerushalmy & Gafni, 1992). The latter, for example, could be discrete or continuous, depending on a particular expression, or function that describes a situation. A major component of symbolic manipulation in algebra consists of tasks involving transformations of expressions, in other words, rewriting the entire given expression in a different algebraic form to get an equivalent expression (Yerushalmy & Gafni, 1992). This ‘cognitive linking’ means that the pre-service teachers should be able to perceive far more than the mere algebraic or mathematical symbols, whether in the form of pencil and paper, or a spreadsheet version.

Examples of instrumented techniques and their associated epistemic values are in order, because they will be useful in the analysis of spreadsheet work. According to Wilson, Ainley, and Bills (2005), to ‘name’ a column or a cell can be defined on a spreadsheet. The letter ‘n’ has been used to ‘name’ column A (See Figure 3).

A B A B

1 1 =2*A1+3 1 1 =2*n+3

2 2 =2*A2+3 2 2 =2*n+3

3 3 =2*A3+3 3 3 =2*n+3

4 4 =2*A4+3 4 4 =2*n+3

Figure 3: An example of ‘naming’ column A

The epistemic value of ‘naming’ column A is that object ‘2n+3’ has become mathematically defined through the equal sign (=) in column B. The equal sign is an instruction for a computation to happen. This object (2n+3) in column B can be ‘filled own’ through a range of cells of columns. What we have here is the process-object duality (Sfard & Linchevski, 1994). In other words, the objects are seen as originating in, or from arithmetical or computational processes, or a conception of school algebra as ‘generalised arithmetic’ (Usiskin, 1988). A pencil and paper algebraic object such as x2 + 3x can be transformed into spreadsheet syntax, where the ‘3’ can be considered as a parameter. In cell E1 the absolute reference of $E$1 is used in column B (See Figure 4).

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A B C D E

1 1 =A1^2+$E$1*A1 3

2 2 =A2^2+$E$1*A2

3 3 =A3^2+$E$1*A3

4 4 =A3^2+$E$1*A4

Figure 4: Absolute referencing in spreadsheets syntax

The findings of this research may be particularly significant, as they have implications for two of the ‘central tasks’ in pre-service teacher learning, namely, building a ‘beginning repertoire’ and ‘subject matter or mathematical knowledge for teaching’ (Feiman-Nemser, 2001). These findings will enable us to comment on what it means to ‘prepare’ our students’ with ‘mathematical knowledge for teaching’ (MKT), in the case of functions in spreadsheet-based algebraic approaches. Over recent years there has been a proliferation of studies on MKT (Ball & Thames, 2008; Adler & Davis, 2006; Ball & Bass, 2000; Ball, Hill & Bass, 2005). The presence of multiple representations in a spreadsheet-based environment means that we can use Adler and Pillay’s (2007) metaphor namely; ‘mathematics for teaching is explanations and representations.’ Their research involved the study of an in-service teacher’s attempt at teaching functions. A ‘beginning repertoire’ thus entails having our students learning to appropriate ways of making spreadsheets a ‘mathematical instrument’, and thereby beginning to function using spreadsheet-based algebraic approaches. In part, this means that they should be able to discern and connect the pragmatic and epistemic values of their instrumented techniques. These are all important considerations in the case of ‘learning to teach’ (Feiman-Nemser, 1983) regarding teachers as learners, with respect to algebra and spreadsheets.

On data and method We have been teaching topics in the school mathematics curriculum using spreadsheets to two cohorts of students in the pre-service teacher education programme. The first of these comprises students in the final or 4th year of the Bachelor of Education (B Ed) programme, while the other is enrolled in a one-year postgraduate certificate in education (PGCE). The former will be certified to teach the middle grades, i.e. grades 4 through 9, while the latter will be certified to teach grades 10 through 12. The PGCE students usually have a Bachelor of Science (BSc) degree with 3rd year university mathematics via the Faculty of Science, while the B Ed students have attended mathematics education modules through the Faculty of Education, with no university or tertiary mathematics (as opposed to the PGCE students.) This is so because the B Ed programme is an ‘integrated’ one wherein both ‘mathematics’ and ‘teaching’ are objects of study in the mathematics education modules. These students are given tasks with a spreadsheet-based algebraic focus, which they have to submit electronically. In these tasks, ‘functions’ are objects of study. For example, they examine linear functions and their connections with quadratic functions, and the role of parameters in these and other functions, such as relationships between perimeter and area, as well as situations where functional relationships can be explored through symbolic, tabular and graphical representations, i.e. multiple representations.

Sally’s work on a task called ‘Factoring’ will be presented for analysis (See Appendix). This task comes from the fourth (final) year mathematics education module (Mathematics Education 478) in the B Ed programme. We present it because through it we hope to begin to answer our research question—what

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problems can be anticipated when pre-service teachers learn to function using spreadsheet-based algebraic approaches? ‘Factoring’ reads as follows:

For which values of a and b is x² + px + q = (x + a) (x + b)?

You can try different values for a and b, and check the corresponding table and graphs. How do you know that you are correct?

In addition, she was asked to produce a written response to the following question:

In a word document, describe briefly what meanings of equivalent algebraic expressions appear in

the task called Factoring. Also, briefly describe the role of technology in the task.

Her allotted task was to make a ‘working replica’ of ‘Factoring,’ meaning that she was asked to reconstruct what was provided to her from a blank spreadsheet. This is a general ‘developmental’ approach we use with the B Ed students, because we are interested in how they learn to make the spreadsheets a mathematical instrument, thereby hopefully deepening their understanding of the algebraic objects with which they work. Another reason for choosing Sally’s response is that the B Ed students were allocated a fair amount of module contact time before completion of the project was required. In fact, four months had passed since the beginning of the academic year, during which the students were exposed to eight contact sessions in the university laboratory, each of which amounted to two hours in length. This was, of course, supplemented by the time they spent working on their own.

Problems in Sally’s answer and assessment can be uncovered by looking at her instrumented techniques, with respect to her knowledge of algebra and spreadsheets, as justified in terms of the theoretical framework shown in Figure 1. (The literature we reviewed alerted us to these.) By ‘opening up’ the different cells in her spreadsheet, we are able to observe the spreadsheet syntax she appropriated. This is one method of analysis. We also took detailed account of her written response. As we must analyse both, in order to answer the research question, the order is not significant, so we will start with the latter, and then proceed to the former.

Findings and discussion In her written response, Sally recorded the following:

In the task, it is expected from the learners to figure out the relationship between x² + px + q and (x + a)(x + b). As they vary the values of a and b, they will notice that the graphs are changing as well as the values in the tables. As soon as they type in the right combinations of numbers, they will notice that the graphs look the same and also that the numbers in the tables correspond. What is expected from learners is that x² + px + q and (x + a) (x + b) are equivalent to each other as soon as the correct values of a, b, p and q are typed in. Although they don’t look the same, x² + px + q is a simplification of (x +a)(x + b).When the two brackets are multiplied out, you get an expression equivalent to x² + px + q. Learners can also deduce from this exercise that by multiplying two straight line formulas the formula for a parabola is obtained.

Technology plays a big role in such activities. Learners are excited about computer work and many of them are better at it compared to the teacher. They can explore the possibilities and not necessarily be wrong, and the fact that it is to be done individually, encourages the learner to keep on trying. Also, learners will not experience it as intimidating mathematics, but as a problem-solving situation.

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From what she wrote, we gain a sense of this student’s spreadsheet knowledge, in addition to her algebraic knowledge. She is aware that a and b, as parameters, can be varied by using the spreadsheets. Evidence of this can be seen where she refers to imaginary schoolchildren; ‘As they vary the values of a and b they will notice that the graphs are changing as well as the values in the tables.’ According to the above, one would assume that she made the spreadsheets a mathematical instrument, where a, b, p and q are parameters in the symbolic representations, or algebraic expressions – x² + px + q and (x + a)(x + b) – and that the tabular and graphical representations are coordinated or connected. She is also aware of what the relationships between these parameters need to be, for the two algebraic expressions to be equivalent. Evidence of this is contained in the line; ‘As soon as they type in the right combinations of numbers, they will notice that the graphs look the same and also that the numbers in the tables correspond’ and the ‘correct values of a, b, p and q are typed in.’ These ‘right combinations’ occur when a + b = p and ab = q. We can therefore safely say that her writing reveals her knowledge of algebra. Evidence of conceptual understanding in her algebraic knowledge is indicated where she articulates a conceptual connection between linear and quadratic functions: ‘Learners can also deduce from this exercise that by multiplying two straight-line formulas, the formula for a parabola is obtained.’ In other words, she knows that ‘parabolas’ are a composition of linear functions. These are the meanings she ascribes to ‘equivalent algebraic expressions’ through generalised arithmetic, which spreadsheets enable. (For example, she knows ‘the numbers in the tables’ should ‘correspond.’)

Her spreadsheet syntax on the parameters a, b, p and q, however, does not correspond with her conceptual understanding of relationships between the linear and quadratic functions found in her written response.

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Figure 5: Sally’s spreadsheet showing correct numerical values for parameters but a disconnect in the arithmetic in the columns

In Figure 5, we see that she has the correct values for a, b, p and q so that p = a + b and q = ab (p = 4; q = 3; a = 1; and b = 3). Her columns C and D, for x² + px + q and(x + a)(x + b), respectively, are not numerically equivalent. Perhaps she was convinced by the fact that the corresponding graphical representations ‘appear to be the same.’ Her instrumented technique of drawing the graphs through spreadsheets in a pragmatic sense seemed ‘technically’ correct. We can see this from a cursory examination of the shapes of the graphs, shown on the right. When we ‘open up’ cell C11 and D11, a syntax showing that there are no absolute references to F3 and F4 is revealed. Sally knows that the generalised arithmetic should produce columns that are numerically equivalent, however, she could not establish a cognitive link between her algebraic knowledge, and how it should be instrumented by means of absolute referencing in the spreadsheets syntax. For example, in C11 and D11 she could have typed ‘=B11^2 + $F$3*B11 + $F$4’and ‘= (B11+ $F$6)*(B11 + $F$7)’, respectively. Afterwards, she had to select these two cells and ‘drag’ them to C21 and D21. The result would have produced numerical and graphical equivalence between the two algebraic expressions, represented in multiple ways as functions. More interestingly, she has ‘named’ cells ‘a’ as F6, ‘q’ as F4 p as the column C11 through C21, as well as ‘naming’ ‘b’ the same column! It was clearly her intention to use the ‘name’ instrumented technique in order to get the ‘generalised arithmetic’ in the columns to correspond, as she stated in her written response. This, however, has not happened.

Concluding remarks As a case-study of a pre-service teacher, Sally has learned to function using the capabilities of spreadsheets in relation to algebra, and vice versa. That learning, however, is superficial rather than profound, as her knowledge of algebra, in conjunction with spreadsheets, is not conceptually well-connected. The latter is not reflected in the former, all of which provide us with information on her individual, instrumental genesis. Whilst Sally indicated her awareness of the fact that ‘technology plays a big role’ in the kind of task she had to perform, in replicating ‘Factoring’, there are other dialectical ‘factors’ that need to come into play, such as the need for her to cognitively link her algebraic knowledge with that of spreadsheets. It is hoped that the interaction with her peers and lecturer regarding these factors will help Sally understand more deeply the purposes behind ‘Factoring’, such as linking generalised arithmetic and functions. Specifically, she needs to learn to keep track of how and where the different ‘pencil and paper’ algebraic objects are written in their corresponding spreadsheet syntax. Examples are the absolute references for parameters needed, in order to ‘drag’ or ‘fill down’, thereby producing functional relationships between numbers in columns. Sally’s difficulties are instances of the epistemic aspects gleaned from her instrumented techniques, which are dependent on pragmatic or technical aspects, i.e. they are dialectically dependent.

There is no doubt that she has the beginnings of a repertoire on MKT, namely, multiple representations of functions and conceptual explanations for connecting linear and quadratic functions. References Adler, J., & Pillay, V. (2007). An investigation into mathematics for teaching: Insights from a case, African

Journal of Research in SMT Education, 11(2):87-102.

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Adler, J., & Davis, Z. (2006). Studying mathematics for teaching inside teacher education. In Proceedings of the SAARMSTE Conference, Pretoria, January 2006.

Ainley, J., Bills, L., & Wilson, K. (2005). Designing spreadsheet-based tasks for purposeful algebra, International Journal of Computers for Mathematical Learning 10, 191-215.

Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245-274.

Artigue, M. (2000). Instrumentation processes and the integration of computer technologies into secondary mathematics Retrieved August 22, 2006, from http://www.lkl.ac.uk/rnoss/MA/ArtigueCAS.pdf

Ball, D. L., & Thames, M.H., G. (2008). Content knowledge for teaching: what makes it special? Journal of Teacher Education 59(5), (Nov-Dec 2008): p389(19).

Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In Boaler, J. (ed.), Multiple perspectives in mathematics teaching and learning (pp.83-104). Wesport, CT: Ablex Publishing.

Ball, D. L., Hill, H.H., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 14-46.

Berger, M. (2007). CAS as a tool for learning mathematics at undergraduate level: some aspects of its use, African Journal of Research in SMT Education, 11(1), 17-28.

Bills, L., Ainley, J., & Wilson, K. (2006). Modes of algebraic communication: Moving from spreadsheets to standard notation, For the learning of mathematics, 26 (1), 36-41.

Department of Education (DoE) 2002. National Revised National Curriculum Statement Grades R-9 (Schools) Mathematics (RNCS).

Dettori, G., Garuti, R., & Lemut, E. (2001). From arithmetic to algebraic thinking by using a spreadsheet. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 191–208). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Dewey, J. (1938). Experience and education. New York: Simon & Schuster.

Feiman-Nemser, S. (2001). From preparation to practice: Designing a continuum to strengthen and sustain teaching. Teachers College Record, 103(6), 1013-1055.

Feiman-Nemser, S. (1983). Learning to teach. In Shulman, L.S. & Sykes, G. (eds.), Handbook of teaching and policy (pp. 150-170). NY: Longman.

Haspekian, M. (2005). An ‘instrumental approach’ to study the integration of computer tool into mathematics teaching: The case of spreadsheets. International Journal of Computers for Mathematics Learning, 10(2), 109-141.

Haspekian, M. (2003) ‘Between arithmetic and algebra: a space for the spreadsheet? Contribution to an instrumental approach’, Proceedings of the Third Conference of the European Society for Research in Mathematics Education, Pisa, Italy, Universita Di Pisa, Thematic Working Group 9, http://www.dm.unipi.it/~didattica/CERME3/proceedings/ (accessed 3/1/06).

Kruss, G van der Heever. (2008).Teacher Education and Institutional Change in South Africa, HSRC Press.

Mbekwa, M., & Julie, C. (2003). Pre-service teachers’ mathematical engagement with the TI-92. The International Journal of Computer Algebra in Mathematics Education, 10(1).

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Mishra, P., & Koehler, M. J. (2007). Technological pedagogical content knowledge (TPCK): Confronting the wicked problems of teaching with technology. In C. Crawford et al. (Eds.), Proceedings of the Society for Information Technology and Teacher Education International Conference 2007, (pp. 2214-2226). Chesapeake, VA: AACE.

Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A new framework for teacher knowledge. Teachers College Record, 108(6), 1017-1054.

Sfard, A., & Linchevsky, L. (1994). The gains and the pitfalls of reification. The case of algebra. Educational Studies in Mathematics, 26, 191-228.

Sutherland, R., & Balacheff, N. (1999). Didactical complexity of computational environments for the learning of mathematics. International Journal of Computers for Mathematical Learning, 4, 1-26.

Sutherland, R., & Rojano, T. (1993). A spreadsheet approach to solving algebra problems. Journal of Mathematical Behavior, 12, 353-383.

Usiskin, Z. (1988). Conceptions of school algebra and uses of variables, in The Ideas of Algebra, K-12, 1988 Yearbook, Coxford A. F. (ed.) National Council of Teachers of Mathematics, Reston, Va.

Vérillon, P., & Rabardel, P. (1995). Cognition and artefact: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology in Education, 9(3), 1-33.

Wilson, K., Ainley, J., & Bills, L. (2005), ‘Naming a column on a spreadsheet: is it more algebraic?’, in D. Hewitt and A. Noyes (Eds), Proceedings of the sixth British Congress of Mathematics Education held at the University of Warwick, pp. 184-191. Available from www.bsrlm.org.uk.

Yerushalmy, M., & Gafni, R. (1992) Syntactic manipulations and semantic interpretations in algebra: The effect of graphic representation, Learning & Instruction, 2(4), 303-319.

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Appendix Factoring

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Breaking up and Making up: a feature of school mathematics pedagogy

Shaheeda Jaffer

Group for the Study of the Constitution of Mathematics in Pedagogic Contexts School of Education, University of Cape Town

This paper analyses observations of teaching and learning of mathematics in two grade 8 classrooms. The first class is situated in a school populated by students from working class backgrounds and the second class is situated in a school populated by students from elite backgrounds. The pedagogy of the two classes appears similar in that both privilege the splitting apart (sundering) and combining (concatenation) of mathematical expressions. This initial analysis tentatively suggests that possibly sundering and concatenation of mathematical expressions are general features of school mathematics and highlights the need to investigate what the pedagogic conditions and individual resources are required to produce a coherent view of mathematics.

Introduction This paper reports on a pilot study of the teaching and learning of mathematics in two grade 8 classes with the aim of identifying a research problem and producing research hypotheses within the general problematic of the constitution of mathematics in pedagogic contexts. The study focuses on two grade 8 classes in contrasting social class contexts. The first class is situated in an ex-DET school populated by ‘African’ children from working class families. The second class is located in an elite school with a student population drawn from elite families across the racial spectrum. The two classes were deliberately chosen to investigate the teaching and learning of mathematics in different social class contexts.

Three international comparative studies, Trends in Mathematics and Science Study (TIMSS) Grade 8 in 2003, Southern and Eastern Consortium for Monitoring Educational Quality (SACMEQ) Grade 6 in 2000 and Monitoring Learner achievement Study (MLA), showed that the majority of South African students are performing well below the mean for other countries including other African countries. The findings are supported by our own National Systemic Evaluation which demonstrated that about 80% of South African learners are performing below the minimum expected standard for their grade. The majority of such students are located in schools populated by students from working class families (Schollar, 2008).

According to the sociology of education (Bernstein, 1996; Holland, 1981) schooling in general reproduces specialised knowledge differentially across social class. Similarly, studies focusing on the reproduction of school mathematics (Dowling, 1998; Hoadley, 2005, Walkerdine, 1988 and Cooper, 1976) illustrate differential reproduction of mathematics in schooling along social class lines. A number of research studies have focused on identifying the pedagogic conditions that would contribute to successful learning outcomes for working class students. Studies conducted by the ESSA group in Portugal (Morais, 2002) on the teaching of science to working class students, for example, and the study by Hoadley (2005: 272) which compares South African Grade 3 classrooms populated by working class students to Grade 3 classrooms populated by middle class students conclude that the ideal pedagogic modality should incorporate a weakening of framing over pacing (allowing students to vary the pace when they need to)

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and strengthening of framing over the evaluative criteria (stronger control by the teacher in making the criteria that mark legitimate from non-legitimate knowledge statements explicit to students) (See Bernstein, 1996: 13).

While these studies have identified general features of pedagogy that seem to contribute to successful learning outcomes for working class students, they limit the study of pedagogic conditions to the study of the pedagogic effects of social relations in classrooms. Consequently the studies do do not adequately provide insight into the nature and generation of the criteria that are transmitted and acquired in successful mathematics classrooms.

This pilot study is situated in a broader research project, the Study of the Constitution of Mathematics in Pedagogic Contexts, investigating two interrelated questions: 1) what is constituted as mathematics in school classrooms? and 2) how is it constituted? The particular focus of this pilot study is to investigate whether there are differences in the constitution of mathematics in different social class contexts and to explore how mathematics is constituted differently across social class contexts.

Video records of lessons in both schools serve as the primary sources for the production of data. The school populated by working class children (School A) is participating in the Mathematics and Science Education Project (MSEP) while the school populated by elite students (School B) participated in the QUANTUM project. In the case of School A, two researchers observed and video recorded three consecutive lessons. The video records of four consecutive lessons in School B were produced as part of the QUANTUM project and made available to this researcher for the purposes of conducting a pilot study.

Empirical contexts School A situated in Cape Town is an ex-DET school. The particular class focused on in this school consisted of about 45 students of mixed gender. The lessons observed took place in a poorly resourced classroom which formed part of a double room with a divider that did not close properly. Consequently, sound travelled between the two classrooms. Due to the number of students in the classroom, desks were crammed next to each other leaving very little space for movement between them. The language of teaching and learning in this classroom was both English and isiXhosa. The teacher switched between English and isiXhosa while the students communicated mostly in isiXhosa.

The lessons took place on three consecutive days and were observed and video-recorded as part of the MSEP project. The first two lessons were single periods and the third lesson a double period. The time-tabled length of a period at the school was 45 minutes. Table 1 below shows the amount of teaching time for each lesson. The table illustrates that approximately 85% of the total time available for teaching and learning was used. This was due to inconsistent timing of periods at the school and the fact that the third lesson was shortened by the teacher who felt that the lesson was too long to hold the concentration of the students.

Lesson Duration (in minutes) Lesson 1 (single period) 40:36 Lesson 2 (single period) 36:09 Lesson 3 (double period) 76:44 Total teaching time 153:29

Table 1: Distribution of time for each lesson conducted in School A

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The lessons in this classroom focused primarily on integer arithmetic. Table 2 below shows the breakdown the content covered in each lesson.

Lesson Content covered

Lesson 1

• Review of the rules for addition and subtraction of integers (although subtraction of integers was not dealt with explicitly)

• Rules for multiplying and dividing integers • Addition and subtraction problems [4 – (-3); -5 + (-2) and -3 + (-2)].

Lesson 2 • Review of the rules for multiplying and dividing integers • Worksheet on problems focusing on integer arithmetic (all four

operations)

Lesson 3

• Review of the rules for multiplying and dividing integers • Order of operations (BODMAS) • Commutative, associative and distributive properties

Table 2: Content covered in lessons conducted in School A School B situated in Gauteng is an elite private secondary school for boys. The observed lessons took place in a well-resourced classroom with ample space for the teacher to reach each student. The classroom was equipped with a computer, a data projector as well as white board (Talasi, 2007: 38-39). The class focused on in this study consisted of 24 male students across the racial spectrum. The lessons in School B were conducted solely in English by the teacher as well as the students.

The lessons which took place on four consecutive days were observed and video-recorded as part of the study conducted by Talasi (2007). The time-tabled length of a period at this school was 45 minutes. Table 3 below shows the amount of teaching time for each lesson. It seemed that only a portion of lesson 4 was video-recorded because the bulk of the lesson focused on the students working on a worksheet while the teacher assisted individual students.

Lesson Duration (in minutes) Lesson 1 (single period) 43:46 Lesson 2 (single period) 39:24 Lesson 3 (single period) 46:03 Lesson 4 (single period) 19:36 Total time 148:49

Table 3: Distribution of time for each lesson conducted in School B

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Introduction to algebra was the content focus of the lessons. The table below shows the breakdown of what was covered in each lesson:

Lesson Content covered

Lesson 1

• Guess my rule as an introduction to algebra • Introduction to addition of like terms • Conventions and terminology (constant, variable, terms, equation,

expression)

Lesson 2

• Applications of the terminology (constant, variable, terms, equation, expression)

• New terminology (monomial, binomial, trinomial and polynomial) introduced

• Problems involving addition of like terms e.g. 3x + 5x + 2x • Homework (Worksheet) on above content

Lesson 3

• Marking homework exercise • Expressing as a power and expanding exponential forms • Multiplication of variables • Integrated exercise covering terminology, addition and multiplication of

variables • Homework (Worksheet) on above content

Lesson 4 • Further work on Worksheet

Table 4: Content covered in lessons conducted in School B

Constitution of mathematics For the purposes of production of data from the lesson transcripts and analysis of data, I take as foundational the following propositions.

In the context of teaching and learning mathematics, pedagogic activity is taken to be generative of criteria for the recognition and realisation of mathematical objects or processes or procedures. Criteria are produced by both the teacher and students and are acquired by students. What comes to be constituted as mathematics and how mathematics is constituted in a particular pedagogic context is rendered visible through the criteria that circulate in that context (Davis & Johnson, 2007).

Further, criteria are understood to be generated through acts of evaluation over time. Evaluation distinguishes for students legitimate from non-legitimate knowledge statements in a particular pedagogic context. The generation of criteria for the recognition and realisation of specific mathematical objects or processes or procedures can be considered as constituting specific evaluative events. An evaluative event is taken as the unit of analysis for investigating the constitution of mathematics in the pedagogic contexts (Davis, Personal communication, Davis, 2005; Davis, 2001). The notion of an evaluative event as a methodological resource has been taken up by others (cf. Adler (2009), Parker (2008), Talasi (2007) and Pillay (2006)). Similarly here, the lesson transcripts for this study were chunked into evaluative events by marking out the time it takes for the presentation of a problem or question to students and the time it takes for the problem or question to be resolved.

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School A The lessons focused on the arithmetic of integers. Specifically, the teacher dealt with addition, multiplication and division of integers. In the lessons observed, the teacher did not explicitly deal with subtraction of integers. Instead a rule for dealing with subtraction problems such as -8 – (-2) was constructed. This point is elaborated below.

Students were introduced to rules for dealing with each operation without any explanation of why the rule works. For example, a rule for multiplying integers was introduced as “a positive multiplied by a negative is a negative”. The teacher then introduced an example such as 6 × -4 = -24. The teacher appealed to students’ experience of working with whole numbers and then superimposed the rules for working with the signs in addition to the calculation of whole numbers. It appeared as if the examples provided justification for the rules. At no point in the lessons, was an explanation provided for why 6 × -4 = -24. The teacher relied on the students’ experience of whole number calculations and the imposition of the rule for signs as a substitute for explaining why 6 × -4 = -24. Multiplication could be explained as repeated addition i.e. that 6 × -4 is the equivalent to 6 groups of -4, in other words (-4) + (-4) + (-4) + (-4) + (-4) + (-4) which equals -24.

The teacher treated the signs as objects independent of the numbers. The signs were disassociated from the numbers and appeared to function as adjectives to whole numbers. The problem was fragmented into steps with the first step focusing on the signs only and the second step focusing on the numbers only. The answer combined the results of step 1 and step 2. Step 1 therefore constituted a splitting apart or sundering of the sign and the number.

Sundering is understood in general as a pseudo-operation that separates a mathematical expression into a series of two or more expressions. The component expressions into which an expression is sundered are not unique and the production of the component expressions is based on the decision of the individual performing the sundering1(Cf. Davis, 2009).

Step 2 constitutes a putting together or the concatenation of the result of ‘operating’ on the sign and the result of operating on the whole number. Concatenation is understood in general as pseudo-operation which links two or more expressions into a composite expression. The order of the concatenated elements is decided by the individual performing the concatenation2 (Cf. Davis, 2009).

The criteria generated by the teacher therefore fragmented the problem into signs and whole numbers and reduced the problem to calculations with whole numbers. The following extract illustrates the splitting apart or sundering of the sign and the number and then the putting together or concatenation of the sign and the number.

Teacher: So there is a negative and a positive [referring to / 210 ÷− /]

Learners: Negative

Teacher: Ten divided by two?

Learners: five

Teacher: Negative five.

[Extract from School A Lesson 2] 1 For example, the expression /-7 +2/1 could be separated or sundered into /-/ and /7 +2/, or even /-/, /7/ and /+ 2/, or even /-/, /7/, /+/ and /2/.

2 For example, the negative sign /-/ and the number /7/ can be glued together to produce /-7/.

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The teacher first focused the attention of the students on the signs associated with the numbers. In fact, in this extract the operation is completely implicit. The word ‘and’ contextually substitutes for division. The students have to infer that the teacher is referring to division which is only visible as a symbol in the expression / 210 ÷− /. The criteria generated here may give students the impression that whenever they see a negative and a positive, the answer will be negative irrespective of the operation on the numbers. The following extract illustrates the confusion of the students.

Teacher: Minus ten minus two? [Teacher refers to /-10 – 2/]

Learners: Minus eight

Teacher: The signs are the same. Senza ntoni xa i-signs are the same? {What do we do when the signs are the same?}3

Learners: Positive

[Extract from School A Lesson 2]

In this extract the teacher again separated the signs from the numbers. When the teacher asked the question, “What do we do if the signs are the same?”, the operation was implicit. The teacher is referring to the addition of two integers with the same sign. The criteria generated by the teacher are ambiguous because it may give students the impression that if the signs are the same the answer must be positive irrespective of the operation referred to in the problem.

Sundering and concatenation appear to be standard features of these lessons. In the first lesson alone, there are 34 evaluative events dealing with the sundering of the number from the sign and, in six of these instances, the students either only give the sign or only give the number as an answer to the problem.

The sundering and concatenation of signs and numbers are evident in the rules for operating on integers. In addition to the rules for multiplication and division, the teacher generated three different rules for problems involving addition and subtraction of integers. The problem types and associated rules are listed blow:

Type 1: -10 + 2: Rule 1: “Subtract the bigger digit from the smaller digit and put the sign of the bigger digit.” (Extract from transcript School A Lesson 1)

Type 2: -8 + (-2): Rule 2: “When we see brackets that means what? Multiplication. But when there is a sign before a bracket we need to multiply what’s ever inside the bracket by that sign before we can add.” (Extract from transcript School A Lesson 2)

Type 3: 20 + � = 12. Rule 3: “When we are adding to a positive number that number is going to increase, but xa-u-addisha kwi positive number then i-answer yakho iya decreasea {when you are adding to a positive number then your answer will decrease} that means u-addisha what? {What are you adding?} What did you add? A negative number.” (Extract from transcript School A Lesson 2). The teacher’s rule can be restated as follows. When we add we expect the answer to get bigger. If the answer is smaller, then you must add a negative number. If the answer is bigger, then you must add a positive number.

The underlying logic of the teacher’s pedagogy is structured by the calculation procedures for dealing with operations on integers. The object to be acquired here is the selection of the correct procedure for the particular problem type, even though the types are merely different ways of expressing the same problem. In fact, type 2 allows the teacher to deal with subtraction of integers without the students being aware that 3 isiXhosa statements in the transcripts are underlined and the translation of the statements are placed in brackets.

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the problem involves subtraction. For example, /10 – (-2)/ is treated as a problem that involves multiplication and addition.

The calculation procedures regulate the production of the texts produced in these lessons. The conception of number dominant in this lesson is whole numbers. Signs are dislocated from the numbers therefore rendering all calculations involving integers into whole number calculations with additional rules for dealing with the signs. The sundering of signs from numbers and the existence of context-dependent rules for integer arithmetic produces a form of pedagogy that fragments mathematics for students.

School B The lessons in this classroom focused on an introduction to algebra. The teacher revisited number patterns through the game ‘Guess my rule’ as an introduction to the topic. The teacher viewed number patterns as a bridge into algebra proper.

In these lessons mathematics is constituted as adhering to conventions and using the correct terminology. Across the four lessons, 13 evaluative events focus on conventions used in mathematics. These included examples such as writing 2n instead of n2, using cursive letters to represent a variable, multiplication conventions e.g. 4.2 means 4 × 2 or 2n represents 2 × n, the convention used when the coefficient of a variable is 1 and convention for writing positive numbers e.g. 2 as opposed to +2. The extract below illustrates the teacher’s emphasis on conventions used in mathematics.

Teacher: Ok. Traditionally we always put up the numbers before the letter. So n2 is correct but we prefer to write the number before the letter. Also we don’t write the times between a number and the letter. When I write 2n what are we assuming? 2 times n. Another thing here with this x why did I write it with a curly x?

Student: We do not want a straight x

Teacher: Don’t call it out.

Student: [Indistinct]

Teacher: Ok. As Thabo said, if I were to write that curly x as a straightforward x what does that look like? Times times 2 or xx2 doesn’t make sense. So guys I’m going to be very strict with you. When you write your xs, it is the only letter that you have to change. I want you to write it as a curly x. You need to get into that habit. Ok. I will take off marks if you don’t.

[Extract from Transcript School B Lesson 1]

The teacher emphasised that she would deduct marks in a test or an examination if students do not use the convention for writing the variable x as a cursive x. This illustrates the importance that she placed on using the correct conventions. The extract also illustrates that the conventions are as important as the mathematical concepts in these lessons and therefore constitute objects to be acquired by students.

In addition to the emphasis on conventions, the lessons also focused on terminology used in algebra such as coefficient, variable, term, the difference between an expression and an equation, monomial, binomial, trinomial and polynomial. Across the four lessons, 18 evaluative events focused on defining terminology and recognising these definitions in the given expressions.

Teacher: Ok term. What do I mean by a term? Anyone came across that word before?

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Students: [No response from students]

Teacher: Ok. Let me give it to you before you come with all your English definitions [Writes the definition on the board.].An example will be better. They are separated by pluses and minuses.

[Extract from Transcript School B Lesson 1]

Although the content covered in the lessons conducted in School B differs from the content covered in the lessons conducted in School A, there is evidence of sundering and concatenation in both sets of lessons. The transcript below illustrates how the teacher in School B separated mathematical expressions into signs, numbers and variables and then dealt separately with each of them.

Student: Do you always do the numbers then the variables?

Teacher: Correct. The way I traditionally do the multiplying. Kori, Oscar you need to have a look at the board. The process I usually do. I first look at the signs. Is my answer going to be negative or positive? Then I look at my numbers, and then I look at my variables [writes /signs→ numbers → variables/]. First thing. Is my answer going to be positive or negative? Then I look at the big numbers and then I look at each of the letters separately.

Teacher: [Writes /(-2x4)(-3x5)/] Minus two x to the four times. Remember between the brackets is times. Minus three x to the five. Who can figure out what that answer could be? [Silence as students work on the task.] Wily, I am going to pick on you.

Wily: Six x to the power of 9.

Teacher: Six x to the power of 9. How many have got that one right? [Learners raise their hands.] Ok. First let’s look at the signs. Negative and negative, positive answer [writes /(-2x4)(-3x5) = +/] You obviously do not need to write that plus. The big numbers, two times three, six. x to the four times x to the five, if you are not so sure on this one, let’s see [writes /x . x . x . x/ ]

[Extract from School B Lesson 3]

There are 9 evaluative events involving sundering and concatenation across the four lessons, indicating that sundering and concatenation may be a feature of this teacher’s pedagogy. Similarly, this teacher uses the word ‘and’ to substitute for the operation. When she says ‘negative and negative’, the operation multiplication is implicit. Although the teacher indicated that (-2x4)(-3x5) means that /-2x4/ is multiplied with /-3x5/, the operation is implicit in the criteria generated by the teacher. The students have to assume that the operation referred to is multiplication when she deals with the signs.

Although the teacher used sundering as a technique to deal with addition of like terms, she tended to focus students’ attention on putting the bits together to produce a final answer. This emphasis on concatenation to produce the final answer is illustrated in the transcript below.

Teacher: Ok. What are you going to group first? [Referring to /6x2 − 8x − 3x2

+ 4 −2x − x/]? Ok. Go for it. Which one do you want to do first? The x squareds or the xs? Ok. That minus. Who does he belong to?

Wily: 8x

Teacher: Correct

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Wily: [Circles /6x2 / and /3x2

/ leaving out the signs]

Teacher: [Makes some warning noise.] Who belongs to the three? And the minus with the group.

Wily: [Includes the signs but seems to be puzzled.]

Teacher: What’s the answer going to be?

Wily: [Indistinct]

Teacher: Six x squared minus three x squared. [Learner silent] Looking at the numbers. Six minus three?

Wily: Three

Teacher: What’s the answer going to be? For that bit there [Pointing to /6x2 / and /-3x2

/] So 6 minus 3 is? What are we grouping?

Wily: x squared

Teacher: x squared [Writes /3x2 /]

[Extract from transcript School B Lesson 2]

When comparing what constitutes mathematics in these two very different classrooms, it appears that sundering and concatenation are common features of the pedagogic modalities in both classrooms. This commonality raises the question of what difference in pedagogy can then be attributed to the elite school producing better performance in mathematics than the working class school. Although this question remains a problem for further investigation, an initial analysis of the lessons conducted in the two schools highlights differences in the social organization and linguistic orientation which is discussed below.

Social organisation and learning The teacher in School B used a number of strategies to identify whether individual students were able to produce the legitimate texts. These strategies included asking individuals to answer questions rather than the whole class, engaging with individual students while they are working on an exercise and asking students to raise their hands if they get an answer right to a particular problem – this provided the teacher with a quick sense of how many students acquired the criteria and how many were still struggling. This form of social organisation is typical of Durkheim’s (1964) description of organic solidarity which places emphasis on individuals rather than on the collective. Organic solidarity focuses on the differences of individuals rather than their similarity. The individualising form of social organisation produces specialisation of members of a particular community. In their paper on three South African schools, Dowling & Brown (2009) compare a school with an individualising social organisation to a school with a collective social organisation. In the school with an individualising social organisation, the production and acquisition of specialised knowledge resides in individuals. In contrast, the collective production and acquisition of specialised knowledge was evident in the school with a collective social organisation.

The transcript below illustrates how the teacher asks individual students to answer questions.

Teacher: Let’s start with a straightforward one. [Writes /Simplify 3x + 5x + 2x/] Three x plus another five x another plus two x. Mpho, what do you think the answer could be?

Mpho: 10x

Teacher: Explain to the class how you got that.

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[Extract from transcript School B Lesson 2]

The above extract also illustrates the teacher’s strategy of asking individual students to explain how they have obtained the answer to particular problems. In this way the teacher has access to whether an individual student has acquired the criteria. However, at the same time, the student’s explanation served as a resource for those students who may not have acquired the particular criteria under discussion. School B can therefore be described as privileging a grammar-oriented culture of learning, i.e. a culture which emphasizes the rules underlying the text and which enables individuals to generate texts independently (Lotman, 1990).

A grammar-oriented culture depends on ‘Handbooks’, while a text-oriented culture depends on ‘The Book’. A handbook is a code which permits further messages and texts, whereas a book is a text generated by an as-yet- unknown rule which, once analysed and reduced to a handbook-like form, can produce new ways of producing further texts. (Eco in Lotman, 1990: xi)

In contrast to the individualizing culture in School B, the teacher in School A mostly asked questions directed at the whole class who chorused the answer. It was therefore very difficult for the teacher to determine whether individual students had acquired the criteria. The teacher was only able to determine whether the class as a collective has acquired the criteria. Mathematical knowledge resides in the collective rather than in individuals and students rely on the teacher as the resource rather than the logic of mathematics to regulate their activity.

The communalising social organization of Classroom A can be described as exhibiting mechanical solidarity - a form of social organization where the effect is to produce sameness rather than differences (Durkheim, 1964). Students are therefore largely undifferentiated thus hampering specialisation of individuals. The linguistic orientation can be described as text-oriented which privileges the reproduction of texts through imitation rather than the rules underlying the text (Lotman, 1990).

The two schools therefore differ in terms of social organization as well as linguistic orientation. It has to be noted that the class sizes are very different and the space in School B allows the teacher to move easily around the classroom to speak to individual students.

Concluding remarks The analysis of the lessons conducted in these schools populated by students with contrasting social class background shows that the general features of pedagogy displayed in these contexts appear to be similar with respect to sundering and concatenation of mathematical expressions. Further, analysis of the lessons illuminates differences with respect to social organization and linguistic orientation across the two schools. The initial analysis generates a few speculative questions for further investigation.

• Given that the reports on learner performance cited earlier in the paper show a difference in the performance of working class students compared to that of middle class students, are the criteria for the reproduction of mathematics generated in the different socio-economic contexts in fact the same? The analysis therefore highlights the need for a description of criteria generated in the two that goes beyond sundering and concatenation.

• In a social context exhibiting mechanical solidarity together with a text-oriented linguistic orientation, it appears that the reproduction of texts is primary. Whereas in a context exhibiting organic solidarity and a grammar-oriented linguistic orientation, the reproduction of rules for the composition of texts is

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primary. As such, the way in which mathematical activity is regulated and self-regulated is different in the two contexts. This raises the question: What is the relationship between the pedagogic use of sundering and concatenation, and the social organisation and linguistic orientation of the pedagogic context?

The above questions will be investigated more systematically in a broader study, involving a greater number of cases.

Acknowledgements This paper arises out of work on the Mathematics and Science Education Project at the University of Cape Town, and also forms part of the QUANTUM research project on Mathematics for Teaching, directed by Professor Jill Adler, at the University of the Witwatersrand, with co-investigators Dr Zain Davis (University of Cape Town) and Dr Diane Parker (National Department of Education). This material is partly based upon work supported by the National Research Foundation under Grant number FA2006031800003. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Research Foundation.

References Adler, J. (2009) A methodology for studying mathematics for teaching. Recherches en Didactique des

Mathématiques, 29 (1)

Bernstein, B. (1996) Pedagogy, Symbolic Control and Identity: theory, research, critique. London: Taylor & Francis.

Cooper, B. (1976). Bernstein’s codes: A classroom study. Brighton: University of Sussex, Education Area

Davis, Z. (2001) Measure for measure: evaluative judgement in school mathematics pedagogic texts. Pythagoras, Volume 56, 2-11.

Davis, Z. (2005) Pleasure and pedagogic discourse in school mathematics: a case study of a problem-centred pedagogic modality. Unpublished Ph. D. thesis. University of Cape Town.

Davis, Z. (2009). Notes towards a formal language for the description and analysis of criteria operative in the generation of statements for the elaboration of school mathematics. Presentation to the Group for the Study of the Constitution of Mathematics in Pedagogic Contexts, 3 April 2009, Cape Town, Mimeo.

Davis & Johnson (2007) Failing by example: initial remarks on the constitution of school mathematics, with special reference to the teaching and learning of mathematics in five secondary schools. In Setati, M., Chitera, N.& Essien, A. (eds) Proceedings of the 13th Annual National Congress of the Association for Mathematics Education of South Arica: The Beauty, Utility and Applicability of Mathematics, 2 - 6 July 2007, Uplands College, Mpumalanga, Volume 1, 121-136.

Dowling, P. & Brown. A. (2009) Pedagogy and community in three South African Schools. Sociology as Method: Departures from the forensics of culture, text and knowledge. Rotterdam: Sense.

Durkheim, E. 1964 (1893}. The Division of Labor in Society. Translated by Simpson. G. 1893 edition. New York.

Hoadley, U. (2005). Social class, pedagogy and the specialization of voice in four South African primary schools. University of Cape Town. An unpublished Ph.D. thesis.

Holland, J. (1981). Social class and changes in orientations to meaning. Sociology, 15, 1-18.

Lotman, Y. M. (1990). Universe of the Mind. A Semiotic Theory of Culture. Translated by Shukman, A. Bloomington and Indianapolis: Indiana University Press.

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Morais, A. M. (2002) Basil Bernstein at the micro level of the classroom. British Journal of Sociology of Education, 23(4), 559-569.

Parker, D. (2008) The specialisation of pedagogic identities in initial mathematics teacher education in post-apartheid South Africa. University of Witwatersrand. Unpublished Ph.D. thesis.

Pillay, V. (2006) An investigation into mathematics for teaching; the kind of mathematical problem–solving a teacher does as he/she goes about his/her work.University of Witwatersrand. An unpublished M.Sc. Thesis.

Schollar, E. (2008). The primary mathematics research project (2004-2007). Towards evidence-based educational development in South Africa. A final project report.

Talasi, T. (2007). Mathematical Knowledge for Teaching. A focus on the mathematical work of a grade 8 teacher when introducing algebra. Unpublished Master’s dissertation. University of Witwatersrand.

Walkerdine, V. (1988). The mastery of reason. London: Routledge.

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From whole numbers to real numbers: applying Rasch measurement to investigate Conceptual complexity in Key

Concepts Caroline Long

The Centre for Evaluation and Assessment, University of Pretoria

The developments in mathematics that take place in Grades 7 to 9 constitute critical nodes in a learner’s schooling. One of the major transitions to be made is from whole numbers to real numbers, which involves the understanding of rational (and irrational) numbers, and concepts such as ratio, proportion and percent. I hypothesize that these concepts are threshold concepts that provide the conceptual gateway to higher order concepts. The research problem is to describe the learning challenges and provide an array of insights and strategies that will inform teaching. The theory of conceptual fields provides the framework for this research (Vergnaud, 1988). The empirical phase builds on research from the TIMSS study (2003). The Rasch measurement model (Rasch, 1960/80) articulates the qualitative and quantitative aspects of the research. This paper provides an overview of the study and reports on an aspect of the data analysis.

From whole numbers to real numbers Usiskin (2005) highlighted the critical mathematical developments that take place during the “transition years”, most notably the transition from whole number to real number. In a detailed mathematical analysis of the unfolding development of number systems from whole numbers to fractional numbers, fractional numbers to rational numbers, and rational numbers to real numbers, Skemp (1971) notes that there are both smooth assimilations and difficult accommodations that take place in this unfolding. Each new system retains elements of the previous system, but introduces new notation, new meanings to operations, and contributes additional rules.

In each new [number] system there are sub-sets which are isomorphic4 with earlier systems. This [isomorphism] allows us to move freely from one number system to another, and also to mix systems provided that each one is operated according to its own methods. The overall result is a conceptual system of enormous power and flexibility (Skemp, 1971, p. 226).

It is this structure of number systems, where aspects of the system, for example the fraction notation used for both fraction and ratio, deceptively alike and yet subtly different, that is the Rubicon for most learners of mathematics as they make the transitions from whole numbers to rational numbers and then to real numbers. The assimilation into existing schemas and accommodating existing schemas to incorporate new concepts is a challenge for most learners. This accommodation is particularly challenging in the topic of percent. As Parker and Leinhardt (1995) note, percent is a dense language, where the changing meanings are not always accommodated into existing schema, and which may in fact be inappropriate schema. In the Chinese curriculum (Cai & Sun, 2002), the term percent ratio is used. This term indicates the close

4 Two number systems are isomorphic if 1) there is a mapping of one into the other that puts them into one-to-one correspondence. 2) under this mapping, sums and products are preserved.

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relationship between percent use and ratio. There is also however, a part whole use of percent which is more akin to fraction use (Parker & Leinhardt, 1995).

Challenges of major proportion in this domain are not new to the mathematics community. One of the major challenges confronted by the mathematicians of antiquity (340 B.C.) was the discovery that not all

numbers could be represented by an integer over an integer, of the type ba

. This threshold concept5

confronting the mathematicians was that there is no magnitude however small that will divide into both the side of a square and its diagonal. Similarly, there is no magnitude that will divide an integral number of

times into both the diameter and the circumference of a circle. The startling fact that numbers like 2 could not be written as a ratio of two whole numbers and the equally startling fact that two line segments exist for which there is no common measure caused great consternation in the Pythagorean community in as far back as 340 B.C. (Eves, 1990). The historical account of how Eudoxus resolved the crisis through formulating a definition of ratio, based on magnitudes that were independent of whether numbers were commensurable or incommensurable, is a telling account of how mathematics developed in response to a question that concerned the mathematical community (Eves, 1990, p. 150). The concept of an irrational number is a threshold concept that is difficult to learn, but once grasped provides the conceptual gateway to higher mathematics.

This critical mathematical node has caused and still causes a measure of consternation among teachers and learners of mathematics today. According to Eves, “the incommensurable case was relegated to an appendix [in some textbooks], to be covered at the instructor’s discretion, and sometimes it was omitted entirely, as being beyond the rigor of the course” (1980, p. 57). The omission of this crisis, and its resolution, from the high school topic of rational (and irrational numbers) is a tragedy, or at least a flaw, in that it leaves a conceptual gap in the understanding of irrational numbers. Irrational numbers are often introduced as non-terminating, non-repeating decimals, whereas the missing link is that the ratio of two incommensurable magnitudes will result in neither recurring nor terminating decimals.

My initial fascination with the concept of incommensurability, and irrational numbers (Long, 2006A), was temporarily replaced by a need to understand why the concept of ratio, and the consequent conceptualization of rational numbers, a necessary precursor to the understanding of irrational numbers, is difficult to learn. Essentially the need to explore the topics of ratio, and the related concepts of fraction, proportion, and percent, for the purposes of teaching and learning ratio, proportion and percent has directed this research study. A further goal is to contribute to an understanding of rational numbers, by careful scaffolding of these pivotal concepts.

Are ratio, proportion and percent difficult concepts? Usiskin (2005) describes one of the conceptual difficulties in the move from whole number to real number as the complexity of rational and real number concepts. For example ‘fraction’ has three different meanings, namely fraction as a number representing a quotient, as a number indicating a position between 0 and 1 and as a number that is not an integer. There are also different notations, fraction, decimal and percent, for the same number (Smith, 2002). Percent has a part whole meaning in addition to a ratio meaning depending on the context (Parker and Leinhardt, 1995). According to Parker and Leinhardt, teachers and learners use the different meanings embedded in each particular problem situation intuitively but the conceptual differences are not necessarily made explicit.

5 The term ‘threshold concept’ and its meaning is attributer to Meyer and Land (2005).

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The complexity of the real number system and indeed much of school mathematics provides a challenge for researchers in that “a single concept does not refer to only one type of situation, and a single situation cannot be analyzed with only one concept” (Vergnaud, 1988, p. 141). Another complexity is that “a single concept develops not in isolation but in relationship with other concepts, through several kinds of problems and with the help of several wordings and symbolisms” (Vergnaud, 1988, p.141). In order to encompass the complexity it is therefore important to study conceptual fields.

A conceptual field is defined as a set of situations, the mastering of which requires mastery of several concepts of different natures (Vergnaud, 1988, p.142). It is broad enough to accommodate the complexity of related concepts and processes that are required to solve a “bulk” of problem situations, but not too unwieldy that it cannot provide a manageable research domain. The multiplicative conceptual field6 is conceptualised as “all situations that can be analysed as simple and multiple proportion problems and for which one usually needs to multiply or divide. Several kinds of concepts are tied to those situations in addition to the thinking required to master [the problems]” (Vergnaud, 1988, p. 141). Concepts include multiplication and division, fraction, ratio, rate, rational number, linear and non-linear functions, vector spaces and dimensional analysis. The development of proficiency in this field begins in the early grades and continues through high school and further. The focus of this study is at Grades 7 to 9.

Vergnaud (1988) notes that the study of cognitive development can only meaningfully take place in problem contexts in that a concept is not a true concept unless it is operationalisable. The complexity of a problem depends “on the structure of the problem, on the context domain, on the numerical characteristics of the data, and on the presentation” (Vergnaud, 1988, p. 143). However, as noted by Vergnaud (1988) the impact of these factors depend on the cognitive level of the student.

TIMSS 1999 and 2003 Results A secondary analysis of the TIMSS 1999 and 2003 results investigating only the items which fell into the domain ratio, proportion and percent, showed that in general South African Grade 8 learners do not have an operational understanding of this domain; they are not able to solve problems in which these concepts are embedded. This predicament is not unique to South Africa as learning the concepts in this domain has provided challenges to learners in many countries (Hart, 1981; Parker & Leinhardt, 1995; Vamvakoussi & Vosniadou, (2007).

The secondary analysis of items was extended to the contextual data provided by the TIMSS study as it manifests in both the intended and the implemented curriculum. A comparison of the South African curricula with the TIMSS framework indicated insufficient attention to content detail (Mullis et al, 2003; Long, 2006B). This may be explained by the South African context, where it is expected that textbooks interpret the curriculum, and provide teachers with additional support, beyond what is stated in the curriculum documents. According to teacher reports, the topics relating to ratio were in some cases only taught to the “more able” children (Mullis et al, 2003; Long, 2006B).

It is well known that the South African education system has a large tail of inefficient and dysfunctional schooling (Taylor, Muller & Vinjevold, 2003), making it difficult to formulate policy or strategies for intervention based on this aggregated data alone. Therefore, in order to investigate the challenges confronting mathematics teaching and learning, selected TIMSS items were administered to learners at two functional schools at Grades 7, 8 and 9, comprising a sample of 330 learners. Subsequently interviews

6 The additive conceptual field includes the set of situations for which additive structures are required and sufficient. The algebraic conceptual field builds on the additive and multiplicative conceptual fields.

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were conducted with selected learners for the purposes of further investigating the cognitive development of learners at these grades (Long, 2008).

ASsessment instrument for ratio, proportion and percent items The requirement for designing any test instrument is to have a clear idea of the domain to be tested, and to define the construct of interest (Wright and Stone, 1979). This requirement is satisfied, more or less, in class tests and the matriculation examination, because experienced teachers have implicit knowledge of the curriculum and the kinds of questions that should be asked. In the case of this study the construct of interest is explicitly construed as the multiplicative conceptual field, comprising the elements, fractions, ratio, proportion and percent.

Secondly, a further requirement for any test is to develop an instrument which would provide reliable information about the domain to be tested. This precondition necessitated developing an instrument7 that could accurately target the population to be tested in terms of difficulty level. This requirement was assisted by having data available on the items from the TIMSS 2003 study so that the difficulty level of items for the South African context could be established (Long, 2008).

The third and fourth requirements, according to Wright and Stone (1979), are to demonstrate that the items when taken by suitable persons are consistent with expectations and that the patterns of learner responses are consistent with expectations. Finding unexpected responses in either the items or the learners requires an investigation of the items used to measure the construct and an investigation of the learner profile on the test as a whole (including curricula and instructional experiences) and consequently, if required, a deeper investigation of the construct being measured.

In all, 36 items8 from the TIMSS released items were identified as located in the multiplicative conceptual field; these items constituted the research instrument. Some of the items had been categorized as belonging to the domain ratio, proportion and percent, however other items also exhibiting multiplicative structures had been categorized as geometry, measurement, data handling (probability), and algebra (patterns). As a proxy for a pilot test, the difficulty levels of the items were checked against the TIMSS 2003 South African results. In order to include some items that could provide information at the lower levels of proficiency, it was decided to include six Grade 4 items.

The Rasch measurement model The prior empirical requirement of the Rasch model (Rasch, 1960/80), concurring with Wright and Stone, is to have an explicit understanding of the latent trait, that is, in the case of this research study proficiency in solving problems requiring the concepts fractions, ratio, proportion and percent, and to construct and refine an instrument made up of items that operationalise this trait. A feature of the Rasch model permits the discovery and amplification of item anomalies, which are inconsistent with the general expectations of the instrument. The test instrument used in this study proved to fit the requirement of the Rasch model of having a clearly defined domain.

7 The development of an instrument implies the development of a measure that makes the outcome meaningful. 8 One item was rejected as a typing error had been overlooked, making only 35 items. This item was a Grade 4 items..

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The researcher is required to further investigate (post-hoc) any particular item shown to be misfitting, and if necessary eliminate the item, while at the same time identify a plausible explanation of the item misfit in terms of its own characteristics. One item, on the topic of probability did not discriminate between learners of greater proficiency and learners of less proficiency, as measured on the test as a whole. On further investigation it was inferred that probability had not been taught and therefore this item was equally likely to be guessed by both groups of learners, or to elicit an intuitive response, by both groups of learners.

Likewise data from learners who for some or other reason do not perform as expected can be eliminated from the analysis, on the basis of explicit reasoned arguments, for the purposes of refining the instrument and establishing probabilistic estimates of item difficulty and learner ability on the same scale. If for example some learners had not taken the test seriously and guessed all the way through, this anomaly would be picked up and for the purposes of item analysis, these learners would be eliminated. The learners eliminated for the purposes of developing the scale, could still be allocated a location on the scale for other purposes.

The Rasch model, by locating both items and individual learners on the same scale, provides an answer to the severe criticism of applications of statistics in some psychological research that statistical methods can only provide information on groups of individuals9. This class of Rasch models enables “individual-centred statistical techniques” where each individual is characterised separately (Rasch, 1960/1980). It is also possible to develop a profile of individual learners. The clear articulation of qualitative and quantitative information is provided by this model.

Interpretation of the Rasch model The results of the assessment were captured and analysed using RUMM software (Andrich, Sheridan and Luo, 2005). The resulting person–item distribution map shown in Figure 1 indicates the difficulty of the items on the right, from high difficulty at the top to low difficult at the bottom. On the left the learners ranked from high proficiency as measured by this test, at the top, to low proficiency at the bottom. The item mean is set at zero. The location of learners is calculated in relation to the items initially through a probabilistic process, but also gauged in relation to each other10. For the purposes of this paper, it suffices to say that a learner at a particular point on the scale, for example -1.359, the locus of Item 4, has a 50% chance of getting Item 4 correct, a less that 50% chance of getting items higher than Item 4 on the scale correct, and a greater than 50% chance of getting items located lower than Item 4, for example Item 20, correct.

The spread of items and learners along the whole scale indicates that the test functions well. Information on the learners at all levels can be obtained. The lower level of the scale provides information on learners with lower proficiency. Even though five items were added from TIMSS Grade 4, there are some learners for whom even these items have little chance of eliciting a correct response.

Data analysis - percent For the purposes of this paper, I have limited the discussion to four items that included the concept of percent and that are located at different levels of difficulty (see Figure 1). From the perspective of teaching and learning, we ask, “What makes these items more or less difficult, and therefore easier or harder to learn?” Vergnaud (1988), drawing from empirical work on elements of the multiplicative conceptual field, 9 The details of this story may be read in the forward by Benjamin Wright in Rasch (1960/1980). 10 The probabilistic process is explained in Andrich and Marais (2008) and Bond and Fox (2007).

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reports that the complexity of items depends on the context, the mathematical structure, the presentation (or notation) and the number range used for the problem. The analysis of the items presented here draws on these four constructs.

In addition to providing a mathematical analysis of the items, additional information is provided by distractor analyses. The reasons for learners choosing specific distractors are inferred. These inferences were explored through subsequent interviews at a subsequent phase of the project. The analysis presented here however will remain at the level of a conceptual analysis of items, with empirical support provided by the Rasch model. The four items represented increasing levels of difficulty and by implication complexity. Table 1 presents the problem description, the context, the mathematical structure and the type of notation used.

Description Context Mathematical Structure Notation

4. At a play, 253

of the people in the

audience were children. What percent of the audience was this?

Everyday context Part-whole

253

children (3 out of 25)

100253

253

==x

x

bxax

ba

=

Fraction notation, percent notation

8. A shop increased its price by 20%. What is the new price of an item which previously sold for 800 zeds? A.640 zeds B. 900 zeds C.960 zeds D. 1000 zeds

Financial context, Price increase

800 + 20% of 800 100% of 800 + 20% of 800 = 120% of 800 Percent

notation

7. When a new highway is built, the average time it takes a bus to travel from one town to another is reduced from 25 minutes to 20 minutes. What is the percent decrease in time taken to travel between the two towns?

Measurement (time) Percentage ratio

25 – 20

255 converted to %

Percent notation

26. A computer club had 40 members, and 60% of the members were girls. Later, 10 boys joined the club. What percentage of members now are girls?

Everyday Part-whole Percentage ratio

60% of 40 = 24, 40 + 10 = 50

5024 convert

250224

××

Percent notation

Table 1: Percent items Item 4 (location -1.359), a relatively easy item was set in an everyday context that is familiar to most learners. The mathematical structure can be described as a part-whole or fraction-related use of percent. The equivalence concept can be used to convert from a fraction to a decimal. This fraction use of percent is the first use encountered in the curriculum, and it is thought by some research studies, reviewed Parker and Leinhardt (1995), that learners remain at this fraction level of understanding, and therefore do not relate to the ratio understanding of percent. Fraction notation is used, which makes the item more difficult than for example items 13, 1, 2, 17 and 22 (see Figure 1, locations -2.584 to -1.877) which used natural language. The next four items, 20, 6, 12, and 11 (locations -1.557 to -1.402) used fraction notation and were according to the empirical results found to be more difficult than the natural language of the first five items. While fraction notation is used in Item 4, learners are also required to operationalise the concept of percent and recognize percent notation.

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The setting for Item 8 (location -0.060) is a financial situation, which learners may encounter in a shopping experience11. The mathematical structure of the problem requires a ratio understanding of percent, and an understanding of percent notation.

The context for the problem in Item 7 (location 0.937), involves a change in time and a percent decrease. The mathematical construction entails picturing the two time periods (related to the distances travelled), finding the difference, and then finding the ratio of the difference to the original time. This ratio (in fraction form) is then transformed into a percentage. The increase in conceptual difficulty from Item 8 to Item 7 is substantial. This is reflected by the one logit difference between these items on the scale (see Figure 1).

The context for Item 26 (location 5.038) involves a calculation, a change in time and a further calculation. The context can be described as an everyday context, but the mathematical construction involves two steps with two different uses of percent. The first step is to find 60% of 40 members (the referent), which is 24 (the number of girls). The referent whole, 40, then changes (10 boys are added, making 50). The next requirement is to convert the 24 (girls) out of 50 (members) into a percentage. This problem involves switching between fraction and percent notations.

Three of the items (items 4, 8, and 7) were in multiple choice formats for which 4 or 5 response options were provided. Item 26 was presented in a constructed response format. Both item 8 and item 26 were included in the interviews, therefore more information is available on these items (See Long, forthcoming). For the present the discussion will be restricted to the information obtained from the test and from the distractor analyses. Item 8 is given as an example.

Figure 1: Person-Item location distribution LOCATION PERSONS ITEMS[locations] o = 2 Persons o | 7.0 | | | 34 | | 6.0 | | | | 5.038 | 5.0 | 26 | | | o | 4.0 o | | | | 27 | 3.0 o |

11 It is however the mathematical gaze which transforms the shopping experience into a mathematical task, rather than the act of shopping (Dowling, 1998).

26. A computer club had 40 members, and 60% of the members were girls. Later, 10 boys joined the club. What percentage of members now are girls?

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| o | o | oo | 35 2.0 | 30 31 ooo | o | oo | oo | 1.0 o | 28 0.937 ooooo | 7 ooooo | 32 oooooooooo | 10 29 ooooooo | 0.0 ooooo | 18 -0.060 ooooooooooooo | 9 8 15 oooooooooooo | 33 16 oooooooooooo | 5 25 14 oooooooooo | -1.0 ooooooooooooooooooo | 3 21 ooooooooooo | 23 -1.359 oooooo | 4 24 ooooooooo | 20 6 12 11 ooooooooo | -2.0 ooooooooo | 22 oooo | ooooo | 2 17 o | 13 1 ooooo | -3.0 | | o | | | -4.0 |

Analysis of Item 8 The RUMM software used to conduct the analysis provides a graph (see Figure 2) showing the choice of options for learners of different proficiency levels. For the purposes of analysis the sample was divided into five quintile groups with approximately 63 learners in each group. The mean of each quintile group is calculated on the performance of learners in that ordinal group on the test as a whole.

On the horizontal axis, the person locations are exhibited. The mean of each group is marked on the horizontal axis by pointers at - 2.09 (lowest quintile); -1.19; -0.66; -0.02; and 1.23 (highest quintile).

On the vertical axis the probability of attaining a correct response is exhibited. For example, a learner located at 1.23 (the highest quintile) on the horizontal scale, has a 0.72, or 72% chance of selecting the correct response (see line marked 3). The frequency of selecting D (line marked 4), is 0.16 or 16% , of selecting B (see line 2) is 0.07 or 7%, and of selecting A (see line 1) is 0.05 or 5%.

7. When a new highway is built, the average time it takes a bus to travel from one town to another is reduced from 25 minutes to 20 minutes. What is the percent decrease in time taken to travel between the two towns?

8. A shop increased its price by 20%. What is the new price of an item which previously sold for 800 zeds?

4. At a play, 253

of the people

in the audience were children. What percent of the audience was this?

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Figure 2: Item 8 distractor analyses for quintile groups

For learners at the low end of the scale, at –2.09, the frequency of selecting the correct option is 0.11, or 11% (see line 3). The frequency of selecting the option D (4) is 57%. This option was also the most likely choice for the second lowest quintile group. The selection of this distractor was due to a part-whole understanding of percentage, possibly influenced by the presence of 1000 as one of the options. The learners at the lower end of the scale, it is inferred have remained at the early cognitive stage of learning percentage, where fractions, decimals and percent notation are used interchangeably.

This disaggregated information shown in Figure 2, provides the researcher, and teacher, with relevant information. The single statistic, that 60% of the class got this item correct, is replaced with more useful information about different subgroups, which range from 72% (the highest quintile), to 11% (the lowest quintile).

The additional information about what kinds of conceptual errors are made is also useful. The information on the distractors as well as inferred reasons is shown in Table 2, which gives the information in numerical form.

Mean locations of 5 quintile groups Choice Inferred mathematical procedure -2.09 -1.19 -0.66 -0.02 1.23 C. 960 zeds Percentage increase 11% 24% 20% 62% 72%

D. 1000 zeds

Part whole 800 to 1000 increase represents 20% of final amount 57% 45% 40% 17% 16%

B. 900 zeds

Estimate a number greater than 800

21% 24% 26% 16% 7%

A. 640 zeds

Decrease by 20% rather than increase - misreading

11% 7% 6% 5% 5%

Table 2: Inferred mathematical procedure for responses by quintile group After option C and D, Distractor B elicited the most responses. This preponderance may have been due to the fact that learners estimated an answer greater than 800. Distractor A may have been selected on

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account of carelessness. Of the incorrect distracters, B is the closest to correct, when analyzing the mathematics involved. It is also interesting to note that there is a tendency to select distracter B by 20% of the lower group. This percentage increases to 26% in the middle group, and thereafter drops to 16% and 7% for the top group. According to Andrich and Styles (2008), this item may be better regarded as a polytomous12 item, with B being scored as partially correct.

Conclusion The transitions from whole numbers to real numbers, requires firstly an understanding of concepts such as fraction and ratio, elements of the multiplicative conceptual field. The shift from the “conceptualization of the ratios of integers from relations between numbers to numbers” (Vamvakoussi & Vosniadou, 2007, p.266) is necessary for understanding rational numbers, a threshold mathematical concept. This understanding of rational numbers is a necessary pre-curser to understanding irrational numbers, and therefore real numbers, which provides the conceptual gateway to higher order concepts.

Parker and Leinhardt (1995) remind us that percent is a complex construct where the underlying referent is not always made explicit. Percent builds on fractions and ratio; in some ways functioning like fractions and decimals, but in many ways differently, for example, you cannot add percentages unless the referent populations are equivalent or equal.

It is tempting to think that access to the historical account of discovering irrational numbers and the Eudoxian resolution of the crisis confronting mathematicians would enable understanding of incommensurability13. The answer emerging from this research study is that this historical insight may contribute to conceptual understanding, but the true concept cannot be attained without some cognitive engagement on the part of the learner assisted by carefully constructed situations that provide a context, with which learners feel familiar but which demand the development of the appropriate mathematical understanding.

In an appeal for greater regard for the complexity underlying the language of percent, Parker and Leinhardt (1995) propose that attention be given to the underlying mathematical structure of percent and the complex relationship between percent and the related concepts of ratio, proportion, fractions and functions. Vergnaud (1988) proposes that research into the learning and teaching of mathematics requires both a mathematical view and a psychological view, and that conceptual analysis of the mathematical situations, and observation of cognitive schemes that are engaged in the solving of these situations, be described using mathematical terminology. This approach enables the teacher and researcher to locate the learner on a mathematical path.

It has been acknowledged that educational measurement as it manifests in testing and statistics may be a rather blunt instrument for the fine qualitative distinctions required to understand teaching and learning.

12 Polytomous is used here as referring to more than two responses, which differs from dichotomous items where answers are right or wrong. In this case Response B would be allocated 1, being partially correct, and Response C, allocated 2, as correct.

13 The introduction via decimal numbers was invented centuries later than the discovery of irrational numbers circumvents the

very crisis and consequent understanding of irrational numbers and incommensurability, these concepts being the precursors to

the understanding of non-terminating and non-repeating decimal numbers.

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The Rasch measurement model however, is sensitive to qualitative nuances and indeed requires subject experts to engage at all levels of the assessment process, from the conceptualization of the construct to the analysis of the data. In this study the Rasch model and the aligned software have enabled finer degrees of understanding of this complex field.

References Andrich, D., Sheridan, B. E. & Luo, G (2007). RUMM2020 (Rasch Unidimensional Measurement Model,

version 4.3) [Computer software]. Perth, Western Australia: RUMM Laboratory.

Andrich, D., & Marais, I. (2008). Introductory Course Notes: Instrument design with Rasch, IRT and Data Analysis, Perth: University of Western Australia.

Andrich, D. & Styles, I. (2008). Distractors with information in multiple choice items: a rationale based on the Rasch model. Paper presented at Third International Rasch Conference. Perth, University of Western Australia.

Bond, T. & Fox, C. (2007). Applying the Rasch Model: Fundamental Measurement in the Human Sciences. New Jersey: Lawrence Erlbaum Associates

Cai, J., & Sun, W. (2002). Developing Students' Proportional Reasoning. In B. Litwiller (Ed.), Making Sense of Fractions, Ratios and Proportions,195-212. Reston, V.A: NCTM

Department of Education. (2002). Revised National Curriculum Statement Grade R-9 (Schools) Mathematics. Pretoria: Department of Education,

Eves, H. (1980). Great moments in mathematics( before 1650). The Mathematical Association of America

Eves, H. (1990). An Introduction to the History of Mathematics. Pacific Grove CA: Brooks/Cole-Thomson Learning.

Hart, K. (1981). Children's understanding of mathematics: 11-16. Oxford: John Murray.

Litwiller, B. (Ed.). (2002). Making sense of Fractions, Ratios and Proportions. Reston, Virginia: National Council of Teachers of Mathematics.

Long, C. (2006A). Beyond the bad news: Diagnostic implications of TIMSS 2003, Association for Evaluation and Assessment in Southern Africa. University of Johannesburg

Long, C. (2006B). The conceptual and historical development of ratio, SAARMSTE Conference. University of Pretoria.

Long, C. (2008). Ratio, proportion and percent: Using the Rasch model to establish focus points. Paper presented at the Third International Rasch Conference. Perth, University of Western Australia.

Meyer, J. H. F., & Land, R. (2005). Threshold concepts and troublesome knowledge (2): Epistemological considerations and a conceptual framework for teaching and learning. Higher Education, 49, 373-388.

Mullis, I. V. S., Martin, M. O., Smith, T. A., Garden, R. A., Gregory, K. D., Garden, R.A.,, Gonzalez, E. J., et al. (2003). TIMSS Assessment Frameworks and Specifications 2003. Chestnut Hill, MA: International Study Centre, Boston College.

Parker, M., & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65(4), 421-481.

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. (Expanded edition (1980) with foreword and afterword by B.D. Wright, (1980) Chicago: The University of Chicago Press. ed.). Copenhagen: Danish Institute for Educational Research.

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Reddy, V. (2006). Mathematics and Science Achievement at South African Schools in TIMSS 2003. Cape Town: Human Sciences Research Council.

Skemp, R. (1971). The psychology of learning mathematics. Harmondsworth: Penguin.

Smith III, J. P. (2002). The Development of Students' Knowledge of Fractions and Ratios. In B. Litwiller (Ed.), Making sense of Fractions, Ratio and Proportions, 3-17. Reston, Virginia: The National Council of Teachers of Mathematics.

Taylor, N., Muller, J. & Vinjevold, P. (2003). Getting schools working: Research and systemic reform in South Africa. Cape Town: Pearson Education.

Usiskin, Z.(2005). The importance of the transition years, Grades 7-10, in school mathematics. UCSMP Newsletter. Spring 2005

Vergnaud, G. (1988). Multiplicative Structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades. Hillsdale, NJ: National Council of Teachers of Mathematics.

Vergnaud, G. (1998). A Comprehensive Theory of Representation for Mathematics Education. Journal of Mathematical Behaviour, 17(2), 167-181.

Wright, B. D., & Stone, M. H. (1979). Best Test Design. Chicago: MESA Press.

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LEARNERS FAIL MATHEMATICS: AN ARGUMENTATIVE ESSAY ON CONTRIBUTING FACTORS

Mdumiseni G. Mabizela

Mbatshazwa Secondary School, Nkandla, KwaZulu-Natal

While mathematics can be considered a mother of all sciences, the way in which learners under-perform in this subject leaves a lot to be desired. The essay presented below is an argumentative exposition which seeks to navigate through factors believed to be contributory in the way learners fare in mathematics. The argument is supported by local and international literature, and this makes the study theoretical rather than empirical. It will be noted in the study that some factors considered are social, systemic, cognitive and emotional. The author is the teacher of mathematics at rural high schools for 14 years; his experience coupled with literature study is helpful in the development of the forthcoming argumentation.

INTRODUCTION Mathematics is considered to be the ‘mother of all sciences’ as it cuts across all sciences. “Today mathematical methods pervade literally every field of human endeavor and play a fundamental role in economic development of a country” (Tella, 2007: 149). While it is true that mathematics is a prerequisite for university entrance in a number of study fields such as engineering and accounting to mention a few, the way learners under-perform in this subject remains a growing concern. People having interest in this subject (i.e. teachers, learners, parents, specialists, administrators and politicians) have a tendency of defending their positions by pointing fingers of judgement at each other. They do this disregarding the fact that the status quo is the result of the interplay of contributing factors. The study presented here seeks to sail through some of these factors with the purpose of understanding why learners fail mathematics.

Makgato and Mji (2006: 253) in their study quote a number of studies by different international groups that indicate the fact that South Africa scores last in terms of mathematics performance among many countries. They attribute this to out-dated methods of teaching, lack of teachers’ knowledge, the under-qualification of teachers and pathetic teaching facilities in classrooms (ibid: 254).

The importance of mathematics cannot be over-emphasized. The learner or student taking mathematics has their dreams centred in this subject. If they fail mathematics or pass it poorly they consider their academic future to have been adversely affected. “So many

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academic difficulties and failures are concentrated in this one subject that it has become a major selective filter in the educational system” (Ignacio, Nieto & Barona, 2006: 17). Take for example 20 learners who have failed mathematics in your class after having applied for admission to engineering, health sciences, accounting, microbiology and statistical mathematics at a university. It does not matter that they have passed all other subjects and have received their matriculation certificates, but the absence of the subject mathematics takes them out of the academic equation, and their dreams are shattered.

Purpose and significance of the study The purpose of this study is to make use of literature sources to understand reasons why learners under-perform in mathematics. The paper presented here does not seek to answer all questions but it reflects on those issues arising from literature study. International studies have been used in addition to local ones; otherwise this paper focuses on the situation as it prevails in South Africa. It remains true that most challenges we encounter on teaching and learning mathematics in South Africa also have an international trend.

The subject mathematics has a history of difficulty. It has always been perceived as a subject that can only be mastered by the gifted few Dörfler (2007: 106). The myths surrounding the learning of mathematics have over the years contributed in implanting fear in many learners and students of the subject. This fear which still prevails in the learners of today ultimately develops the mindset that one cannot be good enough for mathematics. It is incumbent upon teachers of mathematics to understand challenges around the teaching and learning of this subject. This would assist them in working out strategies aimed at helping learners to improve their performance in mathematics. This study will serve as a beneficial reading source for those practitioners who are involved in mathematics education, as well as other stakeholders and researchers.

Theoretical framework

School mathematics fails to incorporate learners’ experiences taken from reality into classroom learning. Learners are often bombarded with the heavy jargon of the subject rather than let them proceed smoothly from the known to the unknown. Learners encounter mathematics in their daily lives and can (if allowed to do so) use their experiences to solve mathematical problems through informal ways. This is what Barnes (2004: 54) calls horizontal mathematisation. Then, according to Barnes (ibid.), using reality can help the learner proceed from horizontal mathematisation to vertical mathematisation, i.e. solving problems using mathematical language. This would hopefully eradicate fear and enhance performance in mathematics. Engaging with the learners in the classroom (Maoto & Wallace, 2004: 67 and Brodie, 2006: 14) would allow them to informally express their ideas, construct their meanings and understanding. That would be a good foundation from which the teacher can build-up and formalize mathematics. The tendency of teachers of

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jumping into formal mathematics from the onset confuses learners and makes the subject more abstract and difficult for them (learners).

ATTITUDE AND INTEREST Learners experience challenges in all subjects, not only in mathematics. They end up developing a negative attitude towards mathematics because of the way in which it is taught as a ‘dry’ discipline which is not connected to reality. The use of the complicated jargon and symbols adds fuel to the fire. The negative attitude developing in learners, results in lack of interest which affects the way learners react or listen to the teacher (Tella, 2007: 150). The teacher as a result becomes demotivated and might end up using ‘chalk and talk’ as the easiest strategy of disseminating knowledge (ibid.). He/she “may not also bother to vary his [/her] teaching styles to suit individuals; therefore the cycle goes on” (ibid.).

Tella (ibid.): “the negative attitude towards the subject is passed down from one generation of pupils to another and therefore the cycle keeps enlarging”. The negative attitude and lack of interest contribute to learners’ poor performance in mathematics. According to Ignacio, Nieto and Barona (2006: 18) attitude is related to academic self-image and motivation for achievement.

ANXIETY AND FRUSTRATION Tella (2007: 149) enumerates factors responsible for learner poor performance in mathematics, and among them mentions anxiety. Anxiety is the feeling of fear which is caused by learner’s continuous failure to successfully complete mathematics tasks. This frustrates the learner and inhibits them from unleashing their innate mathematical abilities. They, thus, end up performing poorly in mathematics. Dörfler (2007: 106) throws a challenge towards mathematics educators and researchers to change to the better the phenomenon of “widespread anxiety and frustration on the part of many learners of mathematics”.

IDENTITY AND SELF CONCEPT With continuous poor performance in mathematics, learners end up not identifying themselves as achievers in the mathematics learning community; they consider themselves as marginally part of it (Anderson, 2007: 8). Learners identify themselves as members of the mathematics learning community “if they can fit together the small pieces of the ‘mathematics puzzle’ delivered by the teacher” (ibid.). Some learners also attribute their inability to perform well in mathematics to nature: that they are born mathematically disadvantaged; but this is not supported by any scientific evidence (ibid: 11).

“Self-concept and performance mutually influence and determine each other” (Ignacio, Nieto & Barona, 2006: 26). A learner with a positive concept will engage positively in

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mathematics tasks and that might boost their performance. On the contrary, if poor performance persists, their self-concept is adversely lowered. Their failure becomes internalized and they begin to believe that it is beyond their ability to do well in mathematics (ibid: 18). Learners identifying themselves as poor performers consider mathematics as too abstract, not making sense and needing “a special aptitude and gift possessed only by a few” and they “consider themselves in principle unable to understand” (Dörfler, 2007: 106).

Tella (2007: 149 – 151) also writes about personality, self-concept, motivation and self-confidence as contributing to mathematics performance. The writer (ibid.) also comments about performance as being associated with the role models that learners choose.

“Those who have high achievers as their models in their early life experience would develop the high need to achieve, while those who have low achievers as their models, hardly develop the need to achieve” (ibid: 151).

LANGUAGE OF TEACHING Mathematics in South Africa is taught either in English or Afrikaans. The majority of South African learners do not speak English or Afrikaans as their home language. Learners in rural schools are faced with a challenge of struggling to master the language and struggling to master the subject mathematics with all its abstractions at the same time. Setati (2006: 97) comments that although learners prefer to be taught in English, the use of the learners’ first language can support students in learning to communicate mathematically. The language of teaching can serve as a filter in the education process since it “can be used to exclude or include people in conversations and decision-making processes” (ibid: 98). It also appears in Civil’s study that language contributes in mathematical understanding (Civil, 2006: 34). In the study conducted by Van der Berg (2008: 9) it appears that those learners who speak English at home and are taught in English perform better than those who do not speak English at home but are taught in English.

FINDING SENSE IN MATHEMATICS Learners with mathematical difficulties find mathematics too abstract and not making sense (Dörfler, 2007: 106). As it has been mentioned earlier on, learners can only find sense in mathematics if it is connected to reality. Teaching learners to use mathematics to solve real-life problems can help in bridging the gap between in-school and out-of-school mathematics in problem-solving (Civil, 2006: 37). The main challenge lies within teachers’ inability “to uncover the mathematics in everyday contexts” and this is attributed to their academic training (ibid: 38).

Ignacio, Nieto and Barona (2006: 17) agree with writers quoted above that most learners perceive mathematics as “difficult, boring, not very practical, abstract, etc.” If the learner

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fails to connect mathematics to reality they will find it difficult and will ultimately perform poorly in it.

TRAINING AND PROFESSIONAL DEVELOPMENT Looking at the way in which mathematics is taught in schools Jaworski (2004: 17) questions the education of teachers. This is to question whether the programmes offered at teacher training institutions (i.e. universities) are adequate in addressing challenges surrounding the teaching of mathematics. Are student teachers adequately prepared to help learners make connections between mathematics and real-life? “The most urgent task in school mathematics education is to produce teachers who are mathematically well-informed” (Wu, 2006: 1). The failure of universities to produce enough teachers who are mathematically well-informed is of greatest concern. This is exacerbated by the fact that most learners good in mathematics do not choose a teaching career (Makgato & Mji, 2006: 254).

Wu (2006: 2) emphasizes the importance of professional development and in-service training for teachers who are already in practice. These programmes should focus more on content knowledge (ibid.). Most programmes on professional development focus more on administrative or management issues and less on content knowledge. During professional development programmes teachers should be made aware that mathematics is not the science of ‘hidden agendas’ but an ‘open book’, and they must spread this message to their students (ibid: 8). This would help eradicate the notion that mathematics is abstract and has nothing to do with reality.

TEACHING METHODS AND KNOWLEDGE The way in which mathematics is taught in schools has a bearing on the way learners perform in it. Therefore “the methodology of teaching has a definite role to play” in learners’ performance in mathematics (Ignacio, Nieto & Barona, 2006: 28). Activities in class should be interesting and relevant to real-life in order to foster learning (Jaworski, 2004: 17). Learners should be allowed to investigate and discover mathematical facts for themselves; thus investigative mathematics teaching is encouraged (ibid: 24). If learners are not actively involved, the teaching and learning process becomes mundane, and learners will not respond to mathematical challenges in an appropriate manner, and the teacher would end up providing answers to his/her questions (ibid: 18). Tella (2007: 149 & 151) also supports investigative teaching as opposed to the “use of traditional chalk and talk methods”.

To motivate learners to perform well, game activities should be used in mathematics teaching (Tella, 2007: 152). Learning as they play makes the process less painful for pupils. When mathematics teaching is made interesting and when learners’ individual differences

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are taken into consideration, their feeling of esteem would be enhanced and attitude towards mathematics improved (ibid: 155).

Wu (2006: 13) suggests that students should be taught to reason logically since “logical reasoning is the back-bone of problem solving”. This would be effective when teachers in lower grades become aware of the global structure and coherence of mathematics (ibid: 14 & 15). Wu talks about longitudinal coherence of school mathematics across all grades (i.e. R – 12) (ibid.). Teachers in lower grades should be aware of the needs of advanced grades (ibid.).

Teacher’s knowledge of the content also has the contribution in the way in which learners perform in mathematics. Makgato and Mji (2006: 254 & 259) mention outdated teaching practices, lack of content knowledge, under-qualification of teachers, overcrowded non-equipped classrooms and syllabus non-completion as factors leading to learners’ poor performance in mathematics.

For teachers to possess content knowledge, and curricular knowledge is not enough; “teachers need a third kind of knowledge…pedagogical content knowledge (PCK)” (Kazima & Adler, 2006: 35). PCK is defined as the subject matter knowledge for teaching, this transforms knowledge “into a form that learners can comprehend” (ibid.). PCK suggests that it is not enough for teachers to be experts in doing mathematics, but should be able to “unpack or decompress the mathematics they know and have learned, so as to be able to make it accessible to learners” (ibid: 36). The teacher’s knowledge of the mathematics content which is not supported by the teacher’s knowledge of how to teach mathematics has a detrimental contribution to learners’ performance in the subject.

SYSTEMIC DEMANDS The teaching of mathematics is embedded in the context of the systemic challenges. Most of those challenges are beyond the teacher’s control. There is very little that a teacher can do about overcrowded non-equipped classrooms (Makgato & Mji, 2006: 254 and Tella, 2007: 149). It is not the mathematics teacher’s responsibility to buy equipment, to build classrooms and to employ additional teachers of mathematics.

Jaworski (2004: 17) talks about challenges and the complexity of the teacher’s job. Those challenges and complexities include classroom forces that are not understood by office-based departmental officials who visit schools only few times a year. Teachers are faced with day-to-day demands on poor physical conditions of the school, poor attitude of learners and shortage of text-books (ibid.). Text-books, even when there is enough supply of them, very few are adequate in terms of addressing all needs as expected by official curriculum documents. The culture of the school also acts as one of the determinants of performance. The outside visitors of the school may come up with many good suggestions and theories, but these are usually counteracted by school cultures (ibid: 19).

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SOCIAL BACKGROUND In their study, the Organisation for Economic Co-operation and Development (OECD) through their Programme for International Student Assessment (PISA) found that “the relationship between performance in mathematical literacy and social background is relatively strong” (McGaw, 2004: 656). Schools situated within poor communities usually perform poorly compared to schools in richer communities. Van der Berg (2008: 4) attributes this to a legacy of historical educational inequalities in South Africa. Van der Berg (ibid: 5) also quotes the findings of SACMEQ (South African Consortium on Monitoring Education Quality) which witnessed an urban-rural gap in domains such as mathematics. This brings in the point that the “socio-economic status of pupils is an important determinant of learning outcomes” (ibid: 7). Other writers who support the notion of the SES (socio-economic status) of the learner as having a direct positive association with mathematics achievement are Kanyongo, Schreiber and Brown (2007: 38).

The family background of the learner can also determine his or her performance, especially in mathematics. This is also supported by Van der Berg: “pupils [who] live with parents [have] a strong advantage in both reading and mathematics” and that mother’s education reflects in the learner’s mathematics scores (Van der Berg, 2008: 9). This is to suggest that learners with educated mothers perform better in mathematics than those whose mothers are not educated.

FINDINGS AND DISCUSSION The literature that has been reviewed in this study indicates that there is no one factor that can be singled out as being responsible for learners’ poor performance in mathematics. The phenomenon is the result of the interplay of contributing factors. In the same way, there is no one individual person that can be singled out to carry the blame. Stakeholders of mathematics education include university student-teacher educators, school-based educators, office-based educators, educational authorities (locally, provincially and nationally), parents and learners themselves. If all these parties can work together hand-in-hand, factors contributing to mathematics poor performance could be overcome.

The tendency of one party pointing fingers at the others can only worsen things by developing the culture of self-protectiveness. Although it is true that all stakeholders should be responsible, it is equally true that the role of parents and learners is not a professional one as it is the case with the others. University educators carry the professional obligation of producing mathematically well-informed teachers. Authorities in the department of education carry a professional obligation to develop the mathematics curriculum that addresses the social and economic challenges of the nation. Office-based educators have a professional obligation to utilize governmental facilities and time they have to offer support not criticism to school-based educators. School-based educators carry a professional

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obligation to interpret and implement the aspirations of the curriculum in the way that will benefit learners of mathematics and carry our nation forward. During implementation they should present mathematics in the way that will inculcate among learners intrinsic motivation to achieve; they should not instill fear. Parents need to work with teachers in ensuring that their children go to school and participate in mathematics activities accordingly. Learners have to cooperate with teachers with clear understanding that it is their future that is being prepared. They are not studying mathematics to please their teachers or parents.

In my literature study I managed to identify six types of issues that contribute to learners’ poor performance in mathematics, viz.: (i) affective and cognitive issues, (ii) methodological and pedagogical issues, (iii) teacher education and professional issues, (iv) systemic issues, (v) linguistic issues, and (vi) socio-economic-political issues.

Affective and cognitive issues include the state of mind that the learners have regarding their ability to perform well in mathematics. This includes the unhealthy attitude, anxiety, frustration and the way they identify themselves as learners of mathematics. Methodological and pedagogical issues include the methods used by teachers to facilitate mathematics learning. Mathematics is usually presented in a fragmented way and coherence is not observed from one class to the next. Teachers also usually fail to make connections between the mathematics of the school and real-life, this complicates mathematics to learners and they end-up regarding the subject abstract and meaningless. Teacher education and professional issues involve the way in which pre-service and in-service education and training are being conducted; as well as the way educators develop themselves through further study and participation in subject-specific conferences, e.g. AMESA (The Association for Mathematics Education of South Africa). Systemic issues are beyond the control of the individual teacher, they are the part of the system. The shortage of teachers, the overcrowded classrooms and the shortage of resources are all systemic issues which should be addressed by provincial and national educational authorities. Linguistic issues have to do with the language of teaching and the barriers it presents to the learners who do not use it as their first language. Finally the socio-economic-political issues are concerned with the social and home environment from which the learner comes. The social milieu of the school also contributes to learner performance. In South Africa the socio-economic status is to a greater extent determined by the historical legacy of oppression and apartheid.

RECOMMENDATIONS My study is theoretical in that it does not include any empirical methods of research. Proper fieldwork can help in challenging the issues I have raised here, or it can also complement the arguments I have presented. I further recommend that teachers of mathematics improve their knowledge of the subject as well as their pedagogical knowledge. They do not only

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need to know mathematics but also how to lead pupils from what they know to what they do not know. This they can accomplish by making connections between out-of-school mathematical experiences and in-school mathematics. Teachers should make and encourage learners to investigate the mathematics they learn at school. This investigative teaching and learning of mathematics would help in making the subject practical rather than theoretical. This would also improve the attitude and eventually the aptitude. To develop themselves professionally, teachers should take part in further educational programmes and be members of professional organizations which are specific to mathematics.

CONCLUSION The study has endeavoured to navigate through the factors that contribute towards the under-performance of learners in mathematics. It has shown that the phenomenon is the result of the interplay of many contributing factors. It would be over-ambitious to claim that all factors have been covered in this study. But it is true that the factors covered here do have some contribution in learners’ mathematics performance. It is also important to note that the study is limited in the sense that it is a theoretical/philosophical study, and no empirical methods have been employed; but articles used are written by people who have conducted empirical studies.

REFERENCES Anderson, R. (2007). Being a Mathematics Learner: Four Faces of Identity. The Mathematics Educator,

17(1), 7 – 14.

Barnes, H. (2004). Realistic mathematics education: Eliciting alternative mathematical conceptions for learners. African Journal of Research in SMT Education, 8(1), 53 – 64.

Brodie, K. (2006). Teaching mathematics for equity: Learner contributions and lesson structure. African Journal of Research in SMT Education, 10(1), 13 – 24.

Civil, M. (2006). ‘Working towards equity in mathematics education: A focus on learners, teachers, and parents’. Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of mathematics Education. Mérida, México: Universidad Pedagógica Nacional. Vol. 1, 30 – 50.

Dörfler, W. (2007). ‘Making mathematics more mundane – a semiotic approach’. Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education. Seoul. Vol. 1, 105 – 108.

Ignacio, N.G., Nieto, L.J.B. & Barona, E.G. (2006). The affective domain in mathematics learning. International Electronic Journal of Mathematics Education, 1(1), 16 – 31.

Jaworski, B. (2004). ‘Grappling with complexity: Co-learning in inquiry communities in mathematics teaching development’. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education.

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Kanyongo, G.Y., Schreiber, J.B. & Brown, L.I. (2007). Factors affecting mathematics achievement among 6th graders in three sub-Saharan countries: The use of hierarchical linear models (HLM). African Journal of Research in SMT Education, 11(1), 37 – 46.

Kazima, M. & Adler, J. (2006). Mathematical knowledge for teaching: adding to the description through a study of probability in practice. Pythagoras, 63, 35 – 48.

Makgato, M. & Mji, A. (2006). Factors associated with high school learners’ poor performance: a spotlight on mathematics and physical science. South African Journal of Education, 26(2), 253 – 266.

Maoto, S. & Wallace, J. (2006). What does it mean to teach mathematics for understanding? When to tell and when to listen. African Journal of Research in SMT Education, 10(1), 59 – 70.

McGaw, B. (2004). ‘Australian Mathematics Learning in an International Context’. Paper presented at the Conference of the Mathematics Education Research Group of Australasia, Mathematics Education for the Third Millennium: Towards 2010. Townsville, 27 – 30 June 2004.

Setati, M. (2006). ‘Access to mathematics versus access to the language of power’. Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague. Vol. 5, 97 – 104.

Tella, A. (2007). The Impact of Motivation on Student’s Academic Achievement and Learning Outcomes in Mathematics among Secondary School Students in Nigeria. Eurasia Journal of Mathematics, Science & Technology Education, 3(2), 149 – 156.

Van der Berg, S. (2008). ‘How effective are poor schools? Poverty and educational outcomes in South Africa’ Revised version of paper delivered at SACMEQ International Invitational Research Conference. Paris, September 2005.

Wu, H. (2006). ‘Professional Development: The Hard Work of Learning Mathematics’. Presented at the special session of the Mathematical Education of Teachers at the Fall Southern Section Meeting of the American Mathematical Society. East Tennessee State University, Johnson City, Tennessee, 16 October 2005.

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Remarks on the inefficient use of time in the teaching and learning of mathematics in five secondary schools

Roger MacKay

Group for the Study of the Constitution of Mathematics in Pedagogic Contexts School of Education, University of Cape Town

This paper focuses on the use of time for teaching and learning mathematics in selected grade 8, 9 and 10 classes of five high schools with a predominantly working-class student population. Similar to the findings of Davis & Johnson (2007), this study confirms that a large proportion of instructional time was spent on worked examples. In addition, the study reports a significant loss of instructional time for teaching and learning and questions the potential impact of the loss of instructional time on student achievement in mathematics, by commenting on the inefficient use of the actual available time.

INTRODUCTION This paper reflects on a series of observations of teaching and learning mathematics in five working class schools in the Western Cape. Three consecutive lessons in Grades 8, 9 and 10 at each of the schools were observed. Two of the schools are ex- Department of Education and Training (DET) schools situated in townships in greater Cape Town, the third is an ex-DET school situated in a Cape Town suburb, the fourth is an ex-Model C school with an entire student population from black and coloured working class families, and the fifth is an ex-House of Representatives (HOR) school with a student population comprising predominantly coloured children from working class families.

This paper builds on a similar study, undertaken by Davis & Johnson (2007), which focused on pacing and what is constituted as mathematics in five Western Cape secondary schools. Three of these schools form part of the present study. Davis & Johnson (2007) concluded that even though teachers worked at a slow pace, learners failed to acquire the requisite mathematical knowledge since the mathematical procedures taught did not foreground mathematical objects. Learner performance records, gathered early in 2007, of individually written tests and examinations showed that learners at these schools generally performed poorly in the assessment tasks; the learner performance records of the schools observed in 2009 show that the afore-mentioned has not changed significantly since 2007. Whilst these observations conducted in 2009 have numerous purposes, this paper using the construct opportunities-to-learn (Consortium on Chicago School Research, Smith, Smith & Bryk, 1998; Reeves & Muller, 2005; Davis & Johnson, 2007) investigates the use of time allocated for instruction and its potential impact on learner achievement.

The discussions of this paper are based on 41 lesson observations across the five schools: three consecutive lessons per teacher in one Grade 8, one Grade 9 and one Grade 10 class per school. The schools will be identified as School 1 to School 5 (Schools 1 to 3 were also observed in 2007).

The series of observations were observed by 5 researchers, one per school, and a number of camera operators. In addition to the video recordings, each researcher made observation notes and generated analytic descriptions of the teaching that had taken place. These video recordings and observation notes were later considered and discussed at a series of regular research meetings, with the purpose of (1) developing a language of description of the constitution of mathematics in pedagogic contexts, (2)

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designing a strand of intervention strategies and activities aimed at improving the quality of mathematics teaching in these five schools, and (3) developing a series of research papers. Prior to the observations, the researchers developed a methodological framework to guide their observations as well as the production of their observation notes. The observation notes recorded the unfolding of the classroom activities and events, with a particular focus on recording time spent.

Opportunities-to-learn The author’s research interest is situated in the Consortium on Chicago School Research literature with regard to time in education. The study explores the relations between the allocated instructional time and actual instructional time. Allocated instructional time is the time made available by the educational authorities whilst actual instructional time refers to the time of the lesson in which actual teaching and learning takes place. Furthermore this paper adopts the Consortium on Chicago School Research’s construct of opportunity-to-learn which attributes success in summative assessments as a direct consequence of the amount of exposure to the content (see, for example, Reeves and Muller, 2005; Smith, Smith and Bryk, 1998; Millot and Lane, 2002).

The literature shows evidence of extensive research (see Smith, 1998; Smith, Smith & Bryk, 1998; Millot & Lane, 2002; Davis & Johnson, 2007) conducted over long periods of time, which attempts to make explicit the pedagogic categories facilitating the recurring theme of slow pacing as a major indicator of poor performance of students in predominantly working-class schools. The work done by the Consortium on Chicago School Research indicates “the amount and quality of time available for instruction directly shapes school outcomes and student achievement” (Smith, 1998, p. 3). Millot & Lane (2002, p. 210) mirror this by indicating that there is clear evidence in education literature of a relationship between time-on-task and educational outcomes at the macro level. Therefore, it is evident that there is a tension between the allocated instructional time, as indicated by education authorities, and the time of teaching and learning actually experienced by learners in the classroom. Whilst it may be difficult for the teacher to manage the difference between the allocated instructional time and the actual instructional time, during which actual teaching and learning takes place, there is extensive recognition for the value of time in learner achievement (Smith, 1998, pp. 7-15).

In a study, conducted in Chicago’s Elementary Schools, of the relationship between an allocated instructional day and the amount of instructional time that elementary school children actually experience, Smith (1998) noted a loss of 60 minutes (or twenty percent) in well-managed instructional settings and a further loss of 44 minutes (or about thirty five percent) in poorly managed instructional settings. Morning directions, class transitions, moving to and from the bathroom, moving to and from non-academic activities and other tasks like sharpening of pencils, getting books out, collection and handing back tasks, and checking homework contribute to this loss of time. Many of Chicago’s charter schools, consequently, opted to extend the school day by 60 to 90 minutes. However, a study by Rossmiller (1983), (quoted in Millot & Lane, 2002, p. 211), showed that during a typical school year of 1 080 hours, children actually received academic instruction for 34 percent of the time. A similar loss of actual instructional time is noted in another study conducted in Peru (Amadio, 1997, quoted in Millot & Lane, 2002, p. 211) where absenteeism (amongst teachers and students), strikes, semi-official holidays, preparation for parades and other non-academic activities account for some twenty to fifty percent loss of actual instructional time. But being able to measure instructional time at the macro level is far less difficult than measuring what is really going on in the classroom at any given moment. Smith (1998) and other studies conducted by the Consortium on Chicago School Research, attempted to address this task but limited their investigations to how much time was used, on any given day, for actual teaching and learning.

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The analysis of time cited above is based on the data collected and reported on by Smith (1998) during a study in 15 Chicago Public Schools (eight elementary and seven high schools) on school and classroom observations conducted from 1994 to 1996. A team of researchers observed more than 200 teachers across more than 1 000 lessons in mathematics, language and social studies in grades two, five, eight, nine and ten. Observation logs of researchers recorded different types of activity that allowed a grouping of the activities into instructional and non-instructional categories. Field notes, weekly and monthly school schedules, and interviews with more than 200 teachers and administrators were additional sources of information. In the analysis of a typical school day, Smith refers to the concepts: allocated instructional day, the actual school day, and the instruction actually experienced.

At this juncture, it may be useful to illuminate these concepts for one of schools in this study and to simultaneously consider the relationship between the time allocated for instruction and the time actually used for the teaching and learning experience. At one of the schools where the observations took place, the school day commenced at 8.00am and ended at 3.00pm, with the official duration of lesson periods being 60 minutes. This constituted a school day of 420 minutes. The day started with a 10-minute register session and was punctuated with two 25-minute breaks and notice bulletins before each break (± 5 minutes in total). The allocated instructional day, which the school officially calls contact time, is 355 minutes. The duration of the actual school day = 355 minutes – (transition time + time at the end of the lesson to gather belongings) = 355 minutes – 6(10 + 5) minutes = 265 minutes. At this stage, thirty seven percent of school time had been lost and lesson periods had been reduced to 45 minutes, a loss of twenty five percent of instructional time. The instruction actually experienced for each lesson period = 45 minutes – (the time taken up by teacher to get the learners settled, hand back tasks, give instructions, hand out worksheets) = 45 minutes – 10 minutes = 35 minutes, which is equivalent to fifty eight percent of the allocated instructional time, that is, of actual lesson time.

The literature concerning time use in schools confirmed that increased time of exposure to the content increased the opportunity for the learner’s acquisition of knowledge. This study attempts to endorse that reduced time-on-task at the observed schools contributes to the poor performance of its learners in summative assessment tasks. But having noted, in the observed schools, how much time was available for teaching and learning, it was also important to consider how effectively and efficiently the available time was used to transmit and acquire mathematical knowledge.

A DESCRIPTION OF THE PROBLEM In order to generate a description of the observed lessons with respect to the use of time, the lesson transcripts were analysed in terms of the time spent on new worked examples and activities unrelated to mathematics. Specifically, the lessons were analysed in terms of the amount of time spent on worked examples. This time included time spent on marking homework and class exercises.

Table 1 shown below illustrates the time used for exposure to standard procedures by means of worked examples, students and teachers working through worked examples, marking of worked examples, and activities unrelated to the topic of discussion or teachers and learners just doing nothing. Having described time in one of the afore-mentioned categories, it was possible to calculate the percentage of total time taken up by each of the activities in each mathematics lesson, as well as across a series of lessons. The duration of the lessons, under current review, was measured according to the school timetables. Consequently, transition time between lessons and assembly outside the classroom were included in the ‘Activities unrelated to the lesson topic’ category. Generally, at all five schools, teaching and learning

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activities were interrupted by the wail of the siren at the end of each lesson, resulting in a continuance of activities beyond the allotted time.

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min min % min % 8 120 103 85,8 17 14,2 9 95 65 68,4 30 31,5 1 10 155 94 60,6 61 39,4 8 45 37 82,2 8 17,8 9 135 112 83,0 23 17,0 2 10 135 112 83,0 23 17,0 8 135 88 65,2 47 34,8 9 135 108 80,0 27 20,0 3 10 135 106 78,5 29 21,5 8 135 111 82,2 24 17,8 9 135 119 88,1 16 11,9 4 10 135 125 92,6 10 7,4 8 135 115 85,2 20 14,8 9 135 117 86,7 18 13,3 5 10 135 125 92,6 10 7,4

Table 1: Time usage per grade per school

The summary description of time utilisation in all 41 lessons across Grades 8, 9 and 10 across the five schools in Table 1, indicates the amount of time used to elaborate mathematics through worked examples. A study of this data shows that an inordinate amount of class time was spent on worked examples, yet learners appeared to be incapable of completing similar tasks successfully. The lesson observations revealed that very little time, if any, was used to develop the mathematical ideas underpinning the procedures. Considering that the observations were conducted in most of the classes across the five schools over a period of three consecutive days, the data summarised in Table 1 begins to suggest that the trend of teaching by means of worked examples is more common than first suggested in study conducted by Davis & Johnson (2007). Observations conducted over a longer continuous period of time should enable the production of more generalised descriptions with regard to the individual classes as well as the individual teachers.

The current observations revealed that mathematics is still taught predominantly by means of exposure to worked examples and learners are generally not provided with mathematics underpinning the procedures used in the worked examples. This means that unless students are exposed to mathematics underpinning the procedures in worked examples, as Davis & Johnson stated in 2007, they will not perform satisfactorily in tests and examinations.

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Figure 1 graphically illustrates that, on average, about eighty one percent of the time was spent on exposition of worked examples (in the form of new problems, marking homework or completing class exercises). The relationship of the bars for Schools 1 through 5 suggests that the trends were similar across the five schools. It was also noted that, on average, almost nineteen percent of the allocated lesson time was lost as a result of activities unrelated to the actual lesson topic.

Figure 1: Comparison of the time used for exposure of worked examples (including time used for marking homework & class exercises) and unrelated activities, per school across the three grades, as

well as the mean across all five schools Figure 2 illustrates a similar pattern for each grade across all five schools. Once again the results are fairly similar with regard to exposure to worked examples in its different forms, with about eighty one percent of the time being spent on dealing with worked examples and about nineteen percent of the time on unrelated activities. A strong correlation between what happened across schools and what happened within grades is suggested by the data in Table 1 and the graphs in Figure 1 and Figure 2.

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Figure 2: Comparison of the time used for exposure of worked examples (including time used for marking homework & working through class exercises) and activities unrelated to the lesson topic,

per grade across the five schools Whilst a detailed analysis of the lessons has not been done, the observations appear to reveal that at these five schools teachers generally use communalism to transmit and verify acquisition of knowledge. The teacher accomplished this communalism by means of whole class activity in which the teacher led the discussion, generally employed public questioning and allowed choral responses by the learners. Learners themselves contributed to this communalism when they recruited answers and assistance from their peers when working on the chalkboard. However, evaluation of the acquired knowledge through tests and examinations is generally conducted as an individual activity.

The study therefore raises a question for further investigation: To what extent does a communalising pedagogy in a pedagogic context characterised by reduced instructional time and slow curriculum pacing contribute to learner performance in mathematics?

CONCLUSION The intention of this research paper was to describe the practices with respect to the teaching and learning of mathematics in five working class schools in the Western Cape with respect to the use of time. The study revealed that even though an inordinate amount of time is spent on the exposition of worked examples, learners still do not achieve success in summative assessments. It was also intended that this research paper would reveal the need for further investigation, particularly in regard to the pedagogic style of the teacher.

This study, whilst it confirms the finding of Davis & Johnson (2007) that a large proportion of instructional time was spent on worked examples, also noted that an inordinate amount of the allocated instructional time, ranging from about twelve to twenty nine percent, was spent on activities unrelated to the lesson topic. In a particular instance mentioned in this paper, only fifty eight percent of the actual lesson period was available for teaching and learning. The unrelated activities included transition time, excessive time spent on recruiting the learners’ attention, learners’ ineptitude at completing simple tasks or

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carrying out instructions, and teachers and learners simply doing nothing. A markedly reduced actual time of instruction precipitated.

Further investigation may lend support to the suggestion of this paper that, in addition to the amount of allocated instructional time that is lost, the actual time available for teaching and learning is also not used efficiently. Thus, acknowledging that other factors may also contribute to a slowing in the pace of transmission and acquisition of mathematical knowledge, this paper suggests that the lack of individualising may be a primary factor affecting learners’ acquisition of mathematical knowledge and curricular pacing, that is, the pacing of the curriculum across different grades..

ACKNOWLEDGEMENTS I acknowledge the work of Shaheeda Jaffer, Zain Davis, Anthea Roberts and Yusuf Johnson, as the other observers and for the critical discussions of earlier drafts of this paper. The review of Kaashief Hassan and comment of other members of the Group for the Study of the Constitution of Mathematics in Pedagogic Contexts, and the hospitality of the teachers and learners at the five schools, are also acknowledged.

References Amadio, M. (1997). Primary Education: Length of Studies and Instructional Time. Educational Innovation and Information 92: pp. 2 – 7. Berliner, D.C. (1979). Tempus Educare, In: Petersen, P.L. and Walberg, H.J. (Eds). Research on Teaching, Berkeley, CA: McCutchan Publishing. Davis & Johnson (2007b). Failing by example: initial remarks on the constitution of school mathematics, with special reference to the teaching and learning of mathematics in five secondary schools. In Setati, M., Chitera, N. & Essien, A. (eds) Proceedings of the 13th Annual National Congress of the Association for Mathematics Education of South Arica: The Beauty, Utility and Applicability of Mathematics, 2 - 6 July 2007, Uplands College, Mpumalanga, Volume 1, 121-136. Millot, B. & Lane, J. (2002). The Efficient Use of Time in Education. Education Economics Vol. 10, No. 2: pp. 209 – 228. Reeves, C & Muller, J. (2005). Picking up the pace: variation in the structure and organisation of learning school mathematics. Journal of Education, 37: pp. 103 – 130. Smith, B. (1998). It’s about time: Opportunities to Learn in Chicago’s Elementary Schools. Improving Chicago’s Schools. ED 439 216, UD 033 439. Available at: http://www.consortium-chicago.org/publications/pdfs/ Smith, J.B., Smith, B. & Bryk, A.S. (1998). Setting the pace: opportunities to learn in Chicago public elementary schools. Consortium on Chicago School Research. Available at: http://www.consortium-chicago.org/publications/pdfs/p0d04.pdf

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ASSESSING THE ACADEMIC BEHAVIOURAL CONFIDENCE (ABC) OF FIRST-YEAR STUDENTS AT THE CENTRAL UNIVERSITY OF

TECHNOLOGY, FREE STATE

SHEILA MATOTI Central University of Technology, Free State

KAREN JUNQUIERA

Central University of Technology, Free State The researchers conducted a study with the aim of assessing the academic behavioural confidence of first-year students enrolled in two B.Ed. (FET) programmes offered by the School of Teacher Education at the Central University of Technology, Free State. Bandura’s (1986) Social Cognitive Theory is the overarching theoretical framework of the self-efficacy construct and therefore also for this study. A quantitative approach was followed and the Academic Behavioural Confidence scale (ABC) designed by Sander and Sanders (2006), was adopted for use in the study. The study sought to determine whether a significant difference in the academic behavioural confidence of the first-year students within the Natural Sciences and Economic and Management Sciences programmes does exist. Based on the findings, some recommendations on dealing with first-year students have been made. INTRODUCTION

Preparedness of both teachers and students has been found to be related to self-efficacy or confidence. Self-efficacy has been identified as an important predictor of teacher effort and persistence (Emmer & Hickman, 1991); instructional effectiveness (Ashton & Webb, 1986); and efficient classroom organisation, planning and practices (Pajares, 1992). With reference to the students, research has shown that differing levels of student preparation affect students’ academic confidence in different subjects and consequently their study patterns and behaviours. Regarding mathematics as a specific subject domain, it has been reported that learners make judgements about their mathematical capabilities based on accumulated knowledge and experiences. Based on these judgements students tend to see themselves as either mathematically inclined or disinclined. Student perceptions of mathematics efficacy are shaped by a number of personal, environmental and behavioural factors. Learners make judgements about their capabilities based on comparisons of performance with peers (Schunk, Hanson, and Cox, 1987) successful and unsuccessful outcomes based on standardised and authentic measures, and feedback from teachers (Bouffard-Bouchard, 1989), parents and peers. These sources of information about their capabilities accumulate within individuals to form perceptions of mathematical competencies. Pajares and Miller (1994) found that self-efficacy has a high explanatory and predictive power for mathematics performance.

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Gender effects in mathematics learning and mathematics self-efficacy have also been reported. Confidence in learning mathematics has consistently emerged as an important component of gender-related differences (Casey et al, 2001, Vermeer et al, 2000). Confidence in mathematics affects learner behaviour differently. Ryan and Pintrich (1997) showed that students who perceived themselves as cognitively competent were less likely to avoid seeking help, whereas students who were unsure of themselves were more likely to feel threatened when asking their peers for help and more likely to avoid seeking help. The self confidence of girls in mathematics has been reported to be lower than boys (Casey et al, Vermeer et al, 2000). First-year university students also make judgements about their capabilities in mathematics which stem from their past school experiences. These judgements of themselves could affect their confidence. Hence they have to be helped and supported during their transition from high school to higher education in all subjects, including mathematics. It is against this background that the researchers decided to conduct a study with the aim of assessing the Academic Behavioural Confidence of first-year students enrolled in two different programmes offered by the School of Teacher Education at the Central University of Technology, Free State (CUT). These programmes are the B.Ed. (FET): Natural Sciences and the B.Ed. (FET): Economic and Management Sciences. Students who enrolled for the programme specialising in Natural Sciences (NS) have to take mathematics as a compulsory major subject during their first year with the option of continuing with the subject during their second and third years of study. This implies that they had to have mathematics at school in grades 10 – 12. Students who enrolled for the programme specialising in Economic and Management Sciences (EMS) have the option of taking mathematics as a major subject, although it is not compulsory. This implies that only some students in the EMS programme had a mathematics background formed in grades 10 – 12. The mathematical focus of the students in the two programmes is thus very different. Relevant literature has been consulted in order to inform the study and develop a suitable theoretical framework. THEORETICAL PERSPECTIVES BASED ON LITERATURE

The Social Cognitive Theory is the overarching theoretical framework of the self-efficacy construct (Bandura, 1986). This theory will now be addressed. The Social Cognitive Theory

Through the Social Cognitive Theory, Bandura advanced a view of human functioning that accords a central role to cognitive, vicarious, self-regulatory, and self-reflective processes in human adaptation and change (Pajares, 2002). People are viewed as self-organising, proactive, self-reflecting and self-regulating rather than as reactive organisms shaped and shepherded by environmental forces or driven by concealed inner impulses. From this theoretical perspective, human functioning is viewed as the product of a dynamic interplay of personal, behavioural and environmental influences. Bandura (1986) calls this three-way interaction of behaviour, personal factors (in the form of cognition, affect and biological events), and environmental influences or situations the “triadic reciprocality.” Within the classroom setting students’ academic performances (behavioural factors) are influenced by how learners themselves are affected

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(cognitive factors) by instructional strategies (environmental factors), which in turn builds itself in a cyclical fashion. Pajares (2002) argues that of all the thoughts that affect human functioning, and standing at the very core of the social cognitive theory, is self-efficacy beliefs. The next section therefore looks at self-efficacy. Self-efficacy Beliefs

Self-efficacy beliefs are defined as “people’s judgements of their capabilities to organise and execute courses of action required to attain designated types of performances” (Bandura, 1986: 391). Self-efficacy beliefs provide the foundation for human motivation, well-being, and personal accomplishment. It is also a critical determinant of self-regulation. In a later edition Bandura (1995: 2) defines self-efficacy as “the belief in one’s capabilities to organise and execute the courses of action required to manage prospective situations” while Pajares (2000) defines it as people’s confidence in their ability to do the things that they try to do. The ideas that come through in these definitions are one’s judgements, beliefs and confidence in one’s abilities to perform a particular task. Bandura’s (1997) key contentions regarding the role of self-efficacy beliefs in human functioning is that “people’s level of motivation, affective states and actions are based more on what they believe than on what is objectively true” (p.2). How people behave can often be predicted by the beliefs they hold about their capabilities rather than by what they are actually capable of accomplishing, as these self-efficacy perceptions help determine what individuals do with the knowledge and skills they have (Pajares, 2002). Pajares (2002) warns that people’s self-efficacy beliefs should not be confused with their judgments of the consequences that their behaviour will produce. They, however, do help determine the outcomes one expects. Confident individuals anticipate successful outcomes. Students who are confident in their social skills anticipate successful social encounters, while those who are confident in their academic skills expect high marks in examinations and expect the quality of their work to reap personal and professional benefits. The opposite is true of those who lack confidence. Students who doubt their social skills often envisage rejection or ridicule even before they establish social contact. Likewise, a lack of confidence in academic skills could lead students to anticipate a low grade or pass in a particular subject or course. Because individuals operate collectively as well as individually, self-efficacy is both a personal and a social construct. Collective systems develop a sense of collective efficacy, that is, a group’s shared belief in its capability to attain goals and accomplish desired tasks. Schools develop collective beliefs about the capability of their learners to learn and of their teachers to teach. As a result of shared beliefs, schools enhance the lives of their students and teachers by creating environments conducive to the desired tasks. Organisations with a strong sense of collective efficacy exercise empowering and vitalising influences on their constituents. How self-efficacy beliefs influence human functioning

The role of self-efficacy on human functioning can be summarised as follows:

• Self-efficacy beliefs can enhance human accomplishment and well-being. • Self-efficacy beliefs help determine how much effort people will expend on an activity, how long

they will persevere when confronting obstacles, and how resilient they will be in the face of

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adverse situations. The higher the sense of efficacy, the greater the effort, persistence and resilience.

• Self-efficacy beliefs influence an individual’s thought patterns and emotional reactions. High self-efficacy helps create feelings of serenity and composure in approaching difficult tasks and activities. People with low self-efficacy may believe that things are tougher than they really are, a belief that fosters anxiety, stress, depression, and a narrow vision of how best to solve a problem. Consequently, self-efficacy beliefs can powerfully influence the level of accomplishment that one ultimately achieves.

Sources of self-efficacy

Self-efficacy stems from four sources: mastery experience, vicarious experience, verbal persuasion and physiological states (Bandura, 1993). The following explanations of these sources are taken from Pajares (2002). Mastery experience refers to how one interprets the results of previous performance, and this has been found to be the most influential source. Individuals engage in tasks and activities, interpret the results of their actions, use the interpretations to develop beliefs about their capability to engage in subsequent tasks or activities, and act in concert with the beliefs created. Outcomes interpreted as successful raise self-efficacy; those interpreted as failures lower it. In addition to interpreting the results of their actions, people form their self-efficacy beliefs through the vicarious experience of observing others perform tasks. This source of information is weaker than mastery experience in helping create self-efficacy beliefs, but when people are uncertain about their own capabilities or when they have limited prior experience, they become more sensitive to it. The effects of modelling are particularly useful in such contexts. Individuals also create and develop self-efficacy beliefs as a result of social persuasions they receive from others. These persuasions can involve exposures to the verbal judgements that others provide. Persuaders play an important part in the development of an individual’s self-beliefs. Effective persuaders must cultivate people’s beliefs in their capabilities while at the same time ensuring that the envisaged success is attainable. Physiological states, referred to as somatic and emotional states, such as anxiety, stress and mood also provide information about efficacy beliefs. People can gauge their degree of confidence by the emotional state they experience as they contemplate an action. Strong emotional reactions to a task provide cues about the anticipated success or failure of the outcome. When they experience negative thoughts and fears about their capabilities, those affective reactions can themselves lower self-efficacy perceptions and trigger additional stress and agitation that help ensure the inadequate performance they fear. The sources of self-efficacy information are not directly translated into judgements of competence. Individuals interpret the results of events, and these interpretations provide the information on which judgements are based. How people select information, integrate it, interpret it and make recollections, influence judgements of self-efficacy.

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Academic confidence

Bandura (2001) uses the terms confidence and self-efficacy interchangeably, while Sander and Sanders (2004) argue that the two concepts are distinct but related. They see academic confidence as a new construct which is distinct from its parent concept, self efficacy. Academic confidence, therefore, has its theoretical foundations in Bandura’s work of self-efficacy. Sander and Sanders (2004) argue that academic confidence is a mediating variable between the individual’s inherent abilities, their learning styles and the opportunities afforded by the academic environment of higher education. In their comparative study of two distinct groups, namely Medical and Psychology students using an Academic Confidence Scale (ACS), they concluded that the scale could be used to identify students who are not coping well with a course of study as well as in the exploration of the impact of teaching and learning methods. We now look into academic behavioural confidence. Academic Behavioural Confidence

Academic behavioural confidence (ABC) is conceptualised as how students differ in the extent to which they have a strong belief, firm trust, or sure expectation of how they will respond to the demands of studying at a higher education institution (Sander and Sanders, 2004; 2006). They further argue that ABC is distinct from the academic performance aspirations that students may have, although the two may be related to some extent. This confidence applies to the demands of the course as a whole rather than to individual module specific issues where self-efficacy measures would be more appropriate (Sander and Sanders, 2007). Sander and Sanders (2006) developed the ABC scale for use as a survey instrument to assess the confidence that higher education students have in their own anticipated study behaviours in relation to their degree programme. It was developed within an ethos of using survey techniques to try and understand students within the large student groups that many higher education lecturers have to teach. The main argument raised for its use was that with large classes, there is little or no opportunity for the informal interactive discourse possible within small groups and which allows the teacher to understand his/her students or help and guide them by effective teaching. This argument holds true for this particular study as well. Lecturers have to deal with large groups of students who come from different home and school backgrounds. The transition from high school to higher education could prove to be a traumatic experience for some. From the literature we deduce that students form self-efficacy beliefs about their own capabilities. These beliefs are reinforced by past experiences, by school, teachers, parents and peers. The beliefs they have about themselves are related to their level of confidence which could be used as a predictor of their academic performance in later years, including academic performance at universities. Gender differences have also been linked to self-efficacy beliefs and academic performance. It is against this background that the study was undertaken to explore the differing levels of academic behavioural confidence among two first-year education student groups enrolled during 2009 in the School of Teacher Education at the Central University of Technology, Free State.

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RESEARCH QUESTIONS

The study sought to answer the following research questions:

• Is there a significant difference in the academic behavioural confidence of the first-year students within the Natural Sciences (NS) and Economic and Management Sciences (EMS) programmes in the School of Teacher Education at the Central University of Technology, Free State?

• Is there a significant difference in the academic behavioural confidence between male and female students within these two programmes?

HYPOTHESES

Two hypotheses have been formulated for the study:

• There is a significant difference in the academic behavioural confidence of the first-year students within the Natural Sciences (NS) and Economic and Management Sciences (EMS) programmes in the School of Teacher Education. It was hypothesised that the Natural Sciences (NS) students have greater academic behavioural confidence than the Economic and Management Sciences (EMS) students.

• There is a significant difference in the academic behavioural confidence between male and female students in the two programmes. It was hypothesised that male students have greater academic behavioural confidence than female students.

RESEARCH METHODOLOGY

The study is mainly quantitative research with qualitative interpretations of the results. It was an exploratory and descriptive survey of the perceptions of the first-year students enrolled in the Natural Sciences and Economic and Management Sciences programmes in the School of Teacher Education at the Central University of Technology, Free State. The Academic Behavioural Confidence scale (ABC) designed by Sander and Sanders (2006), was adopted for use in the study. The ABC scale was used as it had already been tested for internal reliability by its developers. The researchers wanted to determine if the scale, when used in a different context, could yield similar results, as some problems addressed in the scale are context-specific. Minor adaptations regarding the wording of certain statements as well as the reduction of the response categories, were made to comply with the purpose of this study. Statements addressing common areas were also grouped together. The adapted scale was then used as an instrument to collect data from first-year education students in the two mentioned programmes. Academic behavioural confidence was measured on a three-point Likert-type scale comprising of the categories “Very confident”, “Slightly confident” and “Not confident”. The scale was administered to the students by the researchers with the assistance of the subject lecturers at the beginning of February 2009. This point in time at the beginning of the year was chosen as the registration of new students had been completed and it was felt that all students had settled in their class groups according to a given timetable. The questionnaire was administered immediately at the end of a lecture which the whole programme group had to attend and was collected directly after completion.

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Microsoft Excel was used to record and analyse the data and it was done by the researchers themselves. The following analyses of the captured data were made:

• A calculation of the mean scores (MS) for each student’s responses to the 24 questions (statements).

• A calculation of the mean scores (MS) per question (statement) considering all students’ responses. • A calculation of the standard deviation (SD) for each student’s responses to the 24 questions. • A calculation of the standard deviation (SD) per question (statement) considering all students’

responses. • An analysis of the variance between the NS and EMS groups. • An analysis of the variance by gender.

PRESENTATION AND ANALYSIS OF FINDINGS

An analysis of the findings of the study will now be presented. Biographical Information

Table 1 provides a breakdown of the analysis of respondents according to gender and programme that existed among the 235 respondents that formed part of the study. It shows that 112 (47.7%) respondents were male, while 122 (51.9%) were female. One respondent did not indicate his/her gender. Of the 235 respondents, 86 belonged to the Natural Sciences programme and 149 were students in the Economic and Management Sciences programme.

Programme Total Gender Natural Sciences

N = 86 EMS N = 149

Male 58 (67.4%)

54 (36.2%)

112 (47.7%)

Female 27 (31.4%)

95 (63.8%)

122 (51.9%)

No response 1 (1.2%)

- 1 (0.4%)

Total 86 149 235 Table 1: Respondents by gender and programme

Academic Behavioural confidence (ABC) scores for each individual student

Scores obtained by the students for each individual statement across the two programmes were examined and analysed. The values given to the response categories were 1 for Very confident, 2 for Slightly Confident and 3 for Not Confident. The academic behavioural confidence scores thus range from 1 to 3. Table 2 provides a comparison of the mean scores (rounded off to two decimal places) and the standard deviations between the NS and EMS groups. Table 2 furthermore gives an indication of the difference between the means of the two groups.

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Means Standard Deviation

Difference between Means

I am confident that I can:

NS EMS NS EMS NS mean – EMS mean

1. Study effectively on my own 1.40 1.44 0.49 0.56 -0.04 2. Prepare thoroughly for lectures and tutorials. 1.59 1.79 0.58 0.58 -0.20 3. Read the recommended background material with understanding.

1.45 1.51 0.57 0.56 -0.06

4. Attend most lectures. 1.13 1.09 0.34 0.35 0.04 5. Be on time for lectures. 1.21 1.16 0.44 0.44 0.05 6. Attend tutorials (practise classes) when offered.

1.22 1.30 0.49 0.54 -0.08

7. Understand the content explained and discussed during a lecture.

1.67 1.78 0.56 0.50 -0.11

8. Follow the thread of explanation provided by the lecturer during the lecture and not get lost or fall behind.

1.63 1.95 0.67 0.50 -0.32

9. Respond to questions asked by a lecturer in front of my classmates.

1.78 1.85 0.74 0.66 -0.07

10. Ask my lecturer questions about the material he/she is teaching, in a one-to-one setting (in his/her office for example).

1.64 1.66 0.73 0.95 -0.02

11. Ask my lecturer questions about the material he/she is teaching, during a lecture.

1.74 1.89 0.72 0.73 -0.15

12. Ask for help from a classmate or a friend if I don't understand.

1.17 1.19 0.38 0.47 -0.02

13. Seek appropriate support on whatever level, when the need arises.

1.36 1.46 0.55 0.61 -0.10

14. Manage my workload to meet the coursework deadlines.

1.34 1.54 0.50 0.61 -0.20

15. Give a presentation to a small group of fellow students.

1.37 1.56 0.55 0.62 -0.19

16. Engage in sensible academic debates with my peers.

1.57 1.72 0.70 0.72 -0.15

17. Produce course results at the required standard. 1.37 1.48 0.51 0.54 -0.11 18. Plan an appropriate revision schedule. 1.37 1.49 0.55 0.61 -0.12 19. Produce good results under test and examination conditions.

1.24 1.36 0.43 0.55 -0.12

20. Pass assessments at the first attempt. 1.40 1.43 0.54 0.55 -0.03 21. Produce good work when completing homework or assignments.

1.21 1.38 0.49 0.49 -0.17

22. Write in an appropriate academic (mathematical) style.

1.42 1.82 0.56 0.77 -0.40

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Means Standard Deviation

Difference between Means

I am confident that I can:

NS EMS NS EMS NS mean – EMS mean

23. Remain adequately motivated throughout the year.

1.33 1.38 0.56 0.55 -0.05

24. Make the most of the opportunity to study for a degree at this university.

1.10 1.06 0.38 0.27 0.04

Group Mean 1.41 1.51 0.54 0.23 Table 2: A comparison of means and standard deviations [ABC – scale scores per student

group]

If we compare the means obtained per statement by the NS and EMS groups, we notice that the NS group obtained a mean indicating more confidence in 21 of the 24 statements.

Confidence Means Standard Deviations NS 1.41 0.54 EMS 1.51 0.23

Table 3: An indication of the mean scores obtained from the confidence means and standard deviations per programme

The t-test was used to compare the means obtained from the Natural Sciences and the Economic and Management Sciences groups. The t-value was found to be t = 1.97 at a 95% confidence level (with p-value = 0.05) and degrees of freedom equalling df = 233. From this we can conclude that the difference between the means of the two sample groups is considered statistically significant. Recalling our first hypothesis according to which the Natural Sciences students would have greater academic behavioural confidence than the Economic and Management Sciences students, we can now confirm that our hypothesis is supported.

Although we calculated the means and standard deviations and applied the t-test, we now need to show graphically how the students’ individual scores were distributed around the mean, per programme. A graphical representation highlights the level of significance between the means of the two sample groups. A scatter plot was used for this purpose. Now follows two separate scatter graphs of the confidence means obtained by the students in the NS and EMS programmes. Each mean reflects the average of scores per student for all 24 statements of the questionnaire.

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Distribution of Confidence Means

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70 80 90 100

NS students

Con

fiden

ce M

eans

Series1

Figure 1: Scatter graph of the confidence means obtained by students in the Natural Sciences programme

Distribution of Confidence Means

0.00

0.50

1.00

1.50

2.00

2.50

0 20 40 60 80 100 120 140 160

EMS Students

Con

fiden

ce M

eans

Series1

Figure 2: Scatter graph of the confidence means obtained by students in the Economic and Management Sciences programme

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From Figure 1 and Figure 2 the following deductions can be made: • The confidence means obtained by students in the NS programme varies between 1.96 (with a

standard deviation of 0.36) and 1.00 (with a standard deviation of 0.00). • The confidence means obtained by students in the EMS programme varies between 2.33 (with a

standard deviation of 0.56) and 1.04 (with a standard deviation of 0.20). • The confidence means of students in the NS programme therefore has a range of only 0.96 while

that of the students in the EMS programme has a range of 1.29. This gives us a variance ratio of F = 1.34, df = 85 (NS), df = 148 (EMS) and p = 0.05). We can thus deduce that the confidence profiles of the students in the NS programme are more similar than the confidence profiles of the students in the EMS programme.

• Furthermore, the student in the EMS programme with the highest academic behavioural confidence (a confidence mean of 1.04 and standard deviation of 0.20) still has a lower score than the most academic behavioural confident student in the NS group (a confidence mean of 1.00 and standard deviation of 0.00).

• Also, the student in the NS programme with the lowest academic behavioural confidence (a confidence mean of 1.96 and standard deviation of 0.36) still has a higher score than the least academic behavioural confident student in the EMS group (a confidence mean of 2.33 and standard deviation of 0.56).

We now consider a table describing the level of significance of the statements from the questionnaire for which students from either programme obtained higher confidence means.

Statements Means and SD (NS)

Means and SD (EMS)

Tests of difference with df = 233

Level of significance

t-value, p-value

1. Study effectively on my own 1.40 (0.49) 1.44 (0.56) t = 0.55; p = 0.58

Not Significant

2. Prepare thoroughly for lectures and tutorials.

1.59 (0.58) 1.79 (0.58) t = 2.54; p = 0.011

Statistically Significant

3. Read the recommended background material with understanding.

1.45 (0.57) 1.51 (0.56) t = 0.786; p = 0.43

Not Significant

4. Attend most lectures 1.13 (0.34) 1.09 (0.35) t = 0.8527; p = 0.3947

Not Significant

5. Be on time for lectures 1.21 (0.44) 1.16 (0.44) t = 0.8391; p = 0.4023

Not Significant

6. Attend tutorials (practise classes) when offered.

1.22 (0.49) 1.30 (0.54) t = 1.13; p = 0.2592

Not Significant

7. Understand the content explained and discussed during a lecture.

1.67 (0.56) 1.78 (0.50) t = 1.554; p = 0.1215

Not Significant

8. Follow the thread of explanation 1.63 (0.67) 1.95 (0.50) t = 1606; Extremely

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Statements Means and SD (NS)

Means and SD (EMS)

Tests of difference with df = 233

Level of significance

t-value, p-value

provided by the lecturer during the lecture and not get lost or fall behind.

p<0.0001 Statistically Significant

9. Respond to questions asked by a lecturer in front of my classmates.

1.78 (0.74) 1.85 (0.66) t = 0.4547; p = 0.7488

Not Significant

10. Ask my lecturer questions about the material he/she is teaching, in a one-to-one setting (in his/her office for example).

1.64 (0.73) 1.66 (0.95) t = 0.1686; p = 0.8663

Not Significant

11. Ask my lecturer questions about the material he/she is teaching, during a lecture.

1.74 (0.72) 1.89 (0.73) t = 1.5249; p = 0.1286

Not Significant

12. Ask for help from a classmate or a friend if I don't understand.

1.17 (0.38) 1.19 (0.47) t = 0.336; p = 0.7370

Not Significant

13. Seek appropriate support on whatever level, when the need arises.

1.36 (0.55) 1.46 (0.61) t = 1.2541; p = 0.2111

Not Significant

14. Manage my workload to meet the coursework deadlines.

1.34 (0.50) 1.54 (0.61) t = 2.5804; p = 0.0105

Statistically Significant

15. Give a presentation to a small group of fellow students.

1.37 (0.55) 1.56 (0.62) t = 2.3564; p = 0.0193

Statistically Significant

16. Engage in sensible academic debates with my peers.

1.57 (0.70) 1.72 (0.72) t = 1.554; p = 0.1215

Not Significant

17. Produce course results at the required standard.

1.37 (0.51) 1.48 (0.54) t = 1.5348; p = 0.1262

Not Significant

18. Plan an appropriate revision schedule.

1.37 (0.55) 1.49 (0.61) t = 1.5049; p = 0.1337

Not Significant

19. Produce good results under test and examination conditions.

1.24 (0.43) 1.36 (0.55) t = 1.7392; p = 0.0833

Not quite Statistically Significant

20. Pass assessments at the first attempt.

1.40 (0.54) 1.43 (0.55) t = 0.4055; p = 0.6855

Not Significant

21. Produce good work when completing homework or assignments.

1.21 (0.49) 1.38 (0.49) t = 2.5619; p = 0.0110

Statistically Significant

22. Write in an appropriate academic (mathematical) style.

1.42 (0.56) 1.82 (0.77) t = 4.2153; p = 0.0001

Extremely Statistically Significant

23. Remain adequately motivated 1.33 (0.56) 1.38 (0.55) t = 0.6669; Not

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Statements Means and SD (NS)

Means and SD (EMS)

Tests of difference with df = 233

Level of significance

t-value, p-value

throughout the year. p = 0.5055 Significant 24. Make the most of the opportunity to study for a degree at this university.

1.10 (0.38) 1.06 (0.27) t = 0.9388; p = 0.3488

Not Significant

Table 4: A determination of the level of significance of individual statements from the questionnaire

Although the students in the Natural Sciences group were shown to be more confident than those in the EMS group in 21 out of the 24 statements from the questionnaire (refer to Table 3), Table 4 shows that in only six of these statements (2, 8, 14, 15, 21 and 22) the difference between the means is considered statistically significant. The differences between the means obtained in statements 8 and 22 were considered “Extremely Statistically Significant”, while statement 19 was deemed “Not quite Statistically Significant”. According to the results in Table 3, the EMS group appeared to have displayed more confidence in statements 4, 5 and 24, although the t-test for significance proved no significant difference between the means for those statements between the two groups. We will now compare data regarding the responses of male and female students within the NS and EMS programmes. The t-test was used to compare the means of male and female students obtained from the Natural Sciences and the Economic and Management Sciences groups. The t-value of the NS group was found to be t = 0.0008 at a 95% confidence level (with p-value = 0.05) and degrees of freedom equalling df = 83. The t-value of the EMS group was found to be t = 0.1998 at a 95% confidence level (with p-value = 0.05) and degrees of freedom equalling df = 147. Neither differences, between the means of the male and females, in the NS nor the EMS groups were found to be statistically significant, however.

Confidence Means

Absolute difference in the means (male – female)

Standard Deviations

Absolute difference in the standard deviations (male – female)

Natural Sciences programme

Male 1.4073 0.5418 Female 1.4074 0001.00001.0 =− 0.5429 0011.00011.0 =−

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Confidence Means

Absolute difference in the means (male – female)

Standard Deviations

Absolute difference in the standard deviations (male – female)

Economic and Management Sciences programme

Male 1.50 0.60 Female 1.52 02.002.0 =− 0.58 02.002.0 =

Table 5: An indication of the mean scores obtained from the confidence means and standard deviations for male and female students per programme

Recalling our second hypothesis according to which the male students would have greater academic behavioural confidence than the female students, we can now deduce that our second hypothesis was not true. It is interesting to note, however, that although deemed statistically insignificant, the difference in means between the males and females in the NS programme is less than the difference in means between the males and females in the EMS programme. Male and female students in the NS programme therefore have a slightly greater similar academic behavioural confidence profile than male and female students in the EMS programme. FINDINGS AND CONCLUSIONS

The aim of the study was to explore the differing levels of academic behavioural confidence of first-year students in two of the programmes that are offered by the School of Teacher Education at the Central University of Technology, Free State. These are the B.Ed (FET): Natural Sciences and the B.Ed (FET): Economic and Management Sciences programmes. Although the admission requirements to all the programmes is an M-score of 27, students with an M-score between 22 and 27 can be admitted after writing and passing a special selection test. Irrespective of the entry route into the programmes, the students’ level of preparation for higher education is usually found not to be the same and can be traced back to their school backgrounds. Problems that were encountered at school include inadequate teacher preparation, under-qualified or unqualified teachers in mathematics and science subjects, ineffective leadership and management of schools and a lack of resources. All these have an effect on the level of preparedness of the students, their beliefs about their capabilities, and their confidence as they embark on higher education. The results of this study point to differing levels of academic behavioural confidence between the students in the Natural Sciences and the Economic and Management Sciences programmes. It was hypothesised that the NS group would be more academically confident than the EMS group. This was confirmed by the differences in the means of the two groups (1.41 for NS and 1.51 for EMS). This difference was found to be statistically significant. An analysis of individual statements, however, showed that there was a statistically significant difference between the means of the two groups in only six of the 24 statements, namely in statements number 2, 8, 14, 15, 21 and 22.

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The six statements in which the NS group showed more confidence than the EMS group, and in which the difference were deemed statistically significant, are the following:

• prepare thoroughly for lectures and tutorials; • follow the thread of explanation provided by the lecturer during the lecture; • manage my workload to meet the coursework deadlines; • give a presentation to a small group of fellow students; • produce good results under test and examination conditions; and • write in an appropriate academic style.

Although the NS group appeared to be more confident than the EMS group, there are still those students, even in the NS group, who were found to be only slightly confident or not confident at all. This observation calls for intervention in the form of academic support in order to boost their academic confidence. Sander and Sanders (2007) argue that academic confidence mediates between the individual’s ability, learning styles and the academic environment of higher education. Through guidance and support from their lecturers and other institutional structures, students can be helped to adjust quickly to the demands of higher education. Regarding students in the EMS group, it appears that they have difficulties in all six categories into which the 24 statements could be grouped, namely:

• studying; • understanding; • attendance of classes; • attaining good grades; • voicing out their feelings and • seeking clarification when one does not understand.

The fact that they showed more confidence in statements 4, 5, and 24 (although the difference was not statistically significant) could be an indication that they are aware of their deficiencies and are prepared to attend lectures regularly. These statements were:

• attending most lectures; • be on time for lectures; and • make the most of the opportunity to study for a degree at this university.

Attending lectures only is not enough, though. Students need to have the confidence to study, to understand, to ask for help when it is needed and to speak out about their problems. RECOMMENDATIONS

Based on the findings of this study the following recommendations are made in respect of mathematics as a subject domain:

• New students should be given a pre-test that will serve as a diagnostic tool to assess their level of understanding of mathematics.

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• Students’ learning styles should be assessed in order for lecturers to know their learning style preferences. Lecturers will then be able to adjust their teaching methods to cater for all student groups.

• Further academic behavioural confidence tests should be administered later in the year to assess whether there has been improvement during the course of the year.

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An analysis of Mathematical Literacy curriculum documents: cohesions, deviations and worries

Themba M Mthethwa

University of Fort Hare, School of In-Service programmes

This paper reports on the analysis of the National Curriculum Statement for Mathematical Literacy (ML) (Grades 10-12) official curriculum documents. Key statements and issues around content and context as depicted in various ML documents are identified and discussed. Basil Bernstein’s notion of message system-curriculum, pedagogy and evaluation has been used as a theoretical framework in analysing the ML curriculum documents. This analysis found that though coherence is evident to a large extent among ML curriculum documents but there are some deviations and contradictions in the message conveyed in these documents thus creating worries to both teachers and learners. I conclude this paper by pointing to some areas that I argue that they need special attention.

Introduction. South African department of education (DoE) introduced National Curriculum Statement (NCS) in the Further Education and Training (FET) (Grades 10-12) in 2006. Twenty nine (29) subjects were introduced including Mathematical Literacy (ML). Key documents were published by the Department of Education and made available to all schools prior to the implementation of the curriculum (in 2004 & 2005). These documents are:- Overview document; NCS subject document (policy document); Learning Programme Guideline (LPG) and Assessment Guideline document. In addition to these documents, Life Orientation and ML have Teacher Guide document. All these documents have key messages they intend to convey to both educators and learners. The focus of this study is on the analysis of these various ML curriculum documents as provided by the National Education Department. The aim of the study was to determine key message conveyed in these documents and the extent to which these curriculum documents cohere or depart to each other. Overarching question that the study intended to answer was: What are key messages presented in ML curriculum documents and how these messages cohere or depart from each other? This study is part of an ongoing study on teachers’ interpretation and implementation of ML in Grades 10-12.

Mathematical Literacy in South African Context Mathematical Literacy “refers to the competence of individuals” (Christiansen, 2006:6). In the South African context ML refers both to a school subject and to the competency of individuals. The Department of education DoE (2003) defines ML as:

Mathematical Literacy provides learners with an awareness and understanding of the role that mathematics plays in the modern world. Mathematical Literacy is a subject driven by life-related applications of mathematics. It enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems (p.9).

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Similarly, in the international perspective Pugalee (1999) describes a basic model of M L. He contends that the ML model must meet three important aspects namely, (i) embody the five processes through which the students obtain and use their mathematical knowledge; (ii) demonstrate the intricate interrelationships between various processes that are essential in the development of ML; and (iii) specify enablers that facilitate development of the five processes (p.19). These processes (i) valuing mathematics ;( ii) becoming confident in one’s ability to do mathematics, (ii) becoming a problem solvers; (iv) communicating mathematically; and (v) reasoning mathematically (which I believe) are also captured in the definition provided by the Department of Education.

The Department of education (DoE, 2003) further stipulates that the inclusion of ML as a compulsory subject if Maths is not taken in the (FET) curriculum will ensure that future South African citizens are highly numerate consumers of mathematics. However, Christiansen (2007) argues that there are two main reasons for the introduction of ML as a school subject for South Africa, these reasons were: to reach the 200 000 learners leaving Grade 12 yearly without mathematics and 200 000 learners who fail mathematics in Grade 12 every year; and was to teach learners competencies and knowledge which would be in line with the overall intentions of the NCS. These reasons are supported by Venkat and Graven (2006) who contend that the introduction of ML in the FET was aimed at increasing the number of students taking mathematical courses at all levels. These possible reasons resonate with what Brombacher (2007:4) calls “major forces” for the introduction of ML in South Africa and internationally- the democratisation of mathematics and mathematics for democracy (see more details in Brombacher, 2007).

In South African context four important abilities that ML aims to develop are identified in ML curriculum documents. These abilities are: specifies the following:

The ability to use basic mathematics to solve problems encountered in everyday life and in work situations.

The ability to understand information represented in mathematical ways.

The ability to engage critically with mathematically based arguments encountered in daily life

The ability to communicate mathematically. ( DoE (2005: 8).

These abilities are similar to those appear in the definition of ML by The OECD Programme for International Student Assessment (PISA). PISA (2003) defines Math Lit as it follows:-

Mathematical Literacy is an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen (p.23).

Since 2006 much work in research on ML has been done and reported (see Christiansen, 2007; Mthethwa , 2007, Venkat & Graven; 2006; 2007; 2008; North, 2008; Walton, 2008; Mbekwa & Julie, 2009; and Pythagoras special edition 64, 2006). While most of these studies reported focus on important issues pertaining to the ML curriculum and its implementation, none of them report on the cohesion and diversions among ML curriculum documents thus necessitated the need for this study to be carried out. My argument is that ML curriculum documents should carry the same clear message that is not ambiguous.

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Theoretical framework For my theoretical framework I draw from a socio-cultural perspective to analyse the M L Curriculum documents. I largely draw from Bernstein’s (1971, 1982, 1996) work. In South Africa, recent studies on the new curriculum adopted Bernstein’s framework (see Graven, 2002; Taylor & Vinjevold, 1999; Taylor, Muller and Vinjevold, 2003; Haley & Parker, 1999; Parker, 2006; and Venkat &Graven, 2007). Bernstein (1971,1982, 1996) provides useful concepts for curriculum analysis which I have found relevant to the analysis of ML curriculum documents. Bernstein (1971) contends that there are three message systems through which the formal education knowledge can be recognised, i.e. curriculum (defines what counts as valid knowledge), pedagogy (defines what counts as a valid transmission of knowledge) and evaluation (defines what counts as valid realisation of this knowledge on the part of the taught). Bernstein further provides two important concepts (classification and framing) which are useful in the present study. Bernstein (1975) uses the concept of classification to refer to the nature of the differentiation between contents. He writes:

Classification thus refers to the degree of boundary maintenance between contents (p.88).

Bernstein contends that strong classification results to strong boundaries and strong insulation while weak classification is characterised by weaker boundaries and reduced insulation between categories. Framing is about who controls what (Bernstein, 1996). Bernstein (1971) defines frame that it refers to the degree of control teacher and pupils poses (pp 205-206).

In the section that follows I use Bernstein’s framework to describe ML curriculum as it is presented in policy documents.

Curriculum types: Where does Mathematical Literacy curriculum fall? Bernstein (1971, 1975) describes two broad types of curriculum, collection type and integrated type. Theses types are related to what Cornbleth (1990) calls technocratic curriculum and Critical curriculum respectively. According to Bernstein (1975) collection type “exists if the contents are clearly bounded and insulated from each other” (p.87). Collection type is characterised by strong classification and strong framing . He contends that in this type of curriculum the learner has to collect a group of favoured contents in order to satisfy some criteria of evaluation (p.87). Taylor et al. (2003) maintain that in the collection code school, knowledge is organised according to strong insulated subject hierarchies (p.74). Integrated curriculum exists where the various contents do not go their own separate ways, but where the contents stand in an open relation to each other (p.88). Integrated curriculum is characterised by weak classification and weak framing (F-). Taylor et., al. (2003) point that schools in which the integrated code dominates are characterised by weaker subject boundaries, providing teachers with greater discretion and possibilities for experimentation (p.75). In ML, mathematics is presented in real life contexts to ensure that “the subject is rooted in the lives of the learners” (DoE, 2003: 42). According to the DoE (2003:6) “subject boundaries are blurred”. These weak boundaries suggest that ML fall under the integrated curriculum type.

Pedagogic models fore-grounded in Mathematical Literacy curriculum Bernstein (1996) describes two models and three modes within each model. These models are competence model (liberal/progressive, populist and radical modes) and performance model (singular, regional and generic). Competence models are linked to the learner centred. Competence models are directed towards what the learner knows and can do at the end of learning (Taylor, 1999). According to Bernstein (1996) in competence models acquirers have more control over selection, sequence, pace and over the pedagogic practices, which inhere in personalised forms. Performance models focus on specific learning content and

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texts. Bernstein posits that performance models serve primarily economic goals hence considered instrumental. According to Bernstein (1996) in performance models acquirers have less control over selection, sequence, pace and over the pedagogic practices, which inhere in personalised forms. These models are useful in the analysis of the ML curriculum documents.

Within what Bernstein considers it counts as a valid transmission of knowledge he further provides notion of pedagogic discourse. According to Bernstein (1996) pedagogic discourse is an ensemble of rules or procedures for the production and circulation of knowledge within pedagogic interactions. He identifies two rules which are embedded in pedagogic discourse, instructional discourse (the discourse which creates a specialized skills and their relationship to each other) and regulative discourse (the discourse which creates order, relations and identity). Bernstein contends that instructional discourse (ID) is embedded in the regulative discourse (RD) hence RD is the dominant discourse (p.46). He argues that regulative discourse produces the order in the instructional discourse (p.48). According to Bernstein pedagogical discourse is linked to recontextualizing14 fields (p.47). He identifies two important fields which create the fundamental autonomy of education. He labels these fields as pedagogic recontextualizing field (PRF) and official recontextualizing field (ORF). ORF is created and dominated by the state (in the case of this study, all ML policy documents mentioned above) and its selected agents and ministries while PRF consists of pedagogues in schools and colleges, and departments of education, specialized journals, private research foundations (p.48). These concepts of recontextualising fields have been used in the analysis of curriculum document to establish theoretical framework.

Research Methodology The present study seeks to understand key messages conveyed in ML curriculum documents and the extent to which these key messages in curriculum documents cohere or depart to each other. A qualitative approach has been adopted for its relevance to this study. Qualitative approach uses a naturalistic approach to that seeks to understand phenomena in context-specific settings (Hoepfl, 1997). Libarkin and Kurdziel (2002) assert that qualitative research is an unconstrained approach to studying phenomena. They further assert that qualitative studies provide a window into a contextual setting, and logical picture of events within that setting (p 78). Opie (2004) contends that research which seeks to obtain softer facts, and insights into how an individual creates, modifies and interprets the world in which they find themselves, an anti-positivistic approach, would employ qualitative techniques (p.8). On these bases a qualitative approach has been chosen for this study.

DATA COLLECTION AND ANALYSIS Data collection in this study involved documentary analysis of ML official documents. These documents are:

MATHEMATICAL LITERACY DOCUMENTS Pages 1 NCS GRADES 10-12 (GENERAL) POLICY-2003 74pp 2 Subject Assessment Guidelines (SAG)-2008 42pp 3 Learning Programme Guidelines (LPG)-2005 21pp 4 Teacher Guide (TG)-2006 69pp

Table 1: NCS documents for Mathematical Literacy 14 According to Bernstein, recontextualisation refers to the rules or procedures by which educational knowledge is moved from one educational site to another.

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A subject Statement document has four chapters providing information on NCS, ML, Learning Outcomes, assessment standards, contexts and content and assessment. A Teacher guide document provides information to teachers of ML on how to develop ML; this includes resources needed to teach ML, learning units and assessment in ML. LPG focuses on designing of a learning programme for ML, this includes subject framework, work schedule and lesson plan. SAG provides detailed guidelines for assessment in the NCS; this includes continuous and summative assessment, examination papers marks allocation and different taxonomy levels.

Document review involved content analysis. According to Wilkinson and Birmingham (2003) content analysis involves two methods, conceptual analysis and relational analysis. According to Wilkinson and Birmingham (2003) conceptual analysis involves analysis of themes or issues in the text that the researcher intends to analyze (p.70) while relational analysis attempts to explore and identify relationship between themes or issues (p.77). The following are key categories that I have explored: Curriculum design, content and context, progression and teaching and learning strategies.

Findings and discussion After in-depth analysis of various curriculum documents of ML the following were found and discussed below categorically.

Curriculum design ML curriculum for Grades 10-12 has been designed and structured into four Learning Outcomes namely: LO1 Number and operations in context, LO 2 Functional Relationships, LO 3 Space, Shape and measurement and LO4 Data handling. These LO’s very similar to those of NCS mathematics for Grades 10-12. I argue that the way in which ML curriculum has been designed makes many people to compare ML with mathematics since it looks like mathematics. Previous studies on ML (see Christiansen, 2007) confirm that even assessment standards are too mathematical. In the policy document DoE (2003), six –point scale of achievement is adopted (ranging from code 1-inadequate to code 6-outstanding). Contrary these codes, in the Assessment Guideline document (DoE, 2008:9) there are seven codes-slightly different from the six. This could create confusion to the reader.

It is stipulated in the policy document (ML) that ML is for learners who intend to study disciplines which are not mathematically based. DoE (2003) states:

Mathematical literacy should not be taken by those learners who intend to study disciplines which are mathematically based, such as the natural sciences or engineering (p.11).

This statement can be received or interpreted in many ways by both teachers and learners. A learner who intends to pursue a career in commerce, e.g. Bachelor of Commerce degree (B Com), is likely to be confused whether she should take ML or not. While such programs (like BCom) usually require some Mathematics background, recent studies have shown that ML can be recognized for the entry requirements into a B Com. Degree (see, Walton, 2008).

Content and context In ML contexts are considered to be “central to the development of mathematical literacy in learners” (DoE, 2003:42). Analysis shows that various contexts are used in ML to attain Learning Outcomes (LO’s)

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and Assessment Standards (AS) of ML. These contexts are related to the principle15 of the National Curriculum Statement (NCS) such as issues arise in health (HIV/AIDS), human rights, inclusivity, environmental and socio-economic justice (DoE, 2003). Specific contexts that are used in Mathematical Literacy have been identified in Teacher Guide Document for M L (DoE, 2006), see table below. Most of these contexts match with those that were identified by mathematics educators from South Africa, Zimbabwe, Uganda, Eritrea and Norway (see in Julie & Mbekwa, 2005:33).

Main Cluster of contexts

Clusters and examples

Health Contexts that deal with HIV/AIDS issues and Body Mass Index (BMI)

Finances

• Contexts deal with banking related issues such as accounts (eg, Mzansi), investment, loans, interest (simple and compound) and ATM’s

• Contexts that deal with marketing related issues such as income and expenditure, selling price, profit, and breaking even

• Contexts that deal with budgeting Municipal tariffs Contexts that deals with water, electricity and sewerage costs (monthly

charges) Transport and communications

• Contexts that deals with Telkom telephone cards and charges and cell phones;

• Contexts that deals with mailing or mailing (ordinary and fast mail) envelope sizes and postcard etc.

• Contexts that deals with travelling eg a trip with Shosholoza Meyl Sports Contexts deals with Soccer World Cup (soccer stadium and tickets) and

Athletics Mathematics Contexts that deal with mathematics content like linear equations and

algebraic graphs General • Contexts that deal with baking and cooking.

• Contexts that deal with bicycle gears

Table 2: Examples of contexts that are used in Mathematical Literacy

These contexts (and other not mentioned) should be used in all ML lessons. Bowie and Frith (2006) argue that M L teachers will face many challenges to teach Mathematical Literacy, as they are required to understand more than mathematics (content) but also various contexts used in ML. I argue that both competence and performance models are both fore grounded in ML. In the table below I present mathematics content in the ML. As Christiansen (2006) argues that although contexts are fore-grounded in Mathematical Literacy, ML content is “distinctly mathematical” (p.10). It is

15 NINE Principles of NCS: Social transformation; Outcomes –based education; high knowledge and high skills; integration and applied competence; progression; articulation and portability; human rights, inclusivity, environmental and social justice; valuing indigenous knowledge systems; and credibility, quality and efficiency.

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evident that some of the topics dealt with in the ML are strongly classified mathematical, e.g. trigonometry, linear programming, quadratic equations etc (see topics in the table below).

LO’s Grade10 Grade11 Grade12

LO 1

1. Fractions, decimals, percentages 2. Positive exponents and roots 3. The associative, commutative and distributive laws 4. Rate 5. Ratio 6. Direct proportion 7. Inverse proportion 8. Simple formulae 9. Simple and compound growth 10. scientific notation

1. Content involved in Grade10 work but applied to more complex situations 2. Square roots and cube roots 3. Ratio and proportion 4. Complex formulae 5. Cost price and selling price 6. Profit margins

1. Content of Grd. 10 & Grd. 11 but applied to more complex situations 2. Taxation 3. Currency fluctuations 4. Financial and other indices

LO 2

1. Tables of values 2. Formulae depicting relationships between variables 3. Cartesian co-ordinate system 4. Linear functions 5. Inverse proportion 6. Compound growth 7. Graphs depicting relationships between variables 8. Maximum and minimum points 9. Rate of change (speed, distance, time)

1. Content involved in Grade10 work but applied to more complex situations 2. Simple quadratic functions 3. Solution to linear, quadratic and simple exponential equations 4. Solution to two simultaneous linear equations.

1. Content involved in Grd10 & Grd 11 work but applied to more complex situations 2. Simple linear programming (design and planning problems) 3. Graphs showing the fluctuations of indices over time

LO 3

1.Measurement of length, distance, volume, area, perimeter 2.Measurment of time (international time zones) 3.Polygons commonly encountered(triangles, squares, rectangles that are not squares, parallelograms, trapeziums, regular hexagons) 4. Circles 5. Angles (00 3600) 6. Theorem of Pythagoras 7. Conversion of units within the metric system 8. Scale drawings 9. Floor plans 10. Views 11. Basic transformation geometry, symmetry and tessellations

1.Grade 10 content but applied to more complex situations 2.Measurement in 3D ( Angles included, 00 3600) 3. Surface Area and Volumes of right prisms and right circular cylinders. 4.Conversion of measurements between different scales and systems 5. Compass directions 6. Properties of plane figures and solids in natural and cultural forms 7. Location and position on grids 8. Trigonometric ratios: sin x, cos x, tan x.

1. Content of Grade 10 and Grade 11 but applied to more complex situations. 2. Surface areas and volumes of prisms of right pyramids and right circular cones and spheres. 3. Scale models 4. Sine rule, cosine rule and area rule

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LO 4

1.Construction of questionnaires 2.populations 3. selection of a sample 4. Tables recoding data 5.Tally and frequency tables 6. Single and compound bar graphs 7. Pie charts 8. Histograms. 9. Line and broken-line graphs. 10. Mean, median, mode. 11. Range. 12. Relative frequency 13. Probability

1. The content of Grade 10 but applied to more situations 2. Selection of samples and bias 3. Cumulative frequency 4. Ogives (cumulative frequency graphs) 5.Variance (interpretation only) 6.Standard deviation (interpretation only) 7.Quartiles 8. Compound events 9. Contingency tables 10 Tree diagrams

1. The content of Grade 10 and Grade 11 but applied to more complex situations. 2. Bivariate data 3. Scatter plots 4. Intuitively-placed lines of best fit 5. percentiles

Table 3: Mathematics Content in Mathematical Literacy With such topics presented in the above table, teachers should have good mathematics background in order to teach ML confidently. The misconception of ML being a simple version of mathematics is challenged.

Progression Progression is among key principles of NCS (DoE, 2003). The analysis of the curriculum reveals that progression is evident in mathematical content (see a table above) and in the complexity of contexts. Progression in these two indicators necessitates progression in problem solving skills from applying routine procedures to reasoning and reflecting levels. According to DoE (2003) for ML, “the assessment standards do indicate progression from grade to grade” (p.38). The analysis shows that this is not true with some assessment standards (see example LO1 AS 2 below) (DoE, 2008: 18)

10.1.2 Relate calculated answers correctly and appropriately to the problem situation by: • Interpreting answers in

terms of the context;

• Reworking a problem if the initial is not sensible, or if the conditions change;

• Interpreting calculated answers logically in relation to the problem and communicating processes and results

11.1.2 Relate calculated answers correctly and appropriately to the problem situation by: • Interpreting answers in

terms of the context;

• Reworking a problem if the initial is not sensible, or if the conditions change;

• Interpreting calculated answers logically in relation to the problem and communicating processes and results

12.1.2 Relate calculated answers correctly and appropriately to the problem situation by: • Interpreting answers in

terms of the context;

• Reworking a problem if the initial is not sensible, or if the conditions change;

• Interpreting calculated answers logically in relation to the problem and communicating processes and results

Table 4: Example of assessment standards

The above AS’s show no progression from Grade 10 to 12. DoE (2003) however acknowledges that the progression “is not markedly evident in some of the assessment standards” (p.38). The DoE suggests that

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progression should be ensured in mathematical knowledge and complex situations, however there is no example given to show how this progression is done. Curriculum documents seem to be unclear with this regard. The Teacher Guide document only presents Grade 10 work. North (2008) argues that some of the assessment standards in ML are identical, he provides example in which progression could be achieved (see North, 2008 for details).

Teaching and learning strategies.

Teaching and learning ML is a great challenge. As Vithal (2006) notes that “the teacher has to ensure that neither learners’ understanding of the mathematics or that of the context gets compromised” (pp. 40-41). The Policy document for ML (DoE, 2003) suggests the approach that needs to be adopted in developing Mathematical Literacy; that is to “engage with contexts rather than applying Mathematics already learned to context”(p. 42). This message is contrary to the Teacher Guide document (DoE, 2006) where there are twenty six learning units, each unit being expected to take between five and ten days of classroom time. Four of these learning units (units 4, 8, 15 &19) are labelled “direct content teaching”. Some of the examples given are purely mathematics content based with no real life context, example (DoE, 2006: 43)

Solve for “a” (a) 2 X a – 5 = 19 (b) 2 X (a + 5) = 18 (c) 6 + 2 X a = 21 (d) (a + 4) ÷2 = 24

It appears that the approach to teach these units other units like trigonometry could be predominantly content based than context based. Similarly, Venkat and Graven (2006) observe that documents (for ML) are not clear about the issue of contexts and content. They write:

It would appear that there are mixed messages within the Department of Education’s documentation for ML. Whether educators will give more emphasis to context-specific problem solving using mathematics, or to the mathematics involved in solving contextual problems remains unclear at this stage (p.20).

These contradictions in the curriculum documents could lead to dilemmas and worries on the side of the teacher. Further more, these worries could result into a huge gap between pedagogic recontextualizing field (PRF) and official recontextualizing field (ORF) which then can have an impact (either positive or negative) on intended ML curriculum and implemented ML curriculum.

Implications for classroom practice and further research In an attempt to implement the intended ML curriculum in classroom, teachers should carefully study all curriculum documents- as possible as they could, get the latest ML curriculum documents and other supporting documents and be aware of possible contradictions in these documents.

Given the nature and the purpose of ML- in which real life contexts are emphasised, teachers should ensure that the contexts used are appropriate and meaningful to the learners and ensure that while contextualising mathematics to the real life, content knowledge should also be emphasised such that incorporating the everyday in mathematics does not derail mathematical goals. This requires teachers of ML to be well prepared through training and re-skilling programmes.

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A further research needs to be done on how different ML teachers interpret ML curriculum (document) and how their interpretations impact teaching and learning of ML.

Conclusion In conclusion, the National Department of Education has made it possible that the ML curriculum documents are available in schools and even on the websites; however, the department must ensure that these curriculum documents send the same and clear messages that will not cause any contradictions. Further more- the department of education should well prepare ML teachers to implement ML successfully.

References Bernstein, B. (1982) On the Classification and Framing of Educational Knowledge. In T. Horton & P.

Raggatt, (Eds.) Challenge and Change in the Curriculum, 149-176, Milton Keynes, UK: The Open University.

Bernstein, B. (1971). Class, Codes and Control volume 1: Theoretical studies towards sociology of language. London, RKP

Bernstein, B. (1975) Class, Codes and Control Volume 3. London: Routledge and Kegan Paul.

Bernstein, B. (1996).Pedagogy, Symbolic Control and Identity: Theory, Research, Critique. London, UK: Taylor and Frances.

Bowie, L. & Frith, V. (2006) Concerns about the South African Mathematical Literacy curriculum arising from experience of materials development. Pythagoras, 64, 29-36.

Brombacher, A. (2006). First draft of the report on the SAQA Mathematical Literacy Standards at NQF levels 2, 3 and 4: SAQA

Christiansen, I.M. (2006). Mathematical Literacy as a subject: Failing the progressive vision. Pythagoras 64, December, 2006, pp.6-13

Christiansen, I. M. (2007). Mathematical literacy as a school subject: Mathematical gaze or livelihood gaze? African Journal of Research in Mathematics, Science and Technology Education, 11(1), 91-105

Cornbleth, C. (1990). Curriculum in context. London. The Falmer Press. Chap.2, pp.12-44

Department of Education, 2003. National Curriculum statement Grades 10-12 (general) Mathematical Literacy. Pretoria: Government Printers

Department of Education, 2005. National Curriculum statement Grades 10-12 (general). Learning Programme Guidelines for Mathematical Literacy. Derpartment of Education: Pretoria.

Department of Education, 2008. National Curriculum statement Grades 10-12 (general), Subject assessment guidelines for Mathematical Literacy. Department of Education: Pretoria

Department of Education, 2006. National Curriculum statement Grades 10-12 (general). Teacher Guide: Mathematical Literacy. Department of Education: Pretoria.

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Graven, M. (2002). Mathematics Teacher Learning, Community of Practice and the centrality of confidence. Unpublished PhD Thesis, University of the Witwatersrand, Johannesburg.

Harley, K. & Parker, B. (1999). Integrating Differences: Implications of an Outcomes-based National Qualifications Framework for the Roles and Competencies of Teachers. In J. Jansen and P. Christie (Eds.) Changing South Africa.

Hoadley, U. (2006a). The reproduction of social class inequalities through mathematics pedagogies in South African primary schools. Journal of curriculum Studies, Vol. 39, no.6, 679-706.

Hoepfl, M.C. (1997). Choosing Qualitative Research: A Primer for Technology Education researchers. Journal of Technology Education. v.9 no.1

Julie, C., and Mbekwa, M. (2005). What would Grade 8 to 10 learners prefer as context for mathematical literacy? The case study of Masilakele Secondary School. Perspectives in Education, Vol 23 (3), September 2005.

Libarkin, J.C and Kurdziel, J. (2002). Research Methodologies in Science Education: The qualitative-quantitative Debate. Journal of Geoscience Education, v.50, no1. 78-86

Mthethwa, T. M. (2007). Teachers' views on the role of contexts in mathematical Literacy. unpublished M.Sc thesis, University of the Witwatersrand, Johannesburg.

North, M. (2008). Progression in Mathematical Literacy. Proceedings of the 14th Annual National Congress of the Association for Mathematics of South Africa.

Opie, C. (2004) Doing Educational Research. Sage. London.

Parker, D. (2006b) Grade 10-12 Mathematics curriculum reform in South Africa: A textual analysis of new national curriculum Statements. African Journal of Research in SMT Education, 10(2), 59-73.

Pugalee, D. K. (1999). Constructing a model of mathematical literacy. The Clearing House 73(1), 19–22.

Taylor, N.; 1999, “Curriculum 2005: Finding a balance between the everyday and school knowledge” in N. Taylor and P. Vinjevold. (Eds.) Getting learning Right. Joint Education Trust: Johannesburg.

Taylor, N.; Muller, J. and Vinjevold, P. (2003) .Curriculum delivery in the classroom. Chapter 6, in N. Taylor, J. Muller & P. Vinjevold, P., Getting schools working (pp 67-84). Cape Town: Pearson Education South Africa.

Venkatakrishnan, H. and Graven, M. (2006). Mathematical literacy in South Africa and Functional mathematics in England: A consideration of overlaps and contrasts. Pythagoras 64, December, 2006, pp.14-28

Vithal, R and Bishop, A.J. (2006). Mathematical Literacy: A new literacy or a new mathematics? Pythagoras 64, December, 2006, pp.2-5

Walton, M. (2008). Is Mathematical Literacy sufficient for entry into a Bcom degree? Proceedings of the 14th Annual National Congress of the Association for Mathematics of South Africa.

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Two approaches to learning the mathematics of annuities

Craig Pournara

University of the Witwatersrand Two approaches to learning the mathematics of annuities have been identified – an account balance approach and an individual payment approach. Based on the analysis of data from pre-service secondary maths teachers, the author argues for the importance of students’ accepting and using an individual payment approach.

INTRODUCTION In recent years, as I have become more comfortable with teaching annuities, I have become increasingly able to focus on students’ thinking. This has alerted me to a possible mismatch between students' initial models of annuity situations and the typical models based on geometric progressions found in text books. As part of my doctoral study16, I set out to explore this mismatch more systematically and provided students with opportunities to model annuity-based scenarios before dealing with annuity formulae. In this paper I share my findings based on initial analyses of portions of the data. I describe an eight-day period during which students worked on deriving a formula for future value of an ordinary annuity, and on a task involving missed payments.

The broader study This work is drawn from a larger study focusing on the learning of 40 pre-service secondary mathematics teachers in a Bachelor of Education (B.Ed) programme at the University of the Witwatersrand. At the time of data collection, the students were in their third or fourth year of study and were enrolled in a course on introductory financial mathematics for teachers17 which I was teaching in the first semester (February to June) of 2008. The course focused on mathematics, teaching and the world of finance. The content included simple and compound growth, nominal, effective and real interest rates, simple and complex annuities, outstanding balance, micro lending, time value of money, inflation, financial indices, financial mathematics in the school curriculum, task selection and design, and analysis of learners' thinking on financial maths tasks. Active student participation was encouraged throughout the course, with much time spent working in small groups, and in whole class discussions which were frequently led by the students themselves. Much emphasis was also placed on the use of technology, particularly spreadsheets.

A key design feature of the course was a weekly group tutorial where students worked in pre-assigned groups of four, on an extended task for a two-hour period. At the end of most tutorials, groups submitted a report on their work. Two of these groups formed focus groups in the study. Video recordings were made of all lectures, as well as the group tutorial sessions and follow-up interviews with the focus groups. Two individual interviews were conducted with each student in the focus groups - near the middle of the course and six months after the course had been completed. Records of students' work were also collected. These included journal entries, selected classwork activities, tutorial reports, assignments and tests. 16 This research is based on work supported financially by the Thuthuka programme of the National Research Foundation.

Any opinions, findings and conclusions or recommendations expressed are those of the author and therefore the NRF does not accept any liability in regard thereto.

17 The development of this course was funded by the Hermann Ohlthaver Trust.

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Two approaches to modelling annuity situations Based on my previous teaching of annuities, I had identified two different approaches to modelling annuity situations. The first approach focuses on tracking the account balance and so I have called this an account balance (AB) approach. This approach mirrors what goes on in the bank each month – a deposit is made, it is added to the account balance, interest accumulates and the closing balance is calculated at the end of each month. This approach is easy to make sense of and, in my experience, is the first approach students adopt when initially attempting to model an annuities scenario. It depends on simple iterative calculations but when there is a departure from the perfect payment plan, all balances need to be recalculated. This makes it an inefficient and cumbersome approach. The second approach focuses on the behaviour of each individual payment over time, and has thus been termed an individual payment (IP) approach. In this approach each payment is disaggregated from the whole and its contribution to the overall balance is modelled by moving it forward (or backward) in time (by means of compound interest calculations). This approach is more powerful than the AB approach because it gives access to the geometric progression which lies at the heart of the mathematical model for annuity scenarios. For this reason, it is obvious why maths text books use this approach from the outset. In addition to providing access to mathematical structure, the IP approach is also more efficient when there is a departure from the perfect payment plan because here only the changed payments need to be considered when recalculating balances. This will be illustrated later in the paper.

Grappling with future value of annuities The eight-day period under discussion took place in the first half of the course and consists of four two-hour sessions where students began work on annuities. The first session was a group tutorial, followed by two two-hour class sessions, and then another group tutorial. The first of the group tutorials focused on the derivation of a formula for the future value of an ordinary annuity. Students were given a handout consisting mainly of the text from an annuity-based savings product from one of the South African banks. The scenario involved monthly deposits of R250 for one year, and students were required to determine how much money would accumulate over the period. One of the questions required them to produce a formula to generalize their calculations. I deliberately did not use the language of annuities in the task because I did not want them simply to use a formula they might have remembered from a previous course, nor did I want them to google for a formula. Rather I wanted them to mathematise the given scenario and to develop their own formula. To my surprise none of the groups managed to obtain a formula during the tutorial time. I asked them to work further on the problem at home and we agreed to continue in the class session the following day. The next day's session began with students working in their tutorial groups, sharing the independent work done the previous night. The entire two-hour session was spent on the task with some public contributions from students and myself.

One group obtained a formula during the first hour. It modelled an annuity due scenario but they did not realise this at the time. They used an AB approach with an elimination process typical of deriving a formula for the sum of a geometric progression. During the second hour I invited a student from another group to share their approach, which consisted of separate calculations for the growth of each monthly payment. This reflected the beginnings of an IP approach but the group was unable to draw all their calculations together and use the sum of a geometric progression. Her input generated an extended class discussion with several students asking for clarification from various members of that group, and the group members themselves defending their approach with confidence. At that point it was clear that many students were not comfortable with an IP approach although there was much interest in it. One student asked whether the IP approach dealt with all the accumulating interest of the individual payments. Another

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suggested that the separate payments be made into separate accounts. These questions represented a broader concern about whether the IP approach led to a valid model. I left this question with the class and invited another student to share a graphical representation that showed the growth of each individual payment, similar to fig. 1. This student’s group had been working with this representation but were also unable to convert it into an appropriate formula. The diagram shows a twelve-month period with payments of R250 (Pti) being made at the end of each month. The arcs indicate that each payment is growing separately and accumulates interest at the end of each month. The pictorial representation enabled students to visualise the growth in the account more holistically and thus they accepted this contribution more readily than the individual calculations.

I then introduced a spreadsheet, similar to the one shown in fig. 2. that linked closely to the pictorial representation and foregrounded the growth of individual payments. The spreadsheet contains 12 rows representing the monthly payments from January to December. Each 250 represents the value of the deposit at the time it is made. Thereafter each payment grows at 0.5% per month. Each row constitutes a geometric progression. The balance in the account at the end of any month can be calculated by summing that column. The values in the columns form the same geometric progression as the rows and each entry in a column indicates the value that a particular payment is contributing to the account balance at a given month end.

Pt1

Jan

Pt2

Pt3

Pt4

Pt5

Feb Mar Apr May June July Aug Sep Oct Nov

Dec

Fig 1: Representing annuity payments

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Thus all three public inputs during that session focused on IP approaches. At the end of the session, I asked students to work further on the problem and to consider how the ideas shared publicly in class might help them find a formula. We continued with the problem in the next session (four days later). By that stage several more students had found a formula – either on their own or from another source. During class we derived a formula for future value of an ordinary annuity. At that stage I was confident that most had grasped the work on the derivation of the formula, and I sensed that students were relieved to have a formula. My concern was that they should be able to unpack (Ball, Bass, & Hill, 2004) the formula and re-examine how any particular payment impacts the account balance. Frequently, once annuity formulae have been derived, they become new objects whose use is characterised by substitution of values to solve for an unknown. The formulae thus compress the entire process into relatively complex formulae. As a result the formulae may become black boxes that hide the ways in which money grows over time. While the formulae are important for efficient calculation, their structures provide little insight into the ways in which regular payments each gain interest over time, and how these accumulate.

The next session would be another group tutorial. While I wanted to reinforce what had been done in the previous session, I also wanted students to work with the process that had led to the formula. So I decided to work with an annuity due scenario and a missed payment in the same problem. The annuity due scenario meant that students would have to adapt the ordinary annuity formula. The missed payment component would enable me to see their ability to deal with individual payments and to conceptualise the problem using an IP approach. I felt that the notion of missed payment was accessible and that it was within their reach to develop a model for the situation without formal teaching. An advertised product from another South African bank consisted of an initial lump sum payment followed by a series of equal payments. This seemed a good choice because it required a combination of calculations and would reveal the extent to which students could isolate payments to analyse the situation. I expected that they would easily isolate the lump sum payment from the annuity payment but was not sure whether they would isolate each annuity

Fig 2: Spreadsheet showing individual payments

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payment and use an IP approach. The task is given in fig. 3, and the remainder of this paper focuses on the responses of the ten groups to this task.

You open a NedTerm account at Nedbank in April 2008. This account enables you to make monthly deposits. When you open the account, you have to make an initial deposit of at least R1 000 and then monthly payments of R100 or more. You decide to deposit an initial amount of R1 000 and monthly deposits of R300. These payments are made at the start of every month and you do this for 18 months18. The monthly payments start in May 2008. The interest rate is 7.65% p.a. compounded monthly. 1. How much money will you accumulate over the period? 2. Assume you are unable to make the 14th payment. How will this affect the total that you accumulate? 3. Since you missed the 14th payment, you decide to make a double payment for the 15th payment. Will

this enable you to accumulate the same amount as in (1) above? Provide evidence.

Responses to the missed payment task Since the reports were group efforts, it is not always possible to know the extent to which a report represents the thinking of the entire group. In some instances, the contributions of individual students were clearly marked, in others this was not the case. I have analysed this set of reports on two separate occasions, with approximately one month in between. On the first occasion I focused initially on what had been written and tried to make sense of it: tracing the reasoning, identifying sources of incorrect calculations and then summarising each group’s work. As I did this I continually compared the report I was analysing with those I had already analysed. This constant comparison method (Glaser, 1978) enabled me to identify similarities and gaps in the reports, and to build a more detailed and sensitive lens to explore what had been written up. When I re-analysed the reports, I focused more strongly on identifying AB and IP approaches and looking at where these might be in conflict with each other. For the most part the two analyses rendered similar findings. I also summarised the video recordings of the two focus groups and mapped their discussions onto their written reports, noting the different roles that group members had played in the process, and thus the extent to which what was written might represent individual or shared perspectives in the group. I now move to give some general observations and then focus on three issues that emerged from the reports.

General findings All groups displayed some evidence of an IP approach, with eight of the ten reports making use of AB and IP approaches. The most basic form of an AB approach consisted of 19 iterative calculations of the account balance at the end of each month, a typical example of which is given in fig. 4. Question 2 would then require a recalculation of the account balance for months 14 to 18, to deal with the missed payment.

18 This wording was deliberately ambiguous to force students to negotiate their interpretation of the

timeframes of the task.

Fig 3: Tutorial question on Missed Payments

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Similarly question 3 would require further recalculations for months 15 to 18, to deal with the double payment in month 15. The most elegant IP approach consisted of moving the lump sum forward for 19 months and adding the future value of the annuity. In question 2 the entire 14th payment is simply deducted from the amount calculated in question 1. In question 3 the additional R300 is moved forward in time for 4 months and added to the balance from question 2 (see fig. 5). Only one group produced the solution shown in fig. 5, and it is likely that this reflected the thinking of one dominant group member who had previous experience with annuities. There is no evidence to indicate the approaches taken by other group members.

Q1 FV1 = 1000 (1 + 0.006375) = 1006.375 FV2 = (1006.375 + 300)(1 + 0.006375) = 1314.703… FV3 = (1314.70… + 300)(1 + 0.006375) = 1624.976… … FV19 = 6867.49

Q1

FV19 = 1000(1+0.006375)19 + 0.006375

10.006375)(130018 −+ (1.006375)

= 6867.49 Q2 FV19 = 6867.49 - 300(1+0.006375)5 = 6557.80

Q3 FV19 = 6557.80 + 300(1+0.006375)4 = 6865.52

Fig 4: AB approach to Q1 Fig 5: IP approach to all 3 questions

The fact that most groups worked with both approaches suggests that students were not yet convinced of the validity of the IP approach and/or were not yet comfortable with the approach. One of the groups made an explicit shift from an AB to an IP approach after doing three iterative calculations of account balance, noting “we realised that we would have to analyse each amount seperately (sic) using a timeline”. They then proceeded to work with an IP approach in the remaining calculations. Another group demonstrated AB and IP approaches in dealing with the missed and double payments. However, due to several arithmetic errors and at least one modelling error (where they used an incorrect exponent for compounding period) they did not obtain the same answers for the two approaches. Thus they were not in a position to make useful comparisons of their answers.

The remainder of the paper focuses on three issues that emerged from the written reports and the video recordings of one of the focus groups.

Doubting that an IP approach leads to a valid model It seems that a belief in the IP approach is not automatic for some students. I have already mentioned that one student queried whether the IP approach accounted for all interest. Another student asked whether the individual payments should rather be paid into separate accounts. For him the separate accounts would take care of the interest each month because this can be modelled by means of compound interest calculations. Similar concerns were expressed in one of the reports where the group correctly calculated the total accumulated amount in a single calculation with separate formulae for the lump sum (of R1000)

14th payment of R300 compounds for 5 months, so payment with its interest is removed

15th payment of R300 compounds for 4 months, so extra payment with its interest is added

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and the annuity component – similar to the first section of fig. 5. But it seems their thinking was still dominated by an AB approach because they were not convinced that the lump sum could be separated from the annuity:

“So now the question is how does the monthly diposit (sic) of 300 affect the initial deposit of 1000 so we decided to calculate interest accumulated in each and every months because it did not make sense to us that 1000 is compounde[d] alone and the 300's are compounded alone. The 1000 had an effect on the interest accumulated at the end of each and every month"

They then proceeded to do iterative calculations of the account balance for each month. As I see it, the essence of these struggles is that multiple payments are being made into the same account at different times but the students want to focus on the overall balance at each month end. When one works with an AB approach, all the payments must be merged together to calculate the interest on the accumulated balance. So there are separate streams that flow into a single combined stream before interest can be calculated. Thus each payment impacts the overall interest calculation. When one works with an IP approach, the merging process is ignored. Each deposit is treated as a separate stream flowing in parallel with the other streams. Each stream carries it’s own interest and thus the combined total consists of adding individual entries that already bring their interest with them. The group discussed above wanted to account for the impact of the R1000 on the overall balance and were not yet ready to think about the situation in terms of parallel streams. Clearly this is an issue for further research – what does it take for students to be convinced of the validity of the IP approach to generate a valid model?

Power of the spreadsheet and pictorial representation By comparison, one of the focus groups was able to make use of the IP approach in convincing ways. This group worked with an AB approach in solving the problem so they did iterative calculations for each month of the perfect payment plan and the recalculations for all months affected by the missed payment and the double payment. However, in explaining the impact of the missed payment and the double payment, they worked with an IP approach as illustrated in fig. 6.

The diagram, drawn by Shaun, shows the initial deposit as well as the 18 monthly payments. The horizontal arrows indicate the months where interest is gained on the individual payments and above the arrows he has indicated how many months’ interest will be paid. For example, payment 13 will receive interest for 6 months. When dealing with payment 14, the arrow indicates that the payment should receive interest 5 times but as Shaun talks with his group members, they discuss how this entire line must be removed from the calculation since the payment is not made. They indicate the impact of the fifteenth payment with rectangles (albeit unequal in size). The eight “white” rectangles indicate that “double interest” is paid for four months because of the double payment. The shaded rectangle thus shows that only one month’s interest is lost. That there is a relationship between the structure of Shaun’s diagram, the pictorial representation (fig. 1) and the spreadsheet (fig. 2) is obvious. Shaun refers specifically to the structure of the spreadsheet to reason through the missed payment scenario without any need to refer to specific values, and he manages to convince his group members that the impact of departing from the perfect payment plan is the loss of only one month’s interest. They then calculate this amount and use it to explain the difference in account balances between the perfect payment plan and the adjusted scenario. So the initial calculations in their report reflect an AB approach but their justification for the differences draws on an IP approach. Evidence from the group discussion suggests that as they talk about the spreadsheet and the IP approach, they become increasingly convinced that it provides a suitable way to explain the difference in final balances. There is no evidence to suggest that anyone in the group began the

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tutorial with these ideas. Rather, the conviction emerges through the discussion initiated by Shaun. Further analysis of this episode may provide some insight into possible ways of helping students accept a model based on an IP approach. The role of the spreadsheet may be important in this process.

Clustered individual payment approach The third issue focuses on a sophisticated yet inefficient approach. In two groups there is evidence of students using an IP approach where multiple payments are analysed together. I have termed this a clustered individual payment approach. It is more efficient than an AB approach but much more complex

Fig 6: Shaun’s diagram for missed payment problem

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than an IP approach. Students worked with clustered payments in different ways. One group chunked the timeline to deal with the disruption that started in the 14th month. They assumed that the scenario extended over 18 rather than 19 months. They separated the lump sum payment. Then they calculated the accumulated value of the first 12 annuity payments using a single calculation. Thereafter they moved this amount forward in time to the end of the 18-month period, thus adding the appropriate interest. They also clustered the last four payments together because, when taking an IP approach, these payments are not affected by the missed payment. These separate totals were then added together to obtain the account balance. In fig.7 I have reproduced the group’s response and annotated it. Although this approach is complex, students who execute it correctly demonstrate an ability to work with an IP approach, and to distinguish between single and multiple payment scenarios. They also display an understanding of the time value of money which can be seen in recognizing the need to move the first thirteen payments forward to the end of the 18-month period. A clustered IP approach is clearly the most complex of the three approaches discussed in the paper because it requires attention to several separate elements of the scenario as discussed above. However, it is not an elegant approach and does not produce the most efficient model. So while it reveals complex thinking, I would not encourage students to use it.

R1000 deposit ( )18

127,65%11000 +

= 1121,18457

Lump sum deposit of R1000 with interest for 18 months

Months 1 – end of 13

= 0,006375

(1,006375) 1](1,006375) 300[ 12 −

= 3752,72

Future value of first 12 annuity payments at end of month 13

Interest for R3752,72 for remaining months 3752,72 (1,006375)5 – 3752,72 = 121,15

Interest on the above for remaining 5 months

Remaining payments

0,006375(1,006375) 1](1,006375) 300[ 4 −

=1219,25

Future value of last 4 payments at end of month 18

Total: 1121,18 + 3752,72 + 121,15 + 1219,25 = 6214,30 Adding together 4 sub-totals

Fig 7: Clustered individual payment approach

Conclusion Data from this study suggest that an AB approach is likely to be the first approach students use when working with annuities. However, it is important that they are able to use both AB and IP approaches and to make links between them. While an IP approach is more powerful, it is not an obvious starting point for annuity-based problems, partly because it does not mirror what goes on in the bank. But students need to understand how a simple geometric progression models the series of payments. This is the basis of the IP approach. If students don't adopt an IP approach, they are unlikely to see the structure in the problem and therefore cannot avoid iterative calculations. Additional data from the study, not discussed here, suggest that once students accept this model, they retain it and can use it. This means that students need to be

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convinced that the IP approach is an appropriate model. In order to accept the validity of the mathematical model, students need to be convinced that analysing the whole can be done by disaggregating and analysing the parts. When students accept that by investigating the parts they can describe the whole, they gain access to the structure of an annuity. Further research is needed to explore productive ways of helping students accept these ideas.

References

Ball, D., Bass, H., & Hill, H. (2004). Knowing and using mathematical knowledge in teaching: Learning what matters. Paper presented at the Proceedings for the Twelfth Annual Conference of the South African Association for Research in Mathematics, Science and Technology Education (SAARMSTE) Cape Town, South Africa.

Glaser, B. (1978). Theoretical Sensitivity: Advances in the methodology of Grounded Theory. Sociology Press.

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IMPACT OF LANGUAGE ON THE CONSTITUTION OF MATHEMATICS IN PEDAGOGIC CONTEXTS: A CASE DRAWN FROM

A RESEARCH AND DEVELOPMENT PROJECT Anthea Roberts

Schools Development Unit and Group for the Study of the Constitution of Mathematics in Pedagogic Contexts

School of Education, University of Cape Town

This paper considers how the use of language in two high school mathematics classrooms could influence learning. The linguistic theory of systemic functional linguistics (SFL) frames an analysis of lesson vignettes in two Grade 9 classrooms in an attempt to assess whether teachers provide learners with orders of meaning which are globally valid. The content topic for each lesson is the simplification of exponential terms and expressions.

Introduction What is constituted as mathematics at the level of language? How is the learner’s understanding influenced by the language in circulation in the pedagogic context? This paper attempts to address these questions with reference to two Grade 9 lessons. All data used here derives from a research and development project with five working class schools, the aim of which is to improve maths and science performance of learners by addressing teaching practices in the five schools.19 In our consideration of classroom practice in the project we attempt to describe what mathematical objects are in circulation in the pedagogic context, how these objects are related to each other in terms of the logic that defines those relationships.

In the initial stages of the project forty-five lessons across Grades 8 to 10 (three consecutive lessons, per grade, per school) were observed. The topic of simplifying exponential expressions was addressed in various Grade 9 classes and presented an opportunity to do a comparative analysis of how a content topic is developed by different teachers. Through an analysis of a selection of lesson vignettes that exemplify the teaching of procedures for the simplification of exponential expressions, an attempt is made to determine what the grounds of the procedures might be and what the objects are that the teachers and their learners operate on. Within the project the contention is that calculations, when ‘word perfect’, often obscure significant features of the grounds of pedagogic practice, but that the presence of errors enables productive analysis of the grounding of mathematics.

A theoretical framing of the problem Competence with the discourse of mathematics has significant implications for what is constituted as mathematics at the level of the classroom. If learners are unable to access the language of the discourse they are disadvantaged. It is, however, taken to be the case that every specific instance of the pedagogising

19 The Mathematics and Science Education Project in the School of Education at UCT.

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of mathematics constitutes an inter-related collection of semiotic objects and relations as mathematics. Hence the guiding questions: What is constituted as mathematics? And how is it so constituted?

Systemic Functional Linguistics The linguists Halliday and Martin contend that there is a sharp distinction to be drawn between the language of everyday reality and the language of science (Halliday & Martin,1993: 11). They describe the discourse of science, and, by extension, of mathematics, as a discourse of ‘specialised knowledge characterized by systems of technical concepts arranged in strict hierarchies of kinds and parts’ – that is, a discourse of ‘organised knowledge’ (Op.cit.1993: 6).

In this paper I evaluate the discourse in the mathematics classroom, using elements of Halliday’s Systemic Functional Linguistics (SFL). SFL highlights a powerful feature of scientific discourse—its grammatical organisation—that O’Halloran (2005) describes as the interrelation between discourse-semantics, lexicogrammar and phonology. I shall restrict my discussion to the first two.

The discourse-semantics relates to the way the discourse structure is ‘typified by constructions of nominal groups and clauses’. The latter are important because, in the context of working within mathematical and scientific discourses, nominalisation enables its users to ‘construe a particular form of reasoned argument’ (Halliday & Martin,1993: 7), specifically syllogistic reasoning, capable of drawing necessary conclusions about phenomena and objects that lie beyond sensible experience. The construction of nominal groups warrants further discussion. Nominalisation functions at syntactic20 and semantic21 levels. As a syntactic device nominalisation relates to the process of converting verbs and adjectives to nouns, and of converting clauses to nominal phrases. A clause contains a verb and implies an action being performed. A phrase contains no verb and therefore can have an action performed on it, or can perform an action itself. An example of a suitable clause is: “division of a term in the denominator by a term in the numerator facilitates the simplification of expressions”. The nominal group “division of a term in the denominator by a term in the numerator” is a mathematical notion – an object, and as such can be related to another nominal group (object) ‘the simplification of expressions’, by the verb ‘facilitates’.

Lexicogrammar refers to the technical terms which populate the discourse. For mathematics the technical lexis has two features: terms that are distinctly technical like, for example, exponent and denominator, as well as non-technical terms that are rendered technical in the way they are used to describe mathematical processes. Examples of the latter are simplify and integrate (Veel, 1999:193). The lexical density of the discourse is a relative measure of the occurrence of technical terms in a text, where the latter is a typical instantiation of the discourse, or of a region of the discourse. More importantly, we can think of lexical density as a relative measure of explicitly construed and recognised objects of a discourse.

This interrelation among the elements of a discourse is referred to as grammatical metaphor. This term is described by Halliday as appropriate because its function at the level of grammar resonates with the function of a metaphor in language whereby nominalisation converts verbs (actions) and adjectives (descriptions) to nouns (objects) whose meanings are construed as ‘technical abstractions’ and whose semantic status has evolved (1993: 13). Grammatical metaphor operates at the level of meaning (ideational) and the level of social relations (interpersonal). At the level of the ideational the tool is refined for analysis of what is constituted as reality (the field of existents) and the relationships engendered by that constitution (the logical), as well as the manner in which language construction enables the latter (textual)

20 The grammatical construction of the text. 21 The meaning of the text.

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(cf. O’Halloran, 2005: 62). At the level of the interpersonal SFL facilitates an analysis of how ‘interactants negotiate the exchange of information … and what … this reveal(s) about their social relations’ (Op.cit. 2005: 67).

Competence with the discourse of mathematics can be evaluated within the SFL framework. At the ideational level, the lexical density of the text is a syntactic measure of the constitution of existents. That is, lexical density can be taken as an index of the level of ability to define mathematical objects through technical lexis and the process of nominalisation; syllogistic reasoning can be measured by how grammatical metaphor is used as a semantic tool in the construction and extension of relationships between objects.

Remarks on the constitution of mathematics The focal question for this research is what is constituted as mathematics. In order to describe what is constituted as mathematics I draw on the work of Davis & Johnson (2007) who claim that ‘mathematics is constituted through the operation of evaluative criteria’ (2007:7; italics in the original). One of the examples they discuss is how, in a particular lesson, an error resulted from the specific evaluative criteria in play and how the outcome was mathematically non-legitimate. They conclude that when criteria are not consistent with the intended object, the outcome is often the constitution of a mathematically non-legitimate procedure or statement as mathematics. A closely related phenomenon discussed by Davis & Johnson is the strange effect on mathematical objects of criteria regulated solely by procedures, where they note that ‘the ideas of mathematics and its objects may well be constituted as effects of standard procedures’ and that these are ‘likely to be inconsistent and unstable’ (Op. cit. 2007: 10). Teachers commonly establish mathematical ideas and objects through a set of worked examples. Such examples illustrate the procedures and rules for solving problems but do not necessarily provide as resources all the evaluative criteria that might be made available through a discussion of the general case. Davis & Johnson conclude that ‘a fundamental element of appropriate evaluative criteria for school mathematics is knowledge of the supporting ground for the mathematical operations’ (Op. cit. 2007: 11). As part of my analysis I shall consider the extent to which the latter is made available to learners.

Methodological Framework Each lesson was divided into a series of evaluative events22 to facilitate an analysis of a lesson at every stage of its elaboration. Each event usually starts with a question or some introductory remark, and ends with some conclusion. At the first stage of data collection, the mathematical object and evaluative criteria for each evaluative event were identified. At the next stage the outcome, that is, what was constituted as mathematics, was established. The data was then analysed within the SFL framework. When developing the lesson transcripts each utterance was numbered. The numbers are included in brackets before each utterance in excerpts reproduced here. This functions as an indicator of the locality of the excerpt in the lesson as well as an indicator of where transcripts have been edited.

22 A lesson segment with a discrete focus, and set of criteria which determine interaction.

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lesson 1: Laws for operating on exponential expressions

The lesson topic was the rule for division of exponential expressions, , and the teacher’s strategy

was to transform each exponential term to the form , then simplify the expression

using whole number division. After working through four examples in that manner the rule for division was inductively established. Despite having established the rule, the focus on expanding the numerator and denominator, and using division, remained strong. Towards the end of the lesson the teacher gave learners the option of using either method for solving problems of this nature. The lesson could be divided into the six evaluative events indicated in Tables 1 to 6 in the Appendix. I shall use Evaluative Event 2 to illustrate

how I used SFL as a tool for analysis. Prior to addressing the class, the teacher had written on the

board. The excerpt from the transcript indicates the criteria she provided for learners to think about the problem:

(13 )Teacher: Okay. Isn’t two to the power four divided by two to the power two the same .. sorry .. ja

anyway .. the same as .. [writes the equation on board as she

speaks] er .. two times two times two times two? Isn’t it?

(14) Learners: Yes miss.

(15) Teacher: Remember anything raised to the power four .. you write it up four times. Okay? And two to the power two is the same as ..

(16) Learners: Two times two.

(17) Teacher: Okay. And they’re all to the power one. So looking at that, right, can you simplify it .. or how can I say er … … … what is two divided by two equal to?

(18) Learners: One.

The teacher’s aim was to induce learners to simplify the expression by dividing each value 2 in the denominator by a 2 in the numerator. Learners recognised the expanded representation of the exponential term, but did not recognise the operation suggested by the teacher, as can be seen from their responses. The portion of the transcript below has been edited, as can be seen from the line numbers; I have omitted the learner responses which are very similar to those already reflected.

(25) Teacher: … What can I do here? Remember two divided by two is one. Three divided by three is .. one. Four divided by four is .. one. So what can I do here …

(26) Learner 1: Times it.

(31) Teacher: No, I want to simplify this. … How do I do that?

(33) Learner 2: Miss, two goes into four.

(34) Teacher: No, man!

(35) Learner 3: Mustn’t you first add it?

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(36) Teacher: Is there nothing that we can divide by here? What is .. what is similar between our numerator and denominator?

At an ideational level there are two aspects for consideration: the syntactic and the semantic. Table 7 is a summary of my analysis of the teacher’s discourse at the syntactic level. The analysis indicates that the teacher’s speech does not exhibit elements characteristic of scientific discourse. Her speech is closer to that of everyday discourse. From a semantic perspective she has not employed grammatical metaphor to enunciate on the mathematical relationships within the expression or as a vehicle for encouraging syllogistic reasoning from the learners. Instead she has provided a series of examples23 to encourage learners to make a logical conclusion. Learners’ responses24 indicate that her strategy has not worked. Clearly the evaluative criteria the learners were given did not provide them with the supporting ground for the operation the teacher wanted them to perform. Table 8 summarises my analysis of the syntax of the discourse of each evaluative event. For purposes of comparison I have separated the teacher’s speech from that of the learners’. This is indicated in the table by (T) and (L).The data in Table 8 confirms that the pattern evident in Evaluative Event 2 is typical of the rest of the lesson. So what is constituted as mathematics through this discourse? Some interesting anomalies emerge. For example, towards the end of Evaluative Event 2 the teacher summarised the evaluative criteria for simplifying the expression

:

(54) Teacher: So two can be divided by two. Isn’t it?

(55) Learners: Yes.

(56) Teacher: And one two can be divided by another two. Isn’t it?

(57) Learner: Yes.

(58) Teacher: And how many twos are you left with?

(59) Learners: Two.

(60) Teacher: Two twos. Isn’t it?

(61) Learner: Yes.

(62) Teacher: Which is in actual fact two to the power two. Isn’t it?

From the extract we can write the teacher’s criteria as follows: In an expression of the form aan

m

expand

am

an to

a.a.a ... to m factorsa.a.a ... to n factors

then divide sets of aa

until either the numerator or denominator has no more

a’s. The number of remaining a’s is the value of the exponent that a is raised to in the answer. (Implicit is the fact that a, m and n ∈ Ν). Evaluative Event 2-1 is a sub-event where a learner sought clarity on this

criterion. The learner developed an example with as numerator and as denominator. The teacher had invited her to the board:

23 Utterance 25 24 Utterances 26, 33 and 35

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(79) Learner: Miss, say maar so{for example} it’s two and six ne miss … one .. two .. three .. four .. … … Six times ne miss ...[writes / / on the board] and then two and three …[writes / / on the board.] and then must you scratch three out because here’s three?

The learner had produced two separate exponential terms that she transformed by expansion. She did not insert the division line, which signifies that she did not see division as the operation. She had ‘scratched out’ an equal number of twos in each expression and recognised the remaining number of twos as the solution.

The evaluative criteria recognised by the learner led to

being constituted as mathematics. The teacher acknowledged that the learner was using the criterion correctly and, in fact, reaffirmed the use of the criterion at various moments in the lesson.

In Evaluative Event 5 the teacher had introduced the rule , and summarised it in this way:

(133) Teacher: Subtract your exponents when you’re dividing. Right? What do we say in multiplication? You add your exponents and you keep your base. Your base must be the same. Keep that in mind. Your base the same .. add exponents when multiplying. Base the same .. subtract exponents when dividing. Right? Do you understand that?

In Evaluative Event 6 she summarised the lesson topic using both methods to simplify the expression

. Her first method involved expansion of the terms in both numerator and denominator. She

confirmed the evaluative criteria in operation:

(173) Teacher: So a divided by a times a divided by a. How many a’s are we left with?

What was constituted as mathematics was the notion that

implied being left with two a’s and one b, which equals a2b.

As criteria for use of the rule she made the statement:

(178) Teacher: Right. Another way you can do it is by writing a. Remember the rule in dividing. You keep your bases and you subtract your exponents. Right? So, you keep your a and you subtract your exponents .. (a to the power four) minus two times b to the power three minus one. Isn’t it?

By the time learners were given an exercise to work on they were encouraged to use the method they preferred. This is how a learner interpreted the rule to constitute a mathematical statement using the evaluative criteria available to him:

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The learner had used the criteria made available to him, and had performed every operation in an arithmetically correct way, but what is constituted as mathematics is non-legitimate. The teacher had not provided supporting ground for the mathematical process she demonstrated through the worked examples. Consequently the learner did not have a conceptual understanding of the mathematical object he was transforming, which resulted in the solution he produced.

Lesson 2: Brief comments

The same rule taught in Lesson 1, , was being taught in a Grade 9 class at a different school. The teacher used a similar pedagogic strategy to the other teacher. Space prevents a detailed analysis of the lesson, so I restrict my comments to one anomalous mathematical construction that emerged.

The teacher provided learners with both methods for simplifying exponential expressions involving division. Learners seemed confident of their understanding of the procedures until the teacher challenged them:

(49) Teacher: Alright. Number two. If the number that has not been cancelled is below kwi-denominator {in the denominator}. How do you do it? Let’s say eight to the power five divide by eight to the power seven. Ngubani ofuna ukutraya? {Who wants to try?}

A learner came to the board to solve the problem, , using what the teacher had referred to as the ‘primary method’. He ‘cancelled’ all the eights in the numerator with five eights in the denominator. This left two eights in the denominator not scored through, but he could not conclude with a simplified representation of the term.

A second learner came to his rescue. She chose to use the rule which she interpreted thus: 85 ÷87 =28

= 8−2 .

Figure 1 shows both learners’ solutions.

Figure 1: Board image

One can only speculate on why the second learner wrote . Did she mean ‘two eights at the bottom’?

This is possible given the criteria she was working with, which the teacher reiterated by way of correcting the learner:

(59) Teacher: So if there is nothing left from i-numerator {from the numerator} just write one and then i-denominator {denominator} you count how many numbers that are left. Okay?

Followed by (63) Teacher: And then you change it ..the positive exponent … you to change it into a negative

exponent, you write eight to the power minus .. (two)

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The teacher’s discourse was very similar to that of the teacher in the first lesson discussed; the outcome was also the constitution of mathematical objects that are problematic.

Concluding remarks In both classrooms the use of non-specialised language regulated the flow of mathematical work. Most utterances were context-dependent , meaning that both teachers limited their discussion to the examples they were working on, and did not generalize the mathematical notions in circulation in the lesson. They depended on gestures for clarification, for example: “Look at the exponents. That’s four .. that’s two and that’s two. Right? Look at the exponents. That’s three and that’s two and this is one”.

One of the outcomes of each lesson was the constitution of mathematical solutions that were questionable. In neither classroom was there any evidence that learners were able to organise their thoughts in a concise manner to form the relations necessary for logical reasoning. This could be ascribed to the fact that the teachers’ discourse remained at the level of instruction and regulation, which are processes. By implication teachers did not exploit nominalization i.e. convert processes (actions) to mathematical objects or notions (nouns). Martin (1993) contends that while it is fine for commonsense knowledge to be used in classrooms, appealing to commonsense formulations is best restricted to functioning as a starting point since commonsense understandings differ from scientific ones. He recognizes that the links between commonsense understandings and theoretical concepts are what make learning possible, but that schools have a crucial responsibility to induct students into the technical grammar, which will enable them to establish systems of definitions and taxonomies of logical relationships. Undoubtedly both teachers understood the concepts circulating in their respective lessons and were able to use the taxonomy of logical relationships to select criteria for simplifying the exponential terms. What was absent from the lessons was the conscious process of inducting learners into the discourse in a manner that gave them space to practise the discourse and ‘construe the form of reasoned argument expected’ (op. cit. 1993: 7). The question I am left with is how this notion could be achieved. Ultimately the aim has to be the nurturing of learners who are schooled in the technical lexis of mathematics and who can coherently work on mathematical objects in a way that encourages them to select suitable procedures for solving problems and exploring mathematics.

APPENDIX Evaluative Event 1: Revision of the previous day’s lesson topic: the rule for multiplication of exponential expressions 1 This was a general discussion sparked by the teacher’s question

“What do you remember about exponents?”. Table 1: Evaluative event 1 of Lesson 1.

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Evaluative Event 2: Derivation of the rule for dividing exponential expressions 2-1

Simplification of

2-2 A learner’s question. The learner used an example to clarify her

understanding of the procedure for simplifying the expression 2-3

Simplification of

2-4 Simplification of

2-5 Inductive derivation of the rule for division of exponential expressions, by considering each of the examples from sub-events 2 to 4

2-6 Simplification of by expanding each term and using simple

division Table 2: Evaluative event 2 of Lesson 1.

Evaluative Event 3: Using the rule for dividing exponential expressions 3-1

Simplification of using the rule for division of exponential

terms 3-2

Simplification of using both procedures

Table 3: Evaluative event 3 of Lesson 1.

Evaluative Event 4: Derivation of a0 = 1. 4-1 Show, through expansion of each term and using simple division, that

Table 4: Evaluative event 4 of Lesson 1.

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Evaluative Event 5: Simplification of exponential using the two strategies: expansion and the rule for division 5-1

Simplification of through expansion of each term and using

simple division 5-2

Simplification of using the exponential rule for division

Table 5: Evaluative event 5 of Lesson 1.

Evaluative Event 6: Consolidation 6 Learners copy notes from board and do an exercise

Table 6: Evaluative event 6 of Lesson 1.

Utterance #Words # Technical terms Lexical density (%)

# Instances of nominalisation

13 32 6 6/32 ~ 18.8% 0 15 24 3 3/24 ~ 12.5% 0 17 31 4 4/31 ~ 12.9% 0 25 30 3 3/30 ~10.0% 0 31 11 1 1/11 ~ 9.1% 0 34 2 0 0/2 ~ 0% 0 36 19 3 3/19 ~ 15.8% 0 Total 149 20 20/149 ~ 13.4% 0

Table 7: Measuring the lexical density of utterances and the instances of nominalisation in Evaluative Event 2 of Lesson 1

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Utterance #Words # Technical terms Lexical density (%)

# Instances of nominalisation

1 (T) 40 9 22.5 0 1 (L) 14 3 21.4 0 2 (T) 971 126 13.0 0 2 (L) 328 50 15.2 0 3 (T) 176 26 14.8 0 3 (L) 26 3 11.5 0 4 (T) 76 12 15.8 0 4 (L) 6 0 0.0 0 5 (T) 100 13 13.0 0 5 (L) 46 8 17.4 0 Summary (T)

1363 186 13.6 1

Summary (L)

420 64 15.2 0

Table 8: Measuring the lexical density of utterances and the instances of nominalisation across all Evaluative Events in Lesson 1

Acknowledgements I would like to thank Zain Davis, Shaheeda Jaffer, Yusuf Johnson and Roger Mackay for their guidance and support in the production of this paper. I also benefited from discussions with other members of the Group for the Study of the Constitution of Mathematics in Pedagogic Contexts (GSCMPC). This paper has been supported by the Mathematics and Science Education Project at the University of Cape Town. The opinions and arguments recorded here, however, remain the responsibility of the author.

References Bernstein, B. (1975). Class, Codes and Control, Volume 3. London: Routledge & Kegan Paul.

Davis, Z. and Johnson, Y. (2007a). Failing by Example: Initial Remarks on the Constitution of School Mathematics, with special reference to The Teaching and Learning of Mathematics in Five Secondary Schools. In Setati, M., Chitera, N. & Essien, A. (eds.),Proceedings of the 13th Annual National Congress of the Association for Mathematics Education of South Africa Vol. 1, 121 – 136.

Davis, Z. and Johnson, Y. (2007b). What functions as ground for mathematics in schooling? Further remarks on the constitution of school mathematics, with special reference to the teaching and learning of mathematics in five secondary schools. Paper presented at the Association for Mathematics Education of South Africa Western Cape Regional Conference, 19 May 2007, Mowbray Campus, Cape Peninsula University of Technology, Cape Town. Mimeo.

Halliday, M.A.K. and Martin, J. (1993). Writing Science – Literacy and discursive power. London: Falmer.

Hasan, R. (2001). The ontogenesis of decontextualised language: Some achievements of classification and framing. In A. Morais, I. Neves, B. Davies, H. Daniels (eds.),Towards a Sociology of Pedagogy, New York: Peter Lang.

Holland, J. (1981). Social Class and Changes in Orientations to Meanings. Sociology15(1), 1 – 18.

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O’Halloran, K.L. (2005). Mathematical discourse – language, symbolism and visual images. London: Continuum.

Painter, C. (1999). Preparing for school: developing a semantic style for educational knowledge. In F. Christie (Ed.). Pedagogy and the shaping of consciousness: Linguistic and social processes. London: Continuum.

Veel, R. (1999) Language, knowledge and authority in school mathematics. In F. Christie (Ed.). Pedagogy and the Shaping of Consciousness: Linguistic and Social Processes. London: Continuum.

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THE ZONE OF PROXIMAL DEVELOPMENT IN THE LEARNING OF DIFFERENTIAL CALCULUS

SIBAWU WITNESS SIYEPU

Cape Peninsula University of Technology siyepus@cput.ac.za

Globally there is concern about the poor performance of students in mathematics generally and in particular differential calculus among the foundation students in universities of technology. This article uses socio-cultural Vygotskian theoretical framework focusing on the zone of proximal development (ZPD) to explore the effects of self-study activities on students’ understanding of differential calculus among the foundation mathematics students. This study promotes equity and quality in the learning of differential calculus. The author used mixed research integrating quantitative and qualitative research methods and collected data through observations, in-depth interviews, audio-video recordings; field notes and written tests. The findings showed that students improved their study skills; understanding; positive self-esteem; confidence and lack of insight in aspects of mathematics such as substitution, simplification, trigonometry identities, algebraic identities, conceptualization, and derivatives of algebraic expressions and trigonometry functions. The study motivated the students to learn challenging problems and to enjoy the learning of differential calculus. Introduction This paper reports the effects of self-study activities on students’ understanding of differential calculus. The author explored Vygotsky’s (1978) concept of the zone of proximal development (ZPD) to promote students’ understanding of differential calculus among the foundation mathematics students in the field of engineering. The zone of proximal development implies that “at a certain stage in development, students can solve a certain range of problems only when they are interacting with people and in cooperation with peers” (Morris, 2008, p. 1). Morris (2008) explains that once students’ problem solving activities have been internalized; the problems initially solved under guidance and in cooperation with others can be tackled independently. This study uses socio-cultural Vygotskian theoretical framework to explore foundation mathematics students’ understanding of differential calculus using self-study activities in a collaborative approach. Vygotsky’s socio-cultural theory of human learning “describes learning as a social process and the origination of human intelligence in society or culture” (UNESCO, 2002, p. 1). The author discusses related literature with regard to the students’ understanding of differential calculus. I explain the methodology employed to collect data in this study. Lastly there is a discussion of findings, and a conclusion.

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Rationale of the study This study promotes equity and quality in the learning of differential calculus. The term equity refers to giving all students the opportunity to achieve excellence. The quality of mathematics learning refers to the content of mathematics that is worthwhile and valuable for both the students and society in general, and about how to support students so that they can develop this mathematics. The quality of learning emphasizes the meaningful understanding of rules, algorithms and procedures in differential calculus. This study encourages students to develop skills such as independent learning; taking control over their learning; becoming lifelong learners and learning how to evaluate them. Research question: The main question that the study explored is: What is the impact of guided self-study activities on the performance of students with regard to differential calculus? Literature review The poor performance of students in differential calculus has been a problem for many years. The differential calculus component forms a major section of the first year mathematics course (Naidoo & Naidoo, 2007). Differential calculus is fundamental to further study of mathematics for engineering at a University of Technology. Many researchers reveal that students have difficulties in understanding differential calculus. Students have problems “in understanding the meaning of the derivative when it appeared as a fraction or the sum of two parts, application of the chain rule for differentiation, and procedures for solving maxima and minima problems” (Naidoo & Naidoo, 2007 p. 56). This study uses the ZPD to unpack the meaning of the derivative and, to explain and trace meaningful applications of differentiation rules to solve differential calculus problems. Other researchers like Boaler (1997) and Naidoo (1996) claim that students enter tertiary institutions under prepared to grasp the content of differential calculus. Some researchers argue that students know less because they are given less and worse preparation in high schools. As a result they do not enjoy mathematics and are de-motivated (Naidoo & Naidoo, 2007). They also claim that time and attention given to study mathematics is limited. This study encourages students to find extra time to practice and to learn differential calculus Naidoo & Naidoo (2007). Miller (2006) claims that some of the difficulties stem from not thoroughly learning algebra, lack of problem-solving skills, or lack of study skills The self-study learning activities address the difficulties as they cropped up in the learning process of differential calculus through discussions and negotiations in a collaborative approach. Theoretical framework This study uses socio-cultural Vygotskian theoretical framework as a tool to facilitate students’ understanding in the learning of differential calculus. The researcher’s role is that of facilitating learning among the foundation mathematics students using self-study activities with content which go beyond the students’ current capabilities. This occurs through self-study activities for students to step from their current level to a higher level of understanding. The process can be understood in a socio-cultural perspective with reference to Vygotsky’s ZPD, which explains how to advance the students’ learning process.

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In this study the researcher adapted the three ways of analyzing collaborative learning as suggested by Carlsen (2008) to the context of mathematics for engineering. Carlsen (2008) highlights three major ways of organizing collaborative working groups as: expert-novice collaboration, peer collaboration, and group collaboration (three or more participants).

Expert-novice collaboration: In expert-novice collaboration “a student is able to solve mathematical activities with a more competent, adult or student that he or she is not capable of solving on his/her own” (Carlsen, 2008, p. 23). For this type of collaboration a student works with the lecturer or a more competent student to get support to master meaningful understanding of the self-study activities on differential calculus. This happened through discussions, negotiations and sense making of the self-study activities in a mathematics classroom context. The aim is that the student assisted by a more knowledgeable student or a lecturer should be able to solve the problems independently.

Peer collaboration: In peer collaboration students work in pairs of similar levels of understanding with respect to a problem posed to them (Carlsen, 2008). Carlsen (2008) asserts that “collaborative peers were able to solve problems that they were unable to solve on an individual basis” (p. 24). Collaboration stimulates students to try out new and untested ideas to solve challenging problems and it also helps students to explore other solution strategies through trial and error. They discuss and argue to convince one another with regard their strategies. In that way they make mathematical meaning among themselves through explaining their thinking and interpretation.

Group collaboration: Group collaboration refers to three or more students, normally up to five or six, working together to solve problems (Carlsen, 2008). Carlsen (2008) argues that “group collaboration, as an activity can be used in mathematics teaching to foster various aspects of mathematical learning, such as logical reasoning, communication, problem solving, and making connections within the subject area” (p.24). The difficulties experienced by students with regard to differential calculus require students to have logical reasoning and an ability to make connections within differential calculus to develop advanced understanding of calculus. Through debates and discussion in group collaboration students are able to overcome many of their learning difficulties of differential calculus.

Methodology This study used mixed research, that is, research that involves the mixing of quantitative and qualitative research methods. A mixed research method is a research in which the researcher uses quantitative data for one stage of a research study and qualitative data for a second stage of research (Hunt, 2007). A mixed research is used for the conducting of a detailed research (Hunt, 2007). The advantages of a mixed research are: 1) The strength of the research; 2) Use of multiple methods in a research helps to research a process or a problem from all sides; 3) Usage of different approaches helps to focus on a single process and confirms the data accuracy. A mixed research complements a result from one type of research with another one (Hunt, 2007, p. 1). All these paradigm characteristics are mixed in one case study. This method design involves research that uses mixed data (numbers and text) and additional means (statistics and text analysis) (Hunt, 2007).

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Research site This study took place in a university of technology in Cape Town, South Africa among the foundation mathematics students in the field of engineering. Sample of the study The study involved 19 students who were selected based on their poor performance in the two tests written before their involvement in the intervention programme. All students in the sample group are English second-language speakers. Data collection Data was collected through observation, in-depth interviews, audio-visual recordings, field notes and written tests. This paper reports findings based on the tests.

The students in the sample group wrote four tests. The first two tests were written before the intervention and the third and fourth tests were written after intervention. During the intervention period the students in the sample group used self-study activities to learn differential calculus and wrote tutorials as exercises after each lesson presentation. All the four tests were analyzed quantitatively. Test three was also analyzed qualitatively looking at students’ errors and misconceptions in their calculations. Analysis in test three was to give feedback to the students as well as feed forward to prepare students for the final examination treated as test four in this study.

Findings and discussions This is a reflection on students’ performance before and after one intervention programme.

Table 1 below shows the analysis of marks based on the 4 written tests. T1; T2; T3; and T4 are acronyms used for tests 1; 2; 3 and 4 and Avg is used for average.

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Table 1: Analysis of mathematics results of the four written by students in the sample group.

Analysis of mathematics results of the sample group

T1 T 2

T 3 T 4 Avg. marks for T1 &T2

Avg. marks for T3 &T4

Avg. marks for all tests

Difference betweenT4 & T 3

Difference between avg. marks for T1 & T2 and avg. marks for T3 & T4

L1 76 82 82 83 79 82.5 80.75 1 3.5 L 2 40 09 56 62 24 59 41.75 6 35 L 3 46 24 33 38 35 35.5 35.25 5 0.5 L4 38 27 73 58 32 65.5 49 -15 33.5 L5 66 56 85 78 61 81.5 71.25 -7 20.5 L 6 46 31 67 53 38 60 49.25 -14 22 L 7 38 51 82 77 44 79.5 62 -5 35.5 L8 50 50 76 70 50 73 61.5 -6 23 L 9 36 18 89 72 27 80.2 53.75 -17 53.2 L 10 42 24 84 70 33 77 55 -14 44 L11 42 67 73 67 54 70 62.25 -6 16 L12 24 22 56 40 23 48 35.5 -16 25 L 13 44 53 56 50 48 53 50.75 -6 5 L 14 22 33 42 68 27 55 41.25 26 28 L 15 44 51 44 75 47 59.5 53.5 31 12.5 L16 24 24 56 73 24 64.5 44.25 17 40.5 L17 54 36 42 68 45 56.5 50 26 11.5 L18 34 60 71 77 47 74 60.2 6 27 L 19 42 56 60 63 49 61.5 55.25 3 12.5

% Passed 21 47 84 89.5 % Failed 79 43 16 10.5 Av. Mean 42.5 40.8 64.6 65.4

Students’ performance before and after intervention (quantitative findings) In test 1 only 4 students out of 19 passed, that is, students who obtained more than fifty percent but in that four, one student obtained a distinction. This reveals that only 21% of students in the sample group passed and 79 % failed. In test 2; 9 students out of 19 passed with one student having a distinction. This reveals that 47, 4 % passed and 42, 6% failed. The student who obtained distinction in test 1 also obtained a distinction in test 2.

In test 3; 15 students out of 19 passed and 6 students obtained a distinction. This reveals that 84, 2 % passed and 15, 8% failed. One may look at learner 9 who moved from 18% in test 2 to 89% in test 3 see Table 1. In test 4; 17 students passed and 2 students failed. This suggests that the number of passes increased as the number of failures decreased. Table 2 shows a comparison of the average marks for tests

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1& 2 and test 3. The vertical line represents percentages obtained by the students in the tests and the horizontal line represents students abbreviated as L1up to L19.

Table 2: Comparison of the average marks for tests 1&2 and 3

The comparison of the average marks of tests 1 and 2 to the marks for test 3, show an improvement in the scores of the students. It is only learners 3 and 15 that show a decline in marks. Findings pertaining students’ responses in test 3(qualitative findings) In the analysis of test 3, the researcher noticed poor class attendance among the three students who failed the test and the fourth student had inadequate background to understand differential calculus at the first year university level. The poor class attendance may indicate lack of seriousness and/or lack of understanding the importance of regular class attendance and class participation.

The first question was to use the first principle to find the derivative of 24

x. Eighteen (or 94,7%) students

in the sample group were able to write the correct formula for the first principle as

hxfhxf

hxf )()(

0lim)( −+

→= .The fact that the majority of students apply the first principle of

differentiation correctly indicates that this section was handled thoroughly in the classroom teaching as well as on high school level. Some few students had problems with substitution and simplification as a result they could not get the right answer. The majority (or 84, 2 %) of students in the sample group had a problem of ignoring or failure to leave their answers in the simplest form. For instance they leave their

answers as 48 xx− instead of cancelling x to get 38 x− . There was probably no emphasis on leaving answers in the simplest form either in the instruction or in class teaching or in the lower levels. There is also a problem of uncertainty or lack of confidence, lack of insight and practice as one student changed the right application of the first principle to something wrong. The student noticed the error in the calculation

AVG Test 1& 2 and Score Test 3

79

2435 32

61

38 44 50

27 33

54

23

48

27

47

24

45 47

82

56

33

7385

6782 76

89 84 73

56 56 42 44

5642

7

0

20

40

60

80

100

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 L17 L

%

Avg % test 1&2 % Score test 3

Mar

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but instead of recognizing it as a problem in simplification s/he took it as a problem in the formula and as a result changed the correct formula to a wrong one.

The second question was to find the derivative of xxxy ln3)3(ln3ln ++= . Only two (or 10, 5%) students scored full marks for this problem. Some few students have no clue about the procedure to follow in order to get the solution. They confuse logarithmic power rule with differentiation

power rule and they fail to treat the exponent of the first term 3lnx as a constant. Perhaps they do not understand that 3ln is a constant. The majority of the students in the sample group had an understanding of applying differentiation power rule in the first term, treating 3ln as a constant but they do not understand

that they cannot apply differentiation power rule in the middle term x)3(ln as x is not a natural number. Some treat the middle term as if it is the same as the first term. Some had an understanding that the base

3ln is a constant and the exponent x cannot be treated as a natural number but they could not follow the

procedure correctly. The fact that the derivative of xa where the base a is a constant and the exponent

x is any variable is axa ln leads to a solution that the derivative of x)3(ln is x)3(ln )3ln(ln . Most of those

who had an understanding of this procedure wrote x)3(ln 3ln instead of x)3(ln )3ln(ln . This can be seen as lack of insight and it also supports one of the findings mentioned in the literature that most of the students tend to memorize the procedures. This assists the researcher to identify students’ mistakes to find where to explain more and where to put an emphasis in the revision process. During the revision there is a

need of showing the difference between the derivative of nx and the derivative of xa . In the case of nx

we apply the power rule and the derivative is 1−nnx , that is, x is a variable that represents the unknown base and n is a natural number.

The third question was to find out the derivative ofx

xycos1

sin+

= ; this requires the understanding and

application of quotient rule which says the derivative of a quotient equals the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by

the square of the denominator. In symbols it says =⎭⎬⎫

⎩⎨⎧

vu

dxd

2v

vuuv ′−′.The nature of this problem also

requires the knowledge of the derivative of sine and cosine. Eighteen (or 94, 7%) students applied quotient rule correctly and showed an understanding of the derivatives of the two functions, sine and cosine. Only one student who failed to write the derivative of xcos as xsin− instead wrote xcos− . Twelve or (63, 1%) students scored full marks for this question and five or (26, 3%) students lost marks only in the last step

where they could not simplify 2)cos1(

1cos

x

x

+

+ as

xcos11

+.

The incorrect answers were ;1cos +x xcos1

2+

;xcos

1;

xcos11

− and some correct answers were

untraceable, that is, the marker cannot understand the procedures followed to reach the correct answer. Some correct answers were reached unprocedural but by coincidence, for instance one student has

143

2)cos1(

)2sin2cos(cos

x

xxx

+

++ =

x

x2cos1

cos

+=

xcos11

+. The correct answer is

xcos11

+but it is difficult to

understand how the student reached the answer. The error is on the poor understanding of simplification

and cancellation, the student cancelled xcos and cos 2 x . At the same time the student failed to write

12sin2cos =+ xx to give 1cos +x in the numerator in step 2. There is an indication that the majority of students had problems with simplification. This is probably lack of emphasis in simplification in the junior and senior secondary schools. There is a problem of poor understanding of trigonometry identities and the property of the number one. One student tend to confuse the fact that one is an identity element for multiplication and that zero is an identity element for addition. This may assists lecturers teaching first years and the foundation students to revise the elementary algebra required as a prior knowledge in lessons. Students should also be informed of the basic knowledge required as a prerequisite to understand

the new content knowledge to be presented. One student treated 2)cos1( x+ as the difference of two

squares instead of being a square of a binomial as a result resolved 2)cos1( x+ into factors and obtained )cos1)(cos1( xx −+ instead of )cos1)(cos1( xx ++ . This confirms that some of the difficulties among the

foundation mathematics students stem from not thoroughly learning algebra in the elementary level. This finding suggests that some students in the sample group have problems with algebraic identities and factorization. These findings inform the researcher on the problems, errors, misconceptions to be expected among foundation mathematics students in differential calculus. They also help lecturers and teachers to understand the under-preparedness of students from high school when they enter tertiary institutions. The fourth question was to find out the derivative of )ln(tan4cot xxy += . This problem requires students’ understanding of the derivatives of cotangent and tangent, xln and chain rule. Fourteen students (or 73, 4%) in the sample group scored full marks in this question. One student (or 0,05%) copied the problem incorrectly instead of )ln(tan4cot xxy += wrote )ln(tan4cot xxy −= but did the problem correctly showing understanding of the derivatives of cotangent and tangent, applying chain rule and xln correctly. Another student showed a lack of insight in application of chain rule as he/she wrote ecxcos4− as a derivative of x4cot instead of xec4cos4− . One student (or 0,05%) could not find the derivative of

x4cot although he knew the derivative of xtan . Three students (or15, 8%) seemed as if they had no clue of the differentiation of cotangent and tangent, applying chain rule and xln . This indicates the weaknesses of the students and shows the researcher where to focus in remedial work. The researcher needs to apply expert-novice assistance more frequently with the weak students to improve their performance.

The fifth question was to find out the derivative of )32(cos3 += xecxy . This problem requires students’ understanding of the derivative of cosecant and, the ability to apply the product rule, power rule and chain rule. Only three students (or15, 8%) scored full marks in this question. Six students (or 31, 6%) in the sample group had no clue on what to do to solve this question and as a result they obtained zero (or 0%). There is an indication of the lack of insight as ten students (or 52, 6%) failed to do algebraic manipulation correctly. For instance one student failed to multiply exponents correctly. They tend to write

xecx cotcos− as a derivative of )32(cos +xec instead of )32cot()32(cos ++− xxec . This suggests that

they memorized the derivative and as a result they do not replace x with 32 +x . This may be taken as lack of insight, and /or poor conceptualization.

144

This indicates that there is much work to be done to enable the students in the sample group to differentiate problems involving combination of many rules. Students showed that they master derivatives of sine, cosine and tangent but it appears as a problem to understand the derivatives of the trigonometry reciprocals that is cosecant, secant and cotangent. This may be linked to the mathematical exercises and examples that tend to focus only on the three basic trigonometric functions that is, sine, cosine and tangent. This suggests that lecturers should try to balance their examples and exercises through involving the derivatives of the three reciprocals in the tutorials, examples, and sample tests and in the tutorial tests. There is a need to have exercises that integrates the application of many differentiation rules in one problem, for instance a problem that requires application of power rule, chain rule, and understanding of the derivatives of the trigonometry reciprocals, that is, cosecant, secant and cotangent.

The sixth question was to find the derivative of x

e

e

xxe

xey2

233

3

3+++= . This problem requires

students’ understanding of ;e ;xe chain rule; power rule and quotient rule. Eight students in the sample

group (or 42, 1%) showed an understanding of the derivative of ;xe application of power rule, quotient

rule and chain rule involving xe . One student copied the problem incorrectly although did the calculations correctly. Five (or 26, 3%) students in the sample group had no clue on how to differentiate this problem as they scored zero (0%) in this question. This suggests that students should be given many exercises

involving xe in the form of class activities and tutorials to engage them in the calculations involving application of power, chain rule and quotient rule. Another five students in the sample group (or 26, 3%)

lost marks through careless mistakes involving failure to apply chain rule and failure to understand that 2e is a constant. One may argue that they do not understand the application of chain rule. At the same time

they do not know that 2e is a constant. This may be interpreted as failure to understand the necessity to apply chain rule and the problem of conceptualization.

The seventh question was to find the derivative of )25(4cos xy = . This problem requires students’ understanding of the application of power rule, chain rule in differentiation of cosine. Eleven (or 57, 9%) of students in the sample group scored full marks in this problem. This indicates an understanding of the differentiation of a cosine function and application of chain rule. This suggests that some students need to get extra self-study activities to improve their understanding and application of chain rule in the differentiation of cosine functions. Eight (or 42, 1%) of the students in the sample group had no clue on how to solve this question. Conclusion In the analysis the students in the sample group showed improvement moving from 21% pass rate in the first test to 89, 5% pass rate in the fourth test. To put it differently out of 19 students in the sample group 15 students failed in the first test while two students failed in the fourth test. Another different way of putting it is that out of 19 students in the sample group 4 students passed in the first test while 17 students passed in the fourth test. The increase in the number of distinctions from one in the first test to five in the fourth test also showed improvement among the foundation mathematics students in the sample group.

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The findings showed that students in the sample group improved their study skills as they joined groups where they learned to learn from one another and to work in groups. They also gained insight as they learned to make sense of differentiation rules and symbols. The findings also showed that the students in the sample group lack insight in aspects of mathematics such as substitution, simplification, trigonometry identities, algebraic identities, conceptualization, and derivatives of algebraic expressions and trigonometry functions. The students’ improvement in the learning of differential calculus showed that the implementation of ZPD through self-study activities in a collaborative approach may boost their self-esteem. It can also improve the students’ confidence and change their negative attitude in the learning of differential calculus in particular and in mathematics generally. They also learned that they have the potential to succeed and it motivates them to learn more challenging problems and to enjoy the learning of differential calculus and the entire mathematics. References

Boaler, J. (1997). Experiencing school mathematics: teaching styles, sex and setting. Open University Press: Buckingham

Carlsen, M. ( 2008). Appropriating mathematical tools through problem solving in collaborative small-

group settings. Unpublished doctoral thesis, University of Agder, Norway Hunt, O. (2007). A mixed method design. http://www.articlealley.com/article18597522html 24/02/09 Miller, D. (2006). Evaluating the Effectiveness of a Learning System for Technical Calculus. Unpublished

doctoral dissertation, Department of Mathematics, Oklahoma State University Morris, C.J.F. (2008). Lev Semyonovich Vygotsky’s Zone of Proximal Development http://www.igs.net/~cmorris/zpd.html2008/04/18 Naidoo, R. (1996). Technikon students’ understanding of differentiation. SAARMSE. 96 proceedings. Naidoo, K. & Naidoo, R. (2007). First year students understanding of elementary concepts in differential

calculus in a computer laboratory teaching environment. Journal of College Teaching & Learning vol 4, No. 9

Unesco (2002). Vygotsky’s Socio-cultural theory: UNESCO Education http://portal.unesco.org/education/en/ev.php_ID-

26925&URL_DO=DO_TOPIC&URL_SECTION=201.html 26 /02/09 Vygotsky, L.S. (1978). Mind in Society. Cambridge, MA: Harvard University Press.

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45

PLENTY OF PYTHAGORAS PROOFS Sue Southwood

VULA Mathematics Project, Hilton College

This presentation shows various ways of proving Pythagoras’ theorem. All the proofs use school-level mathematics. The first proof uses similar triangles, the second uses a trapezium. The third proof uses a circle on a Cartesian plane and the fourth, transformations. The fifth proof uses parallelograms with the same base and between the same parallel lines. One is attributed to Leonardo da Vinci, another to a president of the United States. Detailed explanations of the proofs are given and illustrated with the aid of Geometers Sketchpad. Introduction If you ask a layman if he remembers any theorems from his school mathematics classroom, he will probably answer “Pythagoras.” If pressed, he will say “It is something to do with the square on the hypotenuse”.

Builders in ancient Egypt used the converse of Pythagoras’ theorem: they stretched a loop of rope with twelve evenly-spaced knots into a 3 4 5 triangle and used it to mark out a right-angle.

Learners memorise a list of Pythagorean triples. This checked picture shows the most common. Educators ‘teach’ the theorem in Grade 8, probably re-teach it in Grade 9 and thereafter assume that its use has become a basic skill.

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2

13

5

4

5

4

3

2

1

D

C

BA

An educator might even have some pieces of coloured card stored away in his/her Pythagoras file and which demonstrate the theorem – hopefully in a ‘wow’ moment. To dissect either of the smaller squares • Find its median point (centre).

• Draw a line parallel to the hypotenuse through this point

• Draw a second line perpendicular to this line – also through the median point But this is not a proof.

The theorem is finally proved in Grade 11 or 12 and probably only if it is a requirement of the current curriculum. Surely mathematics is more than this? Do we not owe our learners an opportunity to experience the rigour and diversity and excitement of proof?

THE PROOFS The similar triangle proof We usually preface this proof with the words “You have always taken Pythagoras’ theorem on trust. Now we are actually going to prove it.”

The similar right-angled triangle theorem proves that

ABC / / / ADC / / / CDBΔ Δ Δ and hence that

( )( )

2

2

2 2

2

AC AB ADBC AB DBAC BC AB AD BD

AB. AB AB

= ×

= ×

+ = +

= =

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1

2

3

b

c

cb

a

a

E (c; 0)D (-c; 0) C

B (b; a)

A

c

b

a

The trapezium proof James A Garfield, before he became President of the United States of America, wrote this proof.

The area of a trapezium is equal to half the sum of the parallel sides multiplied by the distance between them. The area of this particular trapezium is equal to the sum of the areas of the three triangles into which it is divided.

( )( )

( )( )

2

2

2 2 2

2 2 2

1 1 1 1a b a b ab c ab2 2 2 2

a b a b ab c ab

a 2ab b 2ab ca b c

+ + = + +

+ + = + +

+ + = +

+ =

The coordinate geometry proof This proof uses the coordinates of points on a circle, a right-angle in a semi-circle and the fact that the products of the gradients of perpendicular lines is negative one.

( ) ( )( ) ( )

( ) ( )( ) ( )

( )( )

( )

DB EB

2

2 2

2 2 2

2 2 2

m m 1a 0 a 0

1b c b c

a a 1b c b c

a 1b c

a b c

a b c

× = −

− −× = −

− − −

×= −

+ −

= −−

= − −

+ =

Leonardo da Vinci’s proof Da Vinci would be surprised to learn that he is probably best known for his painting entitled the Mona Lisa. Apart from being a painter, he was an extraordinarily gifted sculptor, engineer and mathematician and a prolific inventor.

His proof uses rigid transformations.

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a

b

c

a

b

c

c

a c

b A

a c

b A

The left hand hexagon is made up of a right-angled triangle with sides a, b and c, squares constructed on the shorter sides a and b, and a reflection of the triangle. In the central diagram the ‘top half’ of the figure has been reflected across the other dotted line. The shape of the hexagon has changed, but the area is the same. In the third diagram, the right hand hexagon keeps the same shape and area, but the right-angled triangles have been rotated through 90°. The series of diagrams shows the original two triangles and the two squares with sides a and b being transformed into a new shape containing the same two triangles and a new single square with side c. Therefore a2 + b2 = c2. The parallelogram proof The areas of parallelograms with the same base and between the same parallel lines are equal.

a

b

c

a c

b A

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a c

b A

The smallest square is changed by an area-preserving transformation, called a shear, to the parallelogram in the central diagram. The square on the other shorter side is similarly transformed in the third diagram. In the last diagram the square on the hypotenuse is divided into two rectangles by a line perpendicular to the hypotenuse and through the vertex of the right-angle in the original triangle. These two rectangles are transformed into two more parallelograms. It can be easily shown that the two resulting arrow-shaped hexagons are congruent. Since the one arrow is equal to the sum of the squares on the shorter sides of the right-angled triangle and the second arrow is equal to the square on the hypotenuse, Pythagoras’ theorem is proved yet again. Conclusion There are plenty of proofs of Pythagoras. The proofs outlined here are just some of those accessible to our learners. There is a short and pithy algebraic proof that uses Euler’s Formula: If the sides of the right-angle triangle are a, b and c and the shorter sides are placed along the real and imaginary axes then ia bi ce θ+ = . Multiply this by its complex conjugate ia bi ce− θ− = . The result, as usual, is 2 2 2a b c+ = .

The multiplication of these two binomials follows the normal rules with which the learner is familiar – all one would need would be an explanation of why 2i 1= − . But why stop there? In the past, each new edition of AMESA’s journal Pythagoras traditionally illustrated a different proof of the theorem. Inside, under a heading THE COVER, there was an explanation of the proof. These make fascinating reading. In the April 1991 edition, No 25, the writer poses the questions “Is it necessary that the constructed figures on the sides of the right-angled triangle are squares?” and “Wouldn’t any set of similar figures do?” Bennett, Dan. (1995) Pythagoras Plugged In, Emeryville: Key Curriculum Press.

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Reflections of South African Teachers on teaching Math in the USA

Bridget Cameron

Rustenburg Girls’ High School, Cape Town The following teachers took part in the Fulbright Teacher Exchange Program to the United States of America in 2008. Bridget Cameron Rustenburg Girls’ High School, Cape Town Thotobolo Mdladlamba ex Esilindini Junior Secondary School, Sterkspruit Viniagumul Govender St Oswald’s Secondary School, Newcastle Zacharia Porogo ex Boitlamo Secondary School Parms Frederick Rufetu King Makhosonke II high School, Kwamhlanga They spent a year in the USA teaching Mathematics. Thotobolo and Zacharia taught in different schools in Washington DC. Frederick taught chemistry in a challenging Manhatten School in New York. Viniagumu taught in Los Angeles and Bridget taught in Albany, Oregon. Bridget will outline the challenges and high points of teaching Math as a South African in a US school. She will talk about the school where she taught. She will comment on standards, length of periods, content and assessment – both state and per class. They came back to South Africa realizing that there are significant strengths in the South African System.

For the year 2008 I took part in the Fulbright Teacher Exchange program to the United States of America. I was not aware of this program before 2007. The American teachers can apply for the Fulbright teacher exchange program. They give, in order of preference, 3 countries where they want to teach. The Fulbright office in those countries is then notified and they look around to find a match. There were 5 teachers who wanted to come to South Africa. I was invited as I had participated in a program with Maths4stats and they were looking for 4 Mathematics teachers. The Fulbright representative and the US consulate interviewed me and the principal at my school as they had to see whether the school was willing to host the American teacher. Sharyn Abbes form Albany, Oregon was my match.

I taught Math at West Albany High School in Albany, Oregon. Sharyn Abbes, the West Albany teacher, taught at Rustenburg Girls’ High School in my place for the year. She taught 4 Mathematics classes for the 2008 school year. I started teaching on 7 January which was in the middle of the year for them. I took over Sharyn’s classes and taught them until the end of the school year – in June. In September, the beginning of the new school year, I was given 6 new classes to teach. I taught up to their Christmas Break on 19 December. I arrived back in South Africa on 30 December.

West Albany High School is a well-run coed High School from Grade 9 to Grade 12. It had 1400 students with a teaching staff of about 60 teachers.

The school timetable had only two days - blue and gold days which alternate. The students took up to 8 “classes”, 4 on a gold day and the other 4 on a blue day. The lessons are 90 minutes long and so there are 6

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hours of contact time per day. I always had the first period free. Then I had an algebra class that was weak. I saw them every day in the second period. This is called a block class because they had Math everyday, whereas the other classes only had Math on a gold day or on a blue day. There was a 20 break between periods 1 and 2. Lunch was after the second period at 11 (40 minutes) – but no tea or staff room! There was a 5 minute break between the next 2 periods. School ended at 2.40 pm. On the gold day I had the Algebra 1 class and 2 Geometry classes and on the blue day I had the Algebra 1 class and 2 other Algebra 1 classes.

I taught Algebra 1 and Geometry classes. These are usually the first 2 Math courses at the high school. Students that are accelerated will take them at the middle school. They can start with Algebra 2 at High school in their Grade 9 year. Students that struggle with Math could start with a pre algebra class. From September I taught an Integrated Math class. This was a class for those who had got a C (70% and below – that is low for the US. A (90%) and B (80%) are the acceptable grades.) for Algebra 1 and Geometry. It was felt that they needed more time on those skills before they took Algebra 2. The current requirement is that the students have to have 3 high school Math credits to be able to graduate. The content of the courses is mostly the same as the South African curriculum.

The normal student starts high school in Grade 9. He then takes Algebra 1. Topics covered are: Solving linear and quadratic equations, simultaneous equations, inequalities, straight line graphs, exponents, multiplying binomials, factorizing, statistics and probability.

Having attained a C or above (>70%) in Algebra 1, he then spends his Grade 10 year studying Geometry.

Topics covered are: Basic angle relationships, properties of quadrilaterals, polygons, congruence, areas, Pythagoras, volumes and surface area, similarity, circle geometry, trigonometry

He would then study Algebra 2 in Grade 11 and then PreCalculus in Grade 12.

Algebra 2, topics covered are: Matrices, Quadratic equations and parabolas, functions, powers, roots and radicals, exponential and logarithmic functions, sequences and series.

Students can be accelerated by studying algebra 1 and Geometry in the middle school. They would then start with Algebra 2 in Grade 9, PreCalculus in Grade 10, AP Calculus in Grade 11 which is a college course and the Probability and Statistics in Grade 12.

There were 3 challenges that I had in teaching there;

1. TIME: I found it difficult to have 90 minute lessons and only see the students every second day. The restless Grade 9 students did not find it easy to concentrate on algebra for so long. I struggled to get more than one concept across in each lesson. I am glad to be back to 45 minute periods and to see my students every day.

2. DIVERSE CLASSES: There you have classes of students of different grades (grade 9 and grade 12 students in same class), different abilities (accelerated Grade 9s in the same class with struggling grade 12s), different genders, different expectations (like students with IEPs (Individual Education Plan) who get modified diplomas and only need to pass 50% of the work, others need extra time, or take tests in different venue) in the same class. Some students had social problems, others had learning problems. It was a challenge to teach Math in that environment. The discipline on the whole was much the same as in South Africa. There was a good back up system with the 4 counsellors and 3 administrators.

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3. Students are GRADE VERSKRIK: The grade (mark / symbol) is more important than engaging with the math ideas. Everything is very modular, and generally sections of work are written off. I found the US students not keen to learn at all. They only were interested in doing the homework because it was marked every day and their marks were entered onto the grading system daily.

I have gained in confidence. I have done things that I have not done before. Without warning and with no preparation, I ‘subbed’ for another Math teacher, teaching the lesson from his lesson notes that he had planned for his class. I ran the Oregon standardized mathematics and language (OAKS) test in computer lab and managed the technical mishaps with a reasonable show of outward calm.

I have gained insight into another culture’s ways of teaching. I attended a three–day professional Math conference in Portland, I learned a new mathematics vocabulary (zee for ‘z’, zero not naught, decimal point not comma, parentheses not brackets, Pythagorean not Pythagoras’ theorem, to factor not to factorise, trapezoid not trapezium).

SA students are on the whole, more keen to learn than their American counterparts. As we cover the same topics every year with greater depth SA learners revise a lot of the concepts each year. With our exam system the learners have to remember more for longer and therefore have a broader grasp of the subject. In the US they had to remember a chapter at a time. The final, in Maths, covered the important topics in the semester. A lot of revision was done before the final test – an hour long. We also have a national unified curriculum with external standards. All schools have the same Matric certificate.

I had the opportunity to visit four other schools in the USA:

1. Crescent Valley, Corvallis [Mike and I spoke at a social studies lesson, visited a Math computer lab class]

2. Smokey Hills School, Denver, Colorado, [big school 2500 students; program like ours with green and red days and 90 minute lessons;

3. Rooseveld High, Seattle, WA [attractive, newly remodeled school. 6 x 45 minute lessons, math had 2 levels; ordinary and honours, but they had an integrated curriculum where algebra and geometry were taught in the same year.]

4. Quilcene near Port Townsend, WA [small school 230 students from K-12. There were 6 students in the Geometry class. Teachers and students were friendly and interested in SA and my experience.]

I was made aware of the different approaches the different schools had in the structuring of the school day as well as structuring the Math curriculum. Each state had its own education system. Each school could choose how they plan the curriculum. Freedom to be independent is highly valued. There were 14 Algebra 1 classes at WAHS. I taught 4 of them. I followed the work scheme in a general way. But we all set our own tests and finals and there was no moderation. Thus Algebra 1 grades vary according to the standard that each teacher sets. This works well where you have good teaching. The variation between schools and districts is great.

It was a life-changing year. I have experienced American geography, history, arts, culture and sports. In the US it was a watershed year economically and politically. The economy took a nose dive and a new president was elected. I took every opportunity that I could to listen, learn and participate. I came to the

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end of my time there with a touch of sadness – sad that it had passed so quickly, that I would be leaving whole communities of friends, and knowing that the challenges that face me back home have required courage and new adjustments. I am using the relationship networks I have forged with colleagues, friends and students to promote understanding and cooperation. This short presentation is to help others appreciate what we have in South Africa.

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EFFECTIVE LEARNING OF ALGEBRA AT SCHOOL Gawie du Toit

University of the Free State

This paper examined reasons why effective learning does not always materialize in Mathematics and more specific in algebra at school level. In an attempt to identify possible reasons why effective learning evades learners a qualitative investigation was performed on students enrolled for mathematics educations courses as well as on teachers furthering their studies in mathematics education. The outcomes were compared to possible reasons as portrayed in literature. In this paper the audience attending the lecture will be exposed to the same instrument with the aim to actively involving them in the reflection on these reasons. INTRODUCTION Teaching algebra for effective learning was and still is a big challenge to mathematics teachers. Various reasons contribute to this phenomenon. It can be the way in which teachers execute their roles; it can be the confidence systems of learners based on their perspectives of mathematics; it can be the teaching methods used to teach algebra; it can be misplaced outcomes; it can be the text books (incorrect content; way of unpacking the content; etc.) used to teach the algebra. At first, effective learning and how it fits into the paradigm of social constructivism will be discussed. An investigation into the dichotomy between algorithms and heuristics; procedural knowledge and conceptual knowledge; inductive and deductive reasoning and concept definition and concept image will be done. There is a saying that states that teachers must research their teaching and then teach what they have researched. This contribution can serve as an example of this saying. THEORETICAL BACKGROUND

• Effective learning

De Corte and Weinert (1996) identified a series of characteristics of effective and meaningful learning processes which emerged from research that constitute building blocks that can serve as an educational learning theory. Those characteristics about which there is a rather broad consensus in the literature can be summarized in the following definition of learning:

Learning is a constructive, cumulative, self-regulated, goal-directed, situated, collaborative, and individually different process of meaning construction and knowledge building (De Corte & Weinert 1996: 35-37).

The characteristics of this definition relates to the principles of constructivist teaching as discussed by Muijs and Reynolds (2005). It thus fits perfectly into the framework of reference of social constructivism. Building knowledge during the process of meaning construction is furthermore an indication that the truth is not out there, but that the truth is within each student.

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Various authors (Cobb 1988; Hiebert & Wearne 1988; Nieuwoudt 1989; Schoenfeld 1988) stated that research has shown that teachers can formulate good goals, but despite of that there were still core problems that existed in the teaching of Mathematics at school. The following core problems were identified in the literature by these authors:

- Learners are not seen by educators as constructors of their own knowledge.

- Learners do not apply the methods taught by educators on problems, but rather use their own methods which they developed themselves.

- Learners cannot relate procedures of manipulating symbols with reality. The consequence is that they accept answers that are unrealistic.

- Learners accept methods taught by educators without any criticism and apply it just like that.

- The over emphasizing of the answer, i.e. the product, lead to the situation that learners only concentrate on the methods and techniques taught to them, and that to the cost of understanding and the applicable processes.

- The teaching implies that learners work in a syntax’s way (i.e. procedures based on rules) to execute an exercise instead of analysing the exercise in a semantic (procedures based on understanding) way to generate an answer.

- The teaching suppresses divergent thinking activities and creativity and problem-solving strategies are not established.

If this is true, then no effective learning took place if measured against the definition of effective learning by de Corte and Weinert. The dominant role of the teacher, the perspective of learners, misplaced objectives and the teaching methodology used to teach mathematics were identified by these authors as possible reasons that contributed to the existence of the mentioned core problems.

• Algorithms and heuristics

Researchers (Suydam 1980) distinguish between two methods of problem solving, namely the algorithmic and heuristic methods. An algorithm is defined as "...a recursive specification of a procedure by which a given type of problem can be solved in a finite number of mechanical steps" (Borowski & Borwein 1989:13).

Polya (1985:112-113) describes the heuristic method as follows: The aim of heuristic is to study the methods and rules of discovery and invention ... Heuristic, as

an adjective, means 'serving to discover'. Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem".

It can thus be concluded that self-discovery plays an important role in this method. This is emphasized by Schultze (1982:44-45): The heuristic method attempts to make students find and discover as much as possible, and to

reduce direct information to a minimum.

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Heuristic methods are not rigid frameworks of fixed procedures which provide a guarantee for the obtaining of a solution. The purpose of value thereof lies mainly therein that you search purposefully and systematically for a solution (De Villiers 1986).

When the two methods or problem solving are compared with one another, it becomes clear that the heuristic method differs considerably from the algorithms taught in the mathematics class. For instance, an algorithm ensures success if it is used correctly and also if the correct algorithm is selected and used. Algorithms are problem-specific, while the heuristic method is not problem-specific, because it is normally a combination of strategies. This leads to the fact that a heuristic method is applicable to all types of problems. A heuristic method provides the "road map", a blue print, which leads a person to the solution of a certain problem situation. In contrast to algorithms the heuristic method does not necessarily lead to immediate success. If pupils are, however, taught to approach every problem heuristically, they should be able to solve problems with which they are confronted in the classroom and in life, successfully (Krulik & Rudnick 1984).

It is important to note that, although the heuristic method could serve as guideline in the solution of relatively unknown problems, it cannot replace knowledge of subject contents. Quite often the successful implementation of a heuristic strategy is based on the fixed foundations of subject-specific knowledge (Schoenfeld 1985).

It can be concluded that the heuristic way of doing problem solving should play an ever-increasingly important role in the teaching learning situation where problem solving is the focus of teaching. Algorithms, on the other hand, form part of the subject contents and are therefore also important. What is of cardinal importance, however, is that an algorithm should be part of the package of knowledge after having been constructed in a heuristic manner.

Students will empower themselves if they are capable to apply a range of problem solving strategies when confronted with a problem (Schoenfeld 1988). If one way doesn't work they will find another way. It is, however, important that the heuristic method should not be viewed as a goal in itself. It should rather be seen as a way in which a certain goal is achieved. In Polya's work the heuristic method was seen as the vehicle with which sense is made of mathematical situations. Drawing diagrams, for example, should not be taught as a unit in the mathematics classroom, but must rather be used to solve problems where applicable.

Groves and Stacey (1988) consider the strategies as important, especially at the beginning when actual problems are tackled. These strategies give the pupils a degree of control in the process of problem solving and it is important that they should be able to apply it spontaneously without being dependent on the teacher's support. This is also echoed by Eiselen (Roux 2009) who stated that learners should be supported in the learning and applying of problem solving strategies. One should take care not to reduce the heuristic method to skills, techniques and even, in contrast, algorithms.

• Procedural and conceptual knowledge

Students and even some teachers have a limited conceptual knowledge span of algebra and it was further found that there conceptual knowledge does not correlate with their procedural knowledge (O’Callaghan 1998; Hollar & Norwood 1999; Roux 2009). Procedural knowledge focuses on the development of skills and can it therefore be deduced that it relates more to the use and application of algorithms (O’Callaghan 1998). Conceptual knowledge on the other hand is characterised by knowledge that is rich in relationships

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between variables and also including the ability to convert between various forms of presenting functions, i.e. in table format or in graph format, etc. (Hiebert & Lefevre 1986). Conceptual knowledge lends it more to self discovery which relates more to the use of heuristics and inductive and deductive strategies.

Troutman and Lichternberg (1995) highlighted the use of four types of learning activities in the teaching and learning of mathematics, namely: developmental activities; reinforcement activities; drill and practice activities and problem solving activities. Developmental and problem solving activities lends it more towards the development of conceptual knowledge whereas reinforcement and drill and practice activities lend it more towards the development of procedural knowledge. The advantage to first expose learners to developmental and problem solving activities is that they do get the chance to develop conceptual knowledge before being exposed to procedural knowledge (Davis 2005).

• Concept definition and concept image

Vinner (1991; 1992) differentiates between a verbal entity, namely a concept definition and a non-verbal entity, namely a concept image. A learner develops a mental presentation of a concept, build on what they were exposed to. Learners who create concept images by giving meaning to it are building knowledge which implies that they have learned a new concept. In the process networks are formed and does it follow logically that insufficient networks will result in misrepresentations of concept definitions.

Mechanical applications (i.e. applying algorithms without a clear concept image) can restrict learners’ understanding of concepts which may result in a distorted concept image. This is one of the core problems in the teaching of mathematics debated earlier on. This usually happens when teachers work deductively at first instead of working inductively. If learners need to construct knowledge by means of meaning construction then teachers also needs to work inductively. Polya (1985:vii) stated:

Mathematics presented in the Euclidian way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science.

Some definitions are too complicated and do not really contribute towards the creation of concept images. Learners on the other hand need concept images and not concept definitions to handle concepts which imply that concept definitions are not used and are actually forgotten (Vinner 1983). The ideal process seems to be to work inductively to construct the concept image and then to work deductively to formalise the concept definition. RESEARCH QUESTIONS The following research questions were investigated:

• Is the conceptual knowledge of teachers in line with their procedural knowledge?

• Is the conceptual knowledge of students in line with their procedural knowledge?

RESEARCH METHODOLOGY Qualitative research methods were used to find answers to the mentioned research questions. At first mathematics school textbooks and examination papers were analysed to determine to what extend the

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focus was placed on concept definitions and/or concept images. A questionnaire consisting of mainly algebraic statements was designed based on these findings. These questionnaires were administered by the researcher. The target population consists of different groups of fourth year mathematics education students over the period 2005-2009 as well as practicing teachers who have enrolled for an advanced certificate in mathematics education and/or who have participated in mathematics education workshops (2005-2008).

The questionnaire consisted of ten questions. The respondents completed the questionnaire in class and it took them more or less 10 minutes to do so. The response to each of the questions was either true or false. These responses were noted and the questionnaire was thereafter discussed and debated which contributed towards the reliability and validity of the questions posed in the questionnaire.

During these discussions the researcher continually posed questions, obtained answers, and criticise the answers (i.e. the Socratic Method), to obtain a deeper understanding of the thought processes of the respondents.

RESULTS AND DISCUSSION Over the mentioned period of time (2005-2009) only one student (mathematics on third year level) demonstrated a clear correlation between his procedural and conceptual knowledge. He applied algorithms where applicable but work heuristically when confronted with a situation that seemed unfamiliar to him. The majority of the students applied rules mechanically without reflecting on their answers. Rare evidence of conceptual knowledge was noted. It can thus be concluded that the students focused mainly on concept definitions and did not demonstrated concept images. The demonstration of subject content knowledge was not on a desired level and the same applied to their professional pedagogical knowledge.

The situation was even worse in the case of the teachers. Each group clearly demonstrated that they apply rules mechanically without applying problem solving strategies at all. They in other words did not work heuristically or inductively. A possible reason for this phenomenon could be that these teachers did not receive any mathematics training at post grade 12 level. There training in mathematical content was restricted to grade 12 level because they were all trained at Colleges of Education. Their mathematical factual knowledge was at a substandard. It is evident that they will not be able to apply their professional pedagogical knowledge in full when teaching mathematics due to the lack of mathematical content knowledge.

CONCLUSION Teachers and mathematics education students who participated in this research apply mainly algorithms when solving problems involving algebra. They work deductively and demonstrate procedural knowledge. It can be concluded that the same core problems discussed previously still exist in the teaching of algebra.

It can be concluded that the students and teachers who have participated in this endeavour have a limited conceptual knowledge span of algebra and it was further found that there conceptual knowledge is not in line with their procedural knowledge.

The questionnaire was not discussed in this written report because it is the intention to actively involve those who will attend this presentation by means of the questionnaire. In the discussion of each of the questions the aspects as discussed above will be unpacked and debated.

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Bibliography Borowski, E.J. & Borwein, J.M. 1989. Dictionary of mathematics. Glasgow. Collins.

Cobb, P. 1988. The tension between theories of learning and instruction in mathematics education. Vol. 23(2). Spring: pp. 87-103.

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Groves, S. & Stacey, K. 1988. Curriculum development in problem solving. In: ICME 5 Problem Solving – a world view. Adelaide, 1984. Ed. Brukhardt, H., Schoenfeld, A., Groves, S. & Stacey, K. Notingham: Published by The Shell Centre for Mathematical Education, pp. 199-206.

Hiebert, J. & Lefevre, P. 1986. Conceptual and procedural knowledge for teaching on student achieviement. In: Hiebert, J. Ed. Conceptual and procedural knowledge: the case of mathematics. Hillsdale, N.J.: Erlbaum. pp. 1-27.

Hollar, J.C. & Norwood, K. 1999. The effects of a graphing-approach intermediate algebra curriculum on students’ understanding of function. Journal for research in mathematics education, 30(2):220-226, March.

Krulik, S. & Rudnick, J.A. 1984. A Sourcebook for teaching problem solving. Massachusetts: Allyn and Bacon inc..

Muijs, D. & Reynolds, D. 2005. Effective teaching: evidence and practice. Sage Publications. London.

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O’Callaghan, B.R. 1998. Computer-intensive algebra for students’ conceptual knowledge of functions. Journal for research in mathematics education, 29(1):21-40.

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Schoenfeld, A.H. 1985. Mathematical problem solving. San Diego, California. Academic Press Inc.

Schoenfeld, A.H. 1988. When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist. 23(2), pp. 145-166.

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Suydam, M.N. 1980. Untangling clues from research on problem solving. In: Problem solving in school mathematics (1980). Ed. Krulik, S. & Reys, R.E. Reston: NCTM, pp. 34-50.

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Vinner, S. 1991. The role of definitions in teaching and learning. In: Tall, D., ed. Advanced mathematical thinking. Dordrecht: Kluwer. pp. 65-81.

Vinner, S. 1992. The function concept as a prototype for problems in mathematics learning. In: Harel, G. & Dubinsky, E., eds. The concept of a function: aspects of epistemology and pedagogy. Washington, D.C.: Mathematical Association of America. pp. 195-214.

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The effect of Habitual Use of Calculators on the Arithmetic Proficiency of First Year University Students

 Sunday Faleye David Mogari

UNISA UNISA faleys@unisa.ac.za mogarld@unisa.ac.za

In 2007, marks were obtained from 250 scripts in a mathematics module examination in which the use of electronic calculator is not allowed at the University of South Africa. The analysis of the marks obtained in the students’ scripts showed that students performed woefully in the questions that required arithmetic skills. Further analysis on the strategies and algebraic skills displayed in the questions that required arithmetic skills it was evident that the students performed poorly because they were not allowed to use calculator in the examination. The study was triangulated with a survey in the following year (2008 academic year). The use of analysis of variance model to compare the responses that fall in 3-point scale (Always, Sometimes and Never) with the respective mean mark obtained in the 2008 examination by the respondents lead us to conclude that habitual use of calculator among university undergraduates impact negatively on their arithmetic proficiency.

MOTIVATION FOR THE RESEARCH

The study was conceived when it was noticed that some university first year mathematics students in South Africa could not carry out basic arithmetic operations e.g. addition of fractions, multiplications of fractions, multiplications and division of decimal numbers, without the use of calculators.

The use of calculators in the teaching and learning of mathematics is fast becoming a common practice in South African schools, even at the lower classes. A number of schools prescribe calculators to learners. A simple adding machine is used in lower classes, while a more sophisticated scientific calculator is used in higher classes. This practice has made some learners to be addicted to calculators, to the extent that learners use calculators to compute solutions to basic trivial arithmetic problems, which could be solved mentally. The questions we need ask ourselves are “Should children use calculators to learn basic arithmetic facts? At what stage in mathematics education could we safely allow learners to rely on calculator use in the learning of mathematics such that it will not create a learning difficulty in future?”

It is against this background that an investigation is conducted on the impact of the habitual use of calculators on first year mathematics students’ arithmetic proficiency.

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RESEARCH FOCUS

This paper investigates the effect of habitual use of electronic calculator on the arithmetic proficiency of the first year university students, with a view to seek its appropriate integration into the teaching and learning of mathematics such that it does not create future learning difficulty.

However, two research questions guided our inquiry into the study:

1. For how long have the study participants been depending on the use of calculators in the

learning of mathematics?

2. Are the study participants able to solve algebraic problems without the use of calculators?

THE STUDY THEORETICAL FRAMEWORK

Habitual use of calculators in every simple mathematical computational procedure has made some learners not to be able to solve arithmetic problems without the use of calculators. A similar observation was made by Rustemeyer (2007) among German school-age learners and led her to raise concern that excessive use of calculators reduces the practice of elementary arithmetic. The study carried out by Standing (2006) found that there were undergraduate students in Canada who struggled to work out arithmetic problems derived from Grade 3 syllabus. A report by the United States of America National Research Council (2001) notes that significant reliance on calculators in the learning of mathematics undermines the computational proficiency and impedes the conceptualization of basic numeracy procedures of the learners.

That been the case, Rittle–Johnson and Kmicikewycz, (2008) pose the question: should the school children use calculators to learn arithmetic facts, or should they generate the answers on their own? When using calculators to solve mathematics problems, however, conceptual understanding of basic numeracy manipulation is not a prerequisite (Bridgeman, et al., 1995). The learners do not care about the numeracy manipulation proficiency as long as they could use calculators to compute the answers. It is part of the perennial controversies on the appropriate use of calculators in mathematics teaching and learning. May be in the lower classes, the emphasis should be on understanding of the basic mathematical concepts and engrafting of the arithmetic principles. The learners should be introduced to basic arithmetic facts and computational procedures, so that sound arithmetic and algebraic manipulative skills must not be compromised. It is for this reason that the United States of America’s National Research Council (2001) insists that a strong mathematics cognitive foundation is very important at the early school age for subsequent conceptual development to take place without any difficulties.

Ballheim, (1996, p.6), agues that calculators should be used only after learners have learned how to carry out the necessary mathematics processes and procedures without calculators. In fact, learners should first master the fundamentals of number manipulative procedures before introducing them to calculators. Emphasis should be on the procedural skills to generate answers in problem solving. In a more recent study by Nyaumwe, (2006) conducted in Zimbabwe on the ‘O’ Level calculator syllabus 4028, one of the teachers interviewed argued that the method of solving a problem is more important than the answer. He, therefore, suggested that it is necessary for learners to learn mathematics without calculators up to end of Form Three.

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Extensive habitual use of calculator in the mathematics instruction may hinder mastery of the computational skills. The learners may become lazy to think. Learners’ mental reasoning and computational skills are very essential in mathematics. Brown et al. (2007), quoting Fey (1990), argues that “the most important goal of the teachers’ mathematics instruction is to develop students’ abilities to reason intelligently with quantitative information” (p.103). The teachers’ mathematics instruction ought to promote mental thinking and manipulative reasoning. Mental methods are ‘a first resort when calculation is needed’ and ‘the basis upon which all standard and non-standard written methods are built Ruthven (1998). Mathematics classroom should be an interactive environment which stimulates a discussion among the learners and teacher. In actual fact, the use of calculator in the mathematics classroom should be minimal. Calculators should be used for ordinary arithmetic problems. It was noted in Brown (2007) that teachers’ practices of frequently using calculators for mathematics instruction reduced students’ ability to do well on computational problems in end – of – year tests where calculators were prohibited. According to Fey (1990) some parents have put pressure on schools to avoid using calculators in the mathematics classes as this make their children to rely on them, Fey (1990).

On the other hand, studies that have been carried out on the use of calculators in schools support its use in procedural calculations and consider it important, (Bridgeman et al., 1995; Ruthven, 1998; Jackson et al., 2001). From this, the ensuing argument is that calculators are helpful in the teaching and learning of mathematics, and should therefore be carefully integrated into the schools’ mathematics curriculum, more especially at the lower classes. Aimee (2003), advise that calculators, though generally beneficial, may not be appropriate for use at all times.

In spite of the lots of research work that have been done on the use of calculator in the mathematics instruction in schools, Aimee (2003) remarked that calculator controversy is not yet resolved. Learners are taught basic arithmetic facts with calculators. The primary effect of this phenomenon is continuous reduction in the practice of elementary arithmetic skills among school age learners Rustemeyer (2007). The secondary effect is that at the tertiary institutions, students will have lost touch with arithmetic facts to the extent that they will always perform badly in any task that will require mental arithmetic proficiency, as was found by Standing, L. G. (2006) and Standing, Sproule, and Leung, (2006). To completely resolve the issues surrounding the appropriate application of calculator in the learning of mathematics, seemingly more work still needs to done to find the appropriate position of calculator use in the teaching and learning of mathematics.

METHODOLOGY

Data source

We would have used an experimental design in which study participants are divided into ‘answer generation condition’ and the ‘read answer from the calculator condition’ and carried out some comparison analysis (McNamara’s, 1995; Rittle-Johnson, B. & Kmicikewycz, A.O. 2008). Our study participants are university 1st year undergraduate students who are expected have strong arithmetic knowledge base. Therefore, we used ex-post facto research design in this study. One of the first year mathematics module in which some of the questions requires arithmetic skill and the use of calculator was not allowed in the examination was used to generate data for the study.

The study was carried out in two consecutive academic years; 2007 and 2008 academic sessions. Participants for this study were the first year mathematics students at the University of South Africa. The

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university operates an open distance education system but there are arrangements for contact classes. Contact classes take place at the weekends and sometimes during the week. The study was carried out in the following phases:

(i) In 2007, the students’ examination scripts in one of the first year mathematics modules in which the use of calculator is not allowed in the examination were used to generate the necessary data.

(ii) The method used in 2007 was triangulated with questionnaire survey in 2008 academic year. The study participants in 2008 were different from that of 2007.

The reason for using two different groups for over a period of two years is to strengthen our data. The data collected in each year were treated separately.

RESULTS

(i) The 2007 examination scripts (evaluation of the marks)

We need to remind our readers that the use of calculators in the examination of the module under review is not allowed. 14 scripts were randomly selected from the 250 scripts examined as samples and given in table 1.0 below, since it is not possible for us to display the marks obtained in the 250 scripts. The marks obtained in the scripts in questions 2 to 5, the question category ‘A’ (QCA), and in questions 1, and 6 to 11 the question category ‘B’ (QCB) and in the actual examination marks (AEM) were standardized.

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Table 1.0

participants Performance

(%) in QCA

Performance (%) in

QCB

Actual

Performance in

the

Examination (%)

1. 5 12 14

2. 11 43 39

3. 7 52 46

4. 4 17 15

5. 41 62 63

6. 37 70 69

7. 11 35 32

8. 4 29 25

9. 22 46 45

10. 33 62 61

11. 33 40 43

12. 33 38 41

13. 22 51 49

14. 44 73 74

Generally, the students performed very badly in QCA as compared to QCB. For example, the script on row three in the table above scored 7% in QCA while he scored 52% in QCB and in AEM, the script failed the examination with 46%. It is obvious that this script was written by an average student but he/she failed the examination because of hisher poor performance in QCA. In the same vein, the score on row thirteen is 44% in QCB, 73% in QCB, and it is 74% in AEM. It is clear that the student that wrote this script was above average. He/She could have passed the examination out rightly with upper distinction but his marks in QCA disallowed this.

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(ii) Strategies and algebraic skill deficiencies displayed in question category QCA in 2007 examination scripts

In this section, our focus is on the arithmetic proficiency skills demonstrated in answering the questions. Going through our data collected on the evaluation of the arithmetic skills, in most of the scripts, the students could hardly carry out simple arithmetic algorithm.

Generally the students performed poorly in QCA as compared to QCB.

(a) Handling fraction multiplication

18% of the students could not correctly carry out fraction multiplication in the process of solving question 2.2 of the examination questions. Many of them did not know what to do, they just wrote whatever that came to their minds. For example x 36 =18.

Some multiplied 4 and 36, (most of the answers for this multiplication was wrong) , then divide by 12. The method is correct but they could not follow it up by dividing with 12. An example of this type of error is

x 36 = , and leave the answer as it is.

(b) Division of numbers

This type of arithmetic skill deficiency was noted in the solutions to question 2.2 of the examination questions, but it became more prominent in solution to question 2.3 where it is necessary to define time spent in terms of speed and distance covered. 68% of the participants presented incorrect answer to this question. We observed that 47% of the participants missed the answer partly because they lack the skill of number division, the remaining 21% committed other numeracy computational errors. For (though a

wrong substitution), the participant could not attempt the division perhaps because of the denominator that is a decimal number. In another example the set of participants that fall into this category of

solution left the answer perhaps because of the units.

(c) Multiplication of whole numbers

41% of the study participants could not carry out two digits number multiplication. None of the questions involved direct number multiplication, but in the course of solving the questions 2 and 3 of the examination questions , tagged QCA, this particular multiplication deficiency became obvious. This is noticeable in some of the examples given above. A particular attention is drawn to section (a) above where 4 was multiplied with 12 and got 96.

Some other examples could be found in solutions to questions 3.2 and 3.3 of the examination questions, in which case the participants wanted to simplify the expression = a + (n - 1)d. A particular participant substituted and got = -9+ (9 - 1)6 = -9+ 8(6) = -9 + 54. Though this expression is wrong for what is expected from the question asked, but our argument here is about inability to carry out correct multiplication. We see that the participant gave the result of 8 x 6 as 54. We see that calculator use has taken the place of mental calculations.

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(d) Handling operation of exponential numbers

43% of the participant gave incorrect answers in questions 3.3 as a result of their inability to work with numbers that are raised to powers. They were apparently confused on how to go about simplifying the ensuing exponential numbers in questions 3.3. They recalled the algebraic rules for exponential numbers but could not apply them. For example = 81 . = 81 . = . = 4 x (-7) = -28. 30% of the participants gave variations of this type of approach. 13% divided 81 by could not simplify it further. While the remaining just left it as = 81 . . They remembered they need to write 81 in exponential form to be able to simplify the expression but could go not further than that.

One script was randomly selected among the 500 analysed. The selected script was scanned as given below to show how some of the students struggled in arithmetic skills.

We considered the circled area 1, 2, and 3 in the script above. In circle 1, the participant struggled so much to divide 840 by3 because calculator was not allowed in the examination, otherwise this would have not posed any problem to the participant. He/She tried to use long division method (circle 2) without success. He/She could not also divide 342 by 1.4 manually.

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(iii) The Questionnaire Survey in 2008

Descriptive and inferential statistics were used to analyse the data from the surveys. Data analysis involved finding the frequencies of responses falling in each of the items in the survey. The conclusions made from the surveys were based on the majority of responses and from the result of the ANOVA computed. In the consideration of the majority responses, conclusions were made by considering “Always and sometimes or Sometimes and Never” (Nyaumwe, 2006). In the analysis, items that were similar were grouped together under the four categories enumerated earlier on.

Effect of Long Time Calculator Abuse

Six items (12, 13, 14, and 15) were structured to measure this dimension of the study. We counted the frequency of the responses that fall in ‘sometimes’ and ‘never’ together to realise that majority of the survey respondents may not be able to solve mathematics problem correctly without the use of calculators (92% - item 12), equally most of the respondent indicated that they get confused when they do not use calculator, again we put responses that fall in ‘always’ and ‘sometimes’ together (85% - item 13). Responding to Item 14, majority of the respondents understood solving problems better using calculators (86%) and 50% of the respondent never troubled themselves with mental calculations when they can find a calculator (item 15). 40% indicated that they may not know the correct algorithm in problem solving but they can use the calculators to get the answer. Again, the summary of the responses are given in chart 3.0 below:

Effect of Long time Calculator Abuse

14.6

29.7

70.6

57

13

11.1

1.8

2.2

0% 20% 40% 60% 80% 100%

Get Confuse WhenNot Using Calcul.

Understand TheProblem Solving

Better Using Calcul.

Gro

uped

Item

s

Percentages

AlwaysSometimesNeverVoid Items

Chart 3.0

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One way analysis of variance result The result of the descriptive statistics above was further strengthened by using the analysis of variance (ANOVA) to compare the responses represented on the 3-point scale (Always, Sometimes and Never) with the respective mean marks in each group. The ANOVA was computed with 95% confidence intervals.

For item 12 – Do you solve mathematics problems correctly without the use of a calculator? Levene statistics was significant for homogeneity of variance among the three groups with 0.565. The study participants that indicated Always had N=15, M= 45.00 and SD= 22.25, those that belong to Sometimes group had N=223, M=30.87, and SD= 18.54, and Never had N=11, M=24.18 and SD=19.11. The analysis of variance among the groups indicates that there is a significant difference among the variances of the three groups with F=4.81 and p=0.009 < 0.05. The Post Hoc analysis shows that this significant difference came from the comparison of variance of Always and Sometimes, and Always and Never groups with p=0.005 and 0.006 respectively. There is no significant difference between the variance of Sometimes and Never.

For item 13 – Do you get confused when you do not use a calculator in number operation? Levene statistics was significant for homogeneity of variance among the three groups with 0.159. The study participants that indicated Always had N=37, M= 27.19 and SD= 15.13, those that belong to Sometimes group had N=180, M=31.30, and SD= 19.25, and Never had N=32, M=36.62 and SD=21.73. There is no significant difference among the variances of the three groups with F=2.11 and p=0.123 > 0.05. Even though there is a noticeable difference in the mean mark of Never compare to the means of the other two groups. But this difference is not significant.

For item 14 – Do you feel you understand steps in mathematics problems solving better using a calculator? Levene statistics was significant for homogeneity of variance among the three groups with 0.461. The study participants that indicated Always had N=77, M= 30.70 and SD= 17.94, those that belong to Sometimes group had N=143, M=30.57, and SD= 19.79, and Never had N=29, M=37.14 and SD=18.49. There is no significant difference among the variances of the three groups with F=1.498 and p=0.226 > 0.05. Again there is a noticeable difference in the mean mark of Never compare to the means of the other two groups. But this difference is also not significant.

Discussion The effects of misuse of calculator are the results we got from the mass failure in the questions that required arithmetic skills in 2007 examination scripts. We could not immediately place the reason for this mass failure until when we analysed the number computation expertise demonstrated in the scripts. It was evident that in most of the scripts simple divisions and multiplications could not be carried out mentally, which could have been easily done using calculators. These results concur with Standings (2006) findings who wondered why undergraduate students have difficulty in grade 3 arithmetic problems.

Again, the results of the analysis of variance on the items in the survey which bothers directly on the effects of the habitual use of calculators in the learning of mathematics and the marks scored in the

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examination, shows that habitual use of calculator impact negatively on the arithmetic proficiency of the study participants. The results shows significantly that majority of the study participants could not solve mathematics problems correctly without the use of calculators, with p=0.009 < 0.05. In addition, most of the study participants get confused when they do not use a calculator in number operation. Even though the ANOVA result on this item is not significant but we put the result of responses that fall in Always and Sometimes together, with N=37, M=27.19 and N=180, M=31.30 fell into each group respectively. The responses in each group are justified by the mean marks obtained. Almost all the study participants admit that they understand the steps in mathematics problem solving better using calculators. These responses are also justified by the mean mark obtained in each of the groups.

REFERENCES

Aimee J. Ellington (2003). A metal-analysis of the effect of calculators on students’ achievement and attitude levels in precollege mathematics classes. Journal for Research in Mathematics Education 34 (5): 433-463 Baggett, P. and Ehrenfeucht, A. (1992). What should be the role of calculators and computers in mathematics education? Journal of Mathematics Behavior II, 61-72 Ballheim, C. (1999). How our readers feel about calculators. In Z. Usiskin (Ed.), Mathematics education dialogues (pp 4). Reston, VA: National council of Teachers of Mathematics. Bohrnstedt, G.W. and Knoke, D. (1982). Statistics for social data analysis. F.E. Peacock Publishers, Inc, USA. Bridgeman, B., Harvey, A., and Braswell, J. (1995). Effect of calculator use on score on a test of mathematics reasoning. Journal of Educational Measurement 32 (4): 323-340 Brown, T. E. et. al. (). Crutch or catalyst: Teachers’ beliefs and practices regarding calculator use in the mathematics instruction. School Science Mathematics 107(3) Fey, J. (1990). Quantity. In L.A. Steen (Ed) On the shoulder of giants: New approaches to numeracy. (pp 61-94). Washington, D.C.: National Academy Press. Glorial Dion et. al. (2001). A survey of calculator usage in high school. ProQuest Educational Journal 101(8): 427-438. Golden, D. (2000). Unequal signs: For inner-city schools, calculators may be the wrong answer. The Wall Street Journal, 1, A12. Hembree, R. and Dessart, D.J. (1992). Research on calculators in mathematics education. In J. Fey & C. R. Hirsch (Eds.). Calculators in mathematics education (pp. 23-32). Reston, VA: National Council of Teachers of Mathematics. McNamara, D. S. (1995). Effect of prior knowledge on the generation advantage: Calculators versus calculations to learn simple multiplication. Journal of Education Psychology, 87: 307 – 318.

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National Research Council (2001). Adding it up: Helping children learn mathematics. Washington, D.C.: National Academy Press. Nyaumwe Lovemore (2006). Investigating Zimbabwean mathematics teachers’ dis positions on the ‘O’ level calculator syllabus 4028. South African Journal of Education 26 (1); 39-47 Rittle-Johnson, B. & Kmicikewyce, A. O. (2008). When generating answers benefits arithmetic skill: The importance of prior knowledge. Journal of Experimental Child Psychology, 101:75 – 81. Rustemeyer, R. and Stoeger H. (2007). Does hand calculator use explain why university students cannot perform elementary arithmetic? Psychological Reports. 100: 1270-1272 Ruthven, Kenneth (1998). The use of mental, written and calculator strategies of numerical computation by upper primary pupils within a ‘calculator-aware’ number curriculum. British Educational Research Journal 24(1); ProQuest Education Journals pg. 21 Standing, L. G. (2006). Why Jonny still can’t add: predictors of university students’ performance on an elementary arithmetic test. Social Behaviour and Personality, 34: 151 – 160. Standing, L. G., Sproule, R. A. & Leung, A. (2006). Can business and economics students perform elementary arithmetic? Psychological Reports. 98: 549 -555

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A reflection on the teaching of Fluid Mechanics in some South African Universities

Sunday Faleye David Mogari

University of South Africa University of South Africa

faleys@unisa.ac.za mogarld@unisa.ac.za

This study investigates the method of teaching fluid mechanics to engineering students in some South African universities. Classroom observations and semi structured interview were used to gather data from both students and instructors of fluid mechanics. The results revealed that fluid mechanics is taught in a traditional form of lecturing and in most cases students find it difficult to comprehend what was taught in the lessons.

MOTIVATION FOR THE RESEARCH

In the recent years, the school enrolments in engineering courses have been declining while, at the same time, the market demand for engineers is increasing (USA National Science Foundation, 2004; Li et al., 2008). Low enrolments have obviously led to shortage of experts in the field of engineering. French et al., (2005) notes that while the students’ enrolment in engineering courses in universities are very low, some of those that enrolled drop out or change course.

One of the reasons for declining number of experts in the field of engineering is that larger percentage of students nowadays is weak in mathematics, (Felder & Brent, 2005). However, aspects of mathematics form the fundamental conceptual framework of all the engineering courses. Hence many engineering students find engineering modules too demanding because of their weak background in mathematics.

This study is informed by the acute shortage of skills in South Africa, particularly in the field of engineering. As of now South Africa needs about 13000 engineers whereas the universities can only produce about 2000 (Sunday Times, 8 March, 2009). It is also reported that there have been a considerable number of professional, including engineers, who have been leaving South Africa to go and settle elsewhere. Furthermore, the post apartheid government inherited a crop of primary and secondary school science and mathematics teachers whose majority do not have strong knowledge base of their discipline (Naidoo & Lewin, 1998). This has impacted negatively on the ability to teach mathematics. As a result the learning of mathematics has been problematic, it is for this reason the number of mathematics higher grade pass has been low. On the other hand, there has been a campaign by government to encourage more learners to study science and mathematics in high school, so that they could pursue their university studies in careers such as engineering and other science related courses that play significant roles in the socio- economic advancement of the country. Financial aid scheme have been set up by government to support this course.

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RESEARCH FOCUS

The focus of this study is to investigate the prevailing method of teaching fluid mechanics and how the teaching method affects the mechanical engineering students’ conceptual learning in some South African universities. The study serves as a baseline for the main study which seeks to evaluate the effect(s) of introducing animated computer aided instruction (CAI) into the teaching and learning of fluid mechanics in mechanical engineering classes.

In this study we address the following research questions:

(1) How is fluid mechanics taught to undergraduate mechanical engineering students in some South African universities?

(2) Are there any conceptual learning difficulties experienced by the students as a result of the way fluid mechanics is been taught?

THEORETICAL FRAMEWORK

For students to understand the concepts of fluid mechanics, a strong mathematics background is a prerequisite. In the light of the noted poor performance by learners in mathematics (Felder & Brent, 2005), the teaching of fluid mechanics in engineering classes becomes a difficult task.

Leung et al. (2008) remarked that teaching has two main aspects: what is taught (basic construction of knowledge and skills), and how it is taught (the approach to teaching). They noted that the how-it-is-taught aspect of teaching is one of the main factors influencing students’ ways to learning. Generally, mathematics is taught in the traditional lecturing method in schools and universities. This method of teaching encourages learners to memorize contents without actually understanding it. Most assuredly, the same can be said of fluid mechanics students.

The teaching of fluid mechanics in a tradition classroom setting, using examples from the textbooks to demonstrate the correct application of theories on the classroom board is improper (Philpot et al., 2002). It poses a serious problem to students’ understanding of the basic fluid mechanics concepts because they can not visualize properly the applicability of the concepts. Steif &Naples (2003) pointed out that in mechanics courses, students must learn to apply fundamental principles to help understand problem solving and designs, hence they need to both comprehend the fundamentals and perceive their applicability to new situations.

Students should be able to use the declarative knowledge (the understanding of the definitions, facts, and concepts) gained to master the procedural skill in the practical life problem solving in engineering in other to foster a deeper understanding. This belief was also shared by Taraban et al., (2007). If deeper understanding of theoretical concepts is gained, misconceptions of basic fact of the domain of study would be avoided. To apply theoretical concepts in real life experiences will not pose any problem. Sorby & Baartmans (2000) stated that “visualization of problems is critical for success in engineering education. Explaining physical concepts in abstract form may be difficult. The fluid mechanics lecturer has an understanding of the components and processes that constitute fluid mechanics concepts. He can visualize the application of these concepts in his mind. One of the initial challenges he faces is conveying his visual understanding to the students. If this could be overcome, he can proceed to establishing an understanding and application of the relevant theories to real life problems, and develop the students’ problem solving skills needed to become proficient fluid mechanic experts”.

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Exclusive use of lecturing method creates a passive learning environment. Engineering students learn better by participating, acting, reacting, and reflecting, rather than by watching and listening to lectures (Niirmalakhandan et al., 2007). Adopting a more interactive teaching approach which would be supported by computer animated instruction in a fluid mechanics classroom setting may create an environment conducive to learning and promotes social interaction among the learners. Schunk (2004) observes that teachers should not teach in the traditional sense of delivering instruction to a group of students. Rather, they should define situation in such a manner that learners become actively involved with the content through the manipulation of materials and social interaction.

Computer aided instruction (CAI) is a tool that has the ability to illustrate the fluid mechanics concepts visually and interactively in its three dimensional form. Judson (2006) noted that there is a connection between the use of technology as a teaching tool in the classroom and constructivist. The use of computer in classroom instruction dissemination improves interaction among the students and encourages a social context of learning. It facilitates and improves conceptual understanding (Madhavi and Billy, 2002).

Merino and Abel (2003) argues that many students appreciate computer-mediated learning as a supplement to traditional classroom learning. But the duo prefer not to do away with traditional method, but rather suggested that, the ideal is to combine computer-mediated instructions with traditional in order to harness the beliefs of both.

METHODOLOGY

Data source

Data were collected from eight universities in South Africa that offer B.Eng. degree in mechanical engineering. The choice of a method of data collection is based on the nature of the study, as well as the research question (Pirie, 1997). In this study, there is a need to examine the feelings, perceptions and thought of our study participants.

It should be noted that this is the first phase of a bigger study that seeks to investigate the impact of Animated Computer Aided Instruction on the learning of fluid mechanics by Mechanical Engineering. Therefore, our method of data collection in the first phase follows a descriptive survey approach. A methodology that evolved from an interest in finding the current prevailing fluid mechanics teaching approach, in its natural context, in each of the participating universities. Semi-structured interviews and classroom visits were used to collect data from both students and instructors of fluid mechanics.

Data collection

Data were gathered through classroom visits and semi-structured interviews. Interviews were, in most cases, done after the observed lessons with those students who were willing to be interviewed.

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Findings The emerging data show that fluid mechanics lessons are largely taught through the lecturing method. A typical instructor starts by introducing the lesson then goes on to relates content to students. Minimal interactions could be seen in some of the lessons. Most of these interactions tend to be initiated by students seeking clarifications on aspects of the lesson. There were lessons that started and concluded without an instructor allowing students to actively participate in the lesson development. Largely students spend time taking notes during the lessons.

An example of the observed lesson, was on Control Volume. In this lesson, the instructor started by explaining the resolution of problem on Control Volume, taken from a text book (Fluid Mechanics by Frank White). The problem was on a plate parallel to a flowing stream. After writing the problem on the board, he went on to start solving the problem without any student engagement.

“Given a plate which is parallel to a flow, the stream is a river or a free stream of uniform velocity V= Uoi,” he went on to explain that “Pressure is assume uniform, the no-slip condition at the wall brings the fluid there to a halt” He paused for about five seconds reading through his notes which were in his hand. He then went on to relate that “the slowly moving particles retard their neighbours above so that at the end

of the plate there is a significant retarded shear layer, or boundary layer of thickness y = δ ”. He then moved to the other half of the board and started writing “the viscous stream along the wall can sum to a finite drag force on the plate”. As he was explaining and writing on the board, he pointed at the diagram he had on a PowerPoint screen. He then told the students “to make an integral analysis and find the drag force

D in terms of the flow properties ρ , Uo,, δ and the plate dimension L and b”. The lesson was completed 40 minutes after it started. He then gave a related problem to students to solve after the lesson.

After the lesson, a few students were interviewed. The aim was to find out how they perceived the lesson and what they had learned from the lesson. These are some of the students’ comments:

Interviewer: Do you understand the problems solved in today’s lecture and the exercise given as home work?

S1: I do not understand how to go about with the problem he gave us.

S2: I can see the problem needs basic aspect of fluid flow, mass and momentum balance….. But I am not sure how to proceed.

CONCLUSION

From the preliminary data emerging, it is clear that lessons in universities are still instructor dominated and students participate minimally in them, if they ever do. Furthermore, instructors are more concerned about finishing the course than students conceptually learning the content. The bigger study from which this paper has been derived looks at both the effectiveness of taught lessons and tutorials in mechanical engineering education.

Evidently from the data, students have learned very little from the lesson on Control Volume. Perhaps there is a need to seek ways to ensure that students learn effectively during lessons and tutorials. Tutorials help students consolidate what they were taught in the lessons. At the moment tutorials focus mainly on students solving problems provided in the tutorial books.

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The way lessons are structured and taught probably account for the less success rate of students in their studies. It is then hoped that the next phase of the bigger study will introduce animated computer aided instruction which will engage students actively in knowledge development.

REFERENCES

Day, R. (1996). Case studies of pre-service secondary mathematics teachers’ beliefs: Emerging and evolving themes. Mathematics Education Research Journal, 8, 5 – 22.

Epstein, M. & Ryan, T. (2002). Constructivism: using information effectively in education. Online: http://tiger.towson.edu/~mepste1/researchpaper.htm

Felder, R.M., and Brent, R. (2005). Understanding student differences. Journal of Engineering Education, 94 (1), 57-72 French, B.F., Immerkus, J.C., and Oakes, W.C. (2005). An examination of indicators

of engineering students’ success and persistence. Journal of Engineering Education, 94 (4), 419-425 Judson E. (2006). How teachers integrate technology and their beliefs about learning:

Is there a connection? Journal of Technology and Teacher Education, 14(3), 581 – 597. Krajcik, J.S. (2002). The value and challenges of using learning technologies to support students in learning science. Research in Science Education, 32, 411-414. Leung, M.Y., Lu, X. Chen, D., and Lu M. (2008). Impact of teaching approaches on

learning approaches of construction engineering students: A comparative study between Hong Kong and Mainland China. Journal of Engineering Education, 97 (2); ProQuest Education Journals pg. 135

Madhavi, K. & Billy, L.C. (2000). A review of literature on effectiveness of use of information technology in education. Journal of Engineering Education. Malone, J. A. (1996). Preservice secondary mathematics teachers’ beliefs: two case

studies of emerging and evolving perceptions. In: Puig & A Gutierrez (eds). Proceedings of the twentieth conference of the psychology of mathematics education, 3, Valencia, Spain.

Mapolelo, D.C. (2003). Case studies of changes of beliefs of two in-service primary school teachers. South African Journal of Education, 23 (1), 71 – 77. Merino, D. & Abel, K.D. (2003). Evaluating the effectiveness of computer tutorials

versus traditional lecturing in accounting topics. Journal of Engineering Education, pg. 189 -194.

National science Foundation, (2004). An emerging and critical problem of the science and engineering labour force: A comparison to science and engineering indicators 2004. http://www.nsf.gov/statistics/nsb0407

Nirmalakhandan, N., Ricketts, C., McShannon, J., and Barrett, S. (2007). Teaching

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tools to promote active learning: Case Study. Journal of Professional Issues in Engineering Education and Practice, 133(1), 31-37

Philpot, T. A. et. al., (2002). Interactive learning tools: Animating mechanics of

materials. Proceedings of the 2002 American Society for Engineering Education annual Conference & Exposition.

Philpot, T. A. & Hall, R. H. (2006). Animated instructional software for mechanics of materials: Implementation and assessment. Wiley Periodicals, Inc., p.31 – 43. Philpot, T. A. et. al., (2002). Interactive learning tools: Animating mechanics of

materials. Proceedings of the 2002 American Society for Engineering Education annual Conference & Exposition.

Pirie, S. (1997). Working toward a design for qualitative research. In Qualitative

research methods in mathematics education, ed. A. R. Teppo, 79-97 and 164- 177. Reston, VA.: National Council of Teachers of Mathematics.

Qing Li, McCoach, D.B., Swaminathan, H., and Tang, J. (2008). Development of an

instrument to measure perspectives of engineering education among college students. Journal of Engineering Education 97 (1); ProQuest Education Journals pg. 47

Ramsden, P. (1992). Learning to teach in higher education. London: Routledge. Sabry, S. A., and Baartmans, J. (2000). The development and assessment of a course

for enhancing the 3-D spatial visualization skills of first year engineering students. Journal of Engineering Education, 301-307

Steif, P.S., & Naples, L. M. (2003). Design and evaluation of problem solving

courseware modules for mechanics of materials. Journal of Engineering Education.

 

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Mathematics Teaching at GET 

An explanatory framework – ‘Speedometer’

Prince S Jaca

Walter Sisulu University This paper contributes to the development an explanatory framework for the response patterns of Grade 5

learners on measurement items. Following a quantitative analysis it is reported that the context may have

not played any significant role in shaping the response patterns among these learners. Furthermore it

indicates how a limited development of counting techniques may have contributed to how the Grade 5

learners in this sample have responded. Few suggestions are provided as to how mathematics classroom

activities may be improved.

INTRODUCTION This paper is a further development on the work that I presented in the AMESA Congress 2008. Jaca

(2008) presents an analysis of responses of 523 Grade 5 learners from eight Eastern Cape rural schools on

a 25-item Mathematics Challenge questionnaire written annually by Grade 4 to 7 learners. I am

particularly interested on the responses of these learners on Item 1, which required learners to do

‘speedometer’ reading (figure 1).

Fig1: Item 1 on AMESA Mathematics Challenge 2007

Only 8% of the participating 523 Grade 5 learners gave ‘correct’ response (D) on this item. Furthermore

learners’ responses on this item exhibited some kind of an interesting pattern, for example they tended to

be drawn to a particular distractor (that is A) in this multiple choice question. A notable 78% of the

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‘incorrect’ respondents chose A (120 km/h). The diagram below gives a graphical representation of the

learners’ response pattern.

Fig. 2: Response Pattern on Item 1

This presentation seeks to contribute to the development of the explanatory framework for the pattern of

Grade 5 responses on this item (Item 1).

THEORETICAL BACKGROUND South Africa’s national curriculum makes fundamental shift from the traditional understanding of the

nature of mathematics. In this curriculum mathematics is defined more as process than a collection of

Mathematical objects. Competence in Mathematical process skills such as the investigating, generalizing

and proving is more important than the acquisition of content knowledge for its own sake (National

Curriculum Statement Grade 10-12: Mathematics, pg 9). It is an activity involving observing,

representing and investigating patterns in physical and social phenomena and between Mathematical

objects themselves (Revised National Curriculum Statement Grades R-9 Policy: Mathematics, pg 4). In

drawing from the work of Hans Freudenthal, Grevemeijer and Terwel (2000) writes that Mathematics is

first and foremost an activity. Doing Mathematics is more important than Mathematics as a ready-made

product. Mathematics teaching and learning is a process of doing Mathematics that led to a result (a

mathematics object). In Freudenthal sense mathematical activity is a form of ‘organizing’ everyday reality

or mathematical reality. Young children should aim at mathematizing/ “organizing” everyday reality

because they are yet to develop their mathematical reality to organize. Situations should be selected in

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such a way that they can be organized by the mathematical objects which the learners are supposed to

construct.

The ability to use appropriate measuring units and instruments; to make sensible estimates and to be alert

to the reasonableness of measurements and results is identified as one of the learning outcomes in the

General Education and Training band in South Africa national curriculum.

DEVELOPING AN EXPLANATORY FRAMEWORK There is an obvious concern that the speedometer diagram may have misled some learners. As it can be

seen from the diagram the tip of speedometer reading arm is touching ‘0’ of the 120, and that may have led

some of the learners to ‘Option A’. In the follow-up test questionnaires an ‘improved’ diagrams are

presented.

Item 1 is situated in a particular context (that of car driving). Dey (2002) defines ‘context’ as any

information that can be used to characterize the situation of a person, place, or object that is considered

relevant to the interaction between a user and an application, including the user and applications

themselves (Dey, 2002). Context may therefore also refers to language, tendencies and attitudes associated

with a particular situation. In the context of driving car in South Africa when one talks about car driving it

is common to hear phrases like below ‘60’, at ‘60’, at ‘100’, above ‘100’, at ‘120’, above ‘120’, etc. to

indicate the speed the car was driving. ‘60’ is usually associated with slow moving car and ‘120’ is used

in reference to a fast moving car. On the contrary it is not in our common language to talk of car driving at

‘50’, or at ‘110’, etc. In response to Item 1 one may wonder whether the context has not played a role in

shaping the response patterns of these learners.

The speedometer arm is pointing halfway between 120 and 140, that is 130. In order to be successful in

this item one may require elaborate understanding of counting techniques. The majority of learners (92%)

went for options whose values are between 120 and 140. One wonders whether this is as a result of the fact

that more than 80% of the given options fall in that range. For example it will be interesting to know as to

whether the learners who chose for option B (i.e. 121km/h) understood this value as between 120 and 140,

or 121 was seen as just as next reading after 120 km/h . In this item about 10% of the learners went for

Option B (121). Given the difference between 120 and 121, and 121 and 140 this raises the question as to

whether these learners have considered the ‘reasonableness of their choice’.

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TESTING THE EXPLANATORY FRAMEWORK Three 6-item questionnaires were designed to test the above accounts. These questionnaires included some

modification of the speedometer item according to the developing explanatory framework above.

Questionnaire 1A: contained speedometer item with a much improved diagram (where the

arm is pointing directly and clearly halfway between 120 and 140).

Questionnaire 1B: presented a similar item but a different context. Exactly the same diagram

was used. The new context in this questionnaire was a ‘weighing scale’.

Questionnare1C: in this questionnaire the context was kept as in 1A (the speedometer) but

the diagram is modified to indicate sub-calibration.

1A 1B 1C

Item 1 diagram as it appears in different questionnaires

Each of the questionnaires was presented to a Grade 5 classroom in different schools in the rural areas of

the Eastern Cape. The schools were chosen at random and all have not participated in the study before. The

three schools have been chosen from the same district in the Eastern Cape and they draw learners from one

and the same community, they are less than 6 km apart, and they are more than 40 km from one the semi-

urban centres of former Transkei. The number of participating learners in each school is indicated in the

table below.

School Number of Learners Questionnaire they wrote

1 24 1A 2 20 1B 3 32 1C

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RESPONSES AND ANALYSIS School 1: In each school learners were read each item and an IsiXhosa verbal translation was

provided for each item. In this questionnaire 16% gave a correct response and more than 71% of the incorrect respondents went for Option A (that is 120km/h) and, this proves that their response on this item was not necessarily influenced by the ‘misleading’ diagram.

School 2: 20% of the learners gave correct response on their item, and more than 62% of the incorrect respondents in this questionnaire (1B) still chose Option A (120kg). Comparing School 1 and School 2 one can conclude that there is no significant difference between their response patterns. The difference between this item in Questionnaire 1A and Questionnaire 1B is the contexts (speedometer and weighing scale respectively). Accordingly the context may have not played any significant role in shaping the response patterns in this item.

School 3. 3% of the respondents made the correct choice. The response patterns in this item on Questionnaire 1C was interesting as 65% of the incorrect respondents went for Option C (125km/h). Very few learners chose Option A an educationally significant departure from the response patterns in the other two schools. One can conclude that it is possible the learners just count the calibrated indicators without taking into consideration to the next indicated value.

DISCUSSION AND CONCLUSION The fundamental limitation of this report is that the respondents were not given opportunity to account for

their responses and this could have added the qualitative dimension to the data and the analysis. The

suggestions made should be looked at in with this limitation in mind.

Revised National Curriculum Statement Grade R-9 gives as one of the assessment standards towards

Learning Outcome 1 in Grade 4 and Grade 5

We know this when the learners:

• Count forwards and backwards in a variety of intervals (including 2s, 3s, 5s, 10s, 25s, 50s, 100s between

0 and at least 10 000)

In keeping with the understanding that mathematics is about ‘mathematizing’ from everyday reality or

mathematical reality, I believe that an elaborate understanding of counting techniques must be developed.

And such development should take place within contexts. Our classroom activities should go beyond say

just listing/reciting numbers backwards and forwards like: 10, 20, 30, 40, 50…….etc or 150, 140, 130,

120, etc. A variety of activities can be designed that may consider arithmetic means, and common

differences. The following activities could be used:

(1) Count in 10s from 0 to 150

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(2) Fill in the missing numbers: 10, 20, -, 40, 50, -,

(3) Fill in the missing numbers: 10, 20, -, -,50, 60

The distances, analogue watches and speedometers are useful context in the sense that in young children

these representations appeal to their visual abilities than their analytical abilities. Activities that appeal to

our visual senses are easily judged for the ‘reasonableness’ of their answers. It is easier to convince a

learner that the difference between 120 and 121 is much smaller than that between 121 and 140 if the

numbers are placed on a number line, that is, in terms of the ‘distance’ between the numbers.

(4) Find the number that is halfway between 20 and 40

(5) How far is 20 from 60?

(6) Find two numbers that are same distance from 30

References

Department of Education (DoE), 2003. National Curriculum Statement Grades 10-12

(General). Mathematics. Pretoria: Government Printer.

Department of Education (DoE), 2003. Revised National Curriculum Statement Grades R-9

(Schools)Policy: Mathematics. Pretoria: Government Printer

DEY, A. K. (2002). Understanding and Using Context. Future Computing Environments

Group, Georgia Institute of Technology. Gravemeijer, K. and Terwel, J.(2000). Hans Freudenthal: a mathematician on didactics and

curriculum theory. In Journal for Curriculum Studies, 2000, Vol. 32, No. 6, 777-796

Jaca, P. S. (2008). Mathematics Teaching at GET: Analysis of Grade 5 Performance in

AMESA Mathematics Challenge 2007. In the Proceedings of Association of Mathematics

Education of South Africa 14th National Congress in Port Elizabeth. Volume 1, pg217-225.

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LEARNERS THINKING AND REASONING ABOUT THE CONCEPTS AREA AND PERIMETER OF TWO - DIMENSIONAL SHAPES

Tlou Robert Mabotja

Waterberg District ( Limpopo Province ) Curriculum Advisor “Mathematics”

Robetlou@yahoo.com The concept area and perimeter was the major problem to learners in both primary and secondary schools. Focus of this study is to investigate learners thinking and reasoning about the concept area and perimeter of two dimensional shapes. In particular the study emphasizes the use of concrete objects to determine area and perimeter before the introduction of formulas. The study shows how learners struggled with the application of those concepts. Strategies to improve thinking and reasoning where provided as well as teaching strategies to improve the teaching of concepts area and perimeter.

RATIONALE OF THE STUDY The concepts area and perimeter of two dimensional shapes were discovered to be problematic to learners in both primary and secondary levels, especially on how to reason and justify their conjectures. Dickson (1989) in a study stated that confusion between the concepts is due to inadequate preparation in an early stage. Therefore the project was designed to assess learner’s logical thinking about the concepts as well as its application mathematically. Kilpatrick (2001) explains the ability to reason logically as the acquisition of adaptive reasoning. The child reasoning about the concepts is developed through the acquisition of conceptual understanding. For a child to be able to understand the use and purpose of applying the concepts mathematically he or she has to understand and know the meaning of the concepts. Acquisition of conceptual understanding provides learners with the opportunity to solve problems by thinking effectively. The study was conducted at the right time of transition where New Curriculum Statement (NCS) was introduced and implemented. The new curriculum requires learners to think and reason effectively. Learners are not just required to perform the following: Draw diagram, manipulate numbers but they are required to develop insight conceptually to can think and reason their findings. Therefore to cope with the present standard and situation, classroom instructions should be designed in such a way that it could encourage learners to apply their minds mathematically.

Guiding Questions 1) How do grade 9 learners from rural schools in Limpopo province (Waterberg District) think

about the concept area and perimeter:

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a) When measuring two-dimensional shapes using mathematical tools? b) When solving area and perimeter problems set in a real world context?

2) What difficulties/misconceptions do learners have when solving problems related to area and perimeter?

3) Why do grade 9 learners think and reason about the concept area and perimeter the way they do?

THEORITICAL FRAMEWORK The study is informed by constructivist theories, who believe that learners need to be actively involved in their learning. Learners need to draw upon their previous knowledge and utilise their present situation, they need to structure their own knowledge. If learners reasoning and reasoning skills are developed, they could be able to justify their responses on the basis of previous learned facts. Kilpatrick (2001, p130) states that: students need to use new concepts and procedures for sometime and to explain and justify them by relating them to the concepts and procedures that they have already understand. They may be able to link information that is previously learned with learned facts, because if previous learned facts were learned with understanding it will provide the basis of construction new knowledge. Olivier (1989) in his study on handling of pupils misconception, states that in constructivist perspectives “errors and misconceptions are seen as natural results of students efforts to construct their own knowledge and those misconceptions are intelligent constructions based on correct or incomplete previous knowledge. This constructivist actively involves the interaction of a child existing ideas and new knowledge. The new ideas are interpreted and understood in the light of pupils current knowledge built out of previous experience. Making errors or misconceptions cannot be avoided since in itself forms part of the process of learning. Smith, Desessa and Rosehelle(1993) in their research reject the tabular Rasa view of students before instruction by saying: before learners are taught expect concepts, students have conceptions that explain some of the mathematical and scientific phenomena that expect concepts explain, but these conceptions are different from the currently accepted disciplinary concepts presented in instructions. Many researchers have suggested that learning is a process of replacing misconceptions with appropriate expect knowledge. Smith, Desessa and Rosehelle (1993,p126) suggested the need for energetic classroom discussions in which students take position, make sense of and explain problematic phenomena and articulate justification of their ideas. Problem areas can be identified and confronted only if students are given the opportunity to communicate their ideas by explaining and justifying their learned facts.

LITERATURE REVIEW Numbers of researchers have investigated learners thinking and reasoning in their learning process in relation to are and perimeter and how they are related. Most of those researchers

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show how learner’s construction of knowledge is influenced and how difficulties occur through acquisition of knowledge.

Intuitive understanding Outhred and Mitchelmore (1996) Describe the strategy to solve rectangular area and perimeter measurements task before formally taught about the concepts area and perimeter. It is discovered that difficulties in learners about the concepts area and perimeter is due to overemphasis of formulae. If a formula is not clearly understood the best way to learn how to calculate area and perimeter will be by rote. Connecting to real-world situation (Contextualization) Outhred and Mitchelmore (1996) “To overcome the problem of rote learning of formulae, concrete materials have been widely recommended as the basis on which to build abstract concepts. The material used to measure area and perimeter of dimensional shapes may affect learners thinking and also gives insight into their understanding of the concepts area and perimeter as well as the applications. Hatano (1996) stated that “Procedure for converting real-world problems in mathematical representations and visa-versa and procedure for manipulating these representations helps find mathematical solutions”. Real-world problems motivate learners a great deal in thinking constructively. It also provides the basis of moving from unknown to the known. On the other hand connectivity helps learners to think and reason constructively when learned facts are known and make sense to them. This enables learners to think about the aspects of mathematical problems and to tackle new kinds of problems that lead to new knowledge.

Same A – Same B or Intuitive rule Tsamir and Mandel (2000) “ When increasing the length of two opposite sides of a square by a given number and reducing the length of the other two opposite sides by the same number either area and perimeter will remain the same”. Intuitive rule will only be applied with addition, subtraction, multiplication and division only.

Children’s understanding of Area Dickson (1989) Concern is on the topic area, in particular area of a rectangle and the teaching of area with reference to the formalization aspects of multiplying length by breadth. Surveys were conducted and the findings being ‘there is a lot of confusion between area and perimeter which is due to inadequate preparation in an early stage”. Dickson also encourage educators before the

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introduction of formulae to use dotted shapes and blocks for learners to count the number of blocks for each columns and rows. Example:

In this instance application of multiplicative and additive properties were encouraged. To calculate area of this shape a child counts the number of squares in a row and the number of rows. Solution 1: Seven squares in three rows: 7 + 7 + 7 (Additive) 7 × 3 (Multiplicative) = 21 Solution 2: Three squares in seven columns 3 + 3 + 3 + 3 + 3 + 3 + 3 (Additive) 3 × 7 (Multiplicative) = 21 This provides an advantage in learners understanding of area better because they will start thinking of counting the number of squares not in length and also to avoid glide answers that area is length by breadth. During the study it was discovered that learners will develop:

• Knowledge of matric length in cm, m, mm etc • Ability to measure correctly using ruler or tape • Ability to manipulate simple numbers – realize that multiplication is repeated addition.

DATA COLLECTION

Methodology The study is of qualitative nature, structured mainly on the key questions and clinical interview. Interview was conducted to each learner; same questions posed were based on their previously completed tasks. The interviewer created a good classroom environment where learners talk freely with confidence without intimidation. Each learner answered a task of five written questions aimed at assessing learners understanding of the concepts area and perimeter. Emphasis was on learners constructively justified their answers and how the concepts area and perimeter are related and applied in a real world situation.

The sample The six grade 9 learners from Limpopo province (Waterberg District) consist of three boys and three girls. Learners were selected based on their mathematics performance. A quarterly test written before closed for Easter holiday was used as an instrument to sample the six learners. With the help of their mathematics teacher the selected learners were chosen using these

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criteria: three performed above 50% while the other three performed below 50%. In this case gender was considered. The criterion was used mainly to assess thinking and reasoning of learner’s mathematics ability. Proper consultation was made first with the principal and mathematics teacher. Learners were also consulted and informed that the outcome of the study will not be used for their formal assessment and their names will not be used. Learners received consent letters to be completed with their parents as an allowance to participate in the study.

Rationale of the Questions Question 1: Helps to assess learner’s conceptual understandings. Question 2: Assess superficial understanding, accuracy in counting, comprehensive understanding, strategic competency as well as prospective use of mathematical tools. Question 3: Used to identify connection between mathematical content and real-world contexts. Question 4: Assess learner’s intuitive thinking and reasoning. Question 5: Assess in-depth understanding of area and perimeter of both regular and irregular shapes.

HANDOUT QUESTIONS Question 1 Define the following concepts without stating the formulas. 1.1. Area 1.2. Perimeter Question 2 2.1. Calculate area and perimeter of the shape without the use of formulas

2.2. Measure the Dimension of the shape and calculate area and perimeter of the

shape

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2.3. Calculate area and perimeter of the shape

3

7

2.4. Compare the results of the three and give a brief explanation of your findings. Question 3 3.1. Calculate area of this single tile

1

1

3.2. A four-roomed (RDP) house plan was designed into a kitchen, Dining room, passage and two Bedrooms. Calculate the number of tiles (1m2) required for the whole house plan.

Bedroom 2 Bedroom 1

Kitchen Dinning Room

Passage 1

1 1

2

2

3 3

Question 4 4.1. Calculate area and perimeter of a square 6 by 6. 4.2. If one side of the above square is added by two and the other side subtracted by

two respectively calculate: Area and perimeter of the new shape. 4.3. If one side of the above square is multiplied by two and the other side divided by

two respectively calculate: Area and perimeter of the new shape. 4.4. Explain the effect of area and perimeter in 4.2 and 4.3. In relation to 4.1.

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Question 5 A window is covered with silver and gold paper (Shaded) for a birthday party. Calculate area and perimeter of the shaded part from the window.

11

1

1

2

6

Answer sheet for Khomotso Solutions Marking Solutions Marking Question 1

a) Area: inside part b) Perimeter: the outside part

Question 2 2.1.

a) A = l x b = 6 x 3 = 18 b) P = 2l + 2b = 2(6) + 2(3) = 12 + 6 = 18

2.2. a) A = l x b = 6 x 3 = 18 b) P = 2l + 2b = 2(6) + 2(3) = 12 + 6 = 18

2.2. a) A = l x b

= 6 x 2 = 12 b) P = 2l + 2b

p p c c c c c

Question 4 4.1. A = l x b

= 6 x 6 = 36

P = 2l + 2b = 2(6) + 2(6) = 12 + 12 = 24

4.2. A = l x b = 12 x 4 = 48

P = 2l + 2b = 2(12) + 2(4) = 24 + 8 = 32

4.3. A = l x b

= 6 x 6 = 36

P = 2l + 2b = 2(6) + 2(6) = 12 + 12 = 24

c c w w c w

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= 2(6) + 2(3) = 12 + 6

= 18 Question 3 3.1. A = l x b

= 1 x 1 = 1

3.2. a) Dinning :16 b) Bedroom :16 c) Bedroom :16 d) Kitchen : 9 e) Passage : 1

c c w

4.4. Area of shape is A Perimeter of shape is A

Question 5 A = l x b

= 6 x 2 = 12 P = 2l + 2b = 2(6) + 2(2) = 12 + 4

= 18

w w w w

Data Analysis Table 1: Learners names Tumelo Khomotso John Sellwane Lerato Pinkie Question 1 w p p w p c Question 2 c c c c c c Question 3 p w w c c c Question 4 p w w p p w Question 5 w w n p w w

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Table 2: Number of responses Learners names Correct Partially

correct No response

Wrong

Average %

Tumelo 1 2 0 2 60 Khomotso 1 1 0 3 40 John 1 1 1 2 40 Sellwane 2 2 0 1 80 Lerato 2 2 0 1 80 Pinkie 3 0 0 2 60

Table 1 and 2 shows learners’ performance per questions from the written test. Key symbols shown on table 1 represent the mark allocation. Symbol “c” indicates learner’s correct solutions, “p” indicates learner’s partially correct solutions, “w” indicates learner’s incorrect solutions and “n” indicates no responses.

Graphical representation of the data

0

10

20

30

40

50

60

70

80

90

Tumelo

Khom

otso

John

Sellw

ane

Lerato

Pinkie

AVERAGE %

Analysis of learners responses There is only one learner who showed understanding of the concepts area and perimeter as the surface and distance around the figure respectively, three learners showed little understanding while two have no understanding of the concepts. Question 2 reflects positive outcomes as shown in the table above. All learners answered the question correctly but failed to provide brief explanation of their finding. The expectation in this question was to test learners superficial and comprehensive understanding i.e. accuracy in counting, prospective use of mathematical tools and the ability to compare and justify their findings.

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Question 3 is classified to be a balanced question as three learners answered the question correctly; one learner with partially correct answer and two learners answered the question incorrectly. The aim of this question was to identify connection between content and application of content in a real world situation. Question 4 in this study is regarded the core, because it assesses exactly what the study intent to assess: ‘learners thinking and reasoning’. Three learners partially answered the question correctly and the other three learners answered the question incorrectly. No one answered these questions correctly. This showed that learners have difficulties with interpretation. Question 5 assesses in-depth understanding of the concepts, where by they have to identify and transform regular shapes from irregular shapes. In this question one learner partially answered correctly, four learners answered incorrectly and one learner did attempt to answer.

CONCLUSION The study has implication on how Educators can help learners acquire the skills to think and reason about the concepts area and perimeter. In order for learners to reason constructively, conceptual understanding should be develop at an early stage. Educators should avoid teaching procedural understanding only, since it could not help in developing learners to reason effectively and efficiently. The teaching of the concepts area and perimeter without application of formulae develops learners reasoning abilities and gives an overview of what the concepts measures.

REFERENCE 1. Dickson, L. (1989) Area of a rectangle, in Johnson, D. (Ed) Children’s mathematical

frameworks: A study of classroom teaching, Berkshire: Nfer - Nelson. 76 – 88. 2. Hatano.G. (1996). A conception of knowledge acquisition and its implications for

mathematics. In L.Steffe et al (Eds). Theories of mathematical learning. Lawrence Erlbaum. Hilsdale, pp 197 – 217.

3. Kilpatrick, J; Swafford, J. and Findell, B. (Eds) (2001).Adding it up: Helping Children learn

Mathematics. Chapter 4. The strands of Mathematical proficiency. 115 – 115.

4. Olivier,A.(1989)Handling pupils misconceptions. Pythagorus, No.21. 10 – 19. 5. Outhred, L. and Mitchelmore, M. (1996) Children’s intuitive understanding of area

measurement, in Puig, L. and Guitierrez, A. (Eds) Proceeding of the 20th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, Spain: Valencia, 91 – 98.

6. Smith, J.P, Desessa, A.A and Rosehelle,J.(1993). Misconceptions recovered: A

constructivist analysis of knowledge in transition. The Journey of the Learning Science, 3, 2. 115 –163.

7. Tsamir, P. and Mandel, N. (2000).The intuitive rule (Same A – Same B): The case of

area and perimeter, In Nakahara, T. and Koyama, M. (Eds). Proceedings of the 24th

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Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, Hiroshima: Japan, 225 – 232.

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Mathematics Teacher Professional Development:

A Reflection

Helena Miranda

University of the Witwatersrand Often we read papers reflecting on professional development projects for mathematics teachers. However, such reflections are limited to the reports by the project facilitators or by on-site researchers. The reflection discussed in this paper is of a different nature. It is different in that it discusses the reflections made by the teachers themselves, rather than by the project facilitator or researcher. The project on which these reflections were made involved ten Namibian mathematics teachers in weekly in-service training sessions for a period of three months. The project was part of an Action Research study (Miranda, 2009), that was aimed at the enhancement of the teaching and learning of algebra in Namibian mathematics classrooms.

INTRODUCTION One advantageous opportunity that action guided research studies offer is the ability to be sensitive to issues of subjectivity. That is, one should be able to track personal growth in terms of learning and understanding along the research cycles, instead of being attuned to the learning or its lack in others alone. For instance, in this research study, I came to learn that collaborating with teachers is not about telling or showing them how to teach but about helping them become aware of their own expertise and develop ways of how to explore and use that expertise to its fullest while working with children. I also came to realize that the project was helping both the participant teachers and myself to take the collaborative culture into our classrooms where the learners can also be allowed to work closely together, in the same manner that we did in the in-service sessions.

One focus that the action research literature has put emphasis on is the notion of developing teachers into researchers and reflective practitioners. These two are closely related issues and might play an essential role in collaboration. Collaborative inquiries such as this one, also offer rich possibilities for continuous dialogue between theory and practice. It is also in these kinds of inquiries that teachers are allowed a space to interact with others while investigating their pedagogical practices and finding ways of not only changing but also improving those practices. As one of these teachers, I was able to observe my own learning as I came across surprising moments especially those in which I was triggered to think about learning and teaching algebra differently.

One of the methods of data recording used in this project was to keep reflective journals in which participants could record their ideas, questions and any arising problems, as well as any other incidents that could be considered important for the professional group and its learning. This helped me to develop a richer understanding of the observable trends concerning the learning and teaching of algebra in specific, and issues of teacher education and development in general. It was evident in the journal entries that we

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shared with one another that we were all able to recognize and acknowledge possibilities for improvements in our individual practices.

Apart from my own learning and understanding, the project enabled teachers to achieve a number of goals for their own professional development. These include a redefinition (including historical views) of what algebra is, newer means of interpreting algebraic concepts and problems, the development of ways of assisting students make sense of algebra, and possibilities for collaborating with other teachers and teacher educators in the community.

As an overall observation, I can say that this research study allowed both the participant teachers and myself to realize the need to continue collaborating with others such as teacher educators and teachers, about general educational issues and issues of teaching and learning mathematics. It also opened up opportunities for us to discuss with one another our experiences of the collaboration and its effects on our own practices. This, I believe made provision for a chance to teach one another while at the same time learning from each other. Specific impressions that the teachers had of the project and the professional growth opportunities it offered are discussed in the following sections of this paper.

REFLECTIONS ON THE PROJECT Advocates of action research support the use of collaborative action research where teachers can “reflect on their practice in order to organize their teaching differently and so become more effective” (Angelides, Evangelou & Leigh, 2005, p. 85). The benefit of this is that the participants are able to gain from the collaboration the ability to analyze their own teaching. In most research studies, says Angelides et al. (2005), participant teachers’ voices are commonly neglected. However, in this project the teachers were actively involved in the decision-making process concerning what activities were to be explored and where the project should be headed after each in-service session (See Apendix A for an example of a task that the teachers engaged with).

In order to make sure that the teachers’ voices were at the centre of the happenings of the study, I regularly gave explicit questions as a guide to help teachers in drawing their reflections on the project. One such question invited the participant teachers to explicitly discuss the ways in which the mathematical activities explored contributed to their learning of algebra and how they may later, impact their decisions of what approaches to take when teaching algebra. With regard to the question of the mathematical activities, for example, all the teachers acknowledged that this was the first time they were to see and use algebra tiles. The excerpts in the following paragraphs were the explicit words with which the teachers expressed their experiences with the activities we explored together. This sharing was recorded in individual journals and was verbally shared by reading out loud to the rest of the professional group. Here is one teacher Heidi’s 25sharing:

The exercises we did with the algebra tiles were just amazing. I never even connected area and factorizing. This is a fun way of introducing learners to factorizing, expecting quadratic equations, which tend to give problems to most learners. I will definitely be using it in the future.

Prior to this research project, I facilitated a few sessions on algebra tiles, especially with my undergraduate prospective teachers. I also observed, in other research projects, in-service teachers work with algebra tiles,

25 It should be noted here that for ethical purposes, pseudonyms, but not teachers’ real names, were used.

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and the impressions were similar to what I was witnessing in this particular project. Below is another teacher Mia’s impression of the learning opportunities the study was offering:

I have learned a great deal about algebra, more especially from the sessions about manipulating algebraic expressions using algebra tiles. All activities contributed to my learning of algebra. Giving the polynomials a concrete referent made them real. I have also learned that a wide range of materials are available to make algebra teaching and learning easier and more enjoyable. Learning at any level is easiest if it starts with concrete actions, and then moves to symbolic representations and finally, to formal actions (this is the abstract level of thinking). With manipulating algebraic expressions, we used the tiles as concrete representations and then moved to different levels of learning.

At some point of the project, we, as a group, discussed the importance of the role that multiple representations play in the effectiveness of learning and teaching mathematics. As can be seen in Mia’s words, this study enabled the participant teachers to realize and acknowledge this role, and the use of algebra tiles was observed as the most appropriate way of achieving that goal. Due to this new learning about teaching and learning, the participant teachers, including myself, indicated that having worked on the activities that we did, will, in one way or the other, make a positive change in our approaches to teaching algebra and helping students better understand algebra. As prompted, the teachers shared with the following ways, how they believe their thoughts of their own practices have been impacted by the mathematics activities in this project. Below are two different teachers shared their thoughts.

I think my view of algebra has changed and I will definitely be approaching it in a different way. From the activities we did, I have realized, that algebra is actually fun and not just a set of rules. The algebra tiles were my favourite. [Expanding and] factorizing made easy and fun. (Petcha)

After all the activities that we contacted in this project, I felt the need to re-teach my learners. I felt I did not teach them well since my approach was more based on how I was taught. (Seth)

As a follow-up question to the one above, the teachers were asked whether they will likely change their approaches to teaching algebra. The answers to this question were affirmative and the teachers promised themselves to try out the activities in their own classrooms next time they teach algebra. One particular pledge stood out as one teacher declared that “of course I will approach my algebra teaching differently. I am looking forward to starting next year. I am much more confident now, honestly. I will be more practice oriented in approaching algebra.”

To borrow from Mason (2002), these teachers were becoming more expert in the teaching of algebra—i.e. they began to notice that there are many possible ways of approaching algebra that they could try out in the classrooms in the future. The project was helping them become more expert in the teaching of algebra, specifically in helping students make meaning of algebraic problems with the use of more interactive activities. Hence, the study offered the teachers’ ways to expand the horizons of their professional sensitivities to notice what needs to be emphasized when teaching algebra.

POSSIBLE FUTURE COLLABORATIONS There was more sharing about the mathematics activities and what they offered than about possibilities for future collaborations. By the last of all the reflection questions, the teachers were invited to make suggestions, if they had any, about different ways or activities to be utilized in future projects of this kind.

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Many teachers were hesitant to criticize or make new suggestions, and more certain about being willing to have this kind of project run again in not only their region, but also in other regions in the country. Four of the teachers had this to say:

Mia: My suggestion is to keep this kind of project working from now on, so that mathematics teachers can come together to share ideas about mathematics and different approaches to teaching mathematics. In addition, this project was very interesting and enjoyable because every day we met, I learned new things about algebra. I also learned about different approaches I would like to apply in classroom situations.

Heidi: I think graphing equations is one of the topics that need attention as most teachers in the field do have a hard time teaching it.

Aina: Perhaps the time that sessions begin should be reconsidered in order to accommodate teachers working far from the meeting point.

Set: This project has a potential to cause a tremendous improvement in our mathematics teaching approaches. My wish is to keep the project alive.

Reading these lines, it becomes evident that the participant teachers valued what they gained from participating in this project and that they would like this to continue. Offering professional development to teachers of mathematics has been widely recognized as an important aspect of impacting the students’ learning of mathematics. However, research on teacher communities of mathematics teachers has not yet been adopted as a common practice in mathematics education (Gellert, 2008). Therefore the continuation of the PD projects such as this, will have much to offer to the existing discussions in the literature about professional development. Some important concerns that come up when conducting PD projects include the reluctance to participate among most teachers. Also, those that show interest in participating do not keep up their attendance.

Despite the efforts by professional developers that are determined to bring about positive change to the pedagogy of mathematics, many teachers generally have a tendency to fall back into the old-fashioned practices of teaching. One of the reasons for this, as research has indicated, is the failure to do follow-up and continued activities whenever a professional development project has been initiated (Heck, et al., 2008). Even though my initiated project suffered this consequence because I had to be away from the group for sometime, it is my intent to go back to the group once more in order to revive it and expand on what we have started already. This will be done while taking into considerations what the teachers thought should be changed—as discussed below.

CHALLENGES TO THE PROFESSIONAL DEVELOPMENT PROJECT At the conclusion stage of the research study of which this project was a part, I consulted with the teachers once more. Since the teachers had not met once since I left the research site, I decided to have a discussion with the teacher who was supposed to act as the mediator in arranging the monthly meetings as a PD activity. The discussion was limited to identifying problems that inhibited the teachers from meeting and the steps to be taken in future in order to ensure a successful collaborative project. I came to learn that the organizing teacher had many other responsibilities so that she did not get a chance to keep the meetings going by contacting all other teachers. These responsibilities included being a department head, having registered in-service training courses that “were much too demanding,” as she put it. However, the teachers from within the school where the project was based met once but they did not have opportunity for

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discussion. Even though the teachers knew they could approach me (through the mediator teacher) for any mathematical activities, they reportedly did not have means of contacting me.

One of my questions to the teachers was: In what ways could my absence as a project leader have contributed to the problem of not attending? There were a variety of answers to this question, from which I could learn for future collaborations. For example, when I was available at the research site, I had the responsibility of reminding the teachers on the dates that we were supposed to meet. If any of the teachers could not attend, I would be responsible to reschedule the date or time for meting. Therefore in my absence none of the teachers were prepared to take up this responsibility. Another factor, as the mediator teacher shared, included the fact that I provided refreshments for the meetings and reimbursed teachers for money that they used on transportation to come to the research site26. With refreshments and transportation reimbursement no longer available, the teachers “seemed less interested in coming to the meetings,” concluded the mediator teacher. She suggested that if we have another project of this kind, “money should not be given. The teachers have to come to PD sessions because they feel the need to learn something but not because they are being fed or getting paid for their participation.”

Another arrangement that I made with the teachers was, having left them with one more mathematical activity to try out, that they bring their own mathematical problems to share, or approach me for further activities. However, they could not find any interesting activities and the latter was not possible because they did not have means of contacting me. Their internet connection at the school was non-functional for a long time after I had left the site. If I had “left some extra activities with the teachers, the project might have continued,” added the mediator teacher. “Coming up with activities to do was a major challenge. We needed something interesting that could make us think we are gaining something.”

The mediator teacher also expressed that it was difficult for her to get hold of other teachers. She always had to phone everyone to remind them of the meetings and this is costly. “Internet is not available to everyone,” she said. When I asked her for alternatives, she suggested that “in future we must have a calendar or a year plan that everyone will keep and be reminded of the dates to meet.” However, individual reminders will still be needed since it is easy to skip a meeting of any kind. Suggesting ways in which teachers can be kept interested in attending projects of this kind, the mediator teacher had this to say: “Have activities that capture interest; Give them copies of whatever you are teaching so they can make use of them in the future; involve the teachers in creating and generating mathematical activities and teaching materials such as fractal kits and algebra tiles.”

The last but not least important question I posed to the mediator teacher concerned the teachers who felt that PD activities were not for them because they had enough experience already. To this, the mediator teacher indicated that she, herself, has many years of teaching mathematics experience but feels the need to attend PD sessions. “Knowledge is not static, and if you stop learning you stop being a teacher. To remain a teacher, you need to keep learning in order to update your knowledge. Therefore it is important to attend these activities, as long as it is to our benefit.” To wrap up the project, the mediator teacher was asked to say anything that she thought was necessary to the conclusion of this study. In her own words she stated: “We need such projects in our community not only for senior secondary teachers but we have to start from lower levels too. There is also the need to attend to the professional needs for newer teachers. These include but are not limited to: Interpreting and understanding the syllabus; improve their subject content knowledge; assessment policies: e.g. allocating marks to individual questions, developing marking scheme,

26 The refreshments and transport reimbursement were possible with the sponsorship of the FSIDA research fund.

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and setting up exams and tests. PD should be planned according to teachers’ needs, for example by asking the teachers what exactly they find more challenging to teach in a certain area.” She concluded.

SOME CONCLUDING THOUGHTS Proulx (2007) carried out a professional-based research study—where he offered professional support to secondary mathematics teachers while exploring specific mathematical concepts. I therefore find it useful here to make some comparisons between his study and mine, in terms of conducting professional development projects. Unlike in my own study, Proulx reported some difficulty in finding the research participants for his research study. In my case, there were many teachers who wanted to take part in this study but they could not because of the long distances they would have to travel to come to the research site.

Also, as discussed above about the participant teachers of my study, Proulx also acknowledged that secondary school mathematics teachers have much too busy a schedule to completely devote to professional development projects. Even though it is considered an important factor in the teachers’ professional learning, PD is in an extra load to what the teachers already deal with daily. One has to recognize that apart from being teachers responsible for their teaching and the students’ learning, the teachers are also individuals with personal lives outside the school ground. I therefore appreciate very much the teachers’ efforts to take off most of the afternoon times to come and participate in the in-service sessions offered by the project.

Proulx (2007) further argues that if teachers find the content of the professional development project to be of some value to their understanding of mathematics, they will be more willing to attend the sessions it offers. My claim here is that the participant teachers in my study indeed found this project to be a beneficial factor in their professional lives and that is why they kept up their attendance for as long as I was around providing mathematical activities to engage their thinking. However, the challenge now is keeping the project running and expanding it to other parts of the country. This is one recommendation I make for future collaborations, in which the suggestions discussed above, as given by the mediator teacher, should be taken into consideration.

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References Angelides, P., Evangelou, M., & Leich, J. (2005). Implementing a collaborative model of action research for teacher

development. Educational Action Research, 13(2), 275-290.

Gellert, U. (2008). Routines and collective orientations in mathematics teachers’ professional development. Educational Studies in Mathematics, 67(2), 93-110.

Heck, D. J., Banilower, E. R., Weiss, I. R., & Rosenberg, S. L. (2008). Studying the effects of professional development: The case of the NFS’s local systemic change through teacher enhancement initiative, Journal for Research in Mathematics Education, 39(2), 113-152.

Mason, J. (2002). Researching your own practice: The discipline of noticing. Routledge: Falmer.

Miranda, H. (2009). Enacting algebraic meanings: Educating teachers’ mathematical awareness. Unpublished doctoral dissertation: University of Alberta, Edmonton, Alberta.

Proulx, J. (2007). Enlarging secondary-level mathematics teachers’ mathematical knowledge: An investigation of professional development. Unpublished doctoral dissertation: University of Alberta, Edmonton, Alberta.

APPENDIX A: ACTIVITIES WITH THE ALGEBRA TILES

Joe’s Rectangle Heita’s Rectangle Mia’s Rectangle

Table 1: A Representation of Teachers’ Expansion of ( )( )32 ++ xx

x2 3x 1

+ +

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Figure 2: Tiles with Area Totaling ( )132 ++ xx

Figure 3: Representation of Expressions that do not Factor

+

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Primary school teachers’ mathematical content knowledge on division of proper fractions: Some theoretical illustrations

Lovemore J Nyaumwe

Wits School of Education, University of the Witwatersrand

The perplexity that learners often experience on division of fractions can be reduced when teachers use three modes of presenting the division concept. The modes of real contexts, pictorial and symbolic representation of problems may enable learners to use their intuitions to develop the division algorithm conceptually. The algorithm of “invert and multiply” that is ubiquitously used by some teachers is not easily understood by most learners because the process of invert and multiply is seldom explained. The use of the multiplicative inverse of the divisor can be used to show the logic of the “invert and multiply algorithm”. The paper can provide insight into the mathematical content knowledge that is necessary for primary school teachers to teach division of proper fractions effectively without rushing to introduce procedural algorithms that learners learn by rote.

Introduction Learning Outcome 1 (LO1) emphasize learner understanding of number sense and development of

proficiency to make accurate calculations with different types of numbers. Section 4.1.5 of the Revised

National Curriculum Statements (RNCS) for Grades R to 9 (2005: 5) calls for Grade 4 learners to

“recognize and represent common fractions with different denominators” up to eighths. In section 4.1.11

the RNCS (2005) emphasize learner calculations using appropriate operations and sharing. Sharing is an

intuitive way of introducing division involving both whole and fractional numbers.

Many primary school learners face difficulties in understanding fractions and performing computations

involving division (Bulgar, Schorr & Warner, 2004). The perplexity of division of fractions sometimes

arises from the way that teachers teach. For instance, learners lack conceptual understanding of fractions

when taught division of fractions mostly devoid of meaning, yet at home they get involved in sharing

activities. Lack of activities from learners’ contexts to develop division concepts can result in the teachers

using instructional approaches that emphasize rote memorization and procedural executions of rules and

procedures without understanding them. Teaching division of fractions through inverting the second

fraction and multiplying is typically emphasized in rote learning and memorization of procedures. In the

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absence of the logic and conceptual understanding of the procedures that can lead to “invert and multiply”

the teachers drill the procedure, provide repetitive exercises for learners to practice on them. As the

Chinese proverb alluded, learners listen and forget, they do and they remember, the invert and multiply

algorithm is easily forgotten by most learners as they only listen to teacher talk.

Fractions are important in learners’ daily lives and are essential when they continue to study mathematics

beyond primary school. Conceptual understanding of fractions can facilitate learner cognitive flexibility.

Cognitive flexibility as portrayed by Bulgar, Schorr and Warner (2004) is the ability to restructure one’s

knowledge systems in adaptive response to meet situational demands of non-routine mathematical

problems that learners encounter during problem-solving. For instance, cognitive flexibility on fractions

enables learners to restructure their understanding and apply the arithmetic principles when solving

algebraic fractions. Cognitive flexibility of mathematical procedures also enables learners to use multiple

modes of representation of a concept to solve non-routine problems. When learners develop cognitive

flexibility of mathematical procedures they may change their beliefs from viewing “mathematics as a

confusing and irrelevant subject to their lives and associating it with failure” (Suh, 2007:163) to viewing

the subject as relevant and useful.

This paper explores essential teacher content knowledge on division of fractions. The knowledge

encapsulated here enables the division of proper fractions concept to be taught more meaningfully to

enable learners to develop cognitive flexibility of using fractions in their environment and for further

mathematical studies.

The paper would be valuable to elementary and middle school teachers described by Kilpatrick (as

reported by Chesnek, 2001:1) as having a “shaky grasp of mathematics themselves, and often are unable to

clarify key concepts or solve problems that involve more than basic calculations”. The use of multiple

representations involving real contexts, diagrammatic and symbolic in teaching division of fractions may

provoke teacher thinking on how to use a variety of instructional strategies in order to develop learner

conceptual understanding of division of fraction algorithms.

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Literature review

Using the intuition of sharing of real objects can present learners with a clear context and understanding of

the concepts of numerator and denominator. For instance, real contexts of fractions may enable learners to

deduce that a denominator is the number of recipients when equally sharing a unit and the numerator refers

to the number of each share of the unit. The use of context can complement the concept of equi-division of

fractions definition. The equi-division of a unit into equal parts definition introduces the denominator as

the number of equal parts into which a unit is divided and the numerator as the number of divisions.

Understanding the relationships between quantities that are represented by fractions is sometimes

problematic to some learners. Some teachers commonly introduce fractions theoretically as equi-division

of a unit into equal parts. This approach sometimes confuses some learners because they find difficulties to

concretely visualize the meaning of a fraction (Sadi, 2007). The confusion arises from the concept of

fraction that is arbitrary in that it refers to different quantities. For instance, the statement James and John

spent � of their pocket money does not imply that the boys spent equal amounts as they could have had

different amounts to start with. Equal fractions on different amounts of money present fractions as

relations between quantities unlike the exactness that is expressed by equal parts of whole numbers.

Learners first encounter operations with whole numbers before they engage in operations with fractions

that they sometimes get confused when learning fractions. For instance, multiplication is sometimes

introduced as repeated addition of natural numbers which result in a product that is greater than any of the

numbers multiplied. Introducing division as the opposite of multiplication enables learners to use their

intuition to generalize that division should yield a quotient that is smaller than both the dividend and the

divisor. The division of fraction contradicts learner memory of previously learnt multiplication techniques

in that the quotient is usually greater than the dividend and the divisor. For instance, ½ divided by 61

yields 3, which is far greater than both ½ and61

. This contradiction reveals that fractions do not logically

form a normal part of learner operations with natural numbers, leading the learners to believe that

operations on fractions are abstractly defined (Sadi, 2007). These learner views on operations with

fractions can lead them to develop some misconceptions that are caused by the misunderstanding that can

develop out of a lack of conceptualization of the abstract nature of fractions.

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Teacher content knowledge on division of fractions

Division of fractions poses the greatest challenges to teachers as they commonly use the “invert and

multiply” procedure as the only method of dividing fractions. This method was used on the teachers when

they were students and they were not shown how the algorithm logically arises. The question of � ÷ 2 is

used to illustrate how teachers can handle division of fractions using different modes of presentations.

Real situation

A real context can help learners to use their intuitions to understand the problem �÷2. Using a real context

the problem may be presented as “Share � of a cake between two girls. What fraction would each girl get?”

Pictorial representation

To reinforce the power of the real context to enable learners to understand the problem, diagrams can

provide learners with opportunity to experiment with their ideas of sharing. The following diagrams are

possible:

Fig 1: A cake divided into 3 equal parts Fig 2: � of a cake divided among 2 girls

Using their intuitive knowledge of fractions, learners can draw a cake and show a � as shown in Fig 1.

Sharing a third of a cake among 2 learners means dividing a portion of it by two as shown in Fig 2. From

Fig 2 learners can deduce that sharing a third of a cake among two girls gives each of them 61

of the cake.

Here the solution is obtained from logical reasoning with the diagrams.

� � �

61

61

� �

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Symbolic form

The problem � ÷ 2 can be presented using mathematical symbols and solved using formal mathematical

strategies. Teachers should not use the invert and multiply algorithm straight away, but can develop it with

learners as described below. Learners may recall the meaning of division as a number consisting of a

numerator and a denominator, thus 31

÷ 2 =23/1

. The part 23/1

looks unusual and can be written in familiar

fraction notation by removing 2 in the divisor. Learners may recall that any number divided by 1 does not

change, so it is necessary to find a way of creating 1 in the denominator 2 by writing 23/1

as1/23/1

. The task

now is to create 1 from 2/1 in the denominator. Through scaffolding and probing learners’ thinking, they

can remember that a number times its multiplicative inverse gives 1 as in 12

×21

= 1.

The learners can use the concept of equivalent fractions to recall that multiplying a denominator and a

numerator by the same number does not change the value of a fraction. In the division of1/23/1

, both � and

2/1 are multiplied by the multiplicative inverse of 2 to create 1 in the denominator. Thus

1/23/1

=2/11/22/13/1

××

=31

×21

=61

. A generalization of the process is that dividing a fraction by a whole number

(except 0) is the same as multiplying the fraction by the multiplicative inverse of the number.

This rule also applies to division of a fraction by a fraction. For instance, 74

÷ 53

= 3/55/33/57/4

××

= 74

× 35

=

2120

because the multiplicative inverse of the divisor 3/5 is 5/3. Multiplying a fraction by the multiplicative

inverse of the divisor is commonly known as the inversion method of fractions because the process

involves multiplying the first fraction by the multiplicative inverse of the second fraction or simply

inverting the second fraction and multiplying. Developing the logic behind the inversion method may

enable learners to understand the process involved in carrying out division of fractions. After

understanding the logic behind the inversion algorithm, learners can be given problems on division of

fractions to solve using the short-cut in order to develop procedural competence of using it.

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Conclusion

Fractions, like other mathematical concepts and structures were invented as tools to organize phenomena

in the natural, social and mental world. Teachers can use this insight to link division of fractions with

learners’ contexts and teach them using multiple representation modes of real contexts, pictorial and

symbolic. The real context mode enables learners to use their intuitions to share quantities in given

fractions. The use of intuition can enable learners to develop strategic competence and adaptive reasoning

(Kilparick, Swaford & Findell, 2001). The use of pictorial diagrams can give learners opportunities to use

their reasoning capacity to find solutions without using a mathematical algorithm. The symbolic mode

enables learners to express their understanding of problem-solving tasks in mathematical jargon and to

look for patterns that can lead to generating some mathematical algorithms that they can use to solve

problems.

References

Bulgar, S., Schorr, R. & Warner, L. (2004). Flexibility in solving problems related to division of fractions. Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Canada. October 21.

Chesnek, M. (2001). Overhaul of school mathematics needed to boost achievement for all. Washington, DC: The National Academy Press

Kilparick, J., Swaford, J. & Findell, B. (2001). The strands of mathematical proficiency, Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press, 115 – 155.

Moss, J. & Case, R. (1999).Developing children’s understanding of rationale numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122 - 47.

Sadi, A. (2007). Misconceptions in numbers. UGRU Journal, Volume 5, Fall.

Suh, J. M. (2007). Tying it all together: Classroom practices that promote mathematical proficiency for all students. Teaching Children Mathematics, October. 163 – 169.

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QUARTILES AND PERCENTILES: WHICH FORMULA?

Anelize van Biljon

University of the Free State It is usually assumed that there is a specific formula to calculate a statistic, but one of the cases where it is not true, is in the calculation of the quartiles and percentiles of a set of observations. This paper investigates some of the formulas in use for ungrouped observations. The computer program that is used for the purposes of this paper is Microsoft Excel, since most teachers have access to Microsoft Office packages. There are many other statistical programs available, such as SPSS, Stata, SAS and MiniTab. Although the concept of quartiles is reasonably easy to grasp, different statistical computer programs use slightly different methodologies to calculate it. There are at least 10 different methods which can produce slightly different results. (MeadInKent, 2008)

INTRODUCTION A set of data can be described in terms of measures of central location (central tendency), measures of dispersion (relative standing) as well as some other measures, such as variance, which will not be discussed here.

The Five-Number-Summary include the minimum value, the maximum value and the first, second and third quartiles. The first (lower) quartile will be indicated by Q1, the second (median) by Q2 and the third (upper) by Q3. The lower and upper quartiles along with the median locate points that divide the data into four sets, each containing an equal number of measurements. (Mendenhall, et al, 2009)

The percentiles divide the data set into 100 equal parts. The first or lower quartile, Q1, is equal to the 25th percentile (P25), the second quartile is equal to the 50th percentile (P50) and the third or upper quartile is equal to the 75th percentile (P75). (Keller, 2008)

The quartiles and percentiles are division points.

The Box-and-Whisker plot graphs five statistics: the minimum of the observations, the first, second and third quartiles and the maximum of the observations. These are all measures of relative standing. The second quartile is also known as a measure of central tendency.

The quartiles and percentiles give an indication to a learner of his/her performance in a test, for example. Is his mark is among the upper 5% or in the lower 20% or above half the marks of the learners who also wrote the test? If the average of the class is 25%, but he still is among the upper 5% he/she knows that his/her performance was not too bad considering the other marks. This is an application with which the learners are familiar.

BOX-AND-WHISKER PLOT In short it is called the Box Plot. It can be drawn vertically or horizontally. It consists of a box part in the middle and the whiskers on each side of the box.

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The Box-and-Whisker plot shown below in Figure 1 is done in Excel (Office 2007). The data points are shown just above the plot itself. The process to draw a Box Plot is described as follows: In the Add-Ins you select Data Analysis and Box Plot. (Keller, 2008) This feature is not available in Office 2003.

Figure 1: The structure of a Box-and-Whisker Plot

DESCRIPTIONS OF THE QUARTILES Various school textbooks describe quartiles in slightly different ways although the meaning is similar. Four of these descriptions follow below:

The lower quartile is the median of the lower half of the values. The upper quartile is the median of the upper half of the values. (Laridon et al, 2007)

Quartiles are measures of dispersion around the median. The median divides the data into two halves. The lower and upper quartiles further subdivide the observations into quarters. (Phillips, Basson, Botha, 2007)

The first quartile is the number above 25% of the ordered data values. The second quartile is the median (middle value). The third quartile is the number above 75% of the ordered observations. (Goba, Van der Lith, 2005)

Q1 is the midpoint of the lower half of the data values and Q3 is the midpoint of the upper half of the observations. (Van der Lith, 2006)

THE POSITIONS OF QUARTILES AND PERCENTILES First of all, the data (observations) in the set must be ordered.

The position (location) of a quartile or percentile must be determined before the value can be calculated, because its value depends on its position in the ordered set of observations.

A simple formula, 21

(n+1) = 42

(n+1), is used to find the position of Q2 where n is the number of

observations. It is therefore easy to calculate the position and value of Q2, also known as the median: if the number of values in the set is odd, the middle value is the median; if the number of values is even, the median is the sum of the two middle values divided by two.

Minimum value

Maximum value

Q1 Q2 Q3

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Similarly, the positions of Q1 and Q3 can be calculated with 41

(n+1) and 43

(n+1) respectively. (Pretorius,

Potgieter, Ladewig, 2005)

Since Q1 = P25, Q2 = P50 and Q3 = P75, it is of importance to know something about the percentiles. The position of the pth percentile can be calculated in a similar way as the quartiles, using the formula

100p

(n+1). (Keller, 2008)

An index i = 100

p(n) can also be used to find the position of a percentile where p is the percentile and n the

number of observations. If i is an integer, the pth percentile is the average of the values in positions i and i+1, otherwise the integer just greater than i denotes the position of the pth percentile. (Anderson et al, 2007; Williams, et al, 2009)

Excel uses Lp = )100

–1(+100

pnp where Lp is the location of the pth percentile in its PERCENTILE()

function [14]. The syntax is =PERCENTILE(array, k). The syntax is =PERCENTILE(A1:A12, 0.25) where the observations are in cells A1 to A12 and the 25th percentile is to be calculated.

THE INTERQUARTILE RANGE AND OUTLIERS The interquartile range, IQR = Q3 – Q1, is used to determine which of the observations are outliers. The limits are 1.5 × IQR below Q1 and 1.5 × IQR above Q3.

Any observations outside these limits are considered to be outliers (values that appear to stand out from the rest). (Van der Lith, 2006)

Since not all formulas for the calculation of the quartiles give the same values, it may have an influence on which observations will be considered as outliers. Note that outliers can only be determined after calculation of the quartiles.

THE VALUES OF QUARTILES AND PERCENTILES

Example 1 The data set is: 0, 0, 5, 7, 8, 9, 12, 14, 22, 33 with n = 10. (Keller, 2008)

A. The formulas, 21

(n+1), 41

(n+1) and 43

(n+1) are used to get the positions of the quartiles.

The position of Q1 is 41

(n+1) = 41

(10+1) = 2.75 which is between observations 2 and 3, 43

of

the distance from observation 2 in the direction of observation 3. Its value is therefore 0 + 0.75(5 – 0) = 3.75.

The position of Q2 is 21

(n+1) = 21

(10+1) = 5.5. This means the median lies between observations 5

and 6. Q2 = 21

(8+9) = 8.5.

The position of Q3 is 43

(n+1) = 43

(10+1) = 8.25 which is between observations

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8 and 9. Its value is 14 + 0.25(22 – 14) = 16.

B. The formula, where the index i = 100

p(n), can be used to find the position of the percentile

required which is then also the position of the corresponding quartile. This formula is explained in the previous section. (Anderson, et al, 2007)

The position of Q1 = P25 is i = 100

p(n) = 0.25 × 10 = 2.5 which becomes 3 (the integer just

greater than 2.5) and the value of Q1 is 5.

The position of Q2 = P50 is i = 100

p(n) = 0.5 × 10 = 5 so that i is an integer and the value of Q2 is

21

(8+9) = 8.5.

The position of Q3 = P75 is i = 100

p(n) = 0.75 × 10 = 7.5 which becomes 8 (the integer just greater

than 7.5) and the value of Q3 is 14.

C. A third approach which can be followed, is to find the value of the median which is 8.5 (the sum of the two middle terms divided by 2) as shown in A. The five values to the left of 8.5 are 0, 0, 5, 7, 8 of which the middle value is 5 and is the value of Q1. The five values to the right of 8.5 are 9, 12, 14, 22, 33 of which the middle value is 14 and is the value of Q3. (Phillips, 2007)

If the number of terms to the left and right of the median is even, the sum of the two middle terms of that half divided is taken as Q1 and Q3. (Van der Lith, 2006)

D. Excel formulas

Excel has a QUARTILE() function which may produce results that don't always look obvious and may differ slightly from other calculations but are nevertheless valid and can be accepted. Attempting to exactly duplicate and check Excel's calculation in every circumstance is difficult. (MeadInKent, 2008)

The syntax is =QUARTILE(array, quart), for example =QUARTILE(A1:A12, 1) where the observations are in cells A1 to A12 and the first quartile is to be calculated. The observations (instead of the reference to the array of observations) can also be put directly in the function, for example =QUARTILE({0, 0, 5, 7, 8, 9, 12, 14, 22, 33}, 1). (BetterSolution, 2007)

If the QUARTILE() function of Excel is used the results are Q1 = 5.5, Q2 = 8.5 and Q3 = 13.5. The formula used for these three values is given below on the left in each line (Microsoft Help and Support, 2003). The calculation to find the value for Q1 is shown to the right of each step.

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Formulas Calculation

k1=(q/4)*(n-1)) + 1 k1 = (1/4) * (10 – 1) + 1 = 3.25

k=TRUNC((q/4*(n-1))+1)=TRUNC(k1) k = 3

f=(q/4*(n-1))-TRUNC(q/4*(n-1))

=(k1-1)-TRUNC(k1-1) f = 0.25

Output = a[k]+(f*(a[k+1]-a[k])) 5 + 0.25 * (7 – 5) = 5.5 = Q1

with 5 (a[k]) the 3rd value and 7 (a[k+1]) the 4th value in the data set.

This is the formula as it is given in the reference. Mathematically it can be simplified and made much easier to work with. The Box-and-Whiskerplot of the data set with no outliers is shown in Figure 2.

Figure 2: The Box-and-Whisker Plot for the given Data set As part of the output of the Box-and-Whiskerplot the following values appear in a table to its left: Q1 = 6, Q2 = 9 and Q3 = 18.

Example 2 The following datasets are from (Keller, Chapter 4).

A. Consider the dataset with 10 observations: 18, 28, 28, 29, 30, 33, 33, 35, 37, 38

Using the QUARTILE() functions the values of the quartiles are

Q1 = 28.25, Q2 = 31.5 and Q3 = 34.5.

From the Boxplot these values are

Q1 = 28, Q2 = 31.5 and Q3 = 35.5.

B. Consider the dataset with 100 observations: 9 (2), 11 (2), 15 (2), 16 (2), 17 (2), 18 (5), 19, 20, 21 (5), 22 (3), 23 (5), 24, 25 (4), 26, 27 (5), 28 (4), 29 (4), 30 (7), 31 (5), 32, 33 (5), 34 (3), 35 (3), 36 (4), 37 (2), 38 (4), 39 (2), 40 (3), 42 (2), 43 (2), 44 (5), 45, 46, 48.

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The numbers between brackets are the frequencies of those values.

Using the QUARTILE() functions the values of the quartiles are

Q1 = 22.75, Q2 = 30 and Q3 = 36.

From the Boxplot these values are

Q1 = 22.25, Q2 = 30 and Q3 = 36.

C. For 225 observations the values of the quartiles are

Using the QUARTILE() functions the values of the quartiles are

Q1 = 23, Q2 = 30 and Q3 = 35.75.

From the Boxplot these values are

Q1 = 23, Q2 = 30 and Q3 = 35.

The corresponding PERCENTILE() function gives the same values as the QUARTILE() function in all the cases.

CONCLUSION There are quite a number of formulas available to calculate the values of the quartiles (and percentiles) of which only a few were discussed here.

The important aspect is that it is good practice to be consistent and to use the same formulas as in the prescribed textbook. No formulas are prescribed. According to the Assessment Standard, quartiles are measures of dispersion.

Pocket calculators that can handle data processing may give other values than any of those calculated in the examples. In Excel itself the formulas are not always the same.

Learners can investigate the influence of small and large data sets on the values of the quartiles (and percentiles) found using different formulas. How do outliers influence these values?

REFERENCES Anderson, D.R., Sweeney, D.J., Williams, T.A., Freeman, J and Shoesmith, E. (2007). Statistics for Business and

Economics. Australia: Thomson.

BetterSolutions. (2004 - 2007). Retrieved March 31, 2009 from http://www.bettersolutions.com /excel/EDH113/LI811221611.htm

Goba, B. and Van der Lith, D. (2005). Mathematics Grade 10: Study and Master. Cape Town: Cambridge University Press.

Keller, G. (2008). Managerial Statistics. Australia: Cengage Learing.

Laridon, P., Jawurek, A., Kitto, A., Myburgh, M., Pike, M., Rhodes-Houghton, R., Scheiber, J. and Sigabi, M. (2007). Classroom Mathematics Grade 12. Sandton: Heinemann Publishers.

MathCS.org - Statistics. (2009). Retrieved March 31, 2009 from http://www.mathcs.org/statistics /course/index.html

MeadInKent (2008). Retrieved March 31, 2009 from http://www.meadinkent.co.uk/excel-quartiles.htm

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Microsoft Help and Support. (2003). Revision 3.1. Retrieved March 31, 2009 from http://support.micro soft.com/kb/214072

Mendenhall, W., Beaver, R.J. and Beaver, B.M. (2009). Introduction to Probability and Statistics. Thirteenth Edition. Australia: Cengage Learning.

Phillips, M.D. (2007). Mathematics 10: Textbook and Workbook. Cape Town:Allcopy Publishers.

Phillips, M.D., Basson, J. and Botha, C. (2007). Mathematics 11: Textbook and Workbook. Cape Town: Allcopy Publishers.

Pretorius, J., Potgieter, R and Ladewig, W. (2005). Mathematics Plus: Grade 10 Learner's Book. Cape Town: Oxford University Press.

Van der Lith, D. (2006). Mathematics Grade 11: Study and Master. Cape Town: Cambridge University Press.

Williams, T.A., Sweeney, D.J. and Anderson, D.R. (2009). Contemporary Business Statistics. Third Edition. Australia: Cengage Learning.

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MATHEMATICS FOR TEACHING MATTERS27

JILL ADLER

Marang Centre, University of the Witwatersrand

Department of Education and Professional Studies, King’s College London jill.adler@wits.ac.za

Introduction

This paper explores the notion of mathematics for teaching, and why it matters for the teaching and

learning of mathematics in general, and mathematics teacher education in particular. This exploration

builds on the seminal work of Lee Shulman. In the mid-1980s Shulman argued cogently for a shift in

understanding, in research in particular, of the professional knowledge base of teaching. He highlighted the

importance of content knowledge in and for teaching, criticizing research that examined teaching activity

without any concern for the content of that teaching. He described the various components of the

knowledge base for teaching, arguing that content knowledge for teaching included subject matter

knowledge (SMK), pedagogic content knowledge (PCK) and curriculum knowledge (Shulman, 1986;

1987). Shulman’s work set off a research agenda, with a great deal focused on mathematics. This paper

draws from the mathematical elaboration of Shulman’s work.

The profound insight of Shulman’s work was that being able to reason mathematically, for example, was

necessary but not sufficient for being able to teach others to reason mathematically. Being able to teach

mathematical reasoning involves recognizing mathematical reasoning in others’ discourse, and at various

curriculum levels, being able to design and adapt tasks towards purposes that support mathematical

reasoning, and critically working with or mediating the development of such in others. We could say the

same for being able to solve algebraic or numeric problems. This assertion is not news to any mathematics

teacher or mathematics teacher educator. Yet, in the particular case of mathematical reasoning, its actuality

in curricula texts, classroom practices and learner performances remains a challenge in many, if not most,

classrooms (Stacey & Vincent, 2009). We could say the same for learner performance in many areas of 27 Keynote to be presented at Australian Association of Mathematics Teachers, (AAMT) and Plenary at AMESA, July 09: AAMT_2009_JA

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mathematics, as well as algebra. Despite the longevity and consistency of elementary algebra in school

mathematics curricula worldwide, large numbers of learners experience difficulty with this powerful

symbolic system (Hodgen, Kuchemar, Brown & Coe, 2009).

In this paper I argue that strengthening our understanding of the mathematical work of teaching, what

some refer to as mathematics for teaching, is a critical dimension of enhancing its teaching and learning.

Mathematics for teaching matters, for all our learners, as do its implications for mathematics teacher

education. I will develop this argument through examples from school mathematics classrooms, together

with comment on developments in mathematics teacher education in South Africa. Ultimately, the

argument in this paper poses considerable challenges for mathematics teacher education.

Teaching and learning mathematics in South Africa

The past fifteen years of post-apartheid South Africa can be categorized as a time of rapid and intense

policy and curriculum change in the country. New mathematics curricula are being implemented in schools

across Grades 1-12, where there is greater emphasis than before on sense-making, problem-solving and

mathematical processes, including mathematical reasoning, as well as on new topics related to data

handling and financial mathematics. New education policy and curricula have strong equity goals, a

function of the deep and racialised inequality produced under Apartheid that effected teachers and learners

alike. New policies and qualifications have been introduced into teacher education, with goals for

improving the quality of teachers and teaching, and in the case of mathematics, addressing critical

shortages of qualified secondary mathematics teachers that persist, and indeed have deepened over time.

Tertiary institutions have responded, offering new degree and diploma programmes for upgrading teachers

in service, retraining into teaching, and preparing new teachers.

It is in moments of change that taken-for-granted practices are unsettled, in both inspiring and

disconcerting ways. Moments of change thus provide education researchers and practitioners challenging

opportunities for learning and reflection. Of pertinence to this paper is that the challenge of new curricula

in schools and thus new demands for learning and teaching, on top of redress, bring issues like the

selection of knowledges for teacher education development and support to the fore. Mathematics teacher

educators in all tertiary institutions have had the opportunity and challenge to make decisions on what

knowledge(s) to include and exclude in their programmes, and how these are to be taught/learnt. This has

meant deliberate attention to what mathematics, mathematics education and teaching knowledge teachers

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need to know and be able to use to teach well. This is no simple task: in South Africa, teaching well

encompasses the dual goals of equity and excellence. At the same time as strengthening the pool of

mathematics school leavers entering the mathematical sciences and related professions, high quality

teaching also entails catering for diverse learner populations, and inspiring school learners in a subject that

all too often has been alienating.

Hence the question: what selections from mathematics, mathematics education and teaching28 are needed

for greatest benefit to prospective and in-service teachers?

Shulman’s caterogies provide a starting point to answering this question. Others, particularly Ball and her

colleagues working on mathematical knowledge for teaching in Michigan USA, have argued that these

categories need elaboration; and that elaboration requires a deeper understanding of mathematics teaching,

and hence, teachers’ mathematical work. Ball, Thames & Phelps (2008) have elaborated Shulman’s

categories, distinguishing within subject matter knowledge, between Common and Specialised Content

Knowledge where the latter is what teachers in particular need to know and be able to use. Within

Pedagogic Content Knowledge, they distinguish knowledge of mathematics and students, and knowledge of

mathematics and teaching. These latter are knowledge of mathematics embedded in (and so integrated

with) tasks of teaching, that is, a set of practices teachers routinely engage in or need to engage in. In their

more recent work where they examine case studies of teaching, Hill, Blunk, Charalambos, Lewis, Phelps,

Sleep & Ball, D (2008) note that while their elaboration is robust, compelling and helpful, they

underestimated the significance of what Shulman identified as Curriculum Knowledge. What this reflects

is that all teaching always occurs in a context and set of practices, of which curricula discourses are a

critical element. Ball et al’s elaboration of Shulman’s categories, and particularly that they have been

derived from studies of mathematics classroom practice, are useful. They provide a framework with which

to think about and make selections into teacher education. They suggest, at immediate face value, that

mathematical content in teacher education and for teaching requires considerable extension beyond

knowing mathematics for oneself.

I go further to say we need to understand what and how such selections take shape in mathematics teacher

education practice. As in school, teacher education occurs in a context and set of practices, and is shaped

by these. In addition, as intimated above, in mathematics teacher education, mathematics as an ‘object’ or

28 Mathematics education here refers to the field of research and other texts related to mathematics curricula; teaching refers to the professional practice.

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‘focus’ of learning and becoming, is integrated with learning to teach. The research we have been doing in

the QUANTUM29 project in South Africa (that now has a small arm in the UK) has done most of its work

in teacher education as empirical site, complemented by studies of school mathematics classroom practice.

The goal is to understand the substance of opportunities to learn mathematics for teaching in teacher

education, and how this relates to the mathematical work teachers do in their school classrooms.

In this paper, I select two examples from studies of mathematics classrooms in South Africa. I use these to

illustrate what and how teachers use, or need to use mathematics in their practice: in other words, the

substance of their mathematical work. Similarities and differences across these examples, in turn,

illuminate the notion of mathematics for teaching, enabling a return to, and critical reflection, on

mathematics teacher education.

Designing and mediating productive mathematics tasks

• Example 1. Angle properties of a triangle

The episode discussed below is described in detail in Adler (2001)30, and takes place in a Grade 8

classroom. This teacher was particularly motivated by a participatory pedagogy, and developing her

learners’ broad mathematical proficiency (Kilpatrick, Swaffold & Flindell, 2001). She paid deliberate

attention to supporting her learners’ participation in mathematical discourse (Sfard, 2008), which in

practice involved having them learn to reason mathematically, and verbalise this. It is interesting to note

that the empirical data here dates back to the early 1990s and long before curriculum reform as it appears

today in South Africa was underway.

As part of a sequence of tasks related to properties of triangles, the teacher gave the activity in the box

below to her Grade 8 class. The questions I will address in relation to this task are: What mathematical

work is entailed in designing this kind of task, and then mediating it in a class of diverse learners.

If any of these is impossible, explain why, otherwise draw it.

29 For details on QUANTUM, see Adler & Davis (2006), Davis, Parker & Adler (2007); Adler & Huillet (2008), Adler (2009) 30 The focus of the study reported in Adler (2001) was on teaching and learning mathematics in multilingual classrooms. There I discuss in detail the learners’ languages, and how and why talking to learn worked in this class. I have since revisited this data, reflecting on the teachers’ mathematical work (see Adler, 2006).

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Draw a triangle with 3 acute angles. Draw a triangle with 1 obtuse angle. Draw a triangle with 2 obtuse angles. Draw a triangle with 1 reflex angle. Draw a triangle with 1 right angle.

The task itself evidences different elements of important mathematical work entailed in teaching learners

to reason mathematically. Firstly, this is not a ‘typical’ task on the properties of triangles. A more usual task

to be found in text books, particularly at the time of the research, would be to have learners recognize

(identify, categorise, name) different types of triangles, defined by various sized angles in the triangle.

What the teacher has done here is recast a ‘recognition’ task based on angle properties of triangles into a

‘reasoning’ task (reasoning about properties and so relationships). She has constructed the task so that

learners are required to reason in order to proceed. In so doing, she sets up conditions for producing and

supporting mathematical reasoning in the lesson and related proficiencies in her learners. Secondly, in

constructing the task so that learners need to respond whether or not particular angle combinations are

‘impossible’ in forming a triangle, the task demands proof-like justification, an argument or explanation

that, for impossibility, will hold in all cases. In this task, content (properties of triangles) and processes

(reasoning, justification, proof) are integrated. The question, of course, is what and how learners attend to

these components of the task, and how the teacher then mediates their thinking.

In preparation for this lesson and task, the teacher would have had to think about the mathematical

resources available to this classroom community with which they could construct a general answer (one

that holds in all cases). For example, if as was the case, learners had worked with angle sum in a triangle,

what else might come into play as learners go about this task? What is it about the triangle as a

mathematical object that the teacher needs to have considered and that she needs to be alert to as her

learners engage in reasoning about its properties?

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Before engaging further with the details of the teachers’ mathematical work, let us move to the actual

classroom, where students worked on their responses in pairs. The teacher moved between groups, probing

with questions like: Explain to me what you have drawn/written here? Are you sure? Will this always be

the case? She thus pushed learners to verbalise their thinking, as well as justify their solutions/proofs. I

foreground here, learners’ responses to the second item: Draw a triangle with two obtuse angles.

Interestingly, three different responses were evident.

• Some said ‘it is impossible to draw a triangle with two obtuse

angles, because you will get a quadrilateral’. And they drew:

• Others reasoned as follows: ‘an obtuse angle is more than 90 degrees and so two obtuse

angles give you more than 180 degrees, and so you won’t have a triangle because the

angles must add up to 180 degrees’.

• One learner (Joe) and his partner reasoned in this way: ‘If you start with an angle say of

89 degrees, and you stretch it, the other angles will shrink and so you won’t be able to get

another obtuse angle’. They drew:

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On this part of the task, as with the one on a reflex angle, there was a range of

learner responses - indicative of a further task-based teaching skill. It is designed

in a way that diverse learner responses are possible. In addition, the third,

unexpected, response produced much interest in the class, for the teacher, and

myself as researcher. The first two responses were common across learners and

more easily predicted by the teacher.

Having elicited these responses, it is the teacher’s task to mediate within and

across these responses, and enable her learners to reason whether each of these

responses is a general one, one that holds in all cases. The interesting interactions

that followed in the class are described and problematised in Adler (2001) and will

not be focused on here. In the many contexts where I have presented the study

and this particular episode, much discussion is generated both in relation to the

mathematical status of the responses, and their levels of generality, as well as

simultaneous arguments as to what can be expected of learners at a grade 8 level.

What constitutes a generalized answer at this level? Are all three responses

equally general? Is Joe’s response a generalized one? How does the teacher

value these three different responses, supporting and encouraging learners in their

thinking, and at the same time judging/evaluating their mathematical worth?

These are mathematical questions, and the kind of mathematical work this teacher

did on the spot as she worked to value and evaluate what the learners produced.

The point here is that this kind of mathematical work i.e. working to provoke,

recognize and then mediate notions of proof and different kinds of justification, is

critical to effective teaching of ‘big ideas’ (like proof) in mathematics. In Ball et al’s

terms, this work entails knowledge of mathematics and teaching (designing

productive tasks) and mathematics and students (and mediating between these

and learners’ mathematics).

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We need to still ask questions about subject matter knowledge, or content in this example, and specifically

questions about the angle properties of triangles? The insertion of a triangle with a reflex angle brought

this to the fore in very interesting ways. Some learners drew the following, as justification for why a

triangle with a reflex angle was possible; and so discussion of concavity, and interior and exterior angles.

The tasks of teaching illuminated in this example are: task design where content (angle properties of

triangles) and process (reasoning, justifying) are integrated; mediation of both mathematical content and

processes; and valuing and evaluating diverse learner productions. The mathematical entailments of this

work are extensive, and are illustrative of both subject matter knowledge and pedagogical content

knowledge. The teacher here reflects a deep understanding of mathematical proof, and in relation to a

specific mathematical object and its properties. To effectively mediate Joe’s response and the two above,

she would also need to ask/suggest productive ways forward for these learners, so that their notions of

proof and of the mathematical triangle are strengthened. Indeed, as learners in the class engaged with the

second triangle drawn above, their focus was that the answer was incorrect because there were three not

one reflect angles, and the teacher had a difficult time shifting them off this focus and onto the interior

angles.

In Adler (2001), I show that as the teacher mediated the three different responses to the triangle with two

obtuse angles, she worked explicitly to value each contribution and probe learner thinking. However, her

judgement of their relative mathematical worth was implicit. She accepted the first two responses above,

but pushed on Joe’s, with questions to Joe that implied she was not convinced of the generality of his

argument. I argued there that if teacher judgement of the varying mathematical worth of learner responses

offered is implicit, it is possible that only those learners who can themselves make such judgements, or

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who are able to read the implicit messages in the teachers’ actions, will appreciate and so have access to

what counts mathematically. Sociological theory and empirical research informs us that these kinds of

practices favour students with school cultural capital, and so can reproduce inequality. In Bernstein’s

(1996) terms, implicit practices will connect with learners who already understand the criteria for what are

most legitimate responses; and alienate or pass by those who are not ‘in’ the criteria. Typically these will

be already disadvantaged learners (Parker, 2009).

The example here is compelling in a number of ways, and provokes the question: Where, when and how

does a mathematics teacher learns to do this kind of work, and in ways that are of benefit to all learners?

Before attempting to answer this and so shift back into teacher education, we need to look at additional and

different examples of mathematical work of teaching.

• Example 2: Polygons and diagonals: or a version of the ‘mystic rose’

The second example is taken from a Grade 10 class (see Naidoo, 2008), where the teacher posed the

following task for learners to work on in groups: How many diagonals are there in a 700-sided polygon?

Here too, the teacher has designed or adapted a task and presented learners with an extended problem.

They have to find the number of diagonals in a 700-sided polygon, a sufficiently large number to require

generalising activity, and so mathematical reasoning. I pose the same questions here as for Example 1:

What mathematical work is entailed in designing this kind of task, and mediating it in a class of diverse

learners?

Many teachers will recognise the ‘mystic rose’ investigation in this problem. The mathematical object here

is a polygon and its properties related to diagonals. Yet the problem has been adapted from a well known

(perhaps not to the teacher) mathematical investigation of points on a circle and connecting lines - a

different, though related object. Here learners are not asked to investigate the relationship between the

number of points on a circle and connecting lines, but instead find an actual numerical solution to a

particular polygon, albeit with a large number of sides and so approaching a circle. I have discussed this

case in detail in Adler (2009), where I point out that unlike triangles and their properties, the general

polygon and its properties is not an explicit element of the secondary school curriculum. However, the

processes and mathematical reasoning required for learners to solve the problem are desired mathematical

processes in the new curriculum.

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My concern in this paper is not the merits of the problem and its adaptation in an isolated way, but rather

to reflect on the mathematical work of the teacher in presenting the problem, mediating learner progress,

valuing and evaluating their responses, and managing the integration of mathematical content and

mathematical processes as foci in the lesson. I present selections from the transcript of the dialogue in the

classroom to illuminate these four components of the teachers’ mathematical work.

The teacher (Tr), standing in the front of the class, explained what the class had to do.

Tr: I want you to take out a single page quickly. Single page and for the next five minutes no discussion. I want you

to think about how would you possibly solve this problem? (pointing to the projected problem: How many

diagonals are there in a 700-sided polygon?

After seven minutes, the Teacher calls the class’ attention. (Learners referred to as Lr A, B etc)

00:07 – 00:14 Tr: Ok! Guys, time’s up. Five minutes is over. Who of you thinks they solved the problem? One, two, three, four, five, six. Lr A: I just divided 700 by 2. Tr: You just divided 700 by 2. (Coughs). Lr A: Sir, one of the side’s have, like a corner. Yes … (inaudible), because of the diagonals. Therefore two of the sides makes like a corner. So I just divided by two … (Inaudible). Tr: So you just divide the 700 by 2. And what do you base that on? … [ ] Tr: Let’s hear somebody else opinion. LrB: Sir what I’ve done sir is … First 700 is too many sides to draw. So if there is four sides how will I do that sir? Then I figure that the four sides must be divided by two. Four divided by two equals two diagonals. So take 700, divide by two will give you the answer. So that’s the answer I got. Tr: So you say that, there’s too many sides to draw. If I can just hear you clearly; … that 700 sides are too many sides, too big a polygon to draw. Let me get it clear. So you took a smaller polygon of four sides and drew the diagonals in there. So how many diagonals you get? LrB: In a four sided shape sir, I got two. Tr: Two. So you deduced from that one example that you should divide the 700 by two as well? So you only went as far as a 4 sided shape? You didn’t test anything else. LrB: Yes, I don’t want to confuse myself. Tr: So you don’t want to confuse yourself. So you’re happy with that solution, having tested only one polygon? LrB: Inaudible response. Tr: Ok! You say that you have another solution. (Points to learner D) Let’s hear. [ ] LrA: I just think it’s right. … It makes sense. Tr: What about you LrD? You said you agree. LrD: He makes sense. …He proved it. … He used a square. Tr: He used a square? Are you convinced by using a square that he is right?

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LrE: But sir, here on my page I also did the same thing. I made a 6-sided shape and saw the same thing. Because a six thing has six corners and has three diagonals. LrA: So what about a 5 - sided shape? Then sir. Tr: What about a 5 - sided shape? You think it would have 5 corners? How many diagonals?

I have underlined the various contributions by learners, and italicised the teachers’ mediating comments

and questions. These highlight the learners’ reasoning and the teachers’ probing for further mathematical

justification.

At this point in the lesson, the teacher realises that some of the learners are confusing terms related to

polygons, as well as some of the properties of a general polygon and so deflects from the problem for a

while to examine with learners, various definitions (of a polygon, pentagon, a diagonal …). In other words,

at this point, the mathematical object in which the problem is embedded comes into focus. It is interesting

to note here that at no point was there reflection on the polygons in use in developing responses to the

problem. All were regular and concave. A little later in the lesson, another learner offers a third solution

strategy. The three different solution representations are summarized in the table below, illustrating the

varying orientations students adopted as they attempted to work towards the solution for a 700 sided-

polygon.

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As with example 1, we see four tasks of teaching demanded of the teacher: task design or adaptation;

mediation of learners’ productions; valuing and evaluating their different responses; and managing

mathematics content and processes opened up by the task.

The representations offered by learners give rise to interesting and challenging mathematical work for the

teacher. All responses are mathematically flawed, though the approaches of Learners B and C show

attempts at specializing and then generalizing (Mason, 2002). This is an appropriate mathematical practice,

however, the move from the special case to the general case in both responses is problematic, though in

different ways. Does the teacher (and if so how?) move into discussion about specializing and generalizing

in mathematics? Open ended investigations and problem-solving like the one above opens possibilities for

this kind of mathematical work in class. Such opportunities were not taken up here. Each was negated

empirically, and not elaborated more generally. Should they have been taken up by the teacher, and if so,

how?

Tasks of teaching and their mathematical entailments

In selecting and presenting two different examples from different secondary school classrooms in South

Africa, I have highlighted four inter-related tasks of teaching, each of which entail considerable

mathematical skill and understanding over and above (or underpinning) the teaching moves that will ensue.

The four tasks (two discussed in each of the bulleted sections below) further illustrate categories of

professional knowledge developed by Shulman and elaborated by Ball et al in mathematics.

• Designing, adapting or selecting tasks, and managing processes and objects

In the first example, the process of mathematical reasoning was in focus, as was the triangle and its angle

properties. I will call this an object-and-process-focused task. Angle properties of triangles are the focus of

reasoning activity. Learners engage with and consolidate knowledge of these properties through reasoning

activity, and vice versa. Here the integration of learning content and process appears to keep them both in

focus, and thus provide opportunities for learning both. Example 2 is also focused on mathematical

reasoning. It is a process-focused task, having been adapted (what I would refer to as recontexualised)

from an investigation and reframed as a problem with a solution. The mathematical object of the activity,

the polygon, is back-grounded. At a few points in the lessons, it comes into focus, when understanding

polygons and their properties is required for learners to make progress with the problem: some learners

make assumptions about what counts as a diagonal, perhaps a function of assuming regularity (and so find

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3 diagonals in a hexagon); some generalize from one specific case (a four-sided figure); while others over-

generalise multiplicative processes from number, to polygon properties. The intricate relationship between

mathematical objects and processes has been an area of extensive empirical research in the field of

mathematics education. It appears from studying two examples of teaching that selecting, adapting or

designing tasks to optimize teaching and learning entails an understanding of mathematical objects and

processes and how these interact within different kinds of tasks. The teaching of mathematical content and

mathematical processes is very much in focus today. Reform curricula in many countries promote the

appreciation of various mathematical objects, their properties and structure, conventions (how these are

used and operated on in mathematical practice), as well as how and what counts as a mathematical

argument, and the mathematical processes that support such. In Example 1, we see opportunity for

developing reasoning skills, and understanding of proof at the same time as consolidating knowledge about

triangles. In Example 2, it is not apparent whether and how either proof or reasoning will flourish through

this example and its mediation. The relevance of the mathematical object in use is unclear. Thus the

question: Do we need a mathematics for teaching curriculum that includes task interpretation, analysis

and design with specific attention to intended mathematical objects and processes and their interaction?

In other words, should a mathematics for teaching curriculum include attention to the mediation of

mathematical content and processes as these unfold in and through engagement with varying tasks? If so,

is this to be part of the mathematics curriculum, or part of the teaching curriculum? And hidden in this last

question is a question of who teaches these components of the curriculum in teacher education? What

competences and expertise would best support this teaching?

• Valuing and evaluating with diverse learner productions.

Diverse learner productions are particularly evident in Examples 1 and 2, given their more open or

extended nature. Thus, in each example, the teacher dealt with responses from learners that they predicted,

and then those that were unexpected. In Example 1, the teacher needed to consider the mathematical

validity of Joe’s argument for the impossibility of a triangle with two obtuse angles, and then how to

encourage him to think about this himself, and convince others in the class. Similarly, we can ask in

Example 2: what might be the most productive question to ask Learner C and so challenge the reasoning

that since 700 can be factored into 7X100, finding the diagonals in a 7-sided figure is the route to the

solution to a 700-sided figure? Such questioning in teaching needs to be mathematically informed.

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Together these examples illuminate what and how teachers need to exercise mathematical judgement as

they engage with what learners do (or don’t so), particularly if teachers are building a pedagogical

approach and classroom environment that encourages mathematical practices where error, and partial

meanings are understood as fundamental to learning mathematics. In earlier work I referred to this as a

teaching dilemma, where managing both the valuing of learner participation and evaluation of the

mathematical worth of their responses was important (Adler, 2001); and illuminated the equity concerns if

and when evaluation of diverse responses, i.e. judgements as to which are mathematically more robust or

worthwhile, are left implicit.

And so, a further question needs to be asked of the curriculum in mathematics teacher education, and the

notion of mathematics for teaching. Learner errors and misconceptions in mathematics is probably the

most developed of research areas in mathematics education. We know a great deal about persistent errors

and misconceptions that are apparent in learners’ mathematical productions across contexts. These provide

crucial insight into the diverse responses that can be anticipated from learners. Yet, as Stacey (2004)

argues, the development of this research into contents for teacher education has been slow. We have shown

elsewhere that the importance of learner mathematical thinking in mathematics teacher education is

evident in varying programmes in South Africa (see Davis, Adler & Parker, 2007; Adler, forthcoming;

Parker, 2009). Yet there are significant differences in the ways this is included into such programmes, and

so with potential effects on who is offered what in their teacher education. How should a mathematics for

teaching curriculum then include such?

Mathematics for teaching matters

I have argued that mathematics for teaching matters for teaching and for opportunities to learn. I have

suggested that what matters are task design and mediation, and attention to content/objects and processes

within these. I have played on the word ‘matters’ by suggesting firstly that these are the ‘matter’ or the

content of mathematics for teaching; and at the same time that they matter (have significance) in and for

teacher education. Secondly, I have suggested that there are equity issues at stake.

I now return to the context of teacher education in South Africa where various innovative teacher

education programmes are grappling with a curriculum for mathematics teachers that appreciates the

complexity of professional knowledge for teaching and its critical content or subject basis. I will focus here

on what we have observed as objects of attention (and so meanings) shift from classrooms to teacher

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education and back again, - observations that support the argument in this paper, that we need to embrace

our deeper understanding of the complexities of teaching and so our task in teacher education.

In more activity-based, participative or discursively rich classroom mathematics practice, there is increased

attention to mathematical processes as critical to developing mathematical proficiency and inducting

learners into a breadth of mathematical practices. The examples in this paper illustrate how mathematical

processes are always related to or based on some mathematical object. If the latter is not well understood,

in the first instance by the teacher, in ways that enable her to notice when it goes out of focus or is

completely missed by students, then their reasoning is likely to be flawed or mathematically empty. This

phenomenon is apparent in classrooms in South Africa, and moreso in historically disadvantaged settings,

thus perpetuating rather then attacking inequality. Mathematical objects and processes and their interaction

are the central ‘matter’ of mathematics for teaching. The shift in new curricula to mathematical processes

creates conditions for diminished attention to mathematical objects. Attention to objects and processes

need to be embraced in the context of teaching if access to mathematics is to be possible for all learners.

And herein lies considerable challenge. In each of the two examples in this paper, a mathematical object

was embedded in a task that worked varyingly to support mathematical reasoning processes. What the

teacher in each case faced was different learner productions as responses to the task. These become the

focus of the teachers’ work, requiring integrated and professional based knowledge of mathematics,

teaching tasks and learner thinking. So what then, is or comes into focus in teacher education, and not only

into teacher education, but into school curricula? What we have observed (and I have seen elements of this

in elementary mathematics teacher education in the UK), is that learner thinking and the diversity of their

responses become the focus, with the mathematical objects and tasks that give rise to these, out of focus.

What one might see in the case of the triangle properties is a task that requires learners to produce three

different arguments for why a triangle cannot have two obtuse angles. And there is a subtle but impacting

shift of attention: from how to mediate diverse responses, to multiple answers/solutions being the required

competence in learners; from teachers’ learning to appreciate diverse learner productions and their relative

mathematical worth, and more generally, multiple representations, and how to enable learners to move

flexibly between these, to these being the actual content of teaching. Simply, there are curricula texts that

now require learners to produce multiple solutions to a problem. I leave this somewhat provocative

assertion for discussion and further debate.

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In conclusion, there is an assumption at work throughout this paper: that teacher education is crucial to

quality teaching. In South Africa, all pre-service and formal in-service teacher education has become the

responsibility of universities. Tensions between theory and practice abound. I hope in this paper to have

provided examples that illuminate the mathematical work of teaching, and through these opened up

challenges for mathematics teacher education. Mathematics for teaching, and its place in mathematics

teacher education, particularly in less resourced contexts, matters profoundly. There is much to do.

ACKNOWLEDGEMENTS This paper forms part of the QUANTUM research project on Mathematics for Teaching, directed by Jill Adler, at the University of the Witwatersrand. Dr Zain Davis from the University of Cape Town is a co-investigator and central to the theoretical and methodological work in QUANTUM. The elaboration into classroom teaching was enabled by the work of Masters students at the University of the Witwatersrand. This material is based upon work supported by the National Research Foundation under Grant number FA2006031800003. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Research Foundation.

References Adler, J. (2001). Teaching Mathematics in Multilingual Classrooms. Dordrecht: Kluwer. Adler, J (2006) Mathematics teacher education in post-apartheid South Africa: A focus on the

mathematical work of teaching across contexts. In Borba, M. (Ed.) Trends in Mathematics Education, Brazil (in Portuguese). (pp. 45-64). São: PauloAutêntí.

Adler, J. (2009) A methodology for studying mathematics for teaching. Recherches en Didactique des Mathématiques.

Adler, J. (forthcoming) Look at me, look at yourself, look at the practice: Modelling teaching in mathematics teacher education, and the constitution of mathematics for teaching. In Ruthven, K and Rowland, T. (Eds.) Mathematical Knowledge in Teaching. Routledge.

Adler, J. and Davis, Z. (2006), Opening another black box: Researching mathematics for teaching in mathematics teacher education. Journal for Research in Mathematics Education. 37 (4), 270-296.

Adler, J. and Huillet, D. (2008), The social production of mathematics for teaching. In T. Wood S. & P. Sullivan (Vol. Eds) International handbook of mathematics teacher education: Vol.1. Sullivan, P., (Ed.), Knowledge and beliefs in mathematics teaching and teaching development. Rotterdam, the Netherlands: Sense Publishers. (pp.195-222). Rotterdam: Sense.

Ball, D. L., Thames, M. H. & Phelps, G. (2008), Content Knowledge for Teaching: What Makes It Special? Journal of Teacher Education, 59, 389-407

Bernstein, B. (1996), Pedagogy, Symbolic Control and Identity: Theory, Research and Critique. London: Taylor and Francis.

Davis, Z., Adler, J. & Parker, D. (2007), Identification with images of the teacher and teaching in formalized in-service mathematics teacher education and the constitution of mathematics for teaching. Journal of Education. 42, 33-60.

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Hill, H., Blunk, M., Charalambos, Y., Lewis, J., Phelps, G., Sleep, L. & Ball, D. (2008), Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction. 26, 430-511.

Hodgen, J., Kuchemar, D.,Brown, M. and Coe, R. (2009) Children’s understandings of algebra 30 years on. In Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 28(3) November 2008.

Kilpatrick, J., Swafford, J. and Findell, B. (Eds.) (2001). Adding it up: Helping children learn mathematics. Washington: National Academy Press.

Mason, J. (2002) Generalisation and algebra: exploiting children’s powers. In Haggerty, L. (Ed) Aspects of Teaching Secondary Mathematics: Perspectives on practice. Pp. 105-120. London: Routledge Falmer.

Naidoo, S. (2007), Mathematical knowledge for teaching geometry to Grade 10 learners. School of Education (Johannesburg, The University of the Witwatersrand).

Parker (2009) The specialisation of pedagogic identities in initial mathematics teacher education in post-apartheid South Africa. Unpublished PhD Thesis. Johannesburg: University of the Witwatersrand.

Sfard, A. (2008) Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, UK: Cambridge University Press.

Shulman, L.: 1986, ‘Those who understand: knowledge growth in teaching’. Educational Researcher, 15(2) 4-14.

Shulman, L, (1987). ‘Knowledge and Teaching: Foundation of the New Reform’, Harvard Educational Review, 57(1), 1-22.

Stacey, K (2004) International Perspectives on the Nature of Mathematical Knowledge for Secondary Teaching: Progress and Dilemmas. In M. Johnsen Hoines & A. Berit Fuglestad (Eds.) The proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education – PME 28. 1(1) (pp. 167 – 196). Bergen, Norway

Stacey, K. and Vincent, J. (2009) Modes of reasoning in explanations in Australian eigth-grade mathematics textbooks. Educational Studies in Mathematics. Published online 25 March 2009. Springer.

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THINKING OUTSIDE THE BOX

Johan Meyer

Department of Mathematics & Applied Mathematics University of the Free State

Bloemfontein

It is evident, at least as far as mathematics is concerned, that by far the most of the students that enter our universities are coached to solve any one of a finite number of types of problem. Apparently this coaching usually means working through enough examples of these problem-types prior to the examination. Being fully equipped with an arsenal of methods, recipes, rhymes, calculators, formula-sheets(!), etc., they are as ready as can be to give back what they have absorbed into their brains. This is, to some extent, something that a copy-machine does.

In few cases are learners granted the opportunity to actually think about what they are doing, and to ask intelligent questions - an opportunity to create and enrich an inquisitive mind.

The test is: can you (or even better: do you want to) solve a type of problem that you have never seen before? Are you one of those who are not only fluent in doing mathematics, but also have the urge to think around the problem? If you pass this test, you should be one of those who are interested in Mathematics Olympiads, i.e., those who are necessarily of the think-and-do kind, and are willing to accept challenges.

In this talk we put emphasis on the following:

• What you miss if your definition of mathematics is “a manipulation of numbers and

symbols”. • What you miss if you meet a mathematician, and all you can think of, is “he/she must

be very good at doing sums in his/her head”. • What you miss when you (and your calculator) did get the correct answer to a problem,

but you do not realize that the problem you have just solved could have created other problems of a better type, and open new avenues of investigation.

• What you miss if your primary goal in mathematics is to achieve high marks in the examination.

• What you miss if you do not have access to enough tools and/or experience to solve a problem.

• What you miss if you fail to see the beauty behind what you are doing when you work on a mathematical problem.

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WHEN LANGUAGE IS TRANSPARENT: SUPPORTING MATHEMATICS LEARNING MULTILINGUAL CONTEXTS

Mamokgethi Setati Bheki Duma University of South Africa Inqgayizivele High School

This paper draws on a wider study conducted in Grade 11 classrooms in South Africa to explore what happens when language is transparent in multilingual mathematics classrooms. Based on an analysis of data collected through lesson observations in a Grade 11 class, we argue the use of language as a transparent resource in multilingual classrooms. Through this learners can gain access to mathematical knowledge while gaining fluency in English, which is presently seen by many parents, teachers and learners as a necessary condition for gaining access to social goods such as higher education and employment. INTRODUCTION Teaching mathematics in South Africa to learners who learn in a language that is not their home language is complex. Research shows that teachers and learners in multilingual mathematics classrooms in South Africa prefer that English be used as the LoLT (Setati, 2008). How can the learners’ languages be drawn on to support mathematics learning? In this paper we argue for the deliberate, proactive and strategic use of the learners’ home languages as a transparent resource in the teaching and learning of mathematics. We begin the paper with challenging three prevalent dichotomies in research on teaching and learning mathematics in multilingual classrooms. First, is the dichotomy between using English as LoLT as opposed to using the learners’ home language(s) as LoLT. Second, is the dichotomy about drawing on socio-political perspectives when analysing interactions in multilingual mathematics classrooms as opposed to drawing on cognitive perspectives. The third dichotomy is about gaining access to mathematical knowledge as opposed to access to English. We then discuss the theory that informed the analysis we present in the paper. MULTILINGUALISM IN MATHEMATICS EDUCATION Debates around language and learning in South Africa tend to create a dichotomy between learning in English and learning in the home languages. They create an impression that the use of the learners’ home languages for teaching and learning must necessarily exclude and be in opposition to English, and the use of English must necessarily exclude the learners’ home languages. In an article published in the Science Africa magazine, Sarah Howie of the University of Pretoria in South Africa, argued that the most significant factor in learning mathematics is not whether the learners are rich or poor. It’s whether they are fluent in English. She insisted, “Let’s stop sitting on the fence and make a hard decision. We must either shore up the mother tongue teaching of maths and sciences, or switch

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completely to English if we want to succeed." (Science in Africa, 2003). This she said drawing on her analysis of South Africa’s poor performance in the Third International Mathematics and Science Study of 1995 (see also Howie 2003, 2004). Our argument in this paper is that in a multilingual country such as South Africa the choices are not as simplistic as Howie suggests. Our argument is informed by a holistic view of multilingual learners, which is different from Howie’s monolingual view. In our view multilingual learners have a unique and specific language configuration and therefore they should not be considered as the sum of two or more complete or incomplete monolinguals. The use of the learners’ home languages as a transparent resource that we are exploring in this paper is informed by this holistic view of multilingual learners. We accept that the idea of drawing on the learners’ home languages during teaching is not necessarily new. The use of code-switching as a learning and teaching resource in bilingual and multilingual mathematics classrooms has been the focus of research in the recent past (e.g. Adler 2001; Barwell 2005; Khisty 1995; Moschkovich 1999, 2002; Parvanehnezhad & Clarkson 2008; Setati, 2005). These studies have argued for the use of the learners’ home languages in teaching and learning mathematics, as a support needed while learners continue to develop proficiency in the LoLT at the same time as learning mathematics. All of these studies seem to be in agreement that to facilitate multilingual learners’ participation and success in mathematics teachers should recognise their home languages as legitimate languages of mathematical communication. The practical manifestation of the use of the learners’ home languages in these studies is through code-switching, mainly to provide explanation to learners in their home languages. In all of these studies code-switching is presented as spontaneous and reactive, the learners’ home languages are only used in oral communication and never in written texts. What this paper argues for is the transparent use of the learners’ languages in a deliberate, strategic and proactive manner. This is in recognition of the fact that learners want access to English and thus while we draw on their home languages and foreground the quality of the mathematics tasks used during teaching, we also ensure that English is still available to them and they can continue to develop fluency in it. Past research on multilingualism in mathematics education informed by a cognitive perspective (e.g. Dawe, 1983; Clarkson, 1991) present an implicit argument in support of the maintenance of learners’ home languages, and of the potential benefits of learners using their home language(s) as a resource in their mathematics learning. Multilingualism is becoming the norm in many classrooms all over the world, hence the need to consider treating the multilingual learner not only as the norm but also to view his or her facility across languages as a resource rather than a problem (Baker, 1993). Through our work in this study we have come to recognise that separating cognitive matters from the socio-political issues relating to language and power when exploring the use of language(s) for

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teaching and learning mathematics in multilingual classrooms is not productive. While we accept that cognitively oriented research does not deal with the socio-political issues relating to the context in which teaching and learning takes place, we acknowledge that it is useful in helping us attend to issues relating to the quality of the mathematics and its teaching and learning in multilingual classrooms. In this study we are thus moving against dichotomies, not only of language choices but also of theoretical perspectives. THEORETICAL UNDERPINNINGS This study is broadly informed by an understanding of language as “a transparent resource” (Lave and Wenger, 1991). While the notion of transparency as used by Lave and Wenger is not usually applied to language as a resource nor to learning in school, it is illuminating of language use in multilingual classrooms (see Adler, 2001). Lave and Wenger (1991) argue that access to a practice relates to the dual visibility and invisibility of its resources. Invisibility is in the form of unproblematic interpretation and integration into activity, and visibility is in the form of extended access to information. This is not a dichotomous distinction, since these two crucial characteristics are in a complex interplay, their relation being one of both conflict and synergy (Lave and Wenger, 1991: 103). For language in the classroom to be useful it must be both visible and invisible: visible so that it is clearly seen and understood by all; and invisible in that when interacting with written texts and discussing mathematics, this use of language should not distract the learners’ attention from the mathematical task under discussion but facilitate their mathematics learning. This idea is similar to the use of technology in mathematics learning. The technology needs to be visible so that the learners can notice and use it. However it also needs to be simultaneously invisible so that the learners’ attention is not focussed on the technology but on the mathematics problem that they are trying to solve. As Lave and Wenger argue the idea of the visibility and invisibility of a resource is not a dichotomous distinction, it is not about whether to focus on language or mathematics, it is about recognising that the two are intertwined and are constantly in complex interplay. Lave and Wenger’s concept of transparency was useful in conceptualising language use in multilingual mathematics classrooms. Multilingual mathematics classrooms are characterised by complex multiple teaching demands: the learners’ limited proficiency in the language of learning and teaching (English); the challenge to develop the learners’ mathematical proficiency as well as the presence of multiple languages. The strategy we are exploring is guided by two main principles, which are informed by the theoretical assumptions elaborated in the discussion above. First is the deliberate use of the learners’ home languages. We emphasise the word deliberate because with this strategy the use of the learners’ home languages is deliberate, proactive and strategic and not spontaneous and reactive as it happens with code-switching. Second, is that through the selection of real world interesting and challenging mathematical tasks, learners would develop a different

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orientation towards mathematics than they had and would be more motivated to study and use it (Gutstein, 2003). Many learners in multilingual classrooms in South Africa have what Gutstein (2003: 46) describes as “the typical and well documented disposition with which most mathematics teachers are familiar – mathematics as a rote-learned, decontextualised series of rules and procedures to memorise, regurgitate and not understand”. In this study we selected high cognitive demand tasks (Stein, Smith, Henningsen and Silver, 2000), that present real world problems that the learners can find interesting and useful to engage with. THE STUDY The study presented in this paper focuses on data collected in Bheki’s classroom as part of a wider study on exploring relevant pedagogies for teaching and learning mathematics in multilingual classrooms. In this study Bheki was observed teaching a Grade 11 class in a secondary school in Thembisa, East of Johannesburg. There were 46 learners who were able to communicate in at least four languages and they had the following home languages: Sepedi, Sesotho, IsiZulu and Isixhosa. They were learning English as a subject at second language level as well as their respective home languages as subject at first language level. Data was collected through video recorded lesson observations and individual learner reflective interviews. During the lessons learners were organised into language groups and they were given tasks in two language versions (English and their home language). In this paper we focus only on lesson observation data to explore what happens when language is transparent. WHEN LANGUAGE WAS VISIBLE AND SIMULTANEOUSLY INVISIBLE In our analysis we found that when language was transparent learners’ interactions were conceptual –focused not only on what the solution is but also why it is correct. In this section we draw on Lesson 2 when the learners engaged with the definition of linear programming, which the teacher introduced in Lesson 1. We do this to illustrate how language typically functioned as a transparent resource during interactions. Below is the task, which learners were working on. It was translated into the four home languages of the learners in the class.

Mandla’s cinema hall can accommodate at most 150 people for one show. a) Rewrite the sentence above without using the words “at most”. b) If there were 39 people who bought tickets for the first show, will the show go on? c) Peter argued that if there are 39 people with tickets then Mandla should not allow the show to go

on because he will make a loss. Do you agree? Why do you agree? d) What expenses do you think Mandla incurs for one show? e) Use restrictions to modify the statement above in order to make sure that Mandla does not make

a loss. f) If Mary was number 151 in the queue to buy a ticket for the show, will they accommodate her in

the show? Explain your answer.

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This task generated a lot of interaction in groups and also during the whole class discussion because the mathematical solution that the learners thought to be correct was not consistent with what they had understood Linear programming to be about. During Lesson 1, Bheki defined Linear programming as the maximization or minimization of a specific performance index, usually of an economic nature like profit, subject to a set of linear constraints. He further indicated that for this exercise to qualify as linear programming the performance index should also be linear. Throughout Lesson 1, Bheki emphasised the fact that linear programming is thus about minimising the losses and maximising the gains. During Lesson 2 learners drew on this information while working on the above task. In the extract below the learners are engaging in a whole class discussion with the teacher on the answer to question b).

242 Maseko: Sithe [We said] the show will not go on because he will not benefit profit from thirty nine people

253 Bheki: Akesizwe isipedi kucala, yes [Let us give a chance to Sepedi group first]. 254 Errol: …Hundred and fifty kuya phansi, akanawumela Hundred and fifty kuya phezulu

[150 and below, cannot wait for 150 or above]. 255 Bheki: Okay, okay, alright okey akebajusifaye ipoint labo okay [Okay, let them justify

their argument]. 256 Mkhonza: Mawudefina iloku LP ithini idefinition ye LP. [When defining LP, what is the

definition of LP?] 260 Errol: Ithi [It says] maximising the profit and minimising the loss, so nami ngiz…[I will

also…] 262 Mkhonza: Wena nawubhekile lomuntu lo nakungena abantu abawu thirty-nine kumele

ucabange ukuthi le I-one show. Nakungena abantu abayi-thirty-nine akusiyo nehhafo ka hundred and fifty. Mara iprofit izophuma kanjani nakungasiyo nehhafu ka hundred and fifty. Akusiyo hehhafu, nehhafu ka hundred and fifty. [When you look at this person, when 39 people get into the cinema hall, how is he going to make profit from 39 people since 39 is not even half of 150, it is not even a quarter of 150?]

What is emerging in the extract above is the fact that according to the problem Mandla’s cinemas can accommodate a maximum of 150 and so what this means mathematically is that the number of people in Mandla’s cinema cannot go beyond 150 but it is allowed to be lower than 150. That is if x is the number of people in Mandla’s cinema then 1 ≤ x ≥ 150. This is the explanation that Maseko and Errol are using for their argument. Mkhonza on the other hand is drawing his argument from the definition of linear programming which Bheki emphasised in Lesson 1. This argument continued through the lesson as Bheki found out what other groups were thinking and also challenging them to think about each other’s arguments and not just to focus on why their argument is correct.

276 Sipho: Ibinessman inama risk [A business man has risks] if you are in a business you are going to take the risk. Even when the business… may be kune [there is])…(learners saying yes)

282 Xhakaza: Abantu nakudlala ibhola abantu baya estimate itulu zining. Kunga fika abantu abayi2 ibhola iyachubeka. [When there is a soccer match in a stadium the game continues even when they are two spectators]

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What is evident in the discussion above is the fact that learners are trying to reconcile what they understand to be a mathematically correct answer, their understanding of what linear programming is about and when to apply that and also their understanding of how things work in the everyday world. As Xhakaza explains in utterance 282 above, the situation in the problem is similar to what happens in soccer matches held in big stadiums, where even if not many people came to the stadium the game would still go on as scheduled. Throughout the discussions the learners did not refer to language either as being a resource or a problem. They went on with the mathematics problem as if the language is not there, hence our argument that language functioned as a transparent resource. CONCLUSION While language is a resource that can help advance mathematics learning, it can also be a stumbling block for successful learning depending on how it is used. The major challenge in multilingual contexts such as South Africa is the fact that while the power of English is unavoidable, many learners do not have the level of fluency that enables them to engage in mathematical tasks set in English. In this paper we argued for the use of learners’ languages as a transparent resource. This argument recognises the political nature of languages. The fact that while the learners’ home languages are drawn on they are not presented as being in opposition to English rather as working together with English to make mathematics more accessible to the learners. The question that emerges from this study is whether language is only a resource when it is transparent. REFERENCES Adler, J. (2001). Teaching Mathematics in Multilingual Classrooms. Dordrecht: Kluwer Academic

Publishers. Baker, C. (1993). Foundations of Bilingual Education and Bilingualism. Clevedon: Multilingual

Matters Ltd.

Barwell, R. (2005). Language in the mathematics classroom. Language and Education, 19(2), 97-102.

Clarkson, P. C. (1991). Bilingualism and Mathematics Learning. Geelong: Deakin University Press.

Dawe, L. (1983). Bilingualism and mathematical reasoning in English as a second language. Educational Studies in Mathematics. 14(1), 325 - 353.

Grosjean, F. (1985). The bilingual as a competent but specific speaker-hearer. Journal of Multilingual and Multicultural Development, 6(6), 467 - 477.

Gutstein, E. (2003). Teaching and learning mathematics for social justice in an urban, Latino school. Journal for Research in Mathematics Education, 34, 37–73.

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Howie, S. J. (2003). Language and other background factors affecting secondary pupils’ performance in Mathematics in South Africa. African Journal of Research in SMT Education, 7, 1 - 20

Howie, S. J. (2004). A national assessment in mathematics within international comparative assessment. Perspectives in Education 22(2).

Khisty, L. L. (1995). Making inequality: Issues of language and meaning in mathematics teaching with Hispanic students. In W. G. Secada, E. Fennema, & L. B. Abajian (Eds.), New Directions for equity in mathematics education. . Cambridge: Cambridge University Press.

Lave, J. and Wenger, E. (1991). Situated Learning: Legitimate Peripheral Participation. Cambridge; Cambridge University Press.

Moschkovich, J. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11–19.

Moschkovich, J. (2002). A situated and sociocultural perspective on bilingual mathematics learners. Mathematical Thinking and Learning, 4, 189–212.

Parvanehnezhad & Clarkson (2008). Iranian Bilingual Students Reported Use of Language Switching when Doing Mathematics. Mathematics Education Research Journal, 20(1); 52 – 81.

Setati, M. (2005). Teaching mathematics in a primary multilingual classroom. Journal for Research in Mathematics Education. 36(5) pp. 447 – 466.

Setati, M. (2008) Access to mathematics versus access to the language of power: the struggle in multilingual mathematics classrooms. South African Journal of Education. 28(1) 103 – 116.

Stein, M., Smith, M., Henningsen, M. and Silver, E. (2000) Implementing standards-based mathematics instruction: A casebook for professional development. NCTM. Reston.

Science Africa Magazine (2003). Why don't kids learn maths and science successfully? Available at http://www.scienceinafrica.co.za/2003/june/maths.htm, accessed 10 April 2007.

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TOWARDS A MORE COMPREHENSIVE “KNOWLEDGE PACKAGE” FOR TEACHING PROOF

Andreas J. Stylianides

University of Cambridge, U.K.

The concept of proof is central to meaningful learning of mathematics, but is hard for students to learn. What knowledge might allow teachers to effectively teach proof to their students? Existing research has constructed a significant, albeit incomplete, “knowledge package” for teaching proof. I summarize major elements of this package and contribute to its further development by focusing on elements of knowledge that have not received much attention and relate to teachers’ ability to implement successfully in their classrooms instructional interventions that address major student misconceptions. I exemplify this expanded knowledge package in the context of a research-based instructional intervention that aimed to help students begin to overcome the common misconception that empirical arguments are proofs.

EMPIRICAL ARGUMENTS vs PROOFS

Consider the generalization: “The sum of any two odd numbers is an even number.” What argument would your students offer for it? Would that be a proof?

An overwhelming body of research shows that students of all levels of schooling, including high-attaining secondary students, “prove” mathematical generalizations such as the above by using empirical arguments (e.g., Coe & Ruthven, 1994; Healy & Hoyles, 2000; Senk, 1985). By empirical arguments I mean arguments that purport to show the truth of a generalization by validating the generalization in a proper subset of all possible cases (see Balacheff, 1988; Sowder & Harel, 1998; Stylianides, 2008). Empirical arguments are invalid, because they cannot exclude the possibility of the existence of a counterexample to the generalization. Here are two examples of empirical arguments for the above generalization:

Empirical argument 1: naïve empiricism

I tried many different pairs of odd numbers and their sum was always an even number: 7 + 9 = 16, 15 + 21 = 36, 25 + 27 = 52, etc. So the sum of any two odd numbers is an even number.

Empirical argument 2: crucial experiment

I checked different kinds of pairs of odd numbers: some with small odd numbers (e.g., 1 + 9 = 10), some with big odd numbers (e.g., 213 + 399 = 612), some with the same odd numbers (e.g., 25 + 25 = 50), and some with prime odd numbers (e.g., 17 + 31 = 48). No pair gave me a

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counterexample – the sum was always an even number. So the sum of any two odd numbers is an even number.

Even though both arguments are invalid, the second argument can be considered more advanced than the first, because the search of possible counterexamples in the second involves a strategic selection of cases in contrast to the random (or convenience) sampling of cases in the first. Balacheff (1998) used the terms naïve empiricism and crucial experiment to describe the special kinds of empirical arguments represented by the first and second examples, respectively.

The fact that a generalization is true in some cases does not guarantee, and thus does not prove, that the generalization is true for all possible cases. This is the main limitation of any kind of empirical argument, but many students do not understand it. What would, then, be a proof for the generalization? Figure 1 shows three possible proofs for the generalization on the set of whole numbers.

Figure 1: Three possible proofs (on the set of whole numbers) for “odd + odd = even” (Stylianides & Stylianides, 2008, p. 108).

Notice the correspondences among the three arguments: they are saying the “same thing” using different representations. Notice also how each argument can be used to not only help someone understand why the generalization is true, but also to convince someone that the generalization is true for all cases without requiring that person to make a leap of faith. A proof’s potential to promote understanding (explanation) and conviction (justification) accounts in part for why proof is considered to be fundamental to meaningful learning in mathematics (see, e.g., Ball & Bass, 2000, 2003; Hanna, 2000; Harel & Sowder, 2007; Mason, 1982; Stylianides & Stylianides, 2008). According to Harel and Sowder (2007),

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“[m]athematics as sense-making means that one should not only ascertain oneself that the particular topic/procedure makes sense, but also that one should be able to convince others through explanation and justification of her or his conclusions” (pp. 808–809).

Unless students realize the limitations of empirical arguments as methods for validating generalizations, they are unlikely to appreciate the importance of proof in mathematics. For this learning objective to be achieved, however, it is necessary that teachers have good knowledge in the domain of proof: the quality of learning opportunities students receive in classrooms depends on the quality of their teachers’ knowledge (e.g., Ball, Thames, & Phelps, 2008; Ma, 1999; Shulman, 1986). What knowledge might allow teachers to effectively help their students begin to overcome the misconception that empirical arguments count as proofs?

KNOWLEDGE FOR TEACHING PROOF

Effective teaching of proof requires, at a minimum, three broad kinds of teacher knowledge:

• Mathematical knowledge about proof, i.e., a solid understanding of critical mathematical aspects of proof that are essential for teaching proof to students;

• Knowledge about students’ conceptions of proof, i.e., a solid understanding of common ways in which students think about proof (including misconceptions); and

• Pedagogical knowledge about proof, i.e., a good command of effective pedagogical practices for helping students develop conceptions of proof that better approximate conventional understandings.

Next I summarize what the field of mathematics education currently considers important for teachers to know in relation to these three broad kinds of knowledge, paying particular attention to the distinction between empirical arguments and proofs. Then I use this summary to discuss important aspects of research knowledge that is currently missing for the development of a more comprehensive “knowledge package” for teaching proof. My use of the term “knowledge package” differs from that of Ma (1999). Ma used this term to describe complex networks of relationships among different concepts that a teacher must understand in order to make proper decisions about which concepts are required for the learning of other concepts, which concepts have to (or can) be learned simultaneously, etc. Thus, Ma used knowledge packages to describe concept maps or organizing structures within teachers’ mathematical knowledge for teaching. My use of the term is broader than that. I use the term knowledge package to describe a cluster of related kinds of knowledge (notably, knowledge about mathematics, students, and pedagogy) that a teacher must have in order to teach effectively a particular concept. Accordingly, the knowledge package for

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teaching proof can include, but is not limited to, a concept map that shows the relationships between proof and other closely connected concepts such as pattern, conjecture, and argument.

Mathematical Knowledge about Proof

In Stylianides and Ball (2008), we conducted a comprehensive review of the literature on teachers’ mathematical knowledge about proof for teaching and we contributed to the further development of this literature. An important element of mathematical knowledge about proof that we identified in that article, which is of interest to this article, is that teachers need to understand the distinction between empirical arguments and proofs (as explained in first section of the article). Unless teachers of all levels of education develop good understanding of the distinction between empirical arguments and proofs, it is unlikely that large numbers of students will overcome the misconception that “empirical evidence = proof.” For example, Martin and Harel (1989) noted about elementary teachers: “If [elementary] teachers lead their students to believe that a few well-chosen examples constitute a proof, it is natural to expect that the idea of proof in high school geometry and other courses will be difficult for the students” (pp. 41-42). Furthermore, an elementary teaching practice that promotes or tolerates a conception of proof as an empirical argument instills inaccurate mental habits in students. Dewey (1903) cautioned educators against such practices: he said that, whatever the preliminary approach to learning is, it should not inculcate “mental habits and preconceptions which have later on to be bodily displaced or rooted up in order to secure proper comprehension of the subject” (p. 217).

Knowledge about Students’ Conceptions of Proof

A significant body of research investigated students’ conceptions of proof and has developed various taxonomies of students’ conceptions of proof (some of which have developmental nature) that inform the field’s understanding of what is important for teachers to know in this area (see, e.g., Balacheff, 1988; Sowder & Harel, 1998). Good knowledge about students’ conceptions of proof can help teachers to identify/describe their own students’ understandings about proof. Also, this knowledge can inform the design of instruction that will aim to help students develop conceptions of proof that better approximate conventional understandings.

Below is a taxonomy of students’ conceptions of proof presented in increasing level of mathematical sophistication (see Stylianides and Stylianides [2009] for elaboration on this taxonomy).

• Conception 1: validating mathematical generalizations (e.g., patterns, conjectures) using naïve empiricism;

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• Conception 2: validating mathematical generalizations using crucial experiment;

• Conception 3: recognizing empirical arguments as insecure methods for validating mathematical generalizations (i.e., seeing a need to learn about more secure methods of validation); and

• Conception 4: recognizing and using proofs as secure methods for validating mathematical generalizations.

As I noted earlier, naïve empiricism and crucial experiment are two kinds of empirical arguments (Balacheff, 1988), with the former kind being a less advanced version of the latter. So, although conceptions 1 and 2 are both representative of an empirical approach to validating mathematical generalizations, conception 1 is less advanced than conception 2. These two conceptions are dominant among students of all levels of schooling, including high-attaining secondary students (e.g., Coe & Ruthven, 1994; Healy & Hoyles, 2000; Senk, 1985). Conception 3 is a transitory conception between the empirical approach to validating mathematical generalizations (conceptions 1 and 2) and the conventional approach (conception 4).

Pedagogical Knowledge about Proof

Compared to the body of research that informed the previous two kinds of teacher knowledge, significantly less research attention has been paid to the identification of pedagogical practices that teachers need to know for effective teaching of proof. Existing research in this area has developed a useful research base about general pedagogical practices for engaging students in mathematical reasoning, argumentation, and proof (see, e.g., Ball & Bass, 2000, 2003; Stylianides, 2007a, b; Stylianides & Ball, 2008; Yackel & Cobb, 1996; Zack, 1997). Yet, this research base needs to be developed further before it can inform the design of effective instructional interventions that will help students develop more accurate conceptions of proof; whereby “instructional intervention” refers to a purposeful and cohesive collection of activities, and respective implementation strategies of these activities, for achieving particular learning outcomes.

For example, in Stylianides and Ball (2008) we elaborated the importance of teachers using a variety of proving tasks that can offer students learning opportunities to develop understanding of different proving strategies and reasoning skills. Although the classroom implementation of particular proving tasks can provoke construction of different kinds of arguments, thereby offering to teachers and students the opportunity to discuss and reflect on the differences between empirical arguments and proofs, it is unclear how this discussion/reflection can be organized in a way that will help students overcome their deeply rooted misconception that “empirical evidence = proof.” Indeed, research and practice showed that addressing this misconception is a stubborn problem in mathematics

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education (see, e.g., Goulding & Suggate, 2001), and so this problem cannot be resolved without a carefully designed instructional intervention. Ability to implement successfully such an instructional intervention would be an essential element of teachers’ pedagogical knowledge about proof and, by implication, an important complement to the developing knowledge package for teaching proof.

What might such an instructional intervention look like and what demands does the successful implementation of the intervention place on teachers’ knowledge? Next I will discuss a research-based instructional intervention that was shown to be effective in helping students begin to overcome the misconception that “empirical evidence = proof.” My discussion will also exemplify the point that successful implementation of the instructional intervention presupposes that the teacher who implements the intervention has, in addition to good understanding of the different aspects of the intervention itself, solid knowledge of the mathematical ideas about proof and students’ conceptions of proof that I discussed earlier. The exemplification of this point will emphasize the inextricable relationships among the different kinds of knowledge that comprise the knowledge package for teaching proof.

AN INSTRUCTIONAL INTERVENTION FOR HELPING STUDENTS BEGIN TO OVERCOME THE MISCONCEPTION THAT “EMPIRICAL EVIDENCE = PROOF”

Background

In a four-year design experiment in an undergraduate mathematics course in the United States (see Schoenfeld [2006] for discussion on design experiment methodology), Gabriel Stylianides and I developed an instructional intervention that we showed to be effective in helping undergraduate students begin to understand the limitations of empirical arguments and see an “intellectual need” (Harel, 1998) for learning about more secure methods of validation (i.e., proofs) (Stylianides & Stylianides, 2009).

Next I will present how a secondary mathematics teacher implemented in her class a modified version of the original instructional intervention; the modification was done by me and was discussed with the teacher to ensure (1) the appropriateness of the activities for her class and (2) that she understood the rationale of the different activities that comprised the intervention and was well prepared to implement the intervention in her class. The implementation of the intervention was part of a school-based design experiment that I conducted last year in two high-attaining Year 10 classes in a state school in England (student ages: 14-15 years old). Partially motivated by research findings that showed that even high-attaining secondary students in England possessed limited understanding of proof (Coe & Ruthven, 1994; Healy & Hoyles, 2000; Küchemann & Hoyles, 2001-03), the

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design experiment aimed to generate theoretical and practical knowledge about possible instructional interventions for helping Year 10 students develop their understanding of proof. It is envisioned that the knowledge to be generated by the design experiment will support future studies that will aim to promote similar research goals in more challenging settings (notably, in classes with less advanced students).

The focal instructional intervention at the school level was underpinned by the theoretical framework that we developed in our undergraduate design experiment (Stylianides & Stylianides, 2009). The student learning outcomes from the implementation of the intervention at the undergraduate and school levels were strikingly similar. This observation, together with the fact that the misconception “empirical evidence = proof” was found to be dominant among students of all levels of education and in different countries, offer a reasonable basis for one to hypothesize that appropriately modified versions of the instructional intervention can yield similar results when implemented in other levels of education (notably, in the elementary and middle schools) and in other cultural contexts.

The implementation of the focal instructional intervention at the school level lasted approximately 60 minutes and extended over two consecutive 45-minute lesson periods. The teacher implemented the instructional intervention following a detailed lesson plan that I prepared and discussed thoroughly with her prior to the lessons. The description of the implementation in the following sections of the article is based on video and audio records of the lessons and field notes that I took on the work of a group of six students during the lessons (there was no special reason for the selection of this small group; all small groups were considered to be of mixed ability).

As you read the description of the implementation of the instructional intervention, I invite you consider what knowledge the teacher needed to have for successful implementation of the intervention (I will address this issue after I describe the implementation of the intervention). Also, I invite you to pay attention to how the teacher used each of the activities in the intervention (Squares Problem, Circle and Spots Problem, and “Monstrous Counterexample” Illustration) to facilitate students’ progression along the “learning path” that is summarized in Figure 2: from (1) using naïve empiricism as a method for validating patterns, to (2) using crucial experiment, to (3) seeing a need to learn about more secure methods for validating patterns (i.e., to learn about proofs). Notice that the three stages in this learning path correspond to conceptions 1 through 3 that I discussed earlier under teachers’ knowledge about students’ conceptions of proof for teaching.1

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Figure 2: The three activities in the instructional intervention and the corresponding “learning path.”

Activity 1: The Squares Problem

Kathy, the teacher, introduced the Squares Problem (Figure 3). The hardest part of the problem was the third, because it asked students to find the number of different 3-by-3 squares in a case that was difficult for them to check practically and also to explain whether and why they were sure their answer was correct.

Kathy made sure that the students understood what the problem was saying and then she asked them to work on the problem in their small groups. The small group where I was sitting during the lesson had the following six students: Bob, Calvin, Dan, Lazarus, Robert, and Sharon. These students counted squares to answer parts 1 and 2 of the problem, and then Bob asked his peers: “Have you actually got a formula?” Dan responded: “It’s the number of … it’s n minus 2, and then squared.” Sharon showed excitement and confirmed with Dan that the answer for part 1 would be 4. Robert asked how many 3-by-3 squares there were in a 60-by-60 square (part 3) and Dan used his calculator and the formula he had described earlier to find the answer: (60 – 2)2 = 3364.

At some point Kathy visited the small group and the students explained their work. Kathy then asked the students whether they were sure their answer was correct. Lazarus said with confidence, “yes,” and Kathy posed a new question: “And have you thought about why you are sure?” There was no response from the students. Kathy asked the students to think about this and to write their ideas on paper.

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Figure 3: The Squares Problem (adapted from Zack, 1997).

Dan drew figures for the 4-by-4 and 5-by-5 squares showing the 3-by-3 squares in each of them. He wrote down 582 = 3364 as the answer to part 3 and also the formula (n – 2)2. He concluded: “We realized that if you took 2 away from the number of cubes along the top and then square the answer you will get the number of 3x3 boxes in the grid.” The other students in the small group wrote similar conclusions.

So, what has happened thus far in the small group? The students identified the pattern that the number of different 3-by-3 squares in an n-by-n square was given by the formula (n – 2)2. They verified the pattern for n=4 and n=5 and, based on this evidence, they concluded that the pattern would hold true for all values of n including n=60. Thus the students validated the pattern on the basis of naïve empiricism (cf. Figure 2).

The whole group discussion that followed illustrated further the use of naïve empiricism in the class, as all groups answered the three parts of the problem using the formula (n – 2)2. After some discussion on the meaning of the formula, Kathy asked the class whether and why they could be sure that applying this formula would give the correct answer. Emily said: “We tried it [the formula] for a 6-by-6 square and it worked for that too.” Kathy invited further comments but the students did not have anything to add to what Emily had said.

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Kathy then asked the students to write down individually their thoughts: “I want to know what your feelings are about whether this [the answer to part 3] is correct or not. You may think it is correct, you may not. If you are sure, I want to learn why you are sure.” Someone asked “what if you’re not sure?” and Kathy responded: “Then put not sure, but say why you are not sure – what makes you doubt it?”

In the focal small group the students wrote:

• Bob: “Because we have found a formula and tried it against smaller squares so we can make sure that the formula is right.”

• Calvin: “I am sure that this solution works because it worked for every one we did.”

• Dan: “I am sure that the answer is correct because it has been proved for a number of smaller grids.”

• Lazarus: “I am sure that the answer is correct because it has been tested and proved correct. The pattern will continue to 60x60.”

• Robert: “I am sure it’s correct because we did a test on the 6x6 grid and it worked.”

• Sharon: “We are sure that it is right because we have tried it for a 6x6 square as well. So we assume that it would work.”

Notice that the six students were convinced of the truth of the pattern on the basis of naïve empiricism: the pattern worked for the first few cases and so, according to the students, it would work also for n=60.

Following the students’ individual reflections, Kathy proceeded with the next item in the lesson plan, which was to summarize the students’ written responses. Kathy’s summary below was based on a quick inspection of students’ written responses as she was circulating around during students’ individual reflection time. The accuracy of Kathy’s summary was confirmed by a more careful analysis of students’ responses at the end of the lesson.

I get a feeling that most of you have said – ‘Well, I think we have sort of answered this question that 582 is the right answer: we have found a pattern by checking smaller grid sizes and then we have used that pattern, assuming that it would continue all the way up to 60-by-60.’ That’s the stage where we are right now: we’ve seen a pattern working, somebody said they tried the 6-by-6 and it worked for that too, and so we continued our pattern up to the 582.

The way the students responded to Kathy’s prompt was anticipated in the lesson plan: the state of student conceptions of proof in Kathy’s class as described in the summary above (predominance of naïve empiricism) was consistent with what was reported in the research literature about students’ conceptions of proof.

After Kathy’s summary, Bob asked Kathy whether the pattern was correct and Kathy said that the class would come back to this issue later, but first they would work on a couple of

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other activities. Indeed, according to our lesson plan the issue about the correctness of the pattern in the Squares Problem would remain tentatively unresolved. The class would revisit and resolve the issue after the students had been assisted to realize the limitations of empirical arguments (both naïve empiricism and crucial experiment). Had the issue been resolved at this point of the lesson, this would probably require a lot of “telling” by the teacher, which would be inconsistent with the teachers’ philosophy of how students learn and the theoretical framework that underpinned the design of the instructional intervention. The intention was for the students to realize the limitations of empirical arguments on their own, by experiencing and reflecting on situations where the empirical validation method was inadequate. For the readers’ information, I note that the (n – 2)2 pattern was actually correct.

Activity 2: The Circle and Spots Problem

Kathy introduced the Circle and Spots Problem (Figure 4) and helped the students understand what the problem was saying. Specifically, she discussed with them the meaning of the terms “maximum” and “non-overlapping regions.” Also, she clarified that the phrase “around the circle” referred to the circle’s circumference and that the spots on the circumference did not have to be equidistant. Then Kathy asked the students to work on the problem in their small groups.

Notice that, similarly to part 3 of the Squares Problem, the question in the Circle and Spots Problem (yellow box in Figure 4) was asking the students to make a statement about a case that was difficult for them to check practically. In our planning, we had anticipated that the students, like they did in the Squares Problem, would check simpler cases, identify a pattern, trust the pattern based on naïve empiricism, and apply it to offer a definite answer for n=15 (where n stands for the number of spots). The main difference between the two problems is that the emerging pattern in the Circle and Spots Problem fails for n=6. Our plan was for Kathy to use the anticipated surprise that the students would experience with the failing pattern to help them move from naïve empiricism towards crucial experiment (cf. Figure 2).

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Figure 4: The Circle and Spots Problem (adapted from Mason, 1982).

After about 10 minutes of small group work, Kathy brought the whole class together and said: “Circulating around I think there are some people who think they know what the answer will be for 15 [spots]. Is there anyone who is willing to tell us their number of regions, what it will be for 15 spots?”

Mac said that his group thought the formula for the problem was (n – 1)2 but soon after he corrected himself and said that the formula included powers of 2. Kathy asked the class to say the maximum number of non-overlapping regions they found for different spots, and she constructed a table on the board with the following numbers: 4, 8, and 16, which corresponded to n = 3, 4, and 5, respectively. Then she pointed out that, as Mac had mentioned earlier, the values were all powers of 2 and that, in each case, the power was one less than the number of spots: 22 (for n=3), 23 (for n=4), and 24 (for n=5). Kathy asked: “So what will it be for 15 spots, then?”

Several students offered to answer Kathy’s question. Based on what I had observed during these students’ earlier work in their small groups, I presume they would propose the application of the 2n – 1 formula for n=15. However, Ken, a student sitting in a different group, said loudly: “Can I just say that is wrong, because on 6 [spots] there are only 30 [regions].” Kathy said: “We were about to say that the answer would be 2 to the power of 14. However, you are telling me that for 6 spots it doesn’t work out to be…. With this pattern for 6 six spots it would be 2 to the power of 5, that would be 32, but did anyone manage to find this number of spots?” Some students said they found 31.

Kathy continued:

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When we were back to the Squares Problem, we said that because the pattern worked for some of the different grids, the 5-by-5, 6-by-6 squares, and so on, we were willing to trust it. But this time we have shown that it works for 3, it works for 4, it works for 5, but actually, Ken, you are right: if we had 6 spots on a circle and we joined them all up, the number of non-overlapping regions that we get is not what we expect to get, it’s not 32. It’s actually 31.

As she talked, Kathy used a PowerPoint slide to illustrate a case in which the number of non-overlapping regions for n=6 was 31, explaining that this number of regions was the maximum number of regions one could get with 6 spots.2 She noted also that, if one drew the spots in a regular hexagon, the maximum number of regions would be 30, which is again smaller than 32. Then, following the lesson plan, Kathy asked the students to write down their thoughts about what the Circle and Spots problem had taught them.

The students in the focal small group wrote:

• Bob: “You can’t always trust a formula until you have tested it many times over for lots of different examples.”

• Calvin: “This test has taught us that if you see a pattern doesn’t make it correct.”

• Dan: “The circle and spots tells us that we can’t always trust a formula that works on the first few.”

• Lazarus: “This teaches us that just because something works for one thing, that doesn’t mean it will work for everything.”

• Robert: “You can’t always trust a formula until you have tested many times over for lots of different no’s of spots.”

• Sharon: “You can’t always trust a formula. You shouldn’t presume it is correct because it worked for the first few.”

Notice that the students started to move away from naïve empiricism. For example, Dan, Lazarus, and Sharon started feeling uneasy to trust a pattern based on checks of the first few cases. Also, Bob and Robert’s comments approximated the crucial experiment method of validation, as they appeared to raise a concern about the number (“many”) and quality (“different”) of cases that had to be checked before a pattern could be trusted.

Thus an important issue for many students at this stage of the lesson was how many cases would be enough for them to check before trusting a pattern. We had anticipated this issue in our planning and we prepared a PowerPoint slide with a fictional student comment on it that Kathy used in the lesson to orchestrate a discussion around the issue. The fictional student comment said:

The Circle and Spots Problem teaches me that checking 5 cases is not enough to trust a pattern in a problem. Next time I work with a pattern problem, I’ll check more cases to be sure.

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Kathy invited reactions from her students on this comment. Dan suggested trying spread cases such as for n = 1, 75, and 100. Robert observed that “you can’t always trust the formula, you have to test it.” Kathy asked Robert how many times one had to test a formula and Robert said “more than like 5 times.” Kathy invited more comments and Larry said: “you should test it as many times as you have time to do.” Kathy asked Larry: “So when you have tested it as many times as you have time to do, can you then trust it?” Larry revised: “No … not a 100%!” Then Pauline said: “try it out with smaller numbers and bigger numbers.” Kathy observed that Pauline’s comment was similar to Dan’s earlier comment.

Indeed, the two comments were similar to one another and illustrative of the crucial experiment method for validating patterns (cf. Figure 2). As I noted earlier, crucial experiment can be considered to be a more advanced method than naïve empiricism, but is still an invalid method because it does not exclude the possibility of a counterexample in a case that was not checked. Some students in the class were thinking along similar lines, as illustrated further by their responses to the following question by Kathy: “And then, do we trust it if it worked for all of those [cases, big and small ones]?” Silvia said in a low voice: “No, because you might have missed one.” Another student was heard to say: “You could spend your whole life and still miss one!” These students’ fear that a pattern can fail in a case that was not checked was manifested in the next activity we planned for the students.

Activity 3: The “Monstrous Counterexample” Illustration

Kathy introduced the PowerPoint slide in Figure 5 that shows what I call the “Monstrous Counterexample” Illustration. Kathy did not use this name during the lesson. The slide was presented in segments to give to the students a chance to process the information in it. For example, there was a discussion about how one would check whether a given number was a square number using a calculator. Also, the students confirmed the statement for particular values of n using their calculators.

Once the students checked many different cases and were comfortable with the meaning of the statement, Kathy presented the counterexample. The students were amazed: they had not anticipated that a pattern that held for so many cases (of the order of septillions) could ultimately fail.

Kathy then directed the students’ attention to their previous discussion about the fictional student comment on what the Circle and Spots Problem had taught them: “We said in the Circle and Spots Problem that, okay, it’s not enough to just check a few cases, you need to try different ones. Well, this expression, what does this tell us?” Emily said: “If you kept trying, you might have to go that high until you find one [a counterexample].” Kathy said: “But I can imagine that it took the computer quite a long time to check all of those cases. And when do you stop checking?” Larry said: “When you’ve found one!” Several

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students laughed with what Larry had said. Kathy continued: “And when do you trust a pattern then?” Adam said: “When you cannot find one, until you are dead!”

Figure 5: The “Monstrous Counterexample” Illustration (adapted from Davis, 1981).

Notice that the students began to develop distrust in empirical arguments of any kind, including crucial experiment, because they were proposing checking the pattern indefinitely. Yet, although the students began to realize the limitations of empirical arguments, they lacked knowledge of more secure methods for validating patterns. This caused a feeling of frustration among some of them as illustrated in Adam’s comment: one would die checking cases before being in a position to trust a pattern. Thus we may say that the students reached the point when they felt a need to learn about more secure validation methods (cf. Figure 2).

Following the lesson plan, Kathy responded to this emerging student need, which was an outcome of the instructional intervention, by introducing her students to the notion of proof in mathematics and by engaging them in a discussion of possible criteria that an argument needed to fulfill in order to count as a proof in their class. Also, Kathy took the class back to the Squares Problem and she helped the students to develop a proof for the pattern they had identified earlier. More discussion of what happened next in Kathy’s class is beyond the scope of this article.

What Knowledge did the Teacher Need to Have to Successfully Implement the Instructional Intervention?

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The elements of mathematical knowledge about proof and knowledge about students’ conceptions of proof that I discussed in previous sections were necessary for successful implementation of the instructional intervention. Specifically, if the teacher did not understand that empirical arguments do not qualify as proofs (cf. mathematical knowledge about proof), she would most likely have accepted students’ empirical arguments as proofs in the Squares Problem and she would have seen neither a need nor a purpose to proceed with the rest of the instructional intervention (which aimed to challenge this empirical conception of proof).

Similarly, if the teacher did not know the taxonomy of student conceptions of proof in Figure 2 and the hierarchical structure of these conceptions as stages in a learning path (cf. knowledge about students’ conceptions of proof), she would most likely have difficulty making sense of her students’ conceptions about proof in the different stages of the intervention and how these conceptions compared to one another (in terms of their level of mathematical sophistication). For example, the teacher might have not recognized the crucial experiment approach to validation that surfaced in students’ work on the Circle and Spots Problem as a progression in their learning when compared to the naïve empirical approach that dominated students’ earlier work on the Squares Problem. Accordingly, the teacher would have been limited in her potential to understand the function of the different activities in the instructional intervention, which aimed to help students to develop conceptions of proof that increasingly approximated conventional understandings.

Despite the importance of the teacher’s mathematical knowledge about proof and her knowledge about students’ conceptions of proof as explained above, these two kinds of knowledge are by themselves inadequate to capture the knowledge that was used by the teacher during the implementation of the instructional intervention. Specifically, the teacher was also drawing on a strong pedagogical knowledge about proof that allowed her to understand the rationale that underpinned the design of the intervention and to implement it successfully in her class, thereby managing to help many students in her class progress along the intended learning path. The description of the implementation of the instructional intervention exemplified several elements of the teacher’s pedagogical knowledge about proof. Two important elements of this knowledge were:

• Understanding the general structure of the intervention and how the activities that comprised the intervention could be used to support students’ progression along the main stages of the intended learning path. The three activities in the instructional intervention constituted a coherent sequence of activities that could be used to help students to progress along the intended learning path; the value of the sequence was more than just the sum of the values of the individual activities that comprised it. For example, our prior experience (Stylianides & Stylianides, 2009) suggests that, if the teacher asked her students to prove the pattern in the Squares Problem before she

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introduced to them the Circles and Spots Problem, this would result in a completely different learning experience for the students: they would not see any reason to abandon their naïve empirical arguments. The beginning work on the Squares Problem was designed to provoke students’ use of naïve empiricism (as documented in the literature) and to create a basis on which the subsequent activities in the intervention built to help the students develop increasingly accurate conceptions of proof: the Circle and Spots Problem challenged students’ reliance on naïve empiricism, while the Monstrous Counterexample Illustration challenged their reliance on crucial experiment. Once the class realized the limitations of empirical arguments, the class revisited the Squares Problem and produced a proof for the pattern. Thus, the Squares Problem did not constitute an isolated activity to be completed independently, but rather an integral part of the sequence of activities that comprised the intervention.

• Understanding the nuances of particular implementation strategies in the instructional intervention. It would be difficult for students to abandon, or actively engage in a process of refining, their current conceptions about proof, unless they became more aware of their current conceptions and recognized the limitations of these conceptions. By asking the students to do the individual reflection at the end of the Squares Problem, the teacher helped them to become more aware of the method of validation they used in that problem (naïve empiricism), thereby making it more likely that they experienced a “cognitive conflict”3 in subsequent activities where the use of this validation method proved to be problematic. Similarly, by organizing the discussions of the fictional student comment after the students encountered the counterexamples in the Circle and Spots Problem and the Monstrous Counterexample Illustration, the teacher created a purposeful learning context in which the students reflected on, and began to recognize, the limitations of naïve empiricism and crucial experiment, respectively.4

It is important to recognize that these elements of pedagogical knowledge about proof are based (implicitly or explicitly) on certain premises about how students learn mathematics and how teaching can support that learning. For example, a premise that underpins both elements, and the design of the instructional intervention as a whole, is that deeply rooted student misconceptions cannot be changed simply by “telling” from the teacher, but rather by careful design of rich and purposeful learning environments for students. Thus, a different philosophy of teaching/learning or theoretical perspective on the instructional design would probably support a different instructional intervention, which would in turn require a different pedagogical knowledge about proof from the teacher. Yet, the fact that existing research and practice have not identified thus far other promising instructional interventions for helping students to overcome the deeply rooted misconception that

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“empirical evidence = proof,” highlights the importance of the instructional intervention presented herein and supports the philosophical and theoretical perspectives that underpinned its design as elaborated in Stylianides and Stylianides (2009).

CONCLUSION

In this article, I explained that existing research on teachers’ knowledge for teaching proof has constructed a significant, albeit incomplete, knowledge package for teaching proof. Specifically, this body of research has identified critical mathematical aspects of proof that are essential for teaching proof to students (mathematical knowledge about proof) and has constructed a detailed map of common ways in which students think about proof (knowledge about students’ conceptions of proof). Less emphasis has been placed on teachers’ knowledge of effective pedagogical practices for helping students develop more accurate conceptions of proof (pedagogical knowledge about proof).

In an effort to contribute to the development of a more comprehensive knowledge package for teaching proof, I elaborated on the importance of expanding teachers’ pedagogical knowledge about proof to include ability to implement successfully in their classrooms instructional interventions that address major student misconceptions. I exemplified my proposal in the context of a research-based instructional intervention that aimed to help students overcome the deeply rooted misconception that “empirical evidence = proof.” My discussion of the implementation of the instructional intervention by a secondary mathematics teacher illustrated also the inextricable relationships among the different kinds of knowledge of the knowledge package for teaching proof that I discussed: if the teacher lacked solid knowledge of any of these kinds, she would not have been able to implement the intervention the way she did (i.e., faithfully to the plan and with good results). Furthermore, given the limited progress that practice and research have made thus far to address the pervasiveness of the misconception that “empirical evidence = proof” among students, my discussion of the implementation of the instructional intervention by the secondary mathematics teacher sends the optimistic message that, with the necessary support, typical teachers can make significant progress in addressing this stubborn problem in their classrooms.

The development of a more comprehensive knowledge package for teaching proof, coupled with a support system for helping teachers (both prospective and practicing) to incorporate the elements of this package into their own knowledge, promise major advancements in the teaching of proof and, by implication, in students’ opportunities to develop competency in proof. Yet, a lot of challenges emerge for research and practice. The deeply rooted misconceptions that many students tend to have in the domain of proof make extremely difficult the design and implementation of instructional interventions that can successfully

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address these misconceptions. Indeed, it took us five research cycles of implementation, analysis, and refinement over a four-year period (Stylianides & Stylianides, 2009) in order to develop the instructional intervention whose modification was presented in this article. Thus, it would be unrealistic to expect from teachers to develop such knowledge for teaching proof on their own; rather, teachers need to be offered systematic support to develop this kind of knowledge.

One way in which the field of mathematics education can support teachers to develop their knowledge for teaching proof is by making available to them “educative curriculum materials” (Davis & Krajcik, 2005), which will incorporate, for example, existing research knowledge about instructional interventions for promoting student learning of proof. Specifically, such educative curriculum materials will be concerned not only with a presentation of the activities that comprise the instructional interventions, but also with helping teachers to develop the kinds of knowledge (about mathematics, students, and pedagogy) that will allow them to effectively implement these interventions in their classrooms. Despite the important role that educative curriculum materials can play in promoting teacher knowledge for teaching proof, it is a long way to the development of such materials. For example, research showed that a popular, reform-oriented textbook series in the United States offered limited support to teachers about how to implement in their classrooms proof tasks that were included in the series (Stylianides, 2007c).

Finally, another way in which the field can support teachers to develop their knowledge for teaching proof is by promoting this knowledge in teacher preparation and professional development programs. This goal will have to be integrated into a coherent set of learning experiences for teachers that: (1) will address all components of the knowledge package for teaching proof that I discussed in this article; and (2) will aim, additionally, to develop in teachers beliefs about proof that will help them appreciate the importance of proof in their students’ mathematical education. In regard to the latter, it is important that teacher preparation and professional development programs offer to teachers opportunities to experience themselves as learners instructional interventions that aim to help them develop their own mathematical knowledge about proof (see Stylianides & Stylianides, 2009). Specifically, research on the role of teacher preparation programs in the development of prospective teachers’ beliefs that are consistent with desirable instructional practices and objectives has showed that teacher preparation courses need to engage prospective teachers in activities that reflect the demands of the mathematics classrooms in which they will someday teach (Kagan, 1992).

ACKNOWLEDGMENTS

The work reported herein received funding support from the UK’s Economic and Social Research Council (Grant Number: RES-000-22-2536) and the Spencer Foundation (Grant

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Numbers: 200700100 and 200800104). The opinions expressed in the article are those of the author and do not necessarily reflect the position, policies, or endorsement of either organization. Parts of this article (notably those that describe the implementation of the instructional intervention) appeared recently in a practitioners’ journal (Stylianides, 2009), but their use herein is different and serves a new purpose (to advance an argument about teachers’ knowledge for teaching proof).

NOTES

1. The teacher and student names are pseudonyms.

2. The question in the Circle and Spots Problem was asking whether there is an easy way to tell for sure what is the maximum number of non-overlapping regions in which the circle can be divided for n=15 (see Figure 4). Although the students’ inability to generate 32 regions for n=6 does not guarantee that it is impossible to generate this number of regions with 6 spots, it suggests that the emerging pattern offers an insecure way to find the maximum number of regions for n=15. In this sense, Kathy’s explanation to the class that the maximum number of regions for n=6 was less than 32 should be interpreted as a confirmation of the students’ emerging view that the pattern they identified for n≤5 offered an untrustworthy means to answer the question for n=15, rather than as an authoritative act to impose conviction of a certain truth in her class.

3. See Stylianides and Stylianides (2009, pp. 319-323) for a discussion of how the similar version of the instructional intervention that was implemented at the undergraduate level used “cognitive conflict” as a mechanism to support developmental progressions in students’ knowledge and how it addressed the common challenge faced by the cognitive conflict approach to mathematics teaching of students treating emerging contradictions in their knowledge as exceptions.

4. I helped the teacher to develop these two elements of knowledge in our discussions prior to the implementation of the instructional intervention in her class.

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