Errors made by First Year Students in an Integral Calculus ... · Errors made by First Year Students in an Integral Calculus Course ... Representations in calculus/analysis ... stated

Errors made by First Year Students in an Integral Calculus Course using Web-Based Learning

R. HARIPERSAD; R NAIDOO

Department of Mathematics, Statistics and Physics Durban University of Technology

41/43 Centenary Road Durban Kwa-Zulu Natal South Africa

Abstract

Calculus is extremely important for engineering and science studies at the University of Technology. Recent studies show that first year students exhibit many conceptual errors in calculus. In order to reduce the frequency of errors, blended learning (a combination of conventional lectures with Web based Learning) was used. The WBL course is designed such that it allows for algebraic, numerical and graphical methods of understanding of mathematics and track the student’s progress regularly during which times the lecturer offers recommendations for the student’s progress. In this way lecturers have a one to one tutoring with students in a large class. To determine the effectiveness of the strategy, we compared whether the students make the same category of errors and frequency as in the conventional lecture. A qualitative study was performed on control (traditional) and experimental (WBL) groups each consisting of 33 students. The results indicate WBL students exhibit less structural errors than conventionally taught students.

Keywords : WBL, Calculus, errors, blended learning, frame theory

1 Introduction

This paper is a record of a curriculum development initiative to address some of these areas of concern such as structural, executive and arbitrary errors in a first year engineering mathematics course. Structural errors occur when a student fails to create mental relationships of the group of principles essential to the solution of the problem, an executive error is a failure to carry out manipulations although the principles may have been understood, and an arbitrary error is when students repeat the statement of the problem or fail to take into account the inputs or output of the problem. Previous studies on errors in elementary calculus

courses [1] led to a pragmatic approach in the development of a new strategy to support the learning in mathematics.

Web based learning (WBL) was introduced to the first year engineering mathematics multicultural group with a mixture of high and low achievers. The WBL course was designed using WebCt, a learning management system (LMS) that constituted the different components outlined below.

• Course: the content is presented by either using text or multimedia

Proceedings of the 7th WSEAS International Conference on CIRCUITS, SYSTEMS, ELECTRONICS, CONTROL and SIGNAL PROCESSING (CSECS'08)

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(graphics, audio and video) to accommodate individual learning styles by providing different approaches

• Assessments: the assignments and tests are provided online with immediate feedback/remedy given on their performance

• Communication: allows for asynchronous discussions-this allows students time to reflect on or further research topics to reinforce concepts before responding.

Finally the WBL course attempts to create formal representation of concepts which is necessary in the engineering disciplines as shown in the diagram, fig 1 below.


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Fig 1: Representations in calculus/analysis

Each of these aspects is significant in the student’s study of the concept. These modes of representation can be categorized into three fundamentally distinct ways of operation [2].

Embodied: based on human perceptions and actions in a real world context including but not limited to enactive and visual aspects. Symbolic-proceptual: combining the role of symbols in arithmetic, algebra and symbolic calculus, based on the theory of these symbols acting dually as both process and concept.

Formal-axiomatic: A formal approach starting from selected axioms and making logical deductions to prove theorems.

Various authors, including Piaget [3], Dubinsky [4] and Sfard [5], theorize that growth of human knowledge begins with actions (first on the environment),some becoming repeatable processes and later conceived as objects to be manipulated on a higher level by further mental processes. The manner in which numeric and symbolic representations develop involves an

interesting form of cognitive growth. There are recurring cycles of activity in which a process, such as in the limiting process the secant becomes a tangent for the derivative concept and limit of the Riemann sum becomes the definite integral concept which Tall, [2], terms procept.

WBL is a powerful resource for facilitating and implementing problem solving, investigation and mathematical discussion. The calculus teaching content was designed using frames as defined by Davis, [6]. A frame is an abstract formal structure, stored in memory, which encodes and represents a sizeable amount of knowledge. It allows multiple points of entry, provides flexibility and retrieved almost instantly. Frames possess considerable internal organization with more complex frames developed on existing one. Ausubel, [7], argues that the most important factor influencing learning is what the student already knows as defined by Davis, [4], as pre-knowledge frames.


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Pre-Knowledege Frame: The notion of area

The area under a continuous curve can be visualized and approximately calculated by covering the area with a grid and counting the squares within, as depicted below

y

y = f(x)

x

where area of Rectangle/Square xyΔ=

The retrieval and execution of the appropriate pre-knowledge frames guides mathematical problem solving. The WBL course includes these pre-knowledge frames as a pre- test quiz in

the assessment component before the student engages with the concept or problem in the lessons.

The area from a to x under the graph is a function A(x) called the “area so far” function, [2]. In the embodied arena of physical measurement, a numerical value of the area can be found by drawing small enough squares and measuring accurately. This notion of “area” support Riemann sums and integration. The use of technology to draw strips under graphs and calculate the numerical area is widely used. If the above graph is stretched horizontally we gain insight that the increase in area is approximately f(x) times the change in x, which leads to the fundamental theorem of calculus that the rate of change of area is the original function.

Lecture Frames: Concept of an Integral

The area of the irregular region bounded by the graph of f, the x-axis, and vertical lines x = a and x = b, is obtained by using the area of rectangles placed within the shaded space – the Riemann Method. This requires the retrieval of various and different frames as depicted in the figure below execution of the appropriate pre-knowledge frames guides mathematical problem solving. The WBL course includes these pre-knowledge frames as a pre- test quiz in the assessment component before the student engages with the concept or problem in the lesson.


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Pre-Frame-Piagetian Concrete-operationalstage

y

f(a+h) f(xi) iii xxfA Δ= )( (Approximate Area)

f(a) f(b)

a xi b x

Pre-conceptual Area Frames

Area under curve ≅ summation of area of rectangles

∑ Δ≅n

iii xxf )( (Symbolic frame)

Limit frame

Exact area )()(lim

1i

n

ii xxf

nΔ

∞→= ∑

=

, where n

abxi)( −

=Δ

Formal-operational level: Representation frame Reciprocity frame

Definite Integration associate frame Differentiation

Area = )()(lim

1i

n

ii xcf

nΔ

∞→ ∑=

∫≡b

a

dxxf )( )()( xfdx

xdF=

Frame within a frame

The Fundamental Theorem of Calculus ∫ += CxFdxxf )()(

)()()()( ] aFbFxFdxxfb

a

b

a

−==∫


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Papert , [8], stated that anything is easy if one can assimilate it into one’s collection of mental modes. This is highly applicable and testifies on the importance of frames. The pre-knowledge area frame is linked to the Piagetian concrete-operational development stage. The transition to a higher level is obtained by the presence of previous frames. This serves as ‘assimilation schemas’ for organizing input data. The initial frame recognizes the input data, modifies it and maps it into the limit frame and represents it as an integration frame. At this formal-operational level, students should be able to define the definite integral in terms of the limit of a sum of progressively narrower rectangular areas. The integral frame causes the retrieval of other appropriate frames, viz, differentiation, which is a reciprocity frame. Hypothetical reasoning allows one to consider whether any of these are similar to the present task. The relationship between the integration frame and differentiation frame is formed and a new frame is created-the Fundamental theorem of Calculus.

Research, [9], indicates that lecturers do not have a clear idea of learner’s understanding and misconceptions - this is a serious omission since it creates problems for later stages of learning. Blended learning remedies this problem. Further, from a study on errors in elementary calculus [1], implications are that students have many misconceptions in mathematics due to the lack of sound pre-knowledge frames. Recommendation for and introduction to computer instruction, seemed to have remedied some of the problems such as the pre-knowledge frames [10], while others persist as students have unique problems.

From the two results above, [1, 10], pre-knowledge frames, limits and algebra seem to bother some students. It seems that each student had to be taught on a one to one basis and that each student needed to be monitored on a daily

basis since they had different mathematical misconceptions and backgrounds. We believe that by communicating with them daily on their own time and pace, discovery learning and motivation may result. To build appropriate conceptual mental frames we need to track, appraise and teach with new strategies (each task or concepts and students cognitive abilities and motivation level were unique due to varying Piagetian development stages). This could not be performed at a traditional lecture or computer laboratory class. Hence we attempted the blended learning which includes traditional, computer labs and WBL in an attempt to reduce errors students.

Generally, it seems that WBL research undergoes the same development as computer aided learning or multimedia learning. However, the two approaches are structurally different and not a direct translation from one to another. WBL has the capabilities of tracking individual student’s progress through the communication component technology thereby enhancing the ability to rectify misconceptions timeously and asynchronously. In this continuous process of student-lecturer/student-student interactions, new and adequate pre-knowledge frames are developed. This allows students to actively use cognitive strategies and previous knowledge to deal with cognitive limitations.

The most critical aspects of WBL is whether it can provide reflection support [11] such as in projects, assignments and problem solving with links to other mathematics sites. WBL can provide powerful scaffolding for reflection when process displays, process prompts, process models, or a forum for social discourse is implemented as advocated by Lin, Hmelo, Kinzer, and Secules [12]. The blended learning mathematics course is based on learning patterns and learner responses. A learning cycle model


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which consists of discovery, concept introduction and concept application is frequently used.

2 Qualitative Theoretical Framework

We developed a theoretical framework to analyze student’s responses in terms of errors and to further construct learning materials after the quiz assessments on the WBL. A qualitative theoretical framework was constructed from the analysis of errors in arithmetic by Donaldson [13], cognitive frame theory by Davis [6] and the modified Orton’s [14] tasks. The inadequate concept image of the integral may lead to three types of errors. These errors are structural, executive and “arbitrary”.

The three types of errors may be linked to the sequential processes, critic, frames and the deeper level procedures of Davis [6]. The learning of integration does not require verbatim repeating of verbal statements but the appropriate mental frames to represent the concepts and procedures of calculus. Structural errors are caused by incorrect frame retrieval, sketchy or incomplete frames, deep-level procedures and sub-procedures. Executive errors are caused by incorrect sub procedures and control structure of sub-assemblies (intermediate structures). “Arbitrary” errors are caused by mapping incorrect inputs to the retrieved frame (surface structures). The qualitative theoretical framework refers to the ways students are thinking with respect to the mathematical tasks. This necessitates that one has to get information from students whilst they are engaged in specific mathematical tasks. The frame theory includes metaphors, collages or chunks embedded in the frames. Engineers typically use metaphors, collages or chunks of cognition to explain design or mechanisms. Oertmans’s [15] research on metaphors used in the understanding of the

calculus exhibit a particular aspect of our theoretical frame structure designed in 1996

3 Methodology

The course was implemented for 33 mathematics one students who were exposed to blended learning during 2007-(experimental group), and another group of students who were lectured conventionally-(control group). Both groups selected in the study were comparable in prior knowledge of the topics. The course consisted of four lectures per week (equally divided between conventional and WBL ). Three instructors and tutors were involved in the study. The tutors involved in the course were adjunct (part-time) faculty members who are less technologically sophisticated and pedagogically experimental than our typical full time faculty member – this creates an unexpected external constraint. Even under these less than ideal circumstances the results of the study indicate that WBL mathematics has the potential to improve students understanding of the subject. The control group was taught by the same group of lecturers/tutors and assessed in the traditional manner.

After 12 weeks the students were given a modified Orton’s [14] test as a post test. If there were more than one type of error in a task both errors were reported. If the error could not be easily categorized responses from two experts were sought to give their opinion. The tasks on integration were listed and discussed as to relevance and type of the frame retrieved. The tasks were then itemized according to required skills and concepts. There were 10 items, listed in Table 1:


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Table 1: Item No and Description

Item No Description Related Tasks

1 Area under graph y = k 1

2 Area under graph y = kx 2

3 Height of rectangles under graph 3(i) & 3(ii)

4 Use of previous heights in new situation 4(i) & 4(ii)

5 Calculation of areas of rectangles 4(iii)

6 Simplification leading to sum of area of rectangles 4(iv)

7 Sequence of approximation to area under graph 6(i) – 6(iv)

8

9

Limit of sequence equals area under graph 6(vi) – 6(vi)

Limit of sequence of fraction and from general term 6(viii) – 6(x)

10 Complications of Area Calculation 7

The clinical method was used to obtain responses from the students. The raw data consists of tape-recorded feedback as well as written solutions given by the students. The data was analyzed for structural, executive and arbitrary errors in the control group which was then compared to the experimental group for structural errors only. If there were more than one type of error in a task we included all errors in our assessment. If errors that could not be classified uniquely we looked at the frames in a similar task. From the case studies of WBL calculus students, it appears that students have different understandings of the integral even though they have been through the same learning environment.

The student activity shows that blended learning offers a possible conceptual approach based on visualization that advocates ‘deep-learning’ amongst the experimental group. The control group was unable to integrate complicated

functions analytically or graphically. The modified Orton’s test chosen to suit Engineering and Science students at a University of Technology are linked to numeric, symbolic and visual displays that students can enact with.

RESULTS:

The data consisted of student task protocols. Below in Table 2, errors in student responses were categorized as structural, executive and arbitrary.


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Table 2: Analysis of Errors

Items Structural executive Arbitrary

control Experimental

1 9 2 4 0

2 10 2 6 3

3 32 5 3 2

4 10 2 1 0

5 33 7 3 0

6 31 18 2 0

7 32 26 2 0

8 25 19 2 12

9 28 22 2 0

10 16 11 0 0

The experimental group made fewer structural errors and executive errors than the control group. A graphical representation follows.

Graph 1: Histogram of the error scores for control and experimental groups.


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Exemplars of structural, executive and arbitrary errors using tasks 3 and 7. Other tasks and exemplars can be found on http://olc.dut.ac.za:8900

TASK 3: HEIGHTS OF RECTANGLES UNDER GRAPHS

The diagram shows part of the graph of y = x2. The area shaded is the area under the graph for x = 0 to

x = a

y

y=x2

In order to obtain the shaded area we can use a Riemann method. For example, we can divide AO into six equal bases and then draw rectangles on five of the six bases, shown below. The height of each rectangle is the y-value for the equation at the left hand end of each base.

y

y=x2

x

i) What is the width of the base of each rectangle?

ii) List the height of the five rectangles?

iii) How can you obtain the areas of the rectangles?


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DISCUSSION

In this task the problem concerning areas “under” the curves was introduced. The question led students step by step through the elementary algebra involved in calculating the areas of the rectangles. Further questioning was required to query how initial incorrect answers were obtained; in the course of explaining some students were able to make the corrections. With reference to the additional note above, some students could not obtain answers for Task 3(ii) and would subsequently not be able to answer the identical question in different situations in following tasks.

Arbitrary Error: Answers should be (a/6)2, (2a/6)2, (3a/6)2, (4a/6)2, (5a/6)2 with maybe the initial insertion of 0 at the beginning and/or a2 at the end

Executive Error: either some or all of the answers initially incorrect and of a form different from the elementary form ka2/n

Structural Error: Unable to appreciate that the heights were to be obtained by squaring x- increments, or unable to begin

TASK 7: COMPLICATIONS IN AREA CALCULATIONS

Calculate the shaded areas below, where possible. If it is not possible, explain why not.

(a) y

y = ‐2x + 2

2

x ‐1 0 1


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(b)

y = 2x – x2

0 2 3 x

(c)

y=1/x2

‐1 2 x

(d)

2

x = 2y – y2


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DISCUSSION

The aim of this task was to probe deeply into students’ understanding of using integration to calculate areas under curves. The usual complications to which this procedure gives rise are all included. If students’ were able to compute the final answer once, in part (a), it was thought that it was not necessary to demand this again, and that parts (b), (c) and (d) should test the method and not computation. Supplementary questions were asked to clarify responses. Part (c) goes beyond the scope of the course and was intended for students with a good understanding of areas “under” curves who will see the problem.

Answers:

(a) Yes, dxxxA )22( 2

1

1

+−= ∫−

= [x3/3 – x2 + 2x]-11

= (1/3 – 1 + 2)- (-1/3 – 1-2)

= 14/3

(b) Yes, “But you have to work the areas above and below the axis separately and then add the numeric values …. because areas below the axis are taken as negative, so the integral from 0 to 3 gives area difference and not the sum.”

(c) No, “It is not possible to obtain the answer (by integrating from -1 to 2)…. because of the discontinuity (or because the area is not bounded, finite or enclosed or because the answer is infinite).”

(d) Yes, dyxA ∫=

2

0 , where x = 2y – y2, or

alternate understanding shown.

Arbitrary Errors: All four answers obtained maybe with possible eliminations but with a deep understanding of the integral concept. Executive Errors: Answers to (a) and (b) only correct Structural Errors: No answers obtained due to gaps in knowledge that indicate incorrect/incomplete frame retrieval.

CONCLUSION

Most of the errors were structural and executive and only a small portion, arbitrary. Structural errors found in the study may be due to student’s rote/mechanistic learning of elementary integral calculus – lack of understanding of concepts since pre-knowledge frames were not developed. Students in the blended learning course made fewer structural errors than the control group. The experimental group scores indicate a deep cognitive understanding of integral calculus compared to the control group. This is so, since the WBL course allowed students the resources and flexibility to understand/analyze a concept/problem in different ways which allowed for reflection and discovery learning.


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References

[1] Naidoo, R (1996). Technikon students understanding of differentiation.

SAARMSE. 96 Proceedings, University of Stellenbosch [2] Tall, D. (2002). Using Technology to

Support an Emboided Approach to Learning Concepts in Mathematics, First Coloquio de Historia e Tecnologia no Matema’tica at Universidade do estado do Rio De Janiero

[3] Piaget, J. (1977). The origin of intelligence of a child. (Neuchatel- Paris: Delachaux et Niestle)

[4] Dubinsky, E. (1991). Reflective Abstraction in Advanced Mathematical Thinking, (D. Tall, ed.), Kluwer, 95-126 [5] Sfard, A. (1992). “On the dual nature of mathematical conceptions: Reflections on the processes and objects as different sides of the same coin’. Educational Studies in Mathematics, 22, 1-36 [6] Davis, R.B. (1984). Learning Mathematics. The cognitive science Kent: Croom-Helm Ltd [7] Ausubel, D., John H, (1969). Psychology in teacher preparation: Toronto: The Ontario Institute for Studies in Education [8] Papert, S. (1980) Mindstorms. New

York: Basic Books [9] Smith, D.A. & Moore, L.C., (1991).

Project Calc: An Integrated Laboratory Course. In C. Leinbach et al. (eds.), The Laboratory approach to Teaching Calculus, MAA Notes, 81-92,

Washington DC: MAA

[10] Naidoo, K., and Naidoo, R. (2007). First year students understanding of elementary concepts in differential calculus in a computer teaching environment. Journal of College Teaching and Learning: Vol 4, No. 8, 99-114 [11] Kashihara, A., Uji’I, H., & Toyoda, J. (1999). Reflection support for learning in hyperspace. Educational Technology, 39, 19-22. [12] Lin, X., Hmelo, C., Kinzer, C.K., & Secules, T.J. (1999). Designing technology to support reflection. Educational Technology Research and development, 47, 43-62 [13] Donaldson, M (1963). A study of children’s thinking. London: Tavistok Publications [14] Orton, A. (1981). Understanding

elementary calculus. Phd thesis. University of Leeds.

[15] Oehrtman, M. (2003). Strong and Weak Metaphors for Limits, Psychology of Mathematics Education.


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Errors made by First Year Students in an Integral Calculus ... · Errors made by First Year Students in an Integral Calculus Course ... Representations in calculus/analysis ... stated