LITERACY IN MATHEMATICS – A CHALLENGE FOR TEACHERS IN THEIR WORK WITH PUPILS

Bodil Kleve Oslo University College

This paper focuses upon the importance of aspects of teachers’ mathematical knowledge in their work in making mathematics available for pupils. As theoretical background, Bruner’s theory about two modes of thought, syntagmatic and paradigmatic, is presented. The Knowledge Quartet is used as analytical tool in analysing transcripts from classroom observations before discussing the episodes from a literacy perspective. Finally, suggestions are made to put more weight on mediating tools, rather than on basic skills in the learning process.

BACKGROUND Our project “The Didactic Challenge of New Literacies in School and Teacher Education” focuses on literacy both in classrooms and teacher education in the perspective of didactics in several subjects. In Kleve & Penne (2010), which was a joint presentation in two subjects, Norwegian and mathematics; we explored the expression “Knowledge is made, not found” (Olson, 2001, p. 104), and discussed how meaning is constructed. Drawing on Vygotsky and particularly on Bruner’s distinction between syntagmatic and paradigmatic modes of thinking (Bruner, 1986), we suggested some didactical perspectives which we claim have been obscured and ignored in recent years’ focus on basic skills. Also, in Norway student centred pedagogy with the slogan ‘responsibility for own learning’ has been focused over time. Thus teachers’ expertise and knowledge have been taken less into account. In this paper I will follow this line and discuss the importance of the two modes of thought in teaching and learning mathematics, and how a teacher Cecilie, took her pupils with her in using these modes of thought in the mathematics classroom. To emphasise the importance of teachers’ expertise and knowledge I have used the Knowledge Quartet (Rowland, Huckstep, & Thwaites, 2005) as an analytical tool to find out how aspects of Cecilie’s mathematical knowledge surfaced in her work in taking her pupils with her in using the two modes of thought in mathematics. The importance of these aspects of a teacher’s knowledge is discussed and implications for teacher education are suggested. The Knowledge Quartet has four broad dimensions; Foundation, Transformation, Connection and Contingency. Foundation is the mathematical knowledge the teacher has gained through his/her own education, it is knowledge possessed and which can inform pedagogical choices and strategies. It is the reservoir of pedagogical content knowledge you draw from in planning and carrying out a lesson and thus informs pedagogical choices and strategies. Transformation focuses on the teacher’s capacity to transform his or her foundational knowledge into forms which can help someone

else to learn it. It is about examples and representations the teacher chooses to use. The third category, Connection, binds together distinct parts of the mathematics and concerns the coherence in the teacher’s planning of lessons and teaching over time and also coherence across single lessons. Contingency is the category which concerns situations in mathematics classrooms that are impossible for the teacher to plan for; the teacher’s ability to deviate from what s/he had planned and the teacher’s readiness to respond to pupils’ ideas are important classroom events within this category.

TWO MODES OF THOUGHT Vygotsky made a strict distinction between two levels of mental functioning:

The lower level he characterized as natural, biological, causal and shared with nonhuman animals. The higher mental processes he portrayed as representational, sociohistorical, linguistic, voluntary, conscious and distinctively human (Olson 2001, p. 107).

As a development of Vygotsky’s distinction between “lower” or natural mental functions and “higher” or cultural functions, Bruner (1986) introduced paradigmatic mode and syntagmatic mode as “two modes of cognitive functioning, two modes of thought, each providing distinctive ways of ordering experience, of constructing reality” (p. 11). The syntagmatic mode of thought is basically related to linear time, and has primarily a narrative character. It is told from a subjective point of view and does not require truth, but rather a general probability or verisimilitude, referring to life in culture. This everyday thinking communicates an experienced world, and communicating in this mode means that the narrative structure is the most pervasive cognitive structure. It is merely based on tacit knowledge and acquired as part of the communication in the family and in our everyday life, and thus slips easily into the mind. We all live in an “obvious” discursive world and we are linked together in communities through common discourses. Syntagmatic thinking is useful and necessary in our lives. From a didactical perspective however, this way of thinking can lead us only into obvious and tautological reasoning. We therefore need something more which can make us more conscious of other opportunities than what is obvious. Paradigmatic mode of thought is different. The mind is more resistant this way of thinking. It is to reason about universal aspects of meaning, and conclusions which are valid beyond time and context are focused. Bruner (1986) writes:

[It] attempts to fulfil the ideal of a formal, mathematical system of description and explanation. It employs categorization or conceptualization and the operations by which categories are established, instantiated, idealized, and related one to the other to form a system (p. 12).

The paradigmatic mode deals in general cases and uses procedures in testing for empirical truth, and consistency and non-contradiction are required.

According to Bruner, the narrative structure acts as a mediating tool in the syntagmatic mode whereas in the paradigmatic mode concepts and metaphors are important mediating tools. For mathematics this means that exercising skills and methods (procedural knowledge) takes place in a syntagmatic mode of thinking, whereas generalisations, reasoning beyond time and context and building concepts (conceptual knowledge) takes place in a paradigmatic mode of thinking. Bruner emphasised that the two modes of thought will never act “alone”. We all use them both continuously, however to different extent depending on the context. Science has often started out as hypotheses in everyday discourse and then been developed scientifically through a paradigmatic effort, a perspective which actively and consciously strives to transcend the narrative structure in the syntagmatic discourse. We therefore need continuous interaction between the two modes of thought to understand both everyday life and science. Bruner (1986) put it this way:

We all know by now that many scientific and mathematical hypotheses start their lives as little stories or metaphors, but they can reach scientific maturity by a process of conversion into verifiability, formal and empirical, and their power at maturity does not rest on their dramatic origins (p. 12).

As an example of the importance of both ways of thinking in the process of constructing concepts, David Olson (2001) used a child’s learning to represent nothing with something, a zero. He claimed that concepts cannot pass smoothly from culture to mind, and therefore for a child to compare what is in his consciousness, “no cats”, with what is in the culture, “the zero”, is problematic. Olson suggested that both the knowledge of absence and the knowledge of a sequence of numerals (learned by rote?) have to be available in the child’s consciousness.

Learning then, consists of applying the memorized sequence of the numerals to the prior knowledge of absence. In so doing the child is not merely making explicit the known but forming a concept applicable to all sorts of nothings (Olson, 2001, p. 113).

According to Bruner, knowledge is made and not found, even for scientists. Olson (2001) put it this way: “Children, like adults, make what they find” (p.113). For teaching this means that teaching defines the problem space from which knowledge is constructed by the children and constructivism makes competent teaching significant. Competent teachers are needed to determine what children have found from which knowledge can be made rather than to “cover” a settled curriculum. Mediational means are central in Vygotsky’s (1986) theory. The task for the competent or expert teacher is to make sure that pupils get access to the language or mediational means which can open up new aspects of the world, and develop their thought. But how does transformation from mediational means to inner thoughts take place, because children do not find new knowledge, they make it? Discussing this question Olson (2001) elaborated how knowledge construction takes place:

Knowledge is constructed I have suggested, when the learner can take one set of concepts and use them as a model for thinking about some other sets of events. My example was seeing [ ] nothing [no cats] in terms of a numeral [zero]. The basic mode is metaphor, abduction, narrative construal, inference to the best representations rather than things. These representations, accumulated archivally in maps, charts, books and computer programs provide many of these most important models (p.113).

Thus importance is put on metaphors; through the metaphor, zero became a part of the child’s thinking. According to Olson, it is easier to understand Vygotsky through metaphor theory. In her book Thinking as Communication Sfard (2010) discussed the same aspects from Bruner’s theory and she put weight on discourse and metaphors in science and everyday life:

Jerome Bruner describes the transition from a metaphor to its operationalized, “scientific” version in a beautifully metaphorical way. After stating that metaphors are “Crutches to help us get up the abstract mountain,” The author notes: Once up, we throw them away (even hide them) in favor of a formal logically consistent theory that (with luck) can be stated in mathematical or near-mathematical terms (p.41).

How do teachers ‘re-present’ their mathematical knowledge to pupils in terms of examples, demonstrations, illustrations, activities and questions? What crutches are the pupils given which they can throw away? In the next section of this paper I will present examples from mathematics classrooms and discuss the importance of aspects of the teacher’s mathematical knowledge, with emphasis on the transformation aspect, in encouraging pupils to think in both modes of thought, syntagmatic and paradigmatic. For pupils to learn to think in a paradigmatic mode, to form new mathematical concepts on which they can perform operations, their interaction with more knowledgeable peers (the teacher) who can use the concept in a paradigmatic mode, is of decisive importance. This stresses the importance of both modes of thinking, syntagmatic and paradigmatic, in pupils’ learning process. In our curriculum, LK06 (Kunnskapsdepartementet, 2006) “basic skills” are emphasised. From a literacy perspective, it is even more important for the pupils to learn how to use mediating tools to understand and develop new insight and to form new mathematical concepts. They need tools of the mind to extend mental abilities. In this work, aspects of the teacher’s mathematical knowledge play a crucial role.

EXAMPLES FROM THE MATHEMATICS CLASSROOM I will now present extracts from two mathematics lessons in 10th grade with a teacher, here called Cecilie. The data are taken from Kleve (2007), a study in which I investigated how mathematics teachers in lower secondary school implemented a curriculum reform in Norway. Four mathematics teachers were observed over a

period of three months, and the mathematics lessons were audio recorded and transcribed. The two episodes which I present below are selected because they illustrate how shifts between syntagmatic and paradigmatic modes of thought took place in Cecilie’s lessons. Episode 1 Cecilie had drawn a right angled triangle on the board and asked the pupils to have their calculators ready:

1 Cecilie: What kind of a triangle do we have there, Mikkel? 2 Mikkel: Right angled triangle 3 Cecilie: Then we know the lengths of the two sides. [ ]. How can I find the

third side, Leif? 4 Leif: You have to use Pythagoras 5 Cecilie: Yes, have to use Pythagoras. Let us try to do that with this triangle. If

we call this side for x, Leif? 6 Leif: Must take x2= 3.62 + 4.82 (Cecilie wrote it on the board) 7 Cecilie: Yes, let us calculate that. Three point six squared is? 8 studs: Twelve point ninety six 9 Cecilie: Four point eight squared is? 10 pupils: twenty three point o four 11 Cecilie: Twenty three point of four (wrote it on the board). The sum of these

numbers is? 12 Pupil: Thirty six 13 Cecilie: It is thirty six 14 Pupil: It makes six 15 Cecilie: Yes okay it became six long. This was lots of calculations. If we look

at the numbers here, we could have simplified it. Is it like, here I have added one point two, and if I add another one point two I’ll get the third side? Is that a rule which always works? Let us take another example. New triangle (She drew a new triangle on the board with sides like 7.5 and 10). If that is seven point five and that is ten, will that one (the hypotenuse) be twelve point five? Can you check if it works?

16 Baard: Yes 17 Cecilie: That worked as well. Your exploratory task is now: Does it always

work? Does it work for any length? Later in the lesson, when the pupils had found counterexamples to the teacher’s “rule”, that in a right angled triangle, you can add the difference between the two smaller sides to get the hypotenuse, Cecilie said:

If it had been that easy in all cases, we wouldn’t have had this rule (Pythagoras’ sentence). Then I’d tricked you to calculate a lot. Next question is then: why does it work

on these sides? What is special with the numbers here? Why does it work with my examples? Take a look at the lengths of the sides.

Towards the end of the lesson, when they had worked out that the ratio between the smaller sides had to be ¾ , they investigated further. They went on to Pythagorean Triples, taking 3-4-5 as a starting point and Cecilie presented Euclid’s formula p2-q2, 2pq, p2+q2, which makes Pythagorean triples when p and q are whole numbers, and makes the special 3-4-5 triple when p=2 and q=1. Before discussing this episode from a literacy perspective, I will investigate aspects in the KQ of the teacher’s mathematical knowledge which became evident in the episode and especially when Cecilie took her pupils with her in what I see as a shift from a syntagmatic to a paradigmatic mode of thought. Since there were no contingent moments in this episode, (which may be because she didn’t open up for it), I will focus on the three other aspects, Foundation, Transformation and Connection of the KQ. I will start with the teacher’s foundational knowledge, the knowledge from which she chose examples, illustrations and kinds of questions asked. First of all, Cecilie demonstrated knowledge of Pythagoras’ sentence and the sentence to be investigated: ‘In right angled triangles where the ratio between the two smaller sides is ¾, you can take the difference between the two smaller sides and add to the largest to get the length of the hypotenuse’. Also the teacher’s knowledge of Euclid’s formula was demonstrated. This shows that the teacher in this case had the foundation knowledge required for the topic of this lesson. How did Cecilie make this knowledge accessible for her pupils? A feature of the transformation aspect of Cecilie’s knowledge was that she related what was new to something well known. She started off with a right angled triangle and Pythagoras’ Theorem. That way she used for the pupils a well known theorem to explore a new sentence. From her knowledge of the sentence to be proved (which was part of her foundational knowledge) Cecilie chose examples which made the sentence true. Also when introducing Euclid’s Formula, she used the now for the pupils familiar 3-4-5 triple to exemplify. To incorporate the pupils in this episode, she asked questions for them to answer. In this episode there was a connection from starting off by calculating the hypotenuse in one triangle then in another triangle, through finding a counterexample to explore in what cases the sentence was true. Cecilie then linked to Euclid’s formula which shows that she was in a position which made her able to make connections between Pythagoras’ Theorem, the sentence to be explored and Euclid’s formula. Looking at this episode from a literacy perspective, we will see that the course of the lesson was in a syntagmatic mode until turn 15. Throughout turns 1-14 the teacher had funnelled the pupils through calculations, stage by stage, using a well known

sentence (Pythagoras). Pythagoras’ sentence was used in calculating the hypotenuse in a triangle with the shorter sides being 3.6 and 4.8, and in another triangle with sides like 7.5 and 10. Then in turn 15 it shifted to: “Is that a rule which always works?”And the question was restated in turn 17: “Does it always work? Does it work for any length?” All of a sudden they were challenged with the word why. And she gave them a hint in saying: “Take a look at the lengths of the sides”. One strategy was to find a counterexample, which is a well known way of proving that something is not true. The next stage was to investigate in what cases the “rule” worked. That way, the teacher took her pupils with her towards a paradigmatic mode of thinking which in mathematics incorporates generalisations. In what cases it always works. To prove that something works, it is not sufficient to find many, thousands, examples with numbers in which it works. Using algebra is a way of proving, generalising and reasoning beyond time and context. In recent curricula in Norway, the algebra content is reduced, and more weight is put on “basic skills”. Thus mathematics in school has moved towards a more linear discourse or syntagmatic mode. Less weight has been put on vertical generalisations and paradigmatic thinking. In turn 17 above, “does it always work?” the pupils were challenged into another mode of thought, the paradigmatic, than in which they had been so far. Until then they had only carried out simple calculations on their calculators and answered the teacher’s closed questions. The examples the teacher used, the questions she asked, and the connections she made acted as mediating tools for the pupils. Episode 2 Let us take a look at a similar episode from a mathematics lesson with the same teacher two weeks later. The starting point was a task from a test. Like in episode 1 there was a sudden break in discourse, or a shift in mode of thinking, from syntagmatic to paradigmatic. The generalisation question was in the text. However, on the test the pupils had worked out the task with concrete examples. The teacher took those as a starting point and thus offered the pupils mediating tools to solve the task which was: “The length of a rectangle is increased by 15% and the breadth is reduced by 20%. How many percent does the area of the rectangle change?” Thus the relation to something, for the pupils known, a reified object, was established, before a similar break or shift in discourse as in episode 1, turn 17 took place.

1 Cecilie: The length in a rectangle is increased by 15% and the breadth is reduced by 20% how many percent does the area of the rectangle change? And the way everybody who answered that task did it, was that you chose a rectangle. Let us take this rectangle in which the length is 20 and the breadth is 5 (she drew it on the board). How big is the area of the rectangle?

2 pupils: Hundred 3 Cecilie: It is hundred. The area is hundred. And then the task was: The length

is increased by 15%, how much will the new length be?

4 Leif: Twenty three 5 Cecilie: Very good, Leif, very good mental calculation. The new length

becomes twenty-three, and when the breadth is reduces by 20 %, what is the new breadth then?

6 Baard: Four 7 Cecilie: Yes, and what is the area? 8 Baard: It is ninety-two, right? It’s a guess. 9 Cecilie: Right. It is not a guess, it is a mental calculation. How many percent is

the area reduced? 10 Baard: Eight percent 11 Cecilie: Yes, it is. If it was hundred percent earlier, then it is ninety-two

percent now, an eight percent reduction. Then the question is: are you sure it is applicable for other rectangles as well? This was for one special rectangle.

Also here I will discuss aspects of the KQ before studying the episode from a literacy perspective. In this episode Cecilie demonstrated that she knew that length 20 and breadth 5 would make the area 100, that the change in area and the change in percent then would be the same. Also, 15% of 20 and 10 % of 5 gave two whole numbers (3 and 1) which again gave them two whole numbers with which to calculate further, is worth to notice. This knowledge, which is a feature of the foundation aspect of the teacher’s mathematical knowledge, informed her choice of measures in the example in this episode. Her choice of using a 100 rectangle was informed by her foundational knowledge which included that the change in area and percentage change would have the same value. This demonstrates the transformation aspect of her knowledge. Also that she took the pupils’ answers to the task as a starting point, which had been choosing a concrete rectangle, demonstrates how she transformed or “re-presented” her knowledge of percentage change to be available for the pupils. She offered the pupils a 100 rectangle as a mediational mean to percentage change. Thus a relation within mathematics was used, which demonstrated the KQ’s connection aspect of Cecilie’s knowledge. In this episode, like in episode 1, they started in a syntagmatic mode, working with concrete examples, and as in episode 1 (is that a rule which always works?), we can see a shift in discourse, a shift in thinking: Is it applicable for other rectangles? A move to the paradigmatic mode of thought was initiated. After this episode, later in the lesson, the teacher emphasised that it on the test had not been sufficient to show the percentage change for concrete rectangles. She required a way to find out if it applied to all rectangles. A shift to a paradigmatic mode was taken. In mathematics, generalisations play a crucial role in this mode, and use of algebra is a way of carrying out generalisations. After the episode presented above, a pupil suggested using algebra: 1.15a•0.8b=0.92a•b, which is 0.08=8% change related to a•b. That way they had proved that the change was 8% beyond all concrete examples.

In this lesson Cecilie referred to the lesson two weeks earlier in which they

concluded with that ( )2

22

35

34

=+

aaa . This elucidates another feature of the

connection aspect of Cecilie’s mathematical knowledge. There was a connection, a relation, between the two lessons, a relation the teacher made the pupils aware of. Such connections can act as mediational means in pupils’ learning, and are important in strengthening their paradigmatic thinking.

DISCUSSION In this paper I have showed how Cecilie, in two lessons, took concrete examples as starting point, and the examples she used acted as meditational means for the pupils to generalize and to move from a syntagmatic mode to a paradigmatic mode of thinking. I have used the Knowledge Quartet as analytical tool and suggest that aspects of teachers’ knowledge, with emphasis on transformation and connection are of crucial importance in the work with pupils in both modes, syntagmatic and paradigmatic, of thinking. In episode 1 the well known Pythagoras’ Theorem was used as ‘crutches’ to find lengths of sides. In episode 2 a 100-rectangle was used as crutches to find percentage change. This implies that there is “something” needed which can serve as meditational means to make pupils more conscious for other opportunities than what is obvious. In concrete rectangles it was for the pupils obvious how big the percentage change would be. That they had done on the test. They had been in a syntagmatic mode of thinking. But for the pupils it was not obvious that the percentage change was the same for all rectangles; that the change was valid beyond concrete rectangles. In earlier research I have discussed how pupils’ difficulties, which surfaced in contingent moments, in the conceptual understanding of fractions greater than one can be traced back to the transformation aspect of the teacher’s mathematical knowledge (Kleve, 2009, 2010). From a literacy perspective, I suggest that the examples and illustrations the teacher used in that lesson was more of a hinder than a help for the students to think in a paradigmatic mode. They failed in serving as meditational means between concrete conceptions of something more than one and improper fractions. In Kleve (2009b) I discussed the teacher’s difficulties in illustrating improper fraction, and suggest that the focus on fraction as part of a whole, the “easiest” way for pupils to understand a proper fraction, also acted as a hinder in illustrating an improper fraction. They were stuck in the syntagmatic mode of thinking. Based on this, I suggest that aspects of teachers’ mathematical knowledge, are crucial factors and therefore important to focus upon in teacher education. What are student teachers’ choices of examples and illustrations and questions informed by? How do questions, examples and connections they choose influence pupils’ thinking and learning in mathematics? For teachers it is of great importance being conscious

of the two modes of thinking and consequently not let that work be overshadowed by “responsibility for own learning” and focus on basic skills.

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University Press. Kleve, B. (2007). Mathematics teachers' interpretation of the curriculum reform,

L97, in Norway. Doctoral Thesis, Høgskolen i Agder, Kristiansand. Kleve, B. (2009). Aspects of a teacher's mathematical knowledge on a lesson on

fractions. Paper presented at the British Society for Research into Learning Mathematics, Loughborough.

Kleve, B. (2010). Contingent Moments in a lesson on fractions. [Current Report]. Research in Mathematics Education, 12(2), 157-158.

Kleve, B., & Penne, S. (2010). Literacy - en fagdidaktisk utfordring. Et eksempel med utgangspunkt i norsk og matematikk. Paper presented at the Fou i Praksis, May 10-11, 2010.

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