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Teacher Development: An international journal ofteachers' professional developmentPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/rtde20
The effects of an overt constructivist approach tolearning mathematics and its subsequent effects onclassroom teachingM. Simmons aa University of Stirling , United KingdomPublished online: 19 Dec 2006.
To cite this article: M. Simmons (1999) The effects of an overt constructivist approach to learning mathematics andits subsequent effects on classroom teaching, Teacher Development: An international journal of teachers' professionaldevelopment, 3:2, 173-196, DOI: 10.1080/13664539900200082
To link to this article: http://dx.doi.org/10.1080/13664539900200082
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Teacher Development, Volume 3, Number 2, 1999
173
The Effects of an Overt Constructivist Approach to Learning Mathematics and its Subsequent Effects on Classroom Teaching
M. SIMMONS University of Stirling, United Kingdom
ABSTRACT A Master’s unit in mathematics education was given to three practising primary teachers. Constructivism as a theory on which to develop a flexible pedagogy was a central theme and the style of presentation of workshops was guided by constructivist principles. Contact time was 20 hours spread over 10 weeks. The course contained a number of different aspects relating to mathematics teaching and learning some of which provided research data: (1) discussion about the nature of constructivism in mathematics learning, and its implications for teaching; this centred around seminar papers on aspects of constructivism; (2) a workshop each week that focused on a rich mathematical topic designed to provoke mathematical activity and interaction and to encourage reflection on that activity; (3) an extended investigation of a mathematical situation; and (4) classroom implementation of some aspect of (3) together with a report of its implementation and an evaluation of children’s understanding. A combined individual and social constructivist approach to learning mathematics by primary teachers is described. some evidence is presented of changes in belief held by teachers about the nature of mathematics and the subsequent changes in classroom practice and expectations of children’s learning.
Constructivist principles, whether they derive from a radical or social perspective, appear to have captured the imagination of educational researchers by providing a forum through which teaching and learning can be enhanced. Much philosophical debate has centred around the various forms of constructivism and their implications for educational practice. Radical constructivism, as a theory of knowing and in particular as this relates to mathematics education, has been thrown into the spotlight by von Glasersfeld (1991, 1995). Research on how mathematics pedagogy may by reconstructed
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from a constructivist perspective is also well documented (Steffe, 1983, 1991, 1995; Steffe & Wiegel, 1992; Steffe & Kieren, 1994; Simon, 1995).
Constructivism works from the premise that constructing meaning and understanding are two sides of the same coin; that in making our own personal constructs and reflecting upon them, we build understanding. The theoretical debate has in part focused on how and under what conditions these personal constructs take place. From this debate (see, for example, Steffe & Gale, 1995) some common ground can be found between the radical constructivists and the social constructionists, which can guide good practice. Both, for example, recognise the difficulties in trying to tie the notion of understanding to some absolute mathematical concept. Both recognise the importance of accepting that understandings are ultimately based upon individual constructions whether gained through individual or group experiences, and that teaching should encourage appropriate interactions that allow a sharing of ideas and further constructions to be made. So both share some ideas about how knowledge is generated and both question the view that knowledge is built up through dispassionate observation:
[constructivism] ... starts form the assumption that knowledge, no matter how it be defined, is in the heads of persons, and that the thinking subject has no alternative but to construct what he or she knows on the basis of his or her experience ... The experience and interpretation of language are no exception. (von Glasersfeld, 1995, p. 1)
On the most abstract level, we might say that what we count as knowledge are temporary locations in dialogic space – samples of discourse that are accorded status as ‘knowledgeable tellings’ on given occasions. More concretely, knowledge is in continuous production as dialogue ensues. To be knowledgeable is to occupy a given position at a given time within an ongoing relationship. (Gergen, 1995, p. 30)
Radical constructivism and social constructionism also question the view that individual minds can accurately reflect the true nature of an independent world but in so doing acknowledge a belief about how such a world is assembled:
If ... people look through distorting lenses and agree on what they see, this does not make what they see any more real – it merely means that on the basis of such agreements they can build up a consensus ... (a consensual domain) ... in certain areas of their subjective experiential worlds. (von Glasersfeld, 1991, p. xv)
The personal success of generating accepted contributions – or, as von Glasersfeld has coined it, to arrive at viable constructions – rests on both the participants’ active engagement in the social practice of mathematizing as well as on the qualities of the social interaction. It is only through becoming actively involved that the experienced resistance from the social group can develop selective power and force for the subject. (Bauersfeld, 1995, p. 152)
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For the purposes of this article a consensual domain is taken to be an area of agreement derived from reflexive activity and giving rise to meaningful communication (Richards, 1991). Richards points out that a consensual domain is a necessary but not sufficient condition for successful communication and in a broader sense the consensual domain of school mathematics is put forward as an example about which there is general agreement but in which meaningful communication does not take place. Despite the many initiatives in numerous countries worldwide over the last 15 years this is sadly still true.
As long ago as 1982, Mathematics Counts (the Cockcroft Report) drew attention to the need for discussion between teacher and pupils and between pupil and pupil. The Department of Education and Science (DES) said in 1985, ‘The quality of pupils’ mathematical thinking as well as their ability to express themselves are considerably enhanced by discussion’. Of course, research into the beneficial effects, or otherwise, of grouping has gone on over the years as a means of developing meaningful dialogue between pupils and has not been wholly part of the constructivist school. For example, Barnes (1976), with my own brief interpretation added Simmons (1993), lists five important factors for effective group organisation:
1. Feeling of competence by pupils: this is an outcome of the composition of groups and the pupils’ perception of the teacher’s role in the whole process. 2. Common ground: pupils’ abilities to restate and reorganise presented situations into presentations they can all share (a consensual domain); the use of language at the pupils’ level is all-important at this stage. 3. Focusing: the teacher’s ability to present a problem framework that provides the kind of help needed to start pupils thinking and to start asking their own relevant questions. 4. Pace: basically allowing time for exploratory talk to take place; again a balance is required between allowing time for learning to take place and time to show that the job has been done. 5. Making public: the need to refine language appropriate to a wider audience. For example, groups reporting back to the whole class would require a spokesperson to collect and present findings of the group using more explicit language so that other groups less familiar with the situation can appreciate the points being made.
These five points to my mind sit easily with social constructionism that ‘In its radical form, ... does not commence with the external world as its fundamental concern or with the individual mind but with language’ (Gergen, 1995, p. 23) and where, unlike the radical constructivists, importance is given to other thinkers beyond that given to other viable constructs. The difference between others and objects as offered by the social constructionists is that we construct others out of ourselves and others contribute to the image of ourselves:
Students have to learn mathematics as social knowledge: they are not free to choose the meanings they construct. These meanings must not only be efficient in
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solving problems, but they must be coherent with those socially recognized. (Balacheff, 1991, p. 89)
Clearly, how we conceptualise knowledge affects our choice of educational process. If we take the view that mathematics is ‘... both a cognitive activity constrained by social and cultural processes and a sociocultural phenomenon that is constituted by a community of actively cognizing individuals’ (Wood et al, 1995, p. 402), then we must adopt teaching strategies that fully recognise the opportunities and constraints afforded by such processes. It effectively means that if teachers are to make appropriate choices about the strategies they use then they have to be aware of the intended outcomes as they relate to their particular discipline. As Bereiter (1994) points out, there is nothing incompatible about the ideas put forward by constructivism and socioculturalism ‘except for the practical difficulty of doing both at once’. Cognitive and social process are then seen as complementary and necessary, but according to Bereiter not sufficient; a theme I shall return to in a moment.
When students attempt to solve problems in mathematics, it is clear that there are periods when they need to interact with other students, often in language that indicates uncertainty and is therefore imprecise, but which seeks a forum in which ideas may be validated. Such interaction may be supported to a lesser or greater degree by objects or symbols, the use of objects often giving way to symbolic representation and manipulation as a more refined view of the problem emerges. It is also clear that there are periods when students need time to think and contemplate (interact with?) objects or symbolic representations in order to deal with internally perceived discrepancies or to move forward towards a particular solution. In mathematics, like other subjects, this often means making connections through the development and interpretation of symbols that represent the problem domain. However, the frequency and style with which these circumstances prevail is, I believe, unique to mathematics.
For example, during children’s investigation into Pythagoras’s Theorem by a variety of methods, an aim of the lesson will be to encapsulate the findings in some general way, which might be through the use of language or through the use mathematical notation, viz. c2 = a2 + b2. However, in order to develop a wider concept of the theorem, this representation can be manipulated in a simple way to read kc2 = ka2 + kb2, which when translated back into the geometric problem domain gives rise to variants of the more familiar theorem. However, where is such knowledge located?
The debate between radical and social constructivists does not appear to provide a clear answer, locating it in the head or in the social interaction that takes place. Yet it would be difficult to deny the existence of the Pythagorean Theorem if it were not located in either of these two locations. One could argue that such a mathematical relationship exists whether or not we are capable of understanding or expressing it. ‘Knowledge in this objective sense is totally independent of anybody’s claim to know; it is also independent of anybody’s belief, or disposition to assent; or to assert, or to act. Knowledge in
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the objective sense is knowledge without a knower: it is knowledge without a knowing subject’ (Popper, 1972, p. 109). Perhaps this additional dimension, based on theories of knowledge long debated, in which theoretical systems, problem situations and critical arguments reside, which appears at odds with constructivism but which does not deny their value, is one that is required to help bring about balance in the redirection in school mathematics and to support the natural application of mathematics to real world problems.
It is generally agreed that specific guidelines for teachers wishing to change current practice are not readily available from the literature dealing with ideas about constructivism outlined above: ‘... there is no means by which practical derivatives can simply be squeezed from a theory of knowledge’ (Gergen, 1995, p. 29). It is also generally agreed that guidelines alone would be insufficient in bringing about change in the mathematics classroom; indeed, if we extend constructivist ideas to the professional development of teachers, one could argue that such guidelines are not only unnecessary but undesirable. However, teaching experiments have been conducted which aim to define and develop ‘a constructivist school mathematics’ based on the constructivist principles outlined above (Cobb & Steffe, 1983; Cobb et al, 1990; Cobb et al, 1992; Steffe & Wiegel, 1992; Pirie & Kieran, 1992; Simon, 1995). Simon (1995) develops from research data a Mathematics Teaching Cycle in which teacher knowledge, activity and thinking, and changing ‘hypothetical learning trajectories’ are key components in decision-making. Such trajectories are considered ‘hypothetical’, since they are essentially based upon hypotheses about pupils’ understanding. In a similar vein, Steffe (1995) talks of ‘the subject’ constructing in order to bring about meaning to what is experienced (First-Order Models) and the constructs ‘observers’ make of the subject’s knowledge (Second-Order Models) and argues for the suspending of observer (adult) knowledge, whilst interacting with pupils so that the observer can concentrate on interpreting the pupils’ actions and tease out their knowledge. He also identifies this single yet far from simple act as a crucial aspect in operationalizing the tenets of constructivism in the classroom. ‘Thus children’s activity forms experiential realities of observers, and what the observers make of these realities is the single most important aspect of education. It is also the single most ignored aspect, which is unfortunate because what teachers make from children’s activity is precisely where education is failing world-wide’ (Steffe, 1995, p. 517). In short, what is crucial for effective teaching is not only teacher knowledge, but teachers’ knowledge of children’s knowledge, and the identification and modification of connections between the two.
Theory into Practice
What is self-evident in the above arguments is the vital importance of the teacher’s role in engaging in teaching/learning episodes with pupils and the way in which a teacher’s pre-existing knowledge may influence the learner’s
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construction of new knowledge (see Becker & Varelas, 1995, re: explicit and implicit orientating, p. 441).
A teaching experiment was devised which centred around a unit of study on mathematics education as part of a Master’s degree programme. Its design was guided by constructivist principles and based on the assumptions that changes in the way teachers operated would come about more readily if teachers (referred to as students on the course) were exposed to a number of different experiences relating to mathematics education that included, as well as problem-solving episodes, an ongoing open and developing debate about the ideas put forward by radical constructivism and social constructionism, and whether students could by identifying aspects of practice begin a process of restructuring. Its stated aims and objectives were as follows:
Aim: The unit will provide opportunities for participants to further their own view of mathematics teaching and learning and become more confident and reflective teachers. Objectives: 1. To provide opportunities for students to explore mathematical situations that disturb their current organisation of knowledge. 2. To strive to maximise opportunities for students to construct concepts through individual and group work. 3. To explore constructivist views of learning mathematics through reading and practice, as learner and teacher.
These were to be achieved through a series of seminars, workshops, private study and a school-based assignment. The role of course tutor was as researcher-teacher where, for example, Steffe’s first-order and second-order models would be operating: ‘... the teacher is aware of him or herself acting in and contributing to the development of a consensual domain while interacting with students’ (Steffe, 1995, p. 518).
The purpose from a research aspect was to study the effects of the course on:
(1) the student’s view of mathematics; (2) the student’s approaches to teaching mathematics; (3) the student’s expectations of pupils.
The course consisted of 10 2-hour sessions. The first hour in most sessions was devoted to discussions about constructivism, what it had to say, in this case, to three practising primary school teachers and how it could inform their practice. It was anticipated that regular exposure to and discussion of constructivist ideas together with opportunities for engaging in mathematical discourse, in exploring mathematical situations and solving problems would not only allow students to use the language associated with constructivist ideas (dangerous in itself), but also to experience first-hand what it meant to develop new understandings in ways that allowed free, but structured
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exploration of objects and symbols systems through actions, interaction, reflection and the use of language.
Evidence of the primary school teachers beginning to develop ideas and understandings for themselves as they took on the role of learner and teacher and the subsequent effects on their classroom practice and expectations of children’s learning is provided by the following data:
1. Pre-first session questionnaires. 2. Post-last session questionnaires. 3. Audio tapes of workshop discussions and their transcripts. 4. Copies of:
(a) reports of individual investigations into a mathematical situation; (b) reports of adapting these for implementation in their own classrooms; (c) tutors’ comments on their written analyses.
5. Course evaluations.
Pre-first and Post-last Session Questionnaires
At the beginning of the first session a short semistructured questionnaire was given to the students (Ann, Barbara and Claire) inviting them to put forward their views about mathematics and its teaching in school. It also asked them to indicate their agreement (1–5, 1 being closest in agreement) with the following statements about mathematics:
(i) beautifully formed and perfectly finished; (ii) a body of knowledge to be passed on intact; (iii) mathematics is out there waiting to be discovered; (iv) a way of knowing; (v) all mathematics is constructed and is fallible.
The same questionnaire was given to the students at the end of the last session.
Mathematics per se
It was clear from the initial responses that all three students had their own experiences of mathematics uppermost in their minds. Ann talked about mathematics being ‘challenging, interesting and very frustrating’, recognised it as a ‘lifeskills subject’ and was aware of the difficulties children have with its abstract nature. Barbara saw it as a clearly defined subject where answers were right or wrong – a tidy subject but challenging, which led to frustration and confusion, and that it needed to be taught well. Claire referred to her insecurity with mathematics as a child, particularly with number, but later began to understand the reasons behind the rules. She claimed that her own difficulties had made her more sensitive to the difficulties some children have.
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From the post-last session responses Ann’s view of mathematics was as a complex body of knowledge and understanding, skills and applications to be built by individuals in their own way and at their own pace. ‘It provides challenges and situations for learning for all abilities.’ Barbara saw it as a subject that ‘still has the power to demoralise and upset me when someone else has the answer and understanding and I don’t’, and was concerned that the subject and ‘mathematicians’ created feelings of inferiority with those less mathematically minded. However, she now viewed mathematics as something ‘out there waiting to be discovered and [that] true understanding must be constructed by the learner’ and as a fascinating and challenging subject that can give a real sense of achievement. Claire now saw mathematics as being all about ‘linking concepts, strategies, procedures and perceptions of fact with the development of language permeating through’. This was a use of language and expression of ideas I can not recall using during the course. These post-course responses appear to reflect various developmental states. Some of the words used appear to have moved little from the dialogue of the course sessions, whilst others, particularly the last statement, indicate that ideas have undergone some form of internalisation and come from genuinely held views.
For each of the statements (i)–(v) above the following table indicates their perceptions about mathematics pre- and post-course (pre-course figures in brackets).
Ann Barbara Claire (i) 4 (4) 4 (3) 4 (3) (ii) 5 (3) 5 (4) 5 (5) (iii) 2 (2) 2 (2) 3 (2) (iv) 1 (1) 1 (1) 1 (1) (v) 3 (5) 3 (5) 2 (4)
It appears from these results that mathematics as ‘a way of knowing’ remained the closest (position 1) to describing how each student felt about mathematics. Mathematics as a ‘body of knowledge to be passed on intact’ was placed firmly in fifth position by all three students after the course. Mathematics being ‘constructed and fallible’ was clearly not close to their thinking at the beginning of the course, but was given some credence by the end. There was, by far, the greatest change of belief towards this view.
Mathematics Teaching
Their views (pre-course) about how mathematics should be taught displayed more confident perceptions. Ann recognised that too much emphasis was given to basic computational skills and on getting the right answers, but that children needed to discover their own understandings through investigative work and integration with other areas of the curriculum. At the same time she recognised the pressures of time required to achieve this. Barabara focused
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more on the classroom management aspects, which catered for mixed ability classes, but again adopting a problem-solving approach using concrete materials to develop new concepts. A central issue for Claire was children’s mental capacity in number and the need for children to develop their own strategies for dealing with number, but also recognising the need ‘to teach some standard pencil and paper methods’. She also recognised the importance of problem-solving approaches in developing and applying understanding.
Their post-course views on how mathematics should be taught were again clearly articulated, but they appeared to be able to focus much better on what they considered to be key issues affecting their practice. Ann saw as crucial the providing of opportunities and situations which allow children to build up their knowledge and understanding of mathematical concepts, and that children should be encouraged to discuss and explain their understanding. Barabara saw interaction to be a key element within ability groups so that the teacher could build on previous knowledge and understanding through careful selection of activities. Claire’s response was more philosophical but demonstrates a deeper understanding of a complex process. She thought that mathematics should be taught ‘in a way that experience/teaching/questioning (by teacher and child) are inextricably linked together where children develop confidently because ‘mistakes’ are seen as part of the process of development and provide opportunities for new learning’. The importance of the use of language and frameworks for promoting interaction figure highly in all of these post-course responses.
Workshops Sessions
Workshop sessions occurred each week and transcripts from sessions provided a record of the spoken interactions which took place. At the beginning of Session 1 students were made aware of the nature of the course and the assignment that was partly school-based. They were also made aware that I would be acting as researcher, as well as tutor throughout the course and of the dual-purpose nature of what was to happen. Transcripts from two sessions will be used for analysis and discussion.
The Cube
At the beginning of this session students were introduced to the idea of a concept map and asked to draw one pertaining to a cube. That is, they would draw a box labelled cube in the middle of a blank piece of paper and attach to it, using lines or arcs, concepts or facts related to the cube. For example, Ann attached statements such as ‘12 edges’, ‘8 corners’; Barbara had ‘1000 units’, ‘puzzle’, and some uses and applications; Claire had ‘does not roll smoothly’ and ‘flat faces with four right angles’ amongst others. This was enough to get the conversation going and to compare notes. Clearly, each had something different, but appropriate, to say about a cube, but the situation had to be
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developed in some way. I gave each student eight ‘centicubes’; these are cubes of edge length 1 cm that can be interlocked with each other. It was a natural step to build a bigger cube; no prompting from me was necessary. I then asked about the relative sizes of the large and small cube. This was a deliberately ambiguous question which had the desired effect of producing two different answers and the dialogue began:
Ann: Four.
Barbara: No, eight, eight times bigger because there are eight cubes to make it.
Ann: In volume ... the space it takes up?
Barbara: Because dimensionally, like the sides are only twice ... breadth was twice what it was before. I think it’s twice what it was before and so that cubically it’s eight times bigger.
Ann: You talk about the area that it takes up?
Barbara: The surface area.
Claire: But the dimensions of its faces are twice ...
Barbara: And it’s because every dimension is twice as big, twice as big ...
Claire: And each face is four times as big.
Researcher: Ye.s
Unison: Four times the area.
Barbara: What do you mean by four times as big though because, like the breadths will be twice as big and the heights only twice as big, so it’s the area of each face.
The conversation developed into surface areas and nets and eventually scale factors and the meaning of mathematically similar shapes. The rate and extent of this ‘construction’ could only be guessed at, at the start of this episode, reinforcing Simon’s concept of Hypothetical Learning Trajectories (HLT). The time was right to move on. I asked them if they could put all eight cubes together and make a smaller volume:
Ann: Then it’s got to be the same no matter how you arrange them.
Researcher: Right, so if we spread them out in a long line, it’ll still be ...
Barbara: Eight cubes.
Researcher: Eight cubes, okay.
Ann: It wouldn’t be a cube shape.
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Claire: No.
Researcher: What about surface area then?
Almost 2 hours of intense activity and dialogue followed, where I had only the occasional input. Essentially, they explored the surface areas of different configurations of eight cubes and began to develop abstract reasoning about minimum and maximum surface area configurations, and Ann came forward with a new notion of the ‘number of faces joined’ as an aid to constructing different shapes that satisfied a given surface area:
Ann: With that you’ve got the minimum number of ... of faces joined together that you don’t see because we’re in a straight line.
Barbara: That’s right, so there’s one ...
Claire: That’s the minimum number you could make ... still being joined together.
Ann: So there’s one which ... So that must be the maximum (surface area) that’s ... you’ve got your two ends as well, did you not get that before? Eight. Thir ... fif ... yes, 34 ... You’ve got these end bits.
Researcher: Right you’ve given a good logical reason why that should be a maximum surface area. Okay try some different configurations using all eight cubes.
The group continued to explore different configurations, and at the same time consolidate what they thought they knew about the problem regarding maximum and minimum surface areas. They also began to demonstrate a growing awareness of the conservation of surface area between certain configurations using the concept of ‘joins’ and used it to check accuracy in counting the number of centimetre squares showing. After a short digression into edge lengths they came back to surface area and began to hypothesise about the even numbers appearing in their answers:
Barbara: 24, 28, 32, 34. They’re all even.
Researcher: Can you explain why?
Claire: Has it got to do with the fact that there is an even number of cubes?
Ann: Or an even number of faces?
Barbara: Something to do with an even number if you ... the equation’s somewhere isn’t it? (Old habits die hard!)
They began to look at results from odd numbers of cubes and latched onto a hypothesis that all square numbers are even until Claire pointed out that 25 is a square number. Then things started to come together:
Ann: Well, each time you join two you take away two faces.
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Claire: Right ... and it leaves you with an even number because there was an even number to start with.
Researcher: And if you’ve got an odd number of cubes?
Unison: There’s still an even number of faces.
Researcher: An even number of even ...?
Unison: Is even.
The researcher pointed out that they had not got a configuration which had a surface area of 26 or 30. Twenty-six took Barbara’s attention:
Barbara: I don’t think you could get 26 because if we’ve established that the smallest is 24 which is our cube, and if we take one out we’re immediately exposing ... how many more faces than we had? ...
This would have led to a correct deduction that 26 was not possible if she had pursued it or got support from the others, but Ann began finding 30 as a surface area and was successful. Claire was pursuing her own line of thought with the ‘26’ question by working backwards, which allowed her to make a shape to order satisfying a certain surface area, but without actually having to count the squares:
Claire: If we separated all the cubes out what would be the number of faces?
Barbara: What do you mean ...available?
Claire: Hmm ...
Barbara: But they’ve got to be joined together in some way.
Claire: I know, but just without them joined together.
Barbara: well, if there’s six ... there’s 48. But if they’ve got to be joined at least one face has to match up with another face, that’s what you said, there’s a minimum of two.
Researcher: Aha.
Claire: If you start off with 48 faces altogether and you want to end up with a ... say you want to end up with, say 30 faces ...
Barbara: Surface area?
Claire: Surface area, yes, somehow you’ve got to lose 18 does that mean losing pairs of nine, two pairs of nine, see what I mean if I join them together.
Researcher: Right, she’s working backwards, yes?
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Barbara: Yes.
Claire: See what I mean?
The discussion continued and tried to deal with covering 18 faces by using nine joins to leave 30 faces exposed. Barbara, not Claire, then went back to the ‘simplest’ case again, that is eight cubes joined in a straight line having 7 joins (7 x 2 faces), so 48 - 14 = 34.
Barbara: So let’s do the simplest case again ... one, two, three, four, five, six, seven, is that right?
Claire: Yes.
Barbara: Seven pairs, now you want two more – as well as those seven you want two more.
However, there were difficulties: as soon as the configuration was broken two more faces were exposed:
Barbara: So if I broke one I’d have ... so that’s made it into like that’s six ...
Claire: Yes.
Barbara: Because every time you’re breaking one off you’re taking one ... taking a pair away but you can add on, so I want to take a pair away and add on two pairs.
After further deliberations, which appeared to let Ann catch up with her thinking, Barbara asked if you could get an odd number of pairs:
Claire: Yes, because you can get seven originally ... with the simplest form.
Ann: Yes ... that’s still an even number, if you start with 48 you make your seven pairs.
Barbara: So I want to take away one and gain two.
Ann: Yes.
After yet more deliberation, a configuration containing nine pairs of faces joined was found and they counted the exposed faces to see if it worked. It did! The session continued beyond that scheduled; they had to find a shape with a surface area of 26:
Anne: So you need 11 joins (i.e. 4 - 22 = 26).
Claire: I want to stay with my 24. Where did you get 11 joins from? (after almost two hours fatigue was obviously setting in). I’m going to just do it my way otherwise I’ll get confused.
The ‘26’ problem was left until the following week, but in essence they had arrived at the point, from two different beginnings, where a crucial argument
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could be made regarding a shape with a surface area of 26. That is, seven cubes can be joined to form a shape, which is just one small cube short of the larger initial cube. In its configuration there are nine pairs of faces joined. If the last small cube is placed so that only two faces join then we have 10 pairs and a configuration with a surface area of 48 - 20 = 28. The only other place which gives a different result is for the last small cube to be place so that it makes the larger cube complete but in this case we gain a further 3 pairs of joins giving a surface area of 48 - 24 = 24. Clearly, adding two joins to give 11 joins is not possible so neither is a shape with a surface area of 26 containing eight such cubes. A similar argument can be made by just referring to surface areas as Barbara was attempting to do earlier on.
Responses to the Video ‘The Constructivist Classroom’
In session 6 of the course, the students were invited to view and respond at any time to the ideas presented by Professor Leone Burton in the videotape ‘The Constructivist Classroom’. By this time the students had been exposed to a number of issues and were beginning to relate their own classroom experiences to ideas about learning. The video presents learning as being active, interpretative, collaborative and validating, and that the implications for the classroom centre around meaningful contexts, making connections, making decisions based on reflective processes, establishing coherence, encouraging creativity and developing rigour.
The reactions of the students were at times surprising and robust. For example, after some discussion about the importance of motivation as an underlying factor in the constructive process, the notion, as presented by the video, that reflection by pupils was automatic and continuous came under scrutiny by students:
Barbara: To what extent are they being reflective and in what way are they being reflective and what are they using ... what values are they using to reflect with? If you see there’s a lot of psychology in there and I don’t know if it’s ...
Researcher: She believes children are reflecting all the time?
Ann: I would take argument with that I think.
Barbara: They could be reflecting very much on the surface in as much as ‘I’ve got to write a story and I don’t like writing stories so I can’t do it’. They may be reflecting at that level. They may be reflecting at the level of ‘this is school, I hate school why should I bother doing anything’. I have children in my class who are like that. It depends on the depth of reflection.
Ann: There’s a bit of responsibility in there too ...
Barbara: There is but taking responsibility for their own ...
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Ann: It’s making children aware that they are responsible for their own learning.
Barbara: It’s not something they come to naturally.
Barbara: No, not at all.
Ann: And the crux of it is how ...
Barbara: ... do you get that over to them?
Ann: ... to influence them in that way, to get them to see the value of working hard and actually working things through?
Barbara: What rewards are there for that?
Ann: Hmm.
Barbara: Is the gaining of knowledge or understanding in itself enough of a reward to be an active learner?
Similarly, on notions of social constructivism put forward in the video, the following comments were made:
Barbara: I would agree with what she said to a certain extent with a lot of children but I have a lot of children who are very confident in their learning, if you like, they enjoy learning but they enjoy it on their own. They don’t want to share it, thank you very much.
Researcher: Depends on the stimulus.
Barbara: Depends on the stimulus, depends on the child, depends on the task, depends on so many variables but I don’t, I mean, I think for her to say that she can predict that there will be no activity going on, is a bit sweeping.
Claire: Aha.
Barbara: Do you not agree? Again the depth of activity or the depth of cognitive activity may not be that great but there will be activity. And the child may very well be taking things on ... a lot of children reflect on their own ... and if they’ve got to build their own knowledge, in the end, no matter what anybody else says to them whether its peer discussion, whether it’s me talking to them , whether it’s them ... in the end they’ve got to work through it in here [their heads] for themselves so that they understand it for themselves in their own way, which may be unique to them and sometimes may be better.
The discussion continued in this way with many indications in the dialogue of the students forming, reforming and refining their own views of mathematics learning and teaching.
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The Assignment
This was quite an undertaking by busy school teachers but it had the saving grace of eventually being put to some practical use in the classroom.
The assignment consisted of (1) a report of a mathematical investigation carried out at the level of understanding of the student, and (2) a report of an adaptation of the topic(s) of investigation for classroom use and of its implementation and integration. This part of the report included:
�� a full description of the adaptations; �� the teaching strategies employed; �� a justification of the teacher’s instructional decisions with possibly some
theoretical underpinning; �� some account of interactions with and between pupils; and �� an evaluation of pupils’ understanding.
Discussions about the nature of investigations, investigating mathematically, problem-solving and reporting took place at the beginning of the course as it related to the assignment, but also throughout the course as it related to constructivism, the nature of knowledge and what we might mean by constructivist school mathematics.
The students were asked to choose any one of three investigation starters, ‘Sums and Products’, ‘Isolations’, ‘Walls’ and to develop their own mathematics about it, whilst also being cognisant of their own classroom situation with a view to adaptation of the investigation and subsequent implementation. These investigations were known to be rich in content.
Two students chose ‘Walls’ and the other chose ‘Isolations’. What follows deals with the first of these and the starter as presented to students is given.
Walls A brick is 2 units long, 1 unit wide and 1 unit high. The figure shows a wall 2 units high, 5 units long and 1 unit thick built with five bricks. FIGURE 1 here no legend 50mm
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Can you find some different designs for walls of the same size? Analyse similar problems for different sizes of wall and, possibly, with different sizes of brick.
Some analysis of the ‘Walls’ investigation by the teachers and their adaptation and implementation in the classroom is given here.
Investigations as a Means of Constructing Knowledge
Ann began by using square paper on which to draw different configurations for a 5 x 2 wall, and almost immediately brought in her own notation for vertically and horizontally placed bricks. 2H was to mean two horizontally placed bricks, one fitting exactly on top of another. Armed with this she was then able to look at and identify four different combinations represented by 3V + 2H (three vertically-placed bricks and two horizontally-placed bricks). One such combination is given in Figure 1. The different arrangements of 5V and 4H + 1V of which there was 1 and 3 combinations, respectively, exhausted all the eight possibilities for this size of wall. Barbara’s approach was similar using the V and H notation, but used cuisenaire rods as an initial aid and digressed into using different colours, possibly prompted by the coloured rods, which delayed finding the numbers associated with different brick positions, but which ventured into a different counting problem associated with the positions of different colours for a fixed brick pattern. She was in fact into the sort of counting which leads to the multinomial theorem.
Both were methodical in what they were doing and demonstrated a confidence in the material being presented which belied their feelings of trepidation before they embarked upon the exercise. Both produced a sequence of numbers associated with different wall lengths of two units high, but failed to recognise an underlying pattern in the number sequence so produced, which in fact was the Fibonacci sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ..., a sequence they were clearly not aware of. The above short description does not do justice to the many other aspects covered and processes experienced by the teachers as they worked on these investigations, but the focus here is on the adaptation and implementation in the classroom after their recent experiences of the course.
At this stage the teachers had a number of supporting factors as they attempted something new. Firstly, they had had some supporting theory about constructivism and the nature of knowledge. Secondly, they had a greater awareness about how constructivism might manifest itself in the everyday workings of the mathematics classroom. Thirdly, they had experienced their own constructions as they worked together as a group in the workshops and individually on their investigations. Both of these experiences involved adaptive activities with concrete materials at one level and some sophisticated abstract thinking and symbolic representation at another level. Fourthly, they had some expert knowledge of the mathematical situation they were introducing to their pupils. Finally, they all had some acquaintance with
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educational research strategies from another unit. Their classroom interactions with the children would be audiotaped.
In the Mathematics Classroom
All three teachers decided to use their adapted investigation with a small group from their own class, but in the context of a whole class situation. The ages of the children involved were between 8 and 11 years. Barbara’s pupils were the youngest and she writes, ‘This type of investigation approach is also new and challenging to myself as class teacher. Classroom organisation for effective learning is a daily challenge in this particular school, situated in an area designated as an “Educational Priority Area”’. In her introduction Barbara talks about ‘the constructivist view of mathematics teaching and learning’ and how new knowledge may be acquired. She quotes, amongst others, Steffe & D’Ambrosio (1995, p. 156), which gives emphasis to the crucial aspect of constructivist teaching: ‘we call a teacher who studies the mathematical constructions of students and who interacts with students in a learning space whose design is based, at least in part, on a working knowledge of student’s mathematics a “constructivist” teacher’. She identifies possible difficulties which may result in a poor start to the investigation. For example, she writes, ‘Do pupils have an existing “schema” and share common ground relating to procedures and concepts of tiling, and the language to share these ideas? Do pupils interpret the diagram as a wall? Difficulties could arise with two-dimensional representation of a three-dimensional object. Are pupils able to record their designs on paper?’ Barbara’s assessment of pupils’ pre-knowledge involves consideration of the topic of tiling, recognised as more helpful than essential, which was completed as a whole class activity some months earlier. She also identified subtopics within the investigation which would allow it to be fully integrated into other topic work in a meaningful way. She then went on to consider her role in the investigation in a quite explicit way by identifying scenarios which involve the types of intervention or non-intervention which the teacher could justify based on key elements of constructivist teaching.
Ann’s approach to the same problem was less formal by introducing the problem to a small group of her class as a problem that ‘is causing me some difficulty and that perhaps they could find something that I couldn’t ... I decided that I would tackle the problem alongside the children, rather than take on an obviously instructional role. They appeared quite motivated by the fact that I did not have an answer ...’.
Adaptation of the Problem
Barbara’s adaptation. Barbara adapted the problem by simply focusing on the 5 x 2 wall with her group and providing the actual bricks with which to build various walls. Some teacher input encouraging them to compare and record
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their designs helped them find and record a total of eight different designs. ‘A lot of spontaneous discussion followed while they decided whether all their designs were different and if they could make any more.’ Barbara records in her evaluation of this investigation that she probably influenced the next stage. ‘In retrospect, I probably influenced their decision to then introduce colour by the conditions I imposed (same size and shape), and the wording of the questions. This was no doubt influenced by my own failure to discover an obvious pattern when extending the number of bricks ... possibly on this occasion I needed the security of knowing where they were going.’ The transcript from the tape-recording confirms this:
Teacher: How can we change the design now, still keeping the same number of bricks?
(Silence)
Teacher: How can we make the walls look different?
Madeleine: We could colour them.
Kayleigh: Different colours.
Madeleine: We could do it five times different colours. Can actually do it more than that. Each brick can be any five colours so we’ve got 5 x 5 = 35.
Kayleigh: 5 x 5 is not 35. 5 x 5 is 25 (all agree).
Teacher: How are you going to organise this so that you can record and check it?
Madeleine: Mmm ...
Kayleigh: Photocopy the designs and then we can colour them.
A lot of time was spent colouring designs and Barbara intervened a number of times to encourage them to look for patterns in order to predict the number of designs they could make with five different colours. After a period of working that revealed no breakthrough in the children’s thinking, Barbara continues to work only with their thinking in order to try to move forward:
Teacher: How did you do it so that you knew you were not missing any designs? (no response)
I see you have all the first bricks red at the top of this sheet. What was your plan?
Madeleine: What we were doing was keeping two colours the same and changing the other three around.
Teacher: How many different designs did that give you?
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Madeleine: Six. Then I kept the blue and twirled these around. (She spots a mistake.) That one is wrong. It should be pink. (She finds more mistakes.) [Essentially she has reduced the problem of counting 5 != 5 x 4 x 3 x 2 x 1 to counting 3! = 3 x 2 x 1]
After yet more work with grouping the patterns produced and realising that there were gaps, Barbara felt that the children were ready for the final stage and reminded them of the original problem of finding the number of designs possible with five different colours. The children started to count in sixes as they pointed to each group and came to a total of 102:
Madeleine: We might be able to get more.
(They start to check all the possible colour combinations.) We could get another four groups. Brilliant!
Teacher: How many extra designs will these four give us?
Andrew: Add on six times four equals 24. That will give us 126. (Madeleine starts to check again and they all get confused and start to argue.)
Further intervention was needed to help the children to sort out the source of confusion. The fact that there was confusion at this stage is extremely encouraging; they obviously knew something was wrong, but did not know what. Further checking revealed a repeated group of six, leaving a correct total of 120. To my way of thinking this particular episode has many characteristics of constructivist teaching and learning. In Barbara’s reflective account of events the importance of the right kind of teacher intervention and the need for children to pursue their own activities in this process is not lost. ‘Each time I intervened to move them on they still clearly wanted to continue colouring every design until they had exhausted the possibilities. Once they had produced the data they could see and understand the pattern which emerged, with the help of some teacher input. They were then able to test, find omissions and explain the reasons for the final result.’ She concludes, ‘Clearly this group are at Orton & Wain’s initial stage in developing strategies for problem solving and investigation. Evidence of understanding will be looked for in the way they are able to transfer processes experienced in this investigation to form a strategy for solving their next investigation’.
Ann’s adaptation. Ann made the original investigation starter simpler by reducing the number of bricks to four and the focus of the investigation remained firmly within the confines of counting the number of different positional arrangements for a given number of bricks. The children were given access to pencils, paper, coloured pens, multilink cubes, scissors and glue. To begin with they worked individually and all started by drawing the diagram on the worksheet. All the children were absorbed in making sense of the problem in their own way.
Ann reports:
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The first talk came from Laurie who said: ‘You can’t have three verticals or three horizontals or one horizontal ... Oh, you can only have even numbers of horizontals.’
Before this I had not used the language ‘horizontal’ and ‘vertical’. I asked Laurie to explain the terms to the others. All the children seemed to understand these terms.
All the children found the five solutions to the four-brick problem and Laurie was able to identify a requirement at this stage ‘Verticals have to be together or have two spaces between so that two horizontals can fit in.’ When asked to consider the five-brick problem early incorrect conjectures were made based on too few data, but after further consideration of the problem as a group the following statements were made:
If it’s an odd number (i.e. we start with an odd number of bricks), you can’t have all horizontals. (Gillian)
If you have a certain number of bricks [n] you can’t have the number below it [n – 1] in verticals because there won’t be room for any horizontals (Graeme).
If it’s an odd number of bricks you can only have an odd number of verticals. If you have an even number of bricks you can have only an even number of verticals. (Laurie)
The eight possible arrangements were found by all but one pupil ‘who was quickly enlightened by the others’. Ann then asked them to consider the number of arrangements for six bricks and ideas relating to the pattern in the numbers produced began to take precedence over patterns held within the diagrams:
11, because you added on 3 last time ...
12, because 5 minus 8 is 3, 8 minus 12 is 4. Perhaps you add one each time.
After receiving the confident answers 2, 4, 6 about the number of vertical bricks possible, they were asked to try some designs for six bricks and at this point the need to be more organised in generating designs was apparent. All but one of the children found 12 solutions from six bricks, Graeme found 13 and the others began the search for the missing design. In order to do this the children came up with the idea of cutting out their drawings and putting them into groups depending on the number of vertical bricks in the design. After some time the missing solution was found. This activity spread to a later session where pupils were encouraged to contribute information about the problems and solutions which would be entered onto a large display. Graeme immediately started drawing a mapping connecting the number of bricks in the original wall to the number of distinct configurations possible:
4–-> 5 5–-> 8
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6–-> 13 7–-> ?
Ann then asked the children to extend the table in each direction. The children could visualise the situation for one and two-brick walls, and quickly determined that there were three possibilities for a three-brick wall. Once this part of the table was complete Laurie found a pattern and described it as follows:
If you take the 1 that you got first and added it to the 2 you just got, then it will make 3 and that’s the next number. Then if you take the 2 you had before and the 3 you just got that’s 5 and you just keep going, so 7 bricks should be 8 and 13 and that’s ...
‘21’ finished Graeme.
You might have expected things to finish there; after all, they produced a sequence (Fibonacci) which gives the key to the number of designs possible for a given number of bricks. What, in fact, happened was a reanalysis of their counting procedures to see if 21 really was the answer for the seven-brick wall. They worked very much as a team and diagrams were now quick sketches and terminology such as 2H + 4V and 6H + 1V was used naturally to convey meaning. Eventually, with some teacher direction, the numbers 1, 6, 10, 4 emerged as being the number of arrangements for configurations containing 0, 2, 4, and 6 horizontal bricks, respectively. Further generation of the table continued with the aid of calculators. In her conclusion, Ann talks of how little conversation took place until the children were familiar with the problem. It was also apparent that different children took different lengths of time to become familiar with the problem, and how to give enough time and support for all the children in the group remained for her an unanswered question. From this point on, the children’s thinking in relation to the investigation became ‘very apparent’ and ‘during the course of the investigation the progression from visual to abstract representation could be seen at stages where new understanding was constructed, justified, then applied’. Only when the investigation had progressed did the activity and discussion merge. She continues, ‘Striking the right balance between giving the children enough guidance to enable them to construct productively but not give them so much information that the initiative is taken away is obviously a skill which takes time to develop, particularly as the input required may vary between children within a group. It seems essential to create a climate where children can express what they are thinking so that appropriate support can be given.’ Ann then went on to describe four more possible extensions to this investigation using brick patterns which she could use with her pupils.
Summary and Conclusions
The design of this Master’s unit in mathematics education was driven by the belief that changes in what we do in mathematics classrooms will only come
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about through changes in how teachers experience mathematics either individually or socially. In order to help the teachers relate their experiences to classroom situations and to meaningful theoretical constructs, learning mathematics was experienced as student members of a group, as individuals seeking understanding about unfamiliar mathematical situations, as teachers preparing to move into new areas in their classroom practice. Teachers experienced their own construction of knowledge individually and collectively over a 10-week period. At the same time they developed, through discussion and construction of their own ideas, through a growing awareness of some theoretical underpinning, a metacognition of what was happening during their learning experiences and how this may contribute to a new learning environment in the classroom.
The mathematical investigation and subsequent implementation was a crucial aspect of this development since it not only gave the teachers the opportunity to develop their own expertise in a particular area over an extended period but it also allowed them to prepare mentally for the changes they perceived as necessary in order to create the kinds of interactions fundamental to constructivist teaching. The actual implementations appeared to lead to no sense of loss of efficacy as teachers (Smith III, 1996) as a result. Indeed, quite the opposite appeared to happen where all three teachers could not only report on the positive effects of their recent experiences, but confident in their own knowledge and expectations, were actively planning for more.
Correspondence
M. Simmons, Institute of Education, University of Stirling, Stirling FK9 4LA, United Kingdom.
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