Investigating Mathematics Teaching

A Constructivist Enquiry

Barbara Jaworski

• Publisher: Routledge
• Year: 2002
• eBook ISBN: 9781135716110
• Topic: Education
• Subtopic: Education General

Barbara Jaworski addresses a number of questions that are central to research on reform in mathematics education today. In this volume she attempts to chart critically yet honestly her own developing ideas as she undertakes a several-year-long enquiry into mathematics teaching and gives a very personal account of her developing conceptions, conjectures, thoughts and reflections. The author accounts for her research both genetically and biographically, simultaneously restructuring the development of her ideas and giving a rigorous, critical and reflective account.

Chapter 1

An Investigative Approach: Why and How?

As a classroom teacher of mathematics in English secondary comprehensive schools in the 1970s and early 1980s, I struggled with ways of helping my students to learn mathematics. I enjoyed doing mathematics myself, and I wanted students to have the pleasure that I had in being successful with mathematical problems. I experienced the introduction and implementation of syllabuses involving the ‘new’ mathematics. This gave me a lot of pleasure as I enjoyed working with sets and functions, with Boolean algebra, with matrices, with transformation geometry. Some of my students enjoyed this too, but the vast majority were, I came to realize, as mystified with the ‘new’ maths as with any of the more traditional topics on the syllabus.

Personally and Historically

Teaching mathematics was difficult, because students found learning mathematics difficult. I started to question what it actually meant to learn mathematics. Explanations or exposition seemed very limited in terms of their effect on students’ learning, so I found myself seeking alternative approaches to teaching mathematical topics, especially for students who were not inclined to like or be successful with mathematics.

I gained considerable personal enjoyment from puzzles, problems and mathematical investigations such as those offered by Martin Gardner (1965, for example) and occasionally used some of these, or modifications of them with students. This was ‘extra’ to my teaching of the mathematics syllabus, and I suspect, on reflection, very much in the mode of the teacher described by Stephen Lerman (1989b, p. 73) who ‘went around the classroom offering advice such as ‘no, not that way, it won’t lead anywhere, try this’. According to Lerman ‘there had been no opportunity for the teacher to discuss or examine…how [an investigation] might differ from “normal” mathematics’. In my earlier teaching it was not so much lack of opportunity to examine this issue as a lack of awareness on my part of how mathematical investigations might be linked to the mathematics on the syllabus.

In 1980, dissatisfaction with the mathematical achievement of many students, and a new post as head of a school mathematics department, made me start to look more seriously at the idea of mathematical investigation permeating ‘normal’ mathematics teaching. A number of my colleagues, who enjoyed getting involved in mathematical problems, were willing to start to think, at least in theory, about what an investigative approach to mathematics teaching might involve. As a result of a number of departmental sessions where we enjoyed ourselves in solving problems together, we came to a view that investigative teaching was about ‘opening up’ mathematics, about asking questions which were more open-ended, about encouraging student enquiry rather than straightforward ‘learning’ of facts and procedures.¹

This was not well articulated, and its implications for what we would actually do in our classrooms was far from clear. We were very conscious of the time factor in terms of ‘getting through the syllabus’. We recognized that enquiry and discussion by students took more time than exposition and explanation by the teacher. We accepted, I think with some guilt and some relief, that we would revert to exposition and explanation whenever we felt under pressure because we were secure with these methods and used them with confidence. To a great extent we were floundering with enquiry methods, not being at all sure of what we were trying to achieve, and having little confidence of their success in terms of the students’ ultimate ability to succeed with O’level or CSE examinations.²

At an early presidential address to the UK Association of Teachers of Mathematics (ATM), Caleb Gattegno said:

When we know why we do something in the classroom and what effect it has on our students, we shall be able to claim that we are contributing to the clarification of our activity as if it were a science. (Gattegno, 1960)

This clarification was one of my main objectives when I moved from secondary-school teaching into higher education in the mid-1980s. It was my declared intent to explore further the potential of an investigative approach to mathematics teaching from perspectives of both theory and practice. I wanted to be clearer about what such an approach meant in theoretical terms, but I wanted also to find out how it might be implemented in the classroom, and what implications this had for mathematics teachers.

The Origins of Investigations

Investigational activity in mathematics teaching in the UK was introduced during the 1960s, primarily through publications of, and workshops organized by the ATM, and through teacher-education courses in colleges and universities (Association of Teachers of Mathematics, 1966; Association of Teachers in Colleges and Departments of Education, 1967). A seminal sourcebook was Starting Points, by Banwell, Saunders and Tahta (1972). From the 1970s, in the UK, there are many documented examples of mathematical investigations in classrooms as part of mathematics learning and teaching. For example, Favis (1975) described an occasion when students in a class were ‘investigating rectangular numbers’ and two girls had taken off in a direction different from that proposed by the teacher, with some particularly fruitful results. Irwin (1976), a third-year (Year 9) student, described his own findings resulting from an investigation into combinations of functions which emerged from a question asked by his teacher. Edwards (1974), a first-year student-teacher, described an investigation which she herself undertook concerning the rotation of the numbered pins on a geoboard. Curtis (1975) described the results of two number investigations used with his lower-sixth form, non-examination, option group. In his introduction he commented:

As you will notice, neither investigation is complete. We found that as soon as we had an answer to one question, there were three other new questions to be answered. This continual pushing forward, trying to evaluate what we had already achieved and what was still to be done, gave the group an insight into how professional mathematicians spend much of their time…the fact that none of our conclusions had any direct application meant that we worked hard simply because we found it enjoyable. (Curtis, 1975, p. 40)

Mathematical investigation seemed to involve students in loosely-defined problems, asking their own questions, following their own interests and inclinations, setting their own goals, doing their own mathematics and, moreover, having fun. Eric Love (1988), following early ATM usage, calls this ‘mathematical activity’.

In contrast to the tasks set by the teacher—doing exercises, learning definitions, following worked examples—in mathematical activity the thinking, decisions, projects undertaken were under the control of the learner. It was the learner’s activity. (Love, 1988, p. 249)

The teachers involved ‘viewed mathematics as a field for enquiry, rather than a pre-existing subject to be learned’. Students had ownership of their mathematics. Love concurs with Curtis that in this activity the children’s work may be seen as paralleling that of professional mathematicians, with the teacher’s role involving provision of starting points or situations ‘intended to initiate constructive activity’. According to Paul Ernest (1991a, p. 283) ‘The mathematical activity of all learners of mathematics, provided it is productive, involving problem posing and solving is qualitatively no different from the activity of professional mathematicians.’

John Mason (1978), reflecting on the introduction of investigations to the summer-school activities of an Open University mathematics course, wrote, ‘the main aim, in my view is to reach a state where the initiative to ask questions rests with the student’ (p. 45). In the beginning, people working on some initial problem or starting point could be seen to be investigating it, but over time, the problem or situation on which they worked came to be known as ‘an investigation’. Particular activities or starting points became popular, and potential outcomes began to be recognized. For example, a certain formula could be expected to emerge or a particular area of mathematics might be addressed. Sometimes the outcomes were seen to be valuable in terms of the processes or strategies which they encouraged. Mason’s summer-school investigations were designed to provide ‘a paradigm for investigation’ (1978, p. 43), but he pointed out difficulties arising from differences in perspective between the originator of an investigation and the students working on it. Pirie (1987) recommended certain investigations to teachers as a starting point for introducing investigational work to their classrooms.

Investigational work has been closely allied and contrasted with mathematical problem-solving which had a strong international following in the 1970s and 1980s based on the work of George Polya (e.g., 1945). Lester (1980) cites 106 research references, representing only a small proportion of what had been published up to this time, and illustrating how much research attention was being devoted to the subject.

The Purposes of Investigations

There were many rationales for undertaking investigations in the classroom. Investigations could be seen to be more fun than ‘normal’ mathematical activity. Thus they might be undertaken as a treat, or on a Friday afternoon. They might be seen to promote more truly mathematical behaviour in students than a diet of traditional topics and exercises. They might be seen to promote the development of mathematical processes which could then be applied in other mathematical work. They could be seen as an alternative, even a more effective, means of bringing students up against traditional mathematical topics.

There were differing emphases, depending on which of these rationales motivated the choice of activity. For example, where investigations were employed as a Friday-afternoon activity they were often done for their own sake. What mattered was the outcome of the particular investigation, and the activity and enjoyment of the students in working on it. It was taken less seriously than usual mathematical work (e.g., Curtis, 1975). However, where the promotion of mathematical behaviour, or of versatile mathematical strategies was concerned, the investigation was just a vehicle for other learning (e.g., Mason, 1978).

This other learning might be seen as learning to be mathematical. David Wheeler (1982) speaks of ‘the process by which mathematics is brought into being’, calling it ‘mathematization’:

Although mathematization must be presumed present in all cases of ‘doing’ mathematics or ‘thinking’ mathematically, it can be detected most easily in situations where something not obviously mathematical is being converted into something that most obviously is. We may think of a young child playing with blocks, and using them to express awareness of symmetry, of an older child experimenting with a geoboard and becoming interested in the relationship between the areas of the triangles he can make, an adult noticing a building under construction and asking himself questions about the design etc….we notice that mathematization has taken place by the signs of organisation, of form, of additional structure, given to a situation. (Wheeler, 1982)

Wheeler elaborates by offering clues to the presence of mathematization. For example, he suggests that ‘searching for pattern’ and ‘modelling a situation’ are phrases which grope towards structuration, and that, as Poincaré pointed out, all mathematical notions are concerned with infinity—the search for generalizability being part of this thrust. Others have tried to pin down elements of mathematization, offering the student sets of processes, strategies or heuristics through which to guide mathematical thinking and problem-solving. Most notable was George Polya, in the United States, whose famous film ‘Let us teach guessing’ promoted guess and test routines and encouraged students first to get involved with a problem then to refine their initial thinking. He offered, for example, stages in tackling problems: understanding a problem, devising a plan, carrying out the plan, looking back (Polya, 1945, p. xvi); or ways of seeing or looking at a problem: mobilization; prevision; more parts suggest the whole stronger; recognising; regrouping; working from the inside, working from the outside (Polya, 1962, Vol. 2, p. 73). He advised students that ‘The aim of this book is to improve your working habits. In fact, however, only you yourself can improve your own habits’ (ibid.). In similar spirit, also in the US, were processes or stages of operation offered by Davis and Hersh (1981) and by Schoenfeld (1985). In the UK, much work in this area has been done by John Mason who has suggested that specializing, generalizing, conjecturing and convincing might be seen as fundamental mathematical processes describing most mathematical activity, and has offered other frameworks through which to view mathematical thinking and problem-solving (see for example, Mason, 1978; Mason et al., 1984; Mason, 1988a). Ernest (1991a) reminds us that such heuristics date back to Whewell’s ‘On the Philosophy of discovery’ in the 1830s, and, even earlier than this, to Descartes ‘rules for the direction of the mind’ in 1628.

One problem with such lists of processes, or stages of activity, is that they can start as one person’s attempt to synthesize mathematical operation, and become institutionalized as objects in their own right. It is possible to envisage lessons on specializing and generalizing. Love points to two further problems, first that the particularity of the lists fails to help us decide whether some aspect that is not included in the list is mathematizing or not; and second that the aspects start out as being descriptions, but become prescriptive—things that must happen in each activity (Love, 1988, p. 254).

This focus on processes resulted in some distinction between content and process elements of mathematical activity (Bell and Love, 1980). Alan Bell (1982) made the distinction:

Content represents particular ideas and skills like rectangles, highest common factor, solution of equations. On the other side there is the mathematical process or mathematical activity, that deserves its own syllabus to go alongside a syllabus of mathematical ideas; I would express it as consisting of abstraction, representation, generalization and proof. (Bell, 1982)

Traditionally, in classroom teaching of mathematics, the mathematical topics were overt and any processes mainly covert. Pupils were often encouraged to show the methods of problems or calculations, but otherwise little emphasis had been put on process. Indeed there was little evidence of awareness of process in students’ mathematical work.

Lerman (1989b) suggests that mathematical knowledge has to be seen as integrally involved with the doing of mathematics. Indeed he goes further to claim that, ‘Mathematics is identified by the particular ways of thinking, conjecturing, searching for informal and formal contradictions etc., not by the specific “content”.’ Thus investigational work, through an emphasis on process, might prove to be an effective way of approaching the content of the mathematical curriculum. Yet, Brown (1990), looking at certain investigative tasks wrote ‘It is evident that the child’s work in respect of tasks developed in this way does not fit comfortably into the categories conventionally found in content-oriented syllabi. Although common sense indicates that content and process would most valuably go hand in hand, moves to make process more explicit were in danger of turning process into yet more content to be learned gather than a dynamic means of enabling learners to construct mathematical ideas for themselves’ (Love, 1988).

The Place of Investigational Work in the Mathematics Curriculum

Investigating became seen more widely as a valuable activity for the mathematics classroom, supported by a government commissioned report into the teaching of mathematics, Mathematics Counts (The Cockcroft report, DES, 1982).³ This included investigational work as one of six elements which should be included in mathematics teaching at all levels (par. 243).

The idea of investigation is fundamental both to the study of mathematics itself and also to an understanding of the ways in which mathematics can be used to extend knowledge and to solve problems in very many fields. (DES, 1982, par. 250)

The Cockcroft authors recognized that investigations might be seen as extensive pieces of work, or ‘projects’ taking considerable time to complete, but that this need not be so. And they went on:

Investigations need be neither lengthy nor difficult. At the most fundamental level, and perhaps most frequently, they should start in response to pupils’ questions,… The essential condition for work of this kind is that the teacher must be willing to pursue the matter when a pupil asks ‘could we have done the same thing with three other numbers?’ or ‘what would happen if…?’ (op. cit.)

Despite this advice, investigations in many classrooms were separate pieces of work, almost separate topics on the syllabus. This was supported, legitimized, and to some extent required by the introduction in 1988 of the General Certificate of Secondary Education (GCSE) requiring an assessed...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Dedication
• Preface by Series Editor
• Preface
• Acknowledgments
• Epigraph
• Chapter 1 An Investigative Approach: Why and How?
• Chapter 2 Constructivism: A Philosophy of Knowledge and Learning
• Chapter 3 Working with Two Teachers: Defining the Study
• Chapter 4 The Research Process
• Chapter 5 Interlude 1: From Phase 1 to Phase 2
• Chapter 6 Clare: Origins of the Teaching Triad
• Chapter 7 Mike: Significant Episodes and the Teaching Triad
• Chapter 8 Interlude 2: From Phase 2 to Phase 3
• Chapter 9 Ben: Affirming the Teaching Triad
• Chapter 10 Investigative Mathematics Teaching: Characteristics and Tensions
• Chapter 11 Reflection and Development
• Chapter 12 Epilogue
• References
• Index