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The relevance of my research to the field of mathematics education will be addressed in the main body of the text. Here I would like to mention the relevance of my research to me, and some of my personal assumptions that have motivated me to conduct this research in the way I have.

As a teacher and researcher in the psychology on mathematics education I have become convinced that learning very much depends on students’ prior knowledge, abilities, and beliefs; what is, in the language of constructivism** and Enactivism, called their "structures." Given this, the practice of teaching mathematical reasoning as if it were unconnected to students’ prior experiences of reasoning in other domains, and without considering students’ prior ability to reason deductively, seems nonsensical to me. I feel that the way to teach students to prove must lie between assuming they know nothing and teaching logic as rules of procedure, and assuming they know everything and penalizing them when they fail to apply their abilities to reason to the peculiar contexts of mathematics.

With this assumption in mind, the first step in improving the teaching of proving must be the development of an understanding of how students reason in mathematical situations so we know where to begin. Such an understanding cannot be a general understanding of how all students reason. Students reason in quite individual ways. What this understanding can and must be is a sense of the range of possibilities in individuals’ reasoning, combined with some way of noticing and talking about this reasoning. I believe my research is an important step toward this understanding, combining what is already available in the mathematics education literature with the results of my own empirical studies of proving.

A large part of my motivation to be a teacher is the conviction that schooling can play a role in preparing students to survive and improve the world in which they live. This conviction is also a motivation for my research. The adoption of proving in mathematics as the model for correct thinking in all domains has been seen as marking the beginning of the modern era, and is central to the Rationalist attitude that continues to affect the way decisions are made in our society. Meanwhile, the limitations of proving have come to be understood through work in mathematics, analytic philosophy, and linguistics. It is in these fields, that depend heavily on proving as their method of discovery, that the determination that proving

*Wanderer, the road is your footsteps, nothing else
** Constructivism the theory of learning, not the philosophy of mathematics or the art movement.

has limits first became possible. It is also in these fields that these limits have been analyzed and understood. Teaching students to survive in a Rationalist world must involve teaching proving, and helping them improve this world must involve teaching proving well enough for the limits of proving to be seen and understood.

It is not only students who need to take responsibility for improving the world. Educators and researchers must also engage in the continuous process of considering how our methods arose, and what limits their origins have passed on to them. Research in education has been strongly influenced by the Rationalist attitude, and this influence, although weakening, continues. The particular limits of proving as a model for research in education have been revealed in two ways. Many researchers have noted that the predictive power of proving seems not to apply in educational research, and that there are many aspects of education that seem to be inexpressible in Rationalist terms. This is a partial understanding of the limits of proving as a model for research, as it detects limits, but does not provide any analysis of the origins of those limits. In some cases this has led researchers to advocate a complete rejection of Rationalist methods in educational research. Other researchers have come to an understanding of the limits of Rationalist methods by a careful use of the methods themselves. This process is analogous to the processes applied in mathematics to reveal the limits of proving. It has the advantage of revealing not only the limits of Rationalist methods, but also the origins of these limits. Limits which have not yet been detected can be predicted, and so areas in which Rationalist methods can be reasonably applied can be identified. In my final chapter I will describe a methodology for research in mathematics education that comes out of such a self-reflective analysis of method.

In an earlier draft of this dissertation I structured my chapters and sections as if they were part of a proof. My arguments were broken down into definitions and lemmas, some of them quite involved, that led up to short sections with ambitious titles in which I asserted my conclusions. These sections referred back to the preceding chapters for the lemmas required to support their conclusions. This format was singularly inappropriate to the message I am attempting to communicate. In my final chapter you will find me asserting that research into the thinking of human beings cannot be like proving, and so casting the results of such research into the shape of a proof was not only a confusing act on my part, it was a serious contradiction.

One of my indulgent readers pointed out this problem, and unlike many people who point out problems, she also provided me with a solution. Of course I must take responsibility for the success or failure of this attempt to implement her idea. The restrictions of text and my own lack of creativity led to the linear form of my writing. I admit that I have written the sections with the lower page numbers as the beginning, and have proceeded in the usual way through a middle, to an end. I think it reads pretty well this way, but I will leave it to you to make the final judgment. The structure ends up resembling my own progress in my research. First I considered what I knew and had read about proving by students and by professional mathematicians. Then I observed students proving, and noted what I felt was important in what they did. Next I considered how the proving of the students I observed might relate to the teaching of proving, and to the role of proving in society. Finally, I considered how the ideas of Enactivism informed and were clarified by what I had learned in my research. When I came to write about what I had learned there were inevitably things which needed to be written, but which played supporting rather than central roles in my thinking. Such things have been included in appendices.

I would like to emphasize that there is no reason to read from beginning through middle to end. There is a degree of connection between the end and the beginning, that lends a circular aspect to the whole work. While writing in my mundane linear manner I have tried to make connections forward and backward, so that an adventurous reader might start anywhere and read in either direction.

Before you decide how adventurous you would like to be, let me describe the territory you will be exploring. In the following you will find me setting forth some fundamental questions related to the need to prove, reporting the results of my attempts to answer these questions, relating my results to the teaching of mathematics, discussing the role of proving in society, and offering some ideas on research and the need to prove.

In Chapter I the basic questions underlying my research are introduced. They are "What is proving?" and "Why do people prove?" I take some preliminary steps to answer the first question, suggesting that proving should include deductive reasoning used for any purpose, in order to fit with the role of proving in mathematics. I also report further on proving in mathematics to address the question "Why do mathematicians prove?" The chapter ends with a sketch of the methods I used in my attempt to answer the question, "Why do students prove?"

In Chapter II, I turn to the question "Why do students prove?" and report some results of the research studies I undertook in order to investigate students’ need to prove. This chapter is organized into several sections addressing specific needs; explaining, exploring, verification, and teacher-games. A network of terms is used to clarify the relationships between needs and proving.

In Chapter III, I use the language developed in Chapter II to describe the proving of two students who participated in one of my studies. This example shows both the application of the language and also expands on the relationships between terms.

In Chapter IV I report on circumstances that constrained the use of proving in my research studies. These include individuals’ structures, social constraints, and the problem situations in which the participants found themselves.

In Chapter V, I discuss the teaching of proving. This includes a consideration of the importance of proving seen by curriculum designers, a critique of current teaching practices, a description and critique of several innovative experiments in teaching proving, my own speculations as to ways in which the teaching of proving might be improved, and finally a reinterpretation of the need to teach proving.

In Chapter VI, I turn to the role of proving in society, including the rise and influence of Rationalism, some problems with Rationalism, and alternative modes of thinking.

In my final chapter, Chapter VII, I describe Enactivism, as an extension of Rationalism that acknowledges its limits. I show how Enactivism can be used to provide a theoretic basis, and a methodology, for educational research into proving. I conclude with a summary of my thoughts on proving, in education and in research.

There are several appendices that provide details of my research that did not fit into the structure of the main body of the text, but that some people might find interesting. They include an annotated bibliography of research on teaching proof (as opposed to teaching proving), details of the design of my research studies, and several different summaries of my data.

As noted above, I have written the text in what I feel is the best way for it to be read. If you prefer to read a more traditional dissertation, or a more deductive argument of my points, Table 1 gives a concordance of the Chapter and sections included here, in an order suitable for those two alternate readings.
 

To clarify, a reader who wished to read a traditional dissertation should begin by reading this introduction, the first section of Chapter I, and the first section of Chapter V. Taken together these sections cover much of what is usually presented in the introduction to a dissertation. On the other hand, the first section of Chapter I is not really needed for the deductive argument reading, and can be omitted.

Note that two sections that are usually found at the beginning of a dissertation, a review of related literature and a description of the design of the studies, have been relegated to appendices. The reasons for this move are given at length at the end of Chapter I and at the beginning of Appendix A. Briefly, the traditional exhaustive review of the literature has been rendered superfluous by the introduction of electronic indexes to the literature, so I restrict the references I make in the main text to those that are directly related to the topics under discussion. For example, the extensive work of Balacheff on teaching students to create proofs is not mentioned until Chapter V, when teaching proving is considered. The custom of describing in detail the design of research studies is taken from the style of reporting research used in the sciences, where reproducibility is an important issue. The complexity of human reasoning makes reproducibility in detail impossible, so in the main body of the text I limit my descriptions of my studies to what is needed for understanding the results I report.

In presenting excerpts from the words spoken and the writing of the participants in my research studies I have attempted to balance clarity of presentation with completeness. While I recognize that transcripts and writing pulled out of context are already a long way from the situations in which they occurred, I realize that some readers will wish to consider how the examples I give might be interpreted differently, and I do not wish to discourage them. At the same time, transcripts and reduced images of written work are more difficult to understand than the original voices and full sized writings. I do not wish to make my examples any more difficult to decipher than they need be. With this in mind I

have made some editorial changes to the transcripts and diagrams included as examples.

In the case of transcripts, I have omitted many of the inevitable "hmms," "uhs," and other sounds that punctuate normal speech. Such omissions are marked with ellipses (...). I have used two conventions in an effort to capture some of the rhythm of spoken language in text. Utterances that were interrupted or left unfinished are marked with a short dash (-) at the point of interruption. Long pauses are marked with long dashes (–). Longer pauses are marked with several long dashes. In a very few cases I have omitted several lines from transcripts where they do not contribute directly to the point I am attempting to illustrate. Such omissions are noted in the analyses of the transcripts, and glosses of the omitted matter are provided there.

The participants in the studies were quite careful not to use more paper than absolutely necessary, which resulted in pages covered with writing, often overlapping or oriented in strange directions. As it is impossible, and unhelpful, to reproduce such pages at their actual size in the space defined by my margins, I have either reduced them in size, or selected smaller areas of pages that are of particular interest. I have also erased stray lines, figures, etc. that do not relate to my purpose in providing the illustration. There are cases where my main interest is in the content of the participants’ writings, and they do not include drawings. In such cases I have typed the participants’ writings. Such passages are italicized.