Thus, far from being an exercise in reason, a convincing certification of truth, or a device for enhancing the understanding, a proof in a textbook on advanced topics is often a stylized minuet which the author dances with his readers to achieve certain social ends. What begins as reason soon becomes aesthetics and winds up as anaesthetics.

   back to table of contents

Mathematics education is essential, both in helping students to function in a Rationalist world, and in assisting them in making that world a better place. The deductive reasoning which typifies mathematical proving is the basis for Rationalism, and so is important in the achievement of both of these goals. At present, however, the teaching of proving is largely unsuccessful. This lack of success seems to be related to an incompatibility between the picture of proving portrayed in schools, and the role of deductive reasoning in professional mathematics and in students’ lives. The research reported here is concerned with developing a better understanding of students’ need to prove, with the aim of identifying aspects of teaching which might be improved.

The research studies involved the observation and interviewing of high school and undergraduate university students as they investigated problem solving situations. Their mathematical activity is described using a vocabulary developed during the research that identifies (1) needs which motivate reasoning, (2) types of reasoning, and (3) degrees of formulation of proving and of proofs. Categories of needs include explanation, exploration, and verification. Reasoning can be inductive, deductive, or analogical. Proving can be unformulated, formulated, mechanical, or formulaic. Proofs can be preformal, or semi-formal.

Three main observations are derived from the research studies: (1) The participants were able to reason deductively, and, with help, to formulate their proving. (2) Proving was applied primarily to exploration and explanation. Verification seemed to be a very poor motivation to prove. (3) The reasoning used by the participants was influenced by the activities of those around them, both observers and other participants.

These observations lead to two suggestions for teaching: (1) The current presentation of proving as deductive reasoning employed to verify statements should be expanded to include the use of proving to explain and explore. (2) The organization of class activities should accommodate the development of a "culture of proving," in which students feel that deduction is an appropriate way to reason about mathematics.
 
 
 

  back to table of contents

Writing is a journey. Like any journey the trip is more fun, and one is less likely to get hopelessly lost, if there are others along to help. I have had the good fortune to have a great deal of company on my journey, which has made the resulting dissertation, the metaphorical slide show, better in uncountable ways.

Tom Kieren, as supervisor, co-researcher, reader, and friend, created a climate of fellowship and intellectual activity, an occasion of which I have been fortunate to be a part. Lynn Gordon-Calvert, and Elaine Simmt, the other half of the Enactivism Research Group, offered ideas, comments, and support which was integral to my research and writing. Other members of the wonderful place in which I found myself include: Al Olson and Ted Lewis, whose presence on my supervisory committee somehow both broadened and focused my thinking; Heidi Kass, Daiyo Sawada, Anna Sierpinska, and Sol Sigurdson, who were always willing to lend an ear, and a critical eye, to my ideas; and the people who made the community, Brent, Dennis, Hridaya, Ingrid, John, Judy, Kgomotso, Laura, Leo, Paul, Ralph, Ray, Rebecca, Roshan, Sandra, Tim, Tim, and Vi, the friends and inhabitants of 948. I must also acknowledge the students and teachers who took part in my studies, whose names are hidden, but whose importance is manifest.

My journey has been an emotional one as well as an intellectual one, and the existence of my dissertation owes as much to the support of my friends as it does to the ideas of my colleagues. Of course, some people contributed in both of these roles. Without Constance I would not have begun. Without Patrick, Tim, Kelly, Johwanna, Elaine, Samantha, Gwen, Peter, Gina, Ben, Allison, Drew, Lynn, Chris, Ralph, Sean, Bonnie, Elyse, Penny, and my parents, I would not have been able to go on. Jennifer, Sarah, and Drue were and are always with me, even from far away.

Finally, I must acknowledge the importance of the financial support given to me by the University of Alberta and by the Social Sciences and Humanities Research Council of Canada (Grant #752 93 3268). I have been fortunate to be among the Canadians who have benefited from community support for research and education, support which I hope we will have the wisdom to maintain.