Argumentation and Proof in the Mathematics Classroom
Résumé
This chapter arose out of discussions of the working group on argumentation, logic, and proof and proving in mathematics. It concerns the relationships between argumentation and proof and begins by addressing the question of what we mean by argumentation and whether it includes mathematical proof. For the purposes of education, we regard argumentation as any written or oral discourse conducted according to shared rules, and aiming at a mutually acceptable conclusion about a statement, the content or the truth of which is under debate. It thus includes proof as a special case. Study of the relationships between argumentation and proof holds great potential for helping teachers and students deal with the tension between the process leading up to the development of a student's proof and the requirements placed on the final product. Students need to experience freedom and flexibility during an initial exploratory phase, whilst ultimately producing a proof that conforms to specific cultural constraints involving both logical and communicative norms in the classroom and in the mathematical community. We also discuss whether the activity of developing proofs under a teacher's guidance can be used to introduce students to meta-mathematical concepts, just as ascertaining the truth of a mathematical statement or the validity of a proof can provide an opportunity for increasing students' mastery of the related mathematical concepts. Finally, we use examples from the contributions of the members of the working group to illustrate aspects of these issues and suggest some possible activities to gradually increase students' awareness about proving and proof. We also examine the theoretical perspectives and educational issues involved in choosing and designing such activities.