Bridging Knowing and Proving in Mathematics: A Didactical Perspective

Balacheff, N.


In this chapter, the author points out that leaners have epistemological difficulty in addition to psychological one in a mathematical proof. He argues that a significant role of mathematical proof is not truth but the validity of a statement, and a mathematical proof requires learners to shift practical position to theoretical position when facing mathematical proofs. According to his explanation model, there are three processes in mathematical proof as shown: (1) “Explanation of the validity of a statement” from the learner’s perspective is described by the learner and (2) it will label the status of proof when it gets sufficient support from belonging community, and it turns out the (3) mathematical proof if it meets the current criteria of mathematical practice. The author also illustrates the mapping of mathematical proof by three dimensions (action, formulation, and validation, see the figure), and suggests mathematics teachers should lead their students with the nature of mathematics on shifting from practical stage to theoretical stage. Finally, language plays an important role as a tool to move within this mapping.




Comment

As the author mentions in this article, there are no correct rigor answers in mathematical proofs. Instead, there is the validity of them based on learner/teacher’s sociocultural perspectives. Like sociomathematical norms, mathematical proofs can be judged by the community learners belong to as appropriate one or not. I believe it is not easy for teachers to teach and guide their students what mathematical proof can be appropriate because of no standards, so the easier way should be memorizing what explanations are accepted. As I see both Canadian and Japanese mathematics textbooks, explanations in mathematical proofs are quite similar: they formed by common steps in terms of the proofs in geometry intending to grade 8-9. I have no idea how Canadian teachers introduce those proofs, and I guess many teachers rely on rote.
  The model of mapping of mathematical proof invites me to raise several questions. I am wondering how my students who are bilingual and influenced by two different curriculums, move in this mapping when they confront challenges in mathematical proofs. In practice part (misunderstanding of theory or concepts)? Or language part (the problem of language proficiency)? Or validate part (sociocultural factor)? It should be not clear that which step can be challenging for my students.