Key Ideas: What Are They and How Can They Help Us Understand How People View Proof?

Raman, M. (2003)


This short article analyzes how people interpret mathematical proofs based on the explanations of university students and professors. The author argues that people differently use two aspects of mathematics: private and public. “Private aspect of mathematics” means people think and utilize their own perspective to cope with mathematical proofs whereas “public aspect” means an appropriate way for the mathematical community they belong. The descriptions of participants illustrate three essential ides in the creation and evaluation of a proof. One is “heuristic idea” that people explore the concept or definition of theories but they do not know how to proceed it and go further. The other one is “procedural idea” that people know typical writing form of proofs but they lack understanding about definitions. The last idea is “key idea” which links heuristic idea and procedural idea. The participating professors used the key idea in their descriptions while the university students explained by heuristic or procedural idea. The author points out that a heuristic idea gives an insight of understanding without conviction whereas a procedural idea gives conviction without insight of understanding.

Comment

Like the other article, when people face mathematical proof, they construct the proof based on their perspective or experience, and they need to adjust their explanation to match the idea of the community they belong. In this transaction, sociocultural factors (norms and language) effect for their ideas. While my students should deserve dual resources in language and culture, we are not sure how they work with their resources during mathematical activities especially in writing such as a proof. Also, I wonder when we find the linguistic error in the student’s answer (proof), how we can evaluate it? Is it the issue of language? Or misunderstanding the concept? Or different sociocultural factor? 
  Most of the teenage students who are in my school and were raised in Canada have been studying mathematics in Japanese for over than 6 years. Although they seem to behave like Japanese students in Japan and seem to utilize well both daily-life Japanese and academic Japanese, they face difficulties when working on word problems or mathematical proof. It is not clear where they are struggling.