Key ideas and memorability in proof

Hanna, G. and Mason, J


The authors, in this article, mainly commenting on works of Gower (2007) which suggests the idea of width of proof and memorability. Width of proof is defined by Gower and means how many ideas and process of information can be recorded in the brain in order to understand, remember, and solve mathematical proofs. The authors quoted three examples of key ideas in Gower’s article that make teachers and learners use less their memory. Like many other researchers, Gower argues the importance of key idea in proofs, and the authors introduce another similar idea with referring to other researchers. Mathematical proofs develop student’s mathematical thinking, and the authors suggest that the key idea in a proof can reduces the volume of less important information and width of the proof. In other words, it can increase the memorability while it impacts on their understanding of mathematical ideas.

Comment

As I posted earlier, I was taught to memorize (nearly) whole sentences in a mathematical proof. This article and Gower (2007) says whereas they acknowledge the necessity of rote, how much teachers and students should remember mathematical idea to complete proofs. (I should read Gower’s article) As they mentioned in their work, we should know how we can find key idea in mathematical proofs. Additionally, are there any cultural differences of key idea. For instance, though I am not sure currently, Japanese textbooks focus on the one key idea while Canadian textbooks emphasize the other idea. If so my students can deserve both ideas as resources, but I wonder whether if they get confused.
  The other interesting part of this article is about there are no rigor criteria (or not well-defined) for mathematical proofs to judge. The evaluation might be based on the aesthetic properties: deep, beautiful, elegant, clear, ingenious, or explanatory and so on. I completely agree with this idea, and I think we can see this problem in other than proof. For example, when students solve difficult calculation and write down the calculating procedure and answer, how many the amount of mathematical expressions is counted as appropriate one by a teacher? If the student works on it by mental calculation and writes a few of the expressions, are they accepted as “elegant” or insufficient? How about the excess expressions? I suppose this judgment is biased by teacher’s belief and it might be affected by their learning experiences.