Response to Thurston article - "On Proof and Progress in Mathematics"

In the field of teaching, I feel it is critical to communicate your ideas, work, and development. I would agree with Thurston that this is a shortfall of many teachers and professors. This paper shines a light on why students get frustrated and insist that they dislike mathematics. In particular, the issue of communication reminds me of being lost in Montreal looking for La Banquise. Although I knew my way around, it was difficult to get to my final destination, because the language being used by those around me was not well defined in an already foreign space. I was, certainly, frustrated (and very hungry). It was impossible to find, until someone spoke my language and directed me in a way I could understand. This was the ninth person I asked. My weak foundation of speaking French (European) was insufficient in a substantially different, but still French (Quebecois), environment.

Now suppose this is translated into teaching; if one in nine teachers can actually explain a math concept to a struggling student with a weak foundation in mathematics and in unfamiliar territory, then of course the student will insist they dislike math. It's confusing, and they're only being shown one way to get there, if they can understand the instructions at all! Introducing proofs to such a student, is by far very frustrating.

For this reason, I appreciate that Thurston showed the multiple ways of thinking used to understand a proof, especially visual and kinaesthetic. I think that as a math teacher candidate, I forget that my teaching is not meant for other people in my field, and that I may cause my students frustration in the future. Of course, this does not permit me to call factoring of quadratic trinomials "plus-ing and times-ing", as one of my tutorees decided to call it, but it does emphasize a need for clarity in math teaching. I think that re-inforcing definitions and using them often can be helpful. I feel like a personal word-bank assignment at the end of each chapter could also be useful, where students need to express a definition in two ways, e.g. as words or using a picture. Perhaps it could be virtual, such as in a video, etc, and hence allow them to use multiple intelligences? The one risk with this is that students would take much too long to complete it, but I imagine that creating an example and setting a suggested time frame would be useful.

When it comes to something more complicated, however, like a proof, I would say that it is up for debate whether or not proofs are critical in a high school classroom. I find it interesting that they have made their way into Foundations 11, followed by logic and set theory in Foundations 12. Are proofs part of the curriculum for any other course, or would this become difficult for teachers teaching math without a math background?