The Best Math Advice I Ever Got

Hieroglyphics and Pictures

Math is just a picture which you can understand hidden in hieroglyphics which you don't understand. Your job is to turn the hieroglyphics back into a picture.

- Prof. David Griffeath, UW - Madison

That quote is the best pedagogical math advice I ever I got. It was an off-hand remark in 1-credit elective on cellular automaton (opens in a new tab) (as in Conway's Game of Life (opens in a new tab)), an esoteric course even by math standards. I doubt any of the other students remember hearing it, and probably professor Griffeath doesn't remember saying it. You never know the impact your words are going to have!

But offhand remark or not, this idea of "translating" concepts into pictures resonated with me. Case-in-point, the only concept that I vividly remember from my senior-level abstract algebra class are the diagrams I drew for myself to understand how 3 major classes of functions map between sets; injective (opens in a new tab) (one-to-one), surjective (opens in a new tab) (on-to), and bijective (opens in a new tab) (both surjective and bijective).

Pictures can be powerful tools. When I say this I'm not trying to say I'm a in some way a "visual learner" (the popular theory of "Learning Styles (opens in a new tab)" is widely debunked). Instead, to me drawing a picture is a shorthand for saying you have a deeper understanding of a concept that just parroting a definition. It's when you understand the meaning of an idea enough to translate it into different forms, such as a picture.

This quote has been kicking around in my head a lot over the past 8 months and I've been taking a second pass at some of the subjects from my undergrad coursework, linear algebra in particular. I don't always draw a literal picture, but I am going more slowly than I have in the past and striving for that same level of deeper understanding that allows me to the express concepts in my own language and shorthand. Being able to rewrite something has proven a pretty good heuristic for how well I understand something. And realizing my "pictures" are wrong has been a pretty effective way to correct my understanding!

This brings me to the final thing I've always liked about this quote, which is that I find it encouraging. On the other side of whatever difficult concept you're banging your head against is something friendly and accessible, a picture.

Your job is just to get to that picture.

This work by Alex C. Viana is licensed under CC BY-NC-SA 4