3 levels of learning math – all surprisingly similar

Had an interesting morning with my son working through a challenging problem that required him to pull together some ideas from geometry and from algebra. I’m slightly annoyed with myself because we were running a little behind schedule and ended up choosing to walk him through the solution rather than finding a better path. Still though, it was a good example of how hard it can be to pull together ideas from different parts of math.

But, I also saw to other pieces about learning math today – one was a post about slightly younger kids, and one was a post about math professors struggling to understand a complicated proof. The similarities in the struggles were fascinating to me.

(1) Young kids – from Michael Pershan:

Dissent of the Day by Michael Pershan

“While walking around, I noticed some kids getting lost in their calculations. Lots of great ideas, but constantly losing the thread.”

Not losing the thread is truly one of the biggest challenges in math.

(2) slightly older kid this morning – the struggle is to find the right path through all of the algebra and geometry:

 

(3) Now skip ahead a few years: professional mathematicians struggling to understand a difficult proof.

Notes on the Oxford IUT Workshop by Brian Conrad on Mathbabe.org

I think if you are interested in communicating / teaching math at any level, this article is an important read. For example:

“There was substantial audience frustration in the final 2 days. Here is an example.

We kept being told many variations of “consider two objects that are isomorphic,” or even something as vacuous-sounding as “consider two copies of the category D, but label them differently.” Despite repeated requests with mounting degrees of exasperation, we were never told a compelling example of an interesting situation of such things with evident relevance to the goal.”

Sort of amazing to have run across all three of these examples on the same day.