# Colors of Math and the space of mathematical educations

(This is the fourth and final “official” post as part of Justin Lanier’s smOOC, some short thoughts as I struggle to process a reading.)

Yesterday I had the chance to see a strange movie about mathematics and mathematicians called Colors of Math.

One quote in particular stood out, from a Russian mathematician named Maxim Kontsevich.

At every level, even an abstract theory is some sort of space. Every notion and every action is about geometric objects.

This was at the front of my mind reading through the Seymour Papert article “An Exploration in the Space of Mathematical Educations” (link). The article had a lot of things I didn’t quite buy (high conclusion to evidence ratio) and a lot of ideas I didn’t quite understand, but I did like the idea of the space of possible math educations as a mathematical object itself, and here’s the primary reason.

Suppose we associate with the space ME a function called A(x), which takes a education x in the space ME and returns the awesomeness of that mathematical education.

Is this function even continuous, or are there points where an arbitrarily tiny change effects a qualitatively different result? If I am teacher 1 and you are teacher 2, is $\int_{ME}(A_1-A_2)$ a measure of how much we differ in our views?

Papert’s definition of the space ME is vague enough in the article that I’m not sure where I would sit, but the idea of ME as a mathematical space makes me think of local maxima versus global maxima. A local maximum of A in the space ME is a teaching and education style that is more awesome than all the other very slight variations that are near it in the space; finding one of these maxima seems hard enough.

But simply being at a local maximum means nothing about the existence, nearness, or value of A(x) at other maxima. Does this mean that the benefit of activities like this smOOC, like reading blogs or more “serious research” is to ensure we can better understand ME, and thus move more quickly (even discontinuously) through it in our own teaching? Alone, we would be stuck with finding maxima on our own; we could still definitely improve but the success of our algorithms would rely so heavily on our starting points.