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.


  1. I’m enjoying reading the Papert piece. A few years back, I stumbled across a great archive of his stuff. I thought I had read it all.

    “The first of my oppositional principles is brought out by contrasting the activity of these children with a more common way to introduce probability in schools by using physical materials such as spinners or dice to introduce children to the idea of probability. I ask: what can these children do with this new knowledge besides talk or deal with teacher-initiated problems? How can a child actually use it to do something that has real personal importance now? My Logo kids are excited because they can produce dramatic screen effects and will go on using their growing control over random processes in projects of increasing complexity. What can your spinner spinners do that will give them a sense of empowerment and achievement?”

    But my kid, who can access things like logo and scratch and minecraft, finds the glitz and action of video games pretty addictive. Like me, Papert doesn’t much care for limiting kids. But if you hand them wide-open technology, the older ones are going to know how to get to the glitz, and aren’t necessarily going to give themselves math-rich experiences.


  2. Yeah, I wonder how his argument changes given that now some kids are amazingly tech-literate, while some of their classmates barely have internet access.

    Also, would you happen to remember where that archive was? I really liked playing with Logo, and didn’t know he was an ed researcher…

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