# Have there been attempts to base early math education on category theory?

The New Math curriculum built math eduction on set theory. Have there been any attempts to do something similar with category theory?

I was fortunate to grow up in a relatively enlightened spacetime, such that I got New Math in my earliest primary education. It Worked For Me in the sense that, as a child, math seemed both interesting and straightforward. Though I retained a continuing interest in math (which I tend to attribute to that primary-math pedagogy), my formal study ended with the undergraduate engineering curriculum (differential equations and calculus-based probability) and the more-fascinating-to-me introduction to discrete math taught in undergraduate informatics (misnamed "computer science" in too much of the world).

I have a longer-standing interest in logics and the history and philosophy of logic, and started learning about attempts to "found mathematics" (where found is a verb, not an adjective) via philosophy of math. Hence when I learned about efforts to found math on set theory, I wasn't able to "go too deep" (lacking much pure math), but found it fairly tractable--that was how I was first taught math.

I later learned a bit more about category theory, which I had previously encountered (less than a bit, and much too briefly) in a programming-language-theory class. Lately I've been reading a bit more about category theory; I seem to understand it well enough (though of course without testing, that's just my quale), but (aside from the applications of CT to sets) I still find CT less immediate or intuitive than set theory. My hypothesis is, even though I didn't get taught formal ST until college (which, in my case, was decades after secondary school), New Math wired some deep and persistent neural networks.

Which motivates my question: has anyone attempted a primary-math pedagogy mostly based on category theory, similar to the way New Math was mostly based on set theory? If so, does it have any empirical advantages (in terms of educational outcomes) other than making category theory more intuitive?

• I have edited your post for clarity. Titles should be descriptive, and should not require a paragraph of explanation. I seem to have saved some words by actually spelling out the thing you sought to abbreviate. May 20, 2020 at 12:02
• I have added the (category-theory) tag.
– J W
May 20, 2020 at 12:46
• So we will teach children to regard a light switch as a $\mathbb{Z}/2$ torsor? May 22, 2020 at 10:19
• New Math failed because it was too abstract for young kids. Do you want to fall into the same pit again? May 22, 2020 at 20:44
• @RustyCore, no, it failed because it was too abstract for the teachers. Young kids would be fine with it. May 23, 2020 at 15:12

Great question! Yes, this was Piaget's project after initially working with more classic structuralism (e.g. his text titled Structuralism). He explicitly mentions Mac Lane's work in his book on understanding functions, Epistemology and Psychology of Functions

I discuss this a bit in my dissertation, but am not familiar with other work that compares mathematical styles and models for children's thinking.

• Your answer says yes, but it seems like the answer is really no, isn't it? The OP is asking whether anyone actually subjected children to such an approach. I looked in your thesis and didn't see any connection to this topic.
– user507
May 22, 2020 at 0:04
• My answer is yes, see Piaget. I discuss this connection in my dissertation. Thanks! May 22, 2020 at 3:22
• Another ref: Morphisms and Categories (English tr.). It seems to my inexpert reading that Piaget used category theory to formulate his theory of cognitive development, in particular, the development of mathematical understanding under normal, conventional teaching. It does not seem he tried to found a "pedagogy mostly based on category theory, similar to the way New Math was mostly based on set theory." May 24, 2020 at 12:46
• I discuss this connection in my dissertation. Where should we be looking in your dissertation, and what should we be looking for?
– user507
May 27, 2020 at 1:00

Dan Ghica has a blog post on discovering category theory for primary school children by way of knot theory. Quite charming, I think, albeit not really an answer to your question. Still, I thought it might be worth noting here.

(I came across it when reading Paweł Sobociński’s Graphical Linear Algebra blog.)