I’ve been working on a few longish posts that just aren’t coming together. Lately I haven’t even been working on them much, just staring at them criticizing them, and going on to something else. To get them off my mind, I’m going to give each one of these failed posts a paragraph or two here, then move along.
Category Theory and Early Math Education
John Armstrong wrote a nice little introduction to category theory a few weeks ago, showing how numbers, addition, and subtraction are really just the ‘decategorification’ of finite sets. It sounds fancy, but it is really elegant and simple. When I read it I immediately thought: Wow! kids understand sets before they understand numbers, they understand putting sets together before they understand addition, and they understand making rectangles or groups before they understand multiplication. We should spend more time teaching kids about these set operations before we teach them formal arithmetic algorithms. I’ve been trying to put together recommended exercises, show how this categorified version of arithmetic makes things like FOIL and the distributive property obvious, etc.
I lose my motivation because:
- It seems like I’m wasting a lot of words just to say “Match things, count things, make rectangles out of objects and break them into smaller rectangles.”
- Singapore Math workbooks already do a lot of the things I’d recommend.
- It’s unlike me to recommend that anyone do anything. I’m more comfortable trying to subvert common knowledge, and if I start talking authoritatively, I just want to argue with myself.
- For the time being, I’d rather spend my time reading John’s posts than writing mine.
Work, Faith, and Grace
When I was reading Polanyi a while back I had a little epiphany. The Pauline scheme of works, faith, and grace isn’t just a tool to help Protestants and Catholics throw mud at each other. It isn’t a deep mystery of the Christian faith either. It really seems like a description of how we learn all sorts of things.
- You’re not going to learn much if you aren’t interested, and you aren’t interested in something unless you have some faith that there is something to learn.
- You aren’t going to learn much if you don’t do some hard work, and you won’t do that work unless you are interested and have some faith that the work will pay off.
- When you are interested and work hard, you are often rewarded with new and deep knowledge — knowledge that is now “part of you”, that you use without thinking. Now you are a greater being, and you are able to ask more questions, develop deeper interests, and work even harder to pursue those interests.
So loosely speaking, work is work, interest is faith, and learning is grace. At least as I described it there. I also had this idea that learning to play the piano gives us a very cute example of this process: you won’t learn unless you want to make music and believe you can, this makes you work at it until a piece flows from your fingers without a thought, the piano is an extension of you. At this point you’re ready to learn even more.
Well my thoughts are very unpolished here, and every time I try to write about it I feel like I’ve smashed the subtleties to bits with a blunt hammer. This is what I feel like when I read the paragraph I just wrote, so I retract it. But saying something and retracting it is not the same as saying nothing at all…
Thinking about work, faith, and grace I concluded that often when we really learn something it becomes an extension of us. It is something we no longer need to think about, something that we no longer subject to conscious reason. When I was thinking about teaching arithmetic I was struck by the fact that all of us can use numbers pretty well, but very few of us can give a good answer to the question “what is a number?” Much of our deepest knowledge seems to be subliminal, whether we were wired with the knowledge or learned it. This should make those of us who want to “live a life based on reason” squirm a little bit. We’ve got a lot of introspection to do.
Now maybe I can stop staring at some of those drafts I’ve been accumulating. For now, I’d rather read a little and play with the kids anyway.