Unmotivated
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…
Sublime knowledge
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.
Actually, though I don’t think I said so, the effort to teach arithmetic in terms of set operations was a driving motivation behind the “New Math”. Its appearance in Singapore Math books is an artifact of that history, as are the “number blocks” used in elementary school texts, at least when I was in elementary school.
John Armstrong
August 15, 2007 at 4:00 pm
to say nothing of the fact that the field axioms
get rolled out in every beginning algebra class
and then ignored pretty much altogether …
god forbid we fiddle around with ‘em and
find out “what follows from what” …
that would be far too mathy …
it appears that the creators of these courses
were made to memorise suchlike vocabulary
as “commutative property of multiplication”,
for reasons they’ve never understood,
and they’re bygum gonna make good and sure
that the next generation is made to learn ‘em too.
the kind of thing that gives the “new math”
a bad name. (i came up in that generation
and had well-prepared teachers. a beautiful thing.)
vlorbik
August 15, 2007 at 4:13 pm
Thanks for the comments. I didn’t know that was behind the “New Math”. I still really don’t know what “New Math” is beyond being a good way to start a fight between Math teachers. I came up in the generation that had a lot of timed tests and not much else, so I always hoped that “New Math” meant “Math, not just computation”.
In my limited experience teaching kids, I’ve found that when they actually understand the ring axioms (they don’t have to know the names of the axioms!) then things like multi-digit multiplication come really easily. From my experience teaching college kids, when they’ve memorized a list of statements they often have an awful time doing anything with it.
I’d love to see more playful exploration in elementary Math. That’s what I’m trying to let my kids do, and I’ll put off the “fundamentals” for later.
Rolfe Schmidt
August 15, 2007 at 6:40 pm
Set theory was the “New Math” after I graduated – but further revision occurred later. Set theory was included as Algebra around Grade Nine, with Co-Ordinate Geometry and Trigonometry likely as last year high school options.
Secondary school math didn’t even use college level tools because it meant the “proofs” had to be accepted at face value : something thought contrary to good teaching practice.
opit
August 15, 2007 at 9:51 pm
I should have said what kind. “Cramer’s Constant” was not used to solve Quadratic Equations, for instance.
opit
August 15, 2007 at 9:56 pm
They’ve pretty much dropped sets from elementary school mathematics. Pity.
And we chose an algebra text that has a couple of sections of proof – though I am sure in most places they ignore proof. We don’t. Nothing too complicated. And we don’t keep it up through the entire course (brief revisit in the Spring, for inequalities). But Vlorbik’s right about 98%+ of algebra classes out there.
jd2718
August 17, 2007 at 6:45 am
[...] immediate inspiration goes to Rolfe, with finder’s fee to [...]
Puzzle: A little properties challenge « JD2718
August 17, 2007 at 8:00 am
Algebra seems like a great place to start introducing proof, precisely because it doesn’t have to be anything too complicated. In my experience with Geometry — the only subject where proof was discussed when I was in school — arguments get confused with intuition. It is very easy for students to say things that are true but are far from proven and this gets really confusing. I don’t think this problem would be as bad when discussing simple algebraic proofs.
As for elementary mathematics (something I’ve been thinking about a lot more recently), it seems obvious to me that spending some time on sets and developing some “geometric intuition” is going to make learning more advanced concepts much easier down the road. But I guess it probably won’t help performance on 3rd grade standardized tests so it would be a big risk for teachers to spend time on developing it.
Rolfe Schmidt
August 17, 2007 at 10:28 am
I’m out of the loop for current practice – but was really ‘knocked out’ by the changes that took place solving problems. Unless you were intuitive with math I don’t think many people found the tools taught of much practical use in the ‘old school’ : likely not even then.
opit
August 21, 2007 at 12:55 am
I don’t know what current practice is either. I’d hope it has been slowly progressing but it sounds like the latest standardized testing craze hasn’t done great things.
I struggled with math in the ‘old school’ until I picked up a calculus book and found out I loved math. It didn’t help me with my classes much, and I can’t say calculus (or much anything I learned in school) has practical value for me. But at least it saved my soul — I learned that math was more than dreary worksheets and timed tests. It’s funny, when I finally took calculus in school, it was mostly long dreary problem sets and tests I couldn’t finish in time.
Anyway, I don’t think math is a subject that requires years of painful training before you can appreciate its beauty. I think math has practical value, but finding out when two trains cross paths on nonstop trips from New York to L.A. certainly is not part of that practical value. Nothing I did in school prepared me for the messiness of real practical problems. I also don’t like it when the practical value of math is overemphasized. We have kids read and write fiction even though that has no “practical value”, and nobody sees a problem with that.
Rolfe Schmidt
August 21, 2007 at 7:13 am
Ouch. That has implications for ‘religion’ that are often considered contentious – instead of insight into the difference between a wisdom designed as a gift for man and one promulgated as enslavement. I have an old thought re: the difference between prophets and disciples … the disciple wants to teach you what to think… the prophet wants you to challenge your preconceptions.
opit
August 26, 2007 at 2:40 pm