We live in the shadow of the 1990's "Math Wars", and crafting a new approach to the K-12 math curriculum is fraught with problems — and hate!
On one side: the math traditionalists. They point out, quite sensibly, that learning one particular method for thinking about, say, division makes math much easier to do. They also point out that repetition is required to develop fluency.
On the other side: the math reformers. They point out, equally sensibly, that learning one particular method for doing math doesn't provide actual conceptual understanding of the math at play: a student can learn how to "do" long division without having any glimpse of what's going on. They also point out that mindless repetition is typically at odds with any sort of enjoyment: doing problems 1–30 (odds!) rarely stirs our curiosity.
What follows: a sketch of how a new kind of STEM school could structure its math curriculum, especially in grades K–4.
In brief: embrace extremes. We can recognize that both the traditionalists & the reformers recognize crucial aspects of math, and of the human mind.
Our job isn't to "balance" them into some sort of middling practice. ("Balance" is typically doomed from the start.) Rather, our job is to hold the extremes together.
Math math as easy as possible
Math is hard, and to help students value it we need to help them see that they can understand it.
And no mathematical idea — at least, none in the K–12 curriculum — is beyond students. Virtually all students can perfectly understand everything in the math curriculum.
It's easy to say that, of course, harder to do it! To make math as easy as possible, a new kind of STEM school could use two tools:
- The JUMP Math curriculum. We've explained our love of JUMP in an earlier post, but to distill it: the K-8 JUMP curriculum breaks every math idea into an armful of tiny ideas, which all students can zip through. JUMP works for advanced students, for struggling students, and for everyone in between.
- Deep Practice Books. We've written a little about this before, too, but to distill: students can keep collections of problems they find frustrating. They revisit these problems, asking questions of them, seeking deeper explanations, and re-solving them in diverse ways. After a few days or weeks, each problem becomes easy — and students tend to enjoy them!
Make math as hard as possible
Math is hard, and to help students value it we need to (wait for it) let them struggle.
JUMP's brilliance comes from the fact that it carefully guides students to full understanding. But to cultivate real mathematical thinking, we need to also give students unguided experiences with math: we need to toss them into problems, and let them fend for themselves.
Well, that's an exaggeration: it's not that we should offer no guidance, but that we should offer minimal guidance. Students need to learn how to problem-solve on their own.
Yes, this is the opposite of the above! And both are important.
How should we do it? We know of two tried-and-true methods:
- Host math circles. What's a math circle? A sports team for math.
Math circles can vary profoundly: some are all about preparing for math competitions, others are anti-competitive. Some look like traditional teacher-led courses, others are inquiry based. What they share in common is that they're led by real mathematicians, and lead children into the depths of mathematical ideas through conversation.
- Attempt mathematical puzzles through the "Japanese Method". In The Teaching Gap, James Stigler and James Hiebert lay out how Japanese schools put unstructured problem-solving before guided explanations. A teacher in Japan will post a challenging problem — one which students do not have the tools to easily solve. Working in small groups, students will tackle it, crafting their own tools to do so. And then the groups share their methods, and the teacher leads a conversation comparing them. Which method is easiest? Which is most elegant? Which is the most complex? By puzzling through how (superficially) diverse methods can lead to the same answer, students see into the heart of mathematics. (We've written about this idea before here.)
If there's one thing that everyone in American education can agree about, it's that math is currently taught abysmally. By reconceiving math teaching (and learning) with the insights of all sides of the math wars, a new kind of STEM school can forge a way forward.