In my experience, learning is all about clunking around, learning something, practicing and automatizing that, and chunking it so that you can learn something more complex. Just because we have kids practice and memorize in our classrooms doesn’t mean we “value the faster memorizers over those who think slowly, deeply and creatively.” We should be talking about the proper balance between concepts and practice/memorization (I think it is splitting hairs to distinguish the two), and how to be most effective at both. I think Jo may be thinking this as well, but her rhetoric tends to push people to one side or the other of this common ground.
I drive a hybrid. I am striving to have my math class be as conceptually based as I can at all times. My students write linear equations from situations and data before they do any solving, and then we use bar models so they can direct the process themselves. We learn the formulas for the area of a triangle and area of a circle by drawing and counting and reasoning. It drives me nuts when a student puts a 1 under the 5 to multiply 5 x 2/3.
But every year, I ask my 7th graders (who were also with me in 6th grade doing that great lesson on areas) to find the area of a triangle and about half “forgot”. If they haven’t memorized the formula for the area of a triangle or a circle, the surface area of cylinders and prisms becomes a real challenge. Could they get along without knowing them? Sure, but with very little time investment, I can help them to memorize them. There doesn’t have to be a sacrifice of any conceptual thinking. What’s 7^0? Multiply 2 fractions? Solve 5x =7 or 2/3x = 3/5? Short, frequent, painless practice sessions can make these routines useful later on, long after the conceptual work is done.
Because here is the big aha for me – does the fact that students don’t recall the formula for the area of a triangle mean I did a crummy job of teaching it last year? I don’t think so. During the unit they could all tell me about the related rectangle… and obviously the triangle was half. But I think it means I didn’t finish the job. Practice is the second half of learning, both for facts and concepts. Jill Barshay’s article describes Jo’s thinking: “The human brain is forgetful by nature, she argues, and what she wants is students to develop the number sense to calculate 7 x 8 quickly even when their brains can’t recall the math fact instantly.” But without practice, the students will lose those strategies as easily as they lose the facts.
Think about what happens when you solve any math problem. There are things you have to think through, maybe use trial and error or other problem solving strategies, and things you just know or recognize. Those are two parts of the brain at work, and as math teachers, we need to be effective in serving both.