There are two major thrusts to math education. One is to teach skills–how to combine numbers, for example, and the definitions and rules of things in the mathematical universe. The other is teaching how to think (as it pertains to the mathematical universe, though some would argue this qualification is unnecessary: how to think in math is, in general, helpful in general thinking). This second purpose is about developing mathematical habits of mind: how to approach solving a problem, how to be creative in problem solving, and other deeper thinking skills.
Math education has generally focused almost exclusively on the first of these two themes. I think in part this is because most people don’t know what mathematical thinking looks like, let alone how to teach it. Even many mathematicians would have to ponder a bit to be able to articulate how they think in their field. We are all much more accustomed to talking about math, rather than talking about how to think about math. It’s this second “about” that puts math habits of mind into a kind of ‘meta’ place. It’s a meta concept, and it takes some laborious self-observation to hammer out just what these ways of thinking are.
There is considerable debate over how to teach skills. There is debate over whether to teach skills. There is debate over which skills to teach. Do we teach the division algorithm or chuck it? Do we emphasize multiple techniques for multiplication or do we just give the time-honored standard multiplication algorithm? What are all these skills building towards, or are they building empty, skeletal towers students will abandon as soon as they are out of high school, a waste of considerable effort and energy? As teachers, educators, and specialists have struggled to sort all this out, the second major purpose of math education has been left to wilt.
There is something inherently challenging about teaching a person how to think, rather than just giving facts. When I was little I struggled to understand how light works: to know how it worked, I thought, we had to look at it, but looking at it means using the very thing we were trying to understand. This has the unsettling quality of an infinite loop, doesn’t it? Likewise, thinking about thinking can bring us into a kind of recursive trouble, almost of the Bertrand Russell paradox flavor. Teaching about thinking is so troublesome that in general we avoid doing it.
Sometimes the straight approach isn’t the most productive, it turns out. Few of us develop our habits of mind explicitly. Rather, they arise implicitly, subterraneally, through experiences that apply pressure on our thinking and shape it in near-geologic processes. A good math experience is one that offers this kind of pressure, that can gently sculpt our ways of thinking. While some skill-based work may be happening on the surface of a lesson, underneath we want the lesson to drag an iceberg through our mental habits, shaping them into math habits of mind. We’ll get sharper by going through this repeated scouring.
When we focus all the time on skill building, we lose the beating heart of teaching. You know what they say: give a person a fish, they eat for a day. Teach a person facts, they may or may not remember them and that will be that. Teach a person how to think, sharpen their mind, help them develop their insight and mental agility, and you’ve done everything we want good teaching to be.