Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class. A quote:

Every rule – even the craziest, most arbitrary mandate – has a reason rooted in this essential purpose. (Why leave the dishes with big particles? Because the person is still eating!) And so it is in math class. If you understand slope not as “that list of steps I’m supposed to follow” but as “a rate of change,” things start making more sense. (Why is it the ratio of the coefficients? Because, look what happens when x increases by 1!)

You get to work a lot less, and think a lot more.

Now, conceptual understanding alone isn’t enough, any more than procedures alone are enough. You must connect the two, tracing how the rules emerge from the concepts. Only then can you learn to apply procedures flexibly, and to anticipate exceptions. Only then will you get the pat on the back that every robot craves.

With my students on Friday, I garbled the whole analogy. I tend to do that.

But there’s a simple takeaway. Even if you don’t care about understanding for its own sake; even if you’re indifferent to the beauty and deeper logic of mathematics; even if you care only about test results and right answers; even then, you should remember that the “how” is rooted in the “why,” and you’re unlikely to master methods if you disregard their reasons.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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