Explaining Your Answers In Math

In all my discussions and readings and trainings pertaining to having students "explain" their work in math class, this one is the closest I've found to encapsulating my own thoughts:

When did we decide that maths needs to be explained in words? I am quite insistent on my students providing explanations; a call them ‘workings’ and they are the series of mathematical steps that they have followed to arrive at their answer. This is how things are explained in mathematics.

However, for some reason this does not show understanding. In order to understand mathematics, we need to be able to waffle on about it in English. And yet mathematics was invented in order to make it easier to express notions that are cumbersome to express with words. That’s part of the beauty of it.

It is as if we were to insist that the only way to understand German is to translate it into English; that reasoning in German alone does not show an understanding of German. Clearly, translation is a useful device for novice learners and a key component of teaching, but being able to work entirely within the target language is a sign of sophistication rather than of a lack of understanding.

Every year, I teach my senior physicists about wave particle duality. Light, I suggest, can be thought of as a wave or as a particle, depending on the situation. “But what,” they ask, “actually is it?”

“Ah,” I say, “If you really want to understand what light is, you need to understand it through the maths. It doesn’t translate well into English.”

And the more math you learn, the less easily it translates well into English.