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Fighting with mathematics

Mathematics is not a spectator sport. During my postgraduate years a paragraph helped focus my ideas about learning mathematics. It’s about reading but is applicable more generally.

Don’t just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

Paul Halmos, I Want to be a Mathematician (1985)

How many mathematicians would disagree? Understanding mathematics involves an active struggle — it’s not a passive activity. So what does the standard mathematics lecture look like? Is Halmos’ fight evident. Generally, no. A lecturer stands in front of a board, writes something, the students copy it.

In this standard scenario students are expected to be thinking and copying. But, can people do two things at once? Yes, we can drive cars and sing at the same time but we can’t walk and text. In this situation — copying and thinking — what comes first? Well, students are concentrating on the copying and not the mathematics — more likely thinking, “is that an x?” rather than “why is that an x?”.

So what is the point of this method? It can’t be to give information — that would be grossly inefficient. Students could be given the notes, read a book, or watch a video. The crucial question to evaluate a teaching method is “are the students learning?”. Given that the students are copying rather thinking, the standard lecture is unlikely to foster learning. In fact, there is a very tall stack of evidence to show that the standard lecturing method (teachers writes, students copy) does not work. (I did look up some good references for this but thought it best to refer to Graham Gibbs from nearly 40 years ago: Twenty terrible reasons for lecturing.)

My belief is that we learn mathematics by doing and so lecturing/teaching should aim to get students doing and, in particular, thinking. (After all, I did call my book How to Think Like a Mathematician.) I recently finished reading Small Teaching: Everyday lessons from the science of learning by James Lang. Chapter 3 on interleaving struck me as it is something I already know I should be doing but, to be honest, I don’t. It’s about time I was rethinking what I could/should be doing and how ideas, such as interleaving, can be incorporated effectively into mathematics teaching in higher education.

To facilitate this, I’ll be posting over the next months some thoughts about what I can do to improve my teaching.

Feel free to join in with suggestions and links to your favourite ideas by commenting below!

Let’s see how far I can go with this.

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