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Practice. Practice. Practice.

Practice is generally considered good for learning mathematics. That’s probably not controversial. The disagreements arise when we talk about the type of practice. There is much talk about deliberate practice and the famous 10,000 hours to achieve mastery. (Both fantastic areas for a discussion but I’ll come back to them on a different day.) Following on from yesterday it is probably clear that it is easy to provide practice in exercises but less easy for problems. (Also, one should distinguish between practice and exam practice.)

A problem that arises most years for me is a student who feels that they did a lot of work during the course but did poorly in their exam score and want to know why. In conversation with them, I often find that they have done a lot of work — lots of practice. Digging deeper it emerges that it is the wrong sort of practice. They have focussed on the bits of the course that they like and can do and have done exercise after exercise (and sometimes problem after problem) on those particular areas.

This I think arises from two aspects of human nature. First, we like to do what gives us pleasure. Solving a problem successfully gives a little bit of satisfaction that encourages us to repeat the action. And so we get into a cycle. The second relevant aspect of human nature is a tendency to find reasons to procrastinate: “I’ll need a big block of time to tackle that difficult area I don’t understand. To fill up the short time I have, I’ll do some more exercises on that thing I like.” Doing the mental work on something new, something hard, is less appealing and so that doesn’t receive the attention it needs.

Hence, I believe that students are succumbing to the enjoyment of solving something and the procrastination arising from the prospect of having to think hard.

The trouble for me is that I’ve not found an effective solution to this problem. Clearly I can tell students that they should practice what they can’t do rather than practice what they can but that doesn’t seem to work well. Does anyone have a better idea?

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