Not A Spectator Sport

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One of my university lecturers gave what we all thought were dreadful lectures. Muddled, unclear, chaotic, with no discernible thread. It took ages to reconstruct and rework the material to a point where we could attack the problems and old exam questions.

I got nearly full marks on that exam.

Other lecturers were brilliant. Clear, lucid, entertaining. But I didn't get full marks on their exams, because I found it hard to do the problems, even though I thought I'd understood the material from the lectures.

I hadn't.

Maths is not a spectator sport. You need to get involved, otherwise you're in the situation of someone who has watched a lot of tennis, but never played.

I used to mock the "it is clear that" phrase when it would then take two or three pages to show the result. But I learned that having done the work to show it, I was then equipped to handle the next stage of the work. Having the explanation given to me as to why it was "clear" would not have done that, my understanding would have been meagre and unsatisfactory, and I would have gradually fallen behind and not been able to understand what was missing.

So when used properly, when used well, when used constructively, the phrase "it is clear that" and its friends are not there because:

that's how the game is played (tradition);
it excludes the uninitiated/untalented;
it's neater to leave out the truth of discovery;
it makes the author seem superhuman;
there's satisfaction for the reader in understanding the puzzle.

When done properly it's genuinely for more effective communication.

I'm not saying it's always done well - after all, not everyone writes equally well - and I'm not saying that everyone always has the best of motives. But working on what you see as gaps in the presentation really is the best way to understand the material.

There was an occasion when I said all this, and someone replied:

Someone, somewhere will have to work out the combination - I'm suggesting it is more efficient for it to be the one writer than the many readers.

If your purpose is simply to have it written down, then yes. However, if your purpose is to communicate effectively to the readers, then no. The "doing" is an essential part of the eventual "understanding".

This was prompted by a discussion of "How to Read Mathematics" by Shai Simonson and Fernando Gouvea. The article is here: