How not to get a good mathematics degree
Definition of Abelian
I’ve been marking homework for Group Theory. (Another exciting Sunday afternoon for me, eh?) I was reminded of a question last year from a student. I was discussing mathematics with some students and one asked me about showing a group is abelian. Their homework question asked them to state whether a particular group was abelian or not. The student liked this question because, she said, it didn’t ask her to justify her answer. She had a 50/50 chance of getting it right and that was good — she knew that she would be unable to supply any reason if asked and so would stand no chance of getting marks
At first, I mistakenly thought she we wanted me to explain how to show a group is abelian. I was wrong. What she wanted was a shortcut, some rule of thumb say, that allowed her to tip the chances in her favour, maybe 60/40 or 70/30.
Now, I’m not sure if there is a decent rule of thumb (if you see Sn or Dn it’s not abelian maybe) but this is definitely the wrong way to go about getting a good degree. Trying to memorize all the short cuts without understanding is not a good recipe for success at university-level mathematics — even if that strategy worked reasonably well for A-Level!
The message for any student reading this is simple. If you have a weakness in your understanding — the student above certainly knew she did — then aim to remove it not by using some shortcut but by understanding the problem. Shortcuts will help only in the short run. In the long run they will do more damage than good.
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Excellent stuff-you taken the words out of my mouth. Too often exam based systems forces the student to swallow maths formulas wholesale rather than trying to appreciate its finer workings. This is exactly what happens over here in Singapore, a country of deeply entrenched in rote learning to achieve good grades.
And some of our politicians and Press here in the UK think that Singapore and places like it are absolutely wonderful at maths, their schoolchildren 2-3 years ahead of ours – and so on. I have to admit I’ve often wondered how much of this is owing to high-pressure memorising rather than real understanding. China and South Korea are two other places that spring to mind – do you have any knowledge of these countries?
Talking now about the most basic level of arithmetic I once came across a learning-disabled student who’d evidently been taught to memorise multiplication tables. Fair enough up to a point, they’re useful things to keep in your head. However, it came as a complete revelation to her that (integer) multiplication was in fact repeated addition – a fact inculcated into me by a small but fairly progressive pre-prep (4-8yrs) school via Cuisenaire rods way back in about 1959 or 1960.
Hi Humphrey,
My knowledge of the mathematics in places such as Singapore, China and South Korea is limited. My impression is that their systems do generate quite a bit of understanding but there seems to be a loss in creativity. However, this seems to be changing and China in particular may come to dominate mathematics in the 21st century. Its main problem is that many of its top mathematicians go to the West and hence their universities lack the really big names. This has a knock-on effect in education nearer to the school level in that the teachers there just recycle the education they were given.
Hi,
I do have over two years’ experience of teaching A-level Maths and Further Maths in China, and four years of university maths in Thailand. All I’ll say is that it wasn’t pretty!
When it comes to Maths, as far as I could see their educational systems have absolutely no element of understanding [whether this is the intended effect I do not know, but it is undeniably the result]. The students are treated like battery hens and learn almost nothing BUT silly exam tricks and shortcuts, which only work because the exam questions are deliberately designed to make it work.
Our beloved A-levels, awful though they are in comparison to the real 1970s versions, are still an improvement on the systems of these countries, at least as far as Maths is concerned. We just need to make them harder!
Dear Zen,
Thanks for the comment. I suppose it’s good that we shouldn’t feel too bad about our A Levels. On the other hand as you say, changes need to be made.
Kevin