This year, in my Groups IA course in Cambridge, I am trying to sprinkle my lectures with useful facts about how to think about mathematics, how to learn mathematics, and how to develop good intuitions as well as good working habits. I will try to post these thoughts on this blog as I go along. Most of these ideas come from a brilliant book by Lara Alcock, “How to Study for a Mathematics Degree”. To quote from her introduction: “Part 1 could be called ‘Things that your mathematics lecturer might not think to tell you.'”
As Director of Studies at Newnham College, and through teaching many first year mathematicians, I experience quite a few students making the transition from school maths to university maths.
Mathematics at university is not the same as mathematics at school. Most people who have not experienced maths at university think of maths as calculation, as working things out following certain rules and recipes. There is the expectation that there is always a “right answer”, a very specific end result of a calculation. However, maths at university is not (just) like that. Yes, there are still calculations to be done, but one of the main aspects of mathematics is a logical way of thinking and proving general results. Mathematics students learn to work and think on a more abstract level. They learn to make rigorous arguments leading from some very specific assumptions to a general result. They learn to question these assumptions and to find connections between different mathematical situations. This often has nothing to do with calculation at all.