Here is an extract from the blog of another Fields Medallist, Tim Gowers. These comments are perhaps illustrative of the difficulties faced by mathematicians and give others not similarly endowed a great measure of confidence and boost in their studies.

“This raises an important point. I think what may make some mathematicians learn in a different way from others is that some people find straight memorization easier than others. I myself find it difficult, so I don’t really learn anything properly unless I’ve gone through a sort of personal process of rediscovery. That takes time, and the result for me was that although I did adequately as an undergraduate, I was by no means the top in my year at Cambridge — if you’d like to know, I was 15th in my finals — and after the exams I forgot a lot of what I had crammed into my head. I was drawn to the areas of Banach spaces and combinatorics for two reasons. First, and more obviously, I was brilliantly taught in those areas by Béla Bollobás. Secondly, in both areas there were many interesting problems that one could realistically tackle without having to learn a lot of machinery first. Such success as I have had in those areas is no evidence at all for any ability in learning mathematics, where I think I am pretty average: I’m sometimes quick, especially when I’ve thought along similar lines already, but if the area is completely unfamiliar then I’m not quick at all (relative to other mathematicians).

My undergraduate days left me afraid of many subjects: complex analysis, measure theory, most of algebra and almost all geometry, for example. Little by little I have lost quite a lot of that fear: I was forced to come to terms with complex analysis when I had to teach it, and the same happened with measure theory and some of the more elementary parts of algebra and geometry. Editing the Princeton Companion has helped me a lot too: although I don’t understand all those scary areas like topology, PDEs, Riemannian geometry, and so on, I now know just enough about them to see why they are interesting and important. I think I finally got to grips with the concept of cohomology (of the most elementary kind, I hasten to add) a couple of months ago.

Of course, this is all a side issue really. As it happens, I am quite a good example of the kind of person who is greatly helped by having examples first in order to understand generalities. But my case doesn’t rely on that, so if you don’t believe it then take a look at some of the comments, which suggest that there is a significant percentage of mathematicians who feel the same way, and also take a look at my explanation of why I think it is helpful to have examples first.”

These are the words of Gowers himself.

His web page has a lot of interesting discussions about various aspects of elementary mathematics.

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