Read as much as you can. Read as little as you can.

In math, as in any subject, there is definitely an advantage to being widely and well read, so that every known fact about your topic of study is immediately available for recall. Thus, so long as the result of your reading is that you actually retain new and useful knowledge, the first sentence in the title is indeed sound advice. However, I have noticed that math students often find it very difficult to learn from reading. They spend time with their math book, but come away with…

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Mourning the Death of High-school Geometry

Don't get me wrong. I'm not some hide-bound traditionalist looking for a return of the "good old days." Just because our ancestors studied geometry in school doesn't mean we have to. Neither do I have any arguments to suggest that geometric knowledge is superior to other mathematical knowledge. In fact, in my life as a mathematician, the content of high-school geometry (for example, proving triangles are similar or congruent) doesn't come up all that often. So why this sense of loss? That will come later. First,…

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Körner on Lecturing

This is a second post on T.W. Körner. It started out being the first, as I was reading his home page and found some particularly good advice for students on "How to listen to a maths lecture". But then I wanted some sort of introduction to express why I so appreciate Körner as an author. Well, one idea followed another until the introduction so mushroomed in size my original theme seemed like an appendix. Thus, the introduction became a post all on its own, and here, finally, is my original…

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T. W. Körner and Story-telling in Mathematics

T. W. Körner at Cambridge University is perhaps my favourite author of mathematics, and his Fourier Analysis probably my favourite math book. I think what sets him apart is that he is equal parts mathematician and story-teller. By story-telling I mean something deeper than merely relating interesting anecdotes along the way, which he does very well, by the way. Rather, I mean he unfolds his mathematics as a story. The big theorems of mathematics all came as answers to important questions. There was a problem context in which they were discovered, and, in…

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Extracting square roots by hand

I recall feeling a curious pleasure when I first encountered the Maclaurin series for sine and cosine. With them I was able to compute for myself, by hand, sines and cosines of various angles to any degree of accuracy I wished. Perhaps part of that delight was the feeling of independence. Rather than requiring a table of values someone else prepared, or using a calculator, I could manufacture on my own any value I needed to any accuracy I needed. Curiosity was also part. Constants…

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Computing the length of day anywhere, anytime

The amount of daylight varies with latitude and time of year. But how, exactly? I began thinking about this as an exercise in spherical coordinates for second-year multivariable calculus students. After all, how hard could it be? If you follow the solution to the end, you'll see why I doubt I would ever assign this to students. However, I was happy with my own approach to the problem, and needed some practice using the tikz graphics program, so I thought I would write it up.…

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