One of the reasons mathematics is powerful and useful is that it is abstract. A collection of abstract symbols may seem sterile, but the power is in the hands of the practitioner, for you can give the symbols whatever meanings you please. Consider Boolean algebra, an abstract system of rules for symbolic expressions. As an … Read more
For four consecutive years I taught a fourth-semester course called “Introduction to Analysis,” in which we looked at differential calculus for a second time, stressing the foundations, the logical structure, and proving all the key theorems. We used Stephen Abbott’s excellent book, Understanding Analysis. The course was intended primarily for math majors, although we had … Read more
Nowadays conic sections are not part of the standard high-school mathematics curriculum in Ontario (at least ellipses and hyperbolas are not; of course circles and parabolas are present), but they are interesting and important curves in mathematics, science, and engineering applications. There are two ways to define an ellipse: (1) as the curve of intersection … Read more
One day a graduate student submitted some writing to me, in which she was explaining rates of change at the high school level. She made an interesting statement: The slope of a secant line joining two points $(a, f(a))$ and $(b, f(b))$ on the graph of a differentiable function $f$ is the average of the … Read more