The Return, Part 5

I am preparing to teach a calculus course again after more than a decade. The calculus course I am going to teach is for incoming first year students. I used to teach a version of calculus using Tom Apostol's books. These books were wonderful and had an unconventional sequencing: integration first before differentiation.

I am toying with an approach to calculus now that I am teaching it again. I have been in an on and off relationship with the attempts by probabilists and statisticians to re-imagine and, sometimes, to re-think proofs for existing results in probability theory and statistics. This re-imagining is sometimes called Radically Elementary Probability Theory (Edward Nelson, 1987). The ideas from this re-imagining depend on nonstandard analysis, or if you wish a calculus using infinitesimals. The result is probability and statistics without relying too much on measure theory (unless really needed) for the rigor. If you get this book, you get to a version of a quite sophisticated central limit theorem in just 80 pages or so, with only high school algebra (and some rules about nonstandard analysis). Charles Geyer has also written some extensions of the ideas in Nelson and is really worth taking a look.

That is why I am finally giving myself the excuse to teach calculus using infinitesimals. H. Jerome Keisler wrote an introductory calculus book along these lines in the 70s. I spent the entire day today reading this book slowly, and considering what a syllabus would look like. I am about 300 pages in. I am also toying with the idea of using Tom Körner's book called Calculus for the Ambitious (aimed at high school students). When I saw the book title, I was sold. A new prep, especially this kind of prep, makes you feel young again and somehow pushes you out of the usual bean-counting that sometimes happens in research.

A side-benefit of choosing the approaches I have described is to wean off Chinese students from looking at the Chinese language books (especially when coursework is in English). The nice thing about the books I mentioned is that I am currently unaware of Chinese translations. That should force the student to listen, communicate, and think in English. More importantly, students will be exposed to different styles of writing which would hopefully improve their own writing. Hopefully, these choices will pay off.

I will report on the progress of this preparation as soon as I could. Now, I do not have a lot to say anymore, I believe I was somewhat exhausted by what I wrote down yesterday.

#Calculus #TeachingDuringCOVID