This post is in response to this post by the famed FSP, whose blog you should check out.
I've mentioned in earlier posts that I do change what I talk about when I teach calculus at the SLAC which currently employs me. My calc classes have fewer than 25 students each, and I do know everyone's name and something about what interests most of them.
Calculus is a subject that seems timeless. For better or worse, I don't keep notes on teaching from year to year: I don't have lecture notes, a ton of worksheets to recycle, problems I particularly liked. I kind of wing it each year (although I do remember tips and tricks from past years). I don't know if I should admit that.
I change the syllabus with the textbook required by the subprogram of the institution for which the calculus class is being taught. For instance, at the big R1 at which I did my graduate work, I taught versions of calc for three different programs. It is easiest for me to go with the textbook rather than being proudly focused on the best teaching/best order/best anything.
Within that framework I change the focus of the class considerably from class to class based on the interests of the students. Today we worked on optimization. We optimized the happiness of a pig as a function of corncobs eaten; my students this year seem big on economics but like maximizing happiness and minimizing poverty more than maximizing profits. Last year, as I mentioned, it was chemists. We looked at reaction rates and maximized reaction rate or concentration of particular reactants. I am happy to skip from application to application based on student interest -- it keeps me interested as well.
In introductory classes like this that have strong demands for content coverage by other departments it is really hard to radically change the syllabus. If I switched to an inquiry-based learning model the physics department would go nuts. If I changed to an applications-focused model of the course not vetted by the biologists they'd riot. If I started focusing on proofs the chemists might have heart attacks. Non-mathematicians have some strong feelings about what a first-year calculus ought to cover and how (if they feel that mathematics is useful for anyone at all!).
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