So, I’m kind of done with hearing/reading about that danged blue-and-black dress, but then I read this Business Insider article about the old Radiolab episode on Colors. There’s a great point in this about how people in Namibia, who never learn a word for blue, can’t distinguish a blue square out of a group of green squares. Read the link above.
This distinguishing of colors problem is analogous to something I’ve noticed in teaching calculus: students tend not to notice differences in functions like
1/(x^2 + 1) and x/(x^2+1) or
sin(x^2) and x*sin(x^2) and x^2*sin(x^2)
Over the years, I’ve made sure to include mini-lessons on integration where students try out different methods (integration by substitution and integration by parts) on these functions juxtaposed, then reflect on the question “What details make the function use one method vs. another?”
It’s these experiences over the years that make me realize that I have had to make decision-making into a lesson that has to be scheduled into the overall course at various points.
When did I first realize this? In my second year of calculus, I taught u-substitution to integrate for 3 days (to get them good at it), then integration by parts for 2 days, then partial fractions & separation of variables (Chapter 6 in Finney Demana Waits). And on the Chapter 6 exam, most of my students tried to use Integration by Parts on EVERY integral except the partial fraction ones. How about that? The class average dropped on that test, and I had to go back and explicitly show how to distinguish between the two types of integrals. One of the more astute students managed to make integration by parts succeed on the simple u-sub questions. Ha!