Year 3 of Calculus is the refinement year. The curriculum feels good, now to improve on some major flaws and oversights. A huge part of Calculus that I underestimated is getting students to understand what notation really stands for. For example: f(2) has meaning, f'(2) has a different meaning, dy/dx can have a numerical value, that sort of thing. The real abstract concepts that demonstrate whether you get Calculus in a big picture kind of way. It's very easy to teach kids the quotient rule and have them parrot it back mechanically, but something like this will cause a major roadblock:

There are no defined functions here. Just notation. Symbolic of some meaning. If you really get Calculus, you can see right through it. You think you know the chain rule? product rule? integration? Here's a chance to prove it.

My students last year couldn't prove it. In our brief discussion after the release of this question, only one (out of 49) admitted to having any clue what to do here. The rest punted. That's a failure on my part.

There are other examples, but more abstract discussions are a feature this year. Surprisingly, our first discussion about a week ago (which included a look at this question), went really well. One of my strugglers was beside themselves with excitement over finding h'(1) if h(x) = f(x)*g(x). Blew away my expectations.

AuthorJonathan Claydon