Thanks to Jonathan Claydon for starting some great conversation about tough concepts in calculus, first with his post on avoiding magic tricks, and then starting work on a calculus chapter in Nix the Tricks. I just learned some cool new ways to introduce tricky ideas in calculus, and my teaching will be better for it. Check out this cool visual explanation of the Product Rule!
Jonathan’s post also brought to mind a neat exchange I saw on Twitter recently on explaining where the Chain Rule comes from, also very cool!
Alright, now it’s time for a confession.
I don’t like introducing complex ideas like the chain rule by proving why they work.
I think this type of introduction-by-proof appeals to a subset of my students, but it tends to turn off others, and the kids who turn away are the ones I most want to engage.
Here is a different approach I’ve used for the Chain Rule:
(doc) (Yes, I introduce transcendental functions before the Chain Rule. I realize many calculus teachers follow a different sequence, but this approach still works with different examples.)
I give students the handout and tell them that it shows functions on the left, and each function’s derivative on the right. I ask them what patterns they notice, and how they could use those patterns to find other derivatives in the future.
Students don’t usually figure everything out on their own. That’s not my goal. Instead, students have a chance to think about “inside functions” and “outside functions” and describe this funny derivative rule informally before we describe it formally. I build off of students’ language describing the types of functions they see and their connection with the derivatives. Then, after offering some explicit instruction in informal language, talking about inside and outside functions and multiplying by the inside derivative on the outside, I might offer them a more formal definition of the Chain Rule, like this one:
I ask students to discuss in pairs or groups how this definition is connected to the examples they just looked at. After a bit of informal discussion on their own terms, I ask them to identify both f(x) and g(x) in several of the initial examples and I annotate a few functions on the board, using color to emphasize the different pieces and how they fit together.
The purpose of this sequence is to move from informal to formal, to give students a chance to make sense of an abstract rule on their terms before Leibniz’s, and to use worked examples to illustrate an idea while still putting the thinking on students. I have no illusions that this is sufficient to teach the Chain Rule, but hopefully at this point students are set up to be successful in engaging with some practice.
After this sequence, students are hopefully thinking, “ok, that kinda makes sense, but why is that the case?”. That’s where I think the conceptual explanations I referenced above come in. Once students have a basic grasp of a rule, I think they are in a much better position to grapple with the complexities of where it comes from. Even better, the initial exploration could happen on one day, stew overnight, and the next day I share a way of understanding where the rule comes from. Students’ informal understandings and experiences with a few concrete examples of a concept will hopefully help them better understand and make sense of an explanation of where that concept comes from.
I like this approach because I think it honors Jonathan’s desire, and my desire, to help math make sense to students, while also prioritizing informal thinking before formal thinking. I also like that, at every step, I can give students who struggle with calculus opportunities to engage on their own terms and feel like they can make sense of new ideas. I don’t think this is the right approach for every new mathematical idea, but in calculus lots of concepts have algebraically or computationally complex proofs, and this approach hopefully minimizes the challenges of that complexity.