I think every IB teacher approaches calculus the same way: look, average rate of change. Look, instantaneous rate of change. Secant, tangent, gradient of tangent. Previous years I've tried to make this introduction come alive by giving students a function describing the distance fallen by a parachute-jumper, and having students discover themselves what the average speed would be, and the instantaneous speed at certain moments.

This has worked... well, let's just say students got the main ideas but not at an intuitive level, neither did they retain their discoveries more than a week or so. The whole thing seemed very artificial, pseudo-contextual.

So this year I decided to try graphing stories, choosing one about distance and one about speed. I downloaded graphing paper for graphing stories from Dan Meyer. I played the first video, asked students to graph it, and showed the answer. It was like lighting a fuse and seeing the classroom first fill with expectancy and then erupt in a constructive chaos of discussions about distance, displacement, speed, velocity, mathematical modelling, acceleration... "Look, the steepness is the same on the way from the camera as on the way back, so the guy who made the video assumes the speed of the dog was the same in both directions!" is one memorable quote.

It was a weird and funny experience to have the class completely ignore me and all my efforts to bring order into their discussions simply because they were so incredibly involved in figuring out how the video related to the graph and what we could infer from the graph about the beliefs of the grapher.

Then I showed the second video, about speed. Students immediately connected steepness or graph to acceleration and after a brief discussion when I asked "so what is the distance that the runner covered?" some students immediately replied that we must look at the area under the curve. I asked how we could calculate that area and students proposed dividing it into small sections, rectangles, trapezoids, triangles.

I believe that students developed a really strong intuitive connection to the main concepts of calculus, through their understanding of distance, speed, and acceleration. We will follow this up with more structured work using the graphing stories graphs to figure out approximations to average and instantaneous rates of change. That will take a full lesson. And then we're ready for functions and the limit definition of derivative.

It might seem like a long time (more than one week of class) to build up to the definition of derivative. In my opinion, conceptual understanding of derivative is necessary for everything that follows, especially when using derivatives (and second derivatives) to graph functions and for optimization problems. Without the conceptual understanding nothing else is going to make sense, will just be an imposing set of strange rules and lengthy procedures. With a solid understanding of derivative (and later integral) calculus comes alive, becomes beautiful, and leads students to ask, as one senior student just did: "What profession should I choose that lets me use calculus as much as possible?" :) :) :)

How did you answer that student?

ReplyDeleteThe link on the phrase "more structured work" only shows when you hover over it. I lucked out and saw it just now. Nice handout! (Did you make it, or did someone else? I'd like to tell my students about the teacher who made it.)

ReplyDeleteHi Sue, so sorry for the delay... summer, ya know. :)

ReplyDeleteAnyways, that very cute student got the answer that physics or some branches of economics, or even just pure math could lead to a lot of calculus. I don't know if that's correct but I kept a straight face saying it.

It's great that you liked the handout - I made it, and I thought it worked well with the videos. The link shows up blue on my screen, I don't know why it doesn't on yours?