In light of my recent reflections on my students' weakness in connecting application problems to algebra concepts (see this post), I started my new Calculus unit a different way than I did the last time. This is only the second time I have taught Calculus (in the IB, it's one of the topics and not an entire course), but the first time I did it I had started with the video that introduced the idea of instantaneous rates and worked my way through the idea of limits, then worked through the various traditional differentiation techniques, before we arrived finally at the applications. That was pretty ineffective, I found. The kids, by the time they got to the applications, struggled with putting them together with the algebra skills, and they found the entire topic to be very challenging and were daunted by the complexity and variety of problems/situations presented on different IB exams.
This year, I did something entirely different.
Day 1, I gave the kids a polynomial function and asked them to use the graphing calculator to find the dy/dx derivative values at specified places on the graph. They sketched the graph, wrote down the derivative values next to the appropriate parts of the graph, and then as a class we hypothesized what the derivative values meant. This was good for two reasons: 1. They're expected on the IB to know how to find derivative values on a graph using a GCD, 2. they got to go from concrete numerical examples to a more abstract definition / generalization of the meaning of a derivative, which helped me reach those concrete thinkers. Then, I gave them another sine equation, and they had to find all the places on the graph (within the standard viewing window) where the derivative was zero. Again, we discussed how they did this, and why that made sense based on their previously generated definition of the derivative.
Day 1 was very successful because by the end, the kids had conceptualized the meaning of a derivative value and were looking at me with these "duh! this is so easy!" looks. They learned the notations dy/dx and f'(x), and we went over how to find the derivative function of a polynomial function (justifying it by showing how, graphically, a cubic function "flattens" to a quadratic if you roughly sketch out its derivative values on the same graph).
Day 2 was also very successful. Instead of introducing more derivative rules, I introduced the idea of f"(x), and we linked f(x), f'(x), and f"(x) to physics. I told briefly the legend of Newton inventing Calculus to support physics, and together they figured out that if f(x) tells you the position at time x, then f'(x), its rate function, must be called the speed function, and f"(x) must be called the acceleration function. Great! With this itty-bitty bit of Calculus that they now "know", I threw them headlong into two 15-point application questions from old IB exams on differential Calculus. They struggled, of course. Two word problems took them about 60 minutes, including discussions as a class. But, it was very productive struggle, and in the end it had fully reinforced their understanding of the meaning of f(x), f'(x), and f"(x). Many of them figured out by themselves that if you need to find out the time that an object comes to a stop, you would set f'(x) = 0 and then get x, and then with that x value and the original f(x) formula, you can find the stopping position of the object.
It was lovely, because instead of the traditional approach of stuffing all the differential Calculus algebra skills down their throats at once, we slowed down enough for them to first digest the meaning, and zoomed out of the algebra skills just enough for them to see the bigger point of it before continuing with more detailed derivative rules.
On Day 3, we learned about the derivatives of sine, cosine, and also about the Chain Rule. It was pretty smooth. I had introduced the derivative of sine by asking them to sketch a sine wave, and from that, asked them to determine/sketch the shape of the derivative function graphically by first sketching the places where f'(x) = 0 and then thinking about what happens between those points. This was challenging, but some of them were able to figure it out. We referred back to the IB formula sheet afterwards to confirm our graphical intuition that f(x) = sin(x) --> f'(x) = cos(x), and we did some examples of the Chain Rule together before I threw them into another 3 old IB problems that required some resourcefulness and that involved the new rules learned on this day.
I am going to keep trying this approach in IB, breaking a big algebra concept into smaller and smaller chunks and integrating the end-to-end process very early, to see where the kids are getting stuck and to adjust instruction accordingly. I'll keep you posted on how effective this is, but my gut feeling is that it will increase their overall confidence with trying new problems on their own, because essentially they will be already doing this all the time with me as part of regular instruction.