Last November I attended the DACTM/MDSTA conference. Most of the workshops I went to were either about Modeling Instruction or Standards Based Grading, but I did go to one workshop called "Start Calculus with Calculus". The idea was that, instead of starting with a lot of review and then an in depth study of limits, you start the class with interesting problems that lead students to develop the basic ideas of the derivative and the definite integral and what they physically mean. Shawn Cornally wrote about doing something similar with his calculus class. I was intrigued by the idea, but hadn't really given much thought to it until I was in my Modeling Instruction in Physics workshop this summer.
I have the joy of teaching two courses that are deeply intertwined. Rates of change and the accumulation of change are big ideas in both physics and calculus. So every time we examined the graphs of our data and interpreted the slope or the area under the graph, I thought about my calculus students as much as I thought about my physics students.
How can I change my teaching method so much for physics and not for calculus? Why can't I use the same experiences to teach students calculus as I use to teach them physics?
Now I'm spending a lot of time thinking about the best way to use what I learned this summer to teach derivatives and definite integrals.
Do I start with the constant velocity model? We'd start with a linear position graph, which my students have a very good understanding of. Drawing the velocity vs time graph would be easy, and it would be simple to look at the meaning of the area under the graph. Or, should I start with a uniformly accelerating model and explore the difference between average rate of change and instantaneous rate of change? Do I even look at integrals yet? Do I keep coming back to the same investigation time after time, looking at it through a new lens?
Will teaching this way ease students' troubles with creating mathematical models of situations? Will it lead them to see the big pictures of derivatives and integrals, or will they still loose sight of those ideas in the details of how to find them?
How do I move on into other contexts besides kinematics? How do I move into a formal study of limits? How/when do I come back to those early ideas?
My spring physics students become my fall calculus students. How will all of this change when they'll have already had those experiences in my physics class? Do I repeat the same investigations, or do I find new ones?
Do I really want to be completely reworking the flow of my calc classes when I'm also going to be reworking my physics courses? I have a lot of work to do!!