In two days, books have been transformed in my mind from a finished product with distant authors, to something that is the outgrowth of years of thought and bursts of high energy dialogue between like-minded idea loving professors.
Let’s start at the beginning.
I had the great privilege of watching two amazing math educators meet for two days to discuss the outline of the book they are writing together on the Didactics of Calculus. Both of them have spent years trying to understand how students think about mathematics. How do our ideas develop? What types of instruction lead to productive understandings. Taking a step back, what IS a productive understanding of Calculus. The more I’m here the more I see that just because I can solve the Calculus problems I find in a typical book doesn’t mean that I have fully formed conceptions of what it means to know Calculus.
The meeting started with lots of mathematics. Implicit differentiation for example. I can do it. I’ve even taught it well enough that my students were able to do it. But do we understand it? Derivatives are rates of change. What does it mean to take the derivative of both sides of an equation defined implicitly. Furthermore, how did I not notice that I was not aware of some of the fundamental issues at play. I think that I just applied it in a few cases and found solutions that made sense and accepted the process and moved on.
This was just the beginning. Can you extend the Fundamental Theorem of Calculus to two variables? What types of understandings would allow me to do this? What are the boundaries between analysis and Calculus? What are the major understandings that students should come to Calculus with? What are multiple ways of thinking of the derivative? How does approaching the Fundamental Theorem as accumulation instead of area lead to ability to apply it to new situations? What is an epistemological obstacle? Which concepts in Calculus are hard because they are hard to understand and which ones are hard because they have been taught poorly? How does the historical development of a concept inform our analysis of what it means to learn something now that computers can perform so many computations for us? Students can think of a function as a process, just plug in a number, evaluate and repeat. Or they can think of it as an object, something to be acted upon and reasoned with. What type of thinking goes in between. And for that matter what do we call the stages between tadpoles and frogs? Is understanding continuous, slowly increasing(or not depending on how you spend xmas break I suppose) or do we make leaps in understanding?
And then there was more math. Genuine excitement and joy about sharing interesting problems and unresolved questions. We marveled at what technology plus Pat’s imagination created with graphing calculator and asked ourselves new questions. I wondered about what students can learn when faced with a tool like graphing calculator and asked to be creative. We debated what types of problems we could only solve with conceptual understanding of accumulation. We thought about symbolism and the importance of turning ideas into symbols. I wondered what the proof of the product rule could reveal to me about the reasons it works.
Symbolism and proof can turn a solution for one case of a problem into a solution for an entire class of problems. This is what makes math miraculous. Ideas combine hundreds of problems into an elegant and simple general solution. Over and over again in this meeting math was beautiful and interesting. And exciting. And I couldn’t believe how much there was to understand about Calculus. It made me want to read the rest of the “The Conceptual History of Calculus” by Boyer, to take Real Analysis and then try to apply it to the sciences. And how do functions work in quantum mechanics. One classmate said he’d endure absurd amounts of pain to have this knowledge transfered into his head. I’d have to agree. Wouldn’t it be amazing to be able to understand the fundamental theorem as applied to multiple dimensions in physics problems? Wow.
Necessity. Why should I learn this? I’ve always had the suspicion while teaching that some of the things I was telling my students were not very useful but just ideas from history before the advent of the Calculator. Like rationalizing denominators. But let’s take this a step further. Derivative rules. Integration techniques. In an age of mathematica and wolfram alpha what do we need to learn these for? We talked about promoting ways of thinking. Teaching Calculus is not about repeating history for the sake of getting through a new textbook modeled on an outdated way of looking at a subject. It’s about wondering what we can learn from doing things a certain way and how we should structure a course. Start with optimization. Solve big, hard, messy problems. How can we generalize? How can we describe symbolically, rigorously, general solutions to these problems? Parameters are important in an entirely new way because they help us answer questions that our Calculator can solve in one instance in a whole variety of situations.
The history of ideas. Where did the idea of process and object conceptions of functions come from? I learned today that although they were not well-cited my professor used Piaget’s ideas in a paper in 1985 to flesh out this distinction. All of a sudden the terms I was studying for my qualifier in two days were being discussed, fleshed out and defined before my eyes by the people who had created them. Knowledge isn’t something to memorize, to pass down, it was to be created. And I got to see the creators talking about it. This was the most fun I’ve EVER had studying vocabulary words. Honestly, I really shouldn’t call something a vocabulary word that is the result of years of thought. But it could be reduced to that.
So, I could spend the rest of my time here thinking about the answers to the questions posed in two days. But of course these two men spent years and years and years getting to the place where they could ask them. I can’t wait to read the book!