These past two weeks have reminded me how hard it is to teach a class for the first time. I continually feel like I'm coming up short because I compare my performance in my summer classes to my performance in a typical 16-week class that I've taught many times. I know that's not a fair judgment, but I still do it. I also know you can't learn how to teach a course well if you don't ever teach it that first time. Still, I feel all my energy is spent just trying to get half-way decent lectures ready, to keep up with homework questions, and to write tests. I don't have much time or energy to provide learning experiences outside of the typical lecture. And I hate that.
That said, I'm learning more about the teaching and learning of calculus every day. And I think that's pretty priceless. Also, I think I've been given a set of unusually patient and gracious students this summer. I get thanked about every other day for doing what I do. And for someone who needs pats on the back, that is the biggest reward I could get.
That's my vent. Now for some calculus...
Every semester I have my students write an introduction about themselves. In addition to hobbies and life goals, I ask them to tell me why they're in the class, their math background, and their current feelings towards mathematics. In doing so this semester, I learned that many of my students have had calculus before. So, before we ever talked about the Power Rule, I would ask something like, "How can we find the derivative y=3x-2?" I would, of course, get an eager, "Well, I learned how to find derivatives another way, and you just take the exponent and multiply it by 3 and then reduce the exponent by one...so the derivative is 3." Everyone loves the Power Rule.
|Apparently, we still need to|
work on writing "lim as h
"You're absolutely right, and we'll talk about the Power Rule soon. But can you think of the geometric definition of a derivative and tell me what 3 corresponds to in the linear function?"
All this to say, I wanted to introduce the Power Rule differently, somehow.
I split the class into eight groups and gave each a "Derivative Card." On the top it had "Find f '(x) when f(x)=..." I used four basic functions: f(x)=x^2, x^3, 1/x, 1/x^2. Each function had its own color. When the students were done, I asked them to find the other group with the same color and see if they got the same answer. I loved this because the students were doing all the work. We then created a table on the board using what the groups just found. I started with f(x)=x (which I didn't give to any group). The table looked something like this:
It's nothing new, but the students came up with it themselves, which is the great part. They were able to generalize the rule no problem too. Which makes my heart very happy.