Unit 1: Exploring the Idea of the Derivative
I've worked with my students for 3 weeks on what is mostly contained in one section of the Briggs (3.1). We started working on average velocity activities in Boelkins and on slope of a secant line. We touched on limits just enough to make sense out of the definition of derivative. We use the definition with h -> 0 most of the time, but did look at a few problems using x -> a. There aren't enough really simple homework exercises available to help students see the basic concept. I'd like to have more to provide next semester.
I will be testing them on Monday. The atmosphere in the classroom changed on Thursday from positive, seeming to be engaged, to tension and distress. I talked with one student, C, who seemed particularly unhappy (by his facial expression and body language). He said he was irritated, and I asked him to come to my office so I could help him. I did something that felt to me almost exactly like what I had done in class a dozen times - I talked him through the slope idea. But he suddenly got it, and was much happier. I don't think he'll know how to practice, though, so I'm guessing he'll need to re-take the test - along with many others. I want to have material he can use to get up to speed; more on that below.
Back to my interaction with C. We chatted until we got to my office, then I closed the door so he'd have more privacy and asked him how he felt about the course. He has to take it for business, and I have the impression it's just a hoop he has to jump through. He has felt like it's not making sense since day one. That is not surprising when the students are so used to algebra classes - find out the new procedure the teacher is pushing, learn how to follow it, practice a few times to get the kinks out from all the other algebraic procedures cluttering up your brain, done. He said, "We were doing one thing, and then switched to something totally unrelated." I see the relationships between the activities (all leading toward derivative, or working with it), but the students don't. They are all related by one basic concept. But the students are looking for things to be procedurally related.
As I tried to understand what the two unrelated things were, I started working with him from the beginning. Did he understand f'(x) = the limit as h goes to 0 of (f(x+h)-f(x)) / ((x+h)-x)? His response made me back up more.
"Do you know slopes for straight lines, from algebra?"
"Yeah. y1-y2 over x1-x2."
So I wrote down what he had said, and drew my picture of a curve with one point labelled (x,f(x)), which I call the stable point, and a second point with x-coordinate x+h, a distance h away from the first, labelled (x+h, f(x+h)), which I call the sliding point. I asked if he could see that these were two points on the curve, with their x and y coordinates labelled. Yep. Then I crossed out the y1 and replaced it with f(x+h), and did the same with the other 3 terms in his slope definition. All of a sudden the light went on for him. I think the notation (f(x+h) especially, see this post and comments) gets in the way for students with weak algebra skills; I also think their skills can be strengthened. I was excited that it suddenly clicked for C, and I see 3 possible factors:
- we were working with his definition of slope
- I crossed out and replaced things term by term, so it was clear I was just using different symbols for the same thing
- we were able to zoom in on his stuck point because we were one-on-one
Today and tomorrow I'll be creating the test, framing the next unit, and thinking about how to help students who didn't yet pass the first test. For material that might help them, I think I need exercises they can do on their own. Problems that have less notational issues, but get them thinking in the right direction. Christopher Danielson proposed working with finite differences. I think that might help them see the light. I wonder if there's any already-made materials for this. (If you know of anything, please point me to it.) I'll be trying to come up with something myself.
Unit 2: Calculating the Derivative
We'll be working on all the usual 'rules' of derivatives. I am thinking about which applications to mix in so that the material includes some motivation. Here's where we need to be more interactive. A textbook can tell the 'rule' and show the proof. But that won't stick. Students need to find the rule from a pattern, then see the need for proving it will work in all cases, then be able to come up with parts of the proof, and be able to explain and reproduce other parts.
Right now I'm thinking about how to lead them to the power rule. The pattern is easy to see from a few examples. They'll want to assume that pattern holds. Is there time (and are they ready for) an activity that shows that patterns you see aren't always the right one? (My favorite is the circle where you add a dot and lines to it from all other dots, and count the total regions. But the solution to that is way hard. Ben Blum-Smith has two posts collecting problems with patterns that break. I might try some of those.) I want to show them Pascal's Triangle (1653, but used 600 to 700 years earlier in India and China, by Halayudha and Jia Xian, respectively), and get them to describe the relationship of (x+h)^n with a row of the triangle.
More to come...