Saturday, September 8, 2012

Calculus: Finishing Our First Unit

For the textbook, I told students they could get the official textbook (Briggs, pretty expensive) or one of the open source textbooks (free, under $25 for print copies of both Boelkins and Guichard).

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:
  1. we were working with his definition of slope
  2. I crossed out and replaced things term by term, so it was clear I was just using different symbols for the same thing
  3. we were able to zoom in on his stuck point because we were one-on-one
If #2 was the most important, a small tweak of my lecture will help a lot. But I suspect that each student will need slightly different things. I have 45 students (we started with 50, 5 have already dropped). It would be hard to have individual sessions with each one. But maybe there's a way. They're in groups of 4. I've changed the groups every week so far, so they'd meet lots of other students. Now they get to pick a group and stick with it (at least for the next unit, maybe for the rest of the course). If I talked with one person in each group, maybe they could then help each other. I think I could do that in a week. Hmm...

 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...


  1. Sue,
    Thanks for your great post. Perfect timing as I will be introducing the derivative (with limits intuitive, also) this week; I'll be sure to work with student-generated ideas about slope, and substitute, as you suggest. Constantly reviewing and reinforcing students' algebra skills is essential in the calculus class I teach.

    Hopefully at the end of the week I will be able to send them home with this:

    I also think you are correct about the difference between progress in conceptual vs procedural learning. I see kids in all levels of high school math who desire the ease of the latter. I consider it my job to show them that math is really about encountering concepts, not following procedural rules.
    Thanks again for all you write!

  2. And thank you for letting me know that what I write is helping someone.

    Have you checked out the Boelkins book? Did you use LaTEx to write up that worksheet?

    I look forward to some good back-and-forth about our calculus classes. (I just subscribed to your blog.)

  3. As I try to plan, I feel like I set up units 2 (finding derivatives) and 3 (using derivatives) wrong. If we spend a whole unit on how to calculate, ... o.m.g. ... they'll be so oriented toward procedures, I'll never get them back.

    It would be good to match an application with each new derivative 'rule'. We could do some graphing after the power rule (constant rule and sum rule are so straightforward they hardly count). Hmm...

    Time to map out all the details...

  4. Do you talk about limits as a separate subject at all, or just in the context of the limit of the difference quotient? In my applied calculus classes, we do an intuitive approach to limits, but I spend a week just dealing with limits and continuity. I just don't think it's possible to "get" calculus without really understanding limits.

    Then, before talking about the derivative, I spend one day on average rates of change and the difference quotient. This makes a huge difference, because the students most of the mistakes students make in finding derivatives are in finding the difference quotient. And we get to have some good conversation about average rates of change that sets us up well for talking about the instantaneous rate of change.

    A key part of this lesson starts with me saying we want to create a new slope formula that emphasizes two things: that we are working with a function, and the horizontal distance between the points. (I make the point strongly that the reason for this is so that we can build calculus on top of it, so even though the old slope formula still works, the new one will be more attuned to our future needs.) I draw a diagram of two points on a line, and tell them that x_1 is now going to be called x. Then I get them to rename the other coordinates, keeping in mind out emphasis goals. Then I tell them to use the old-school slope formula to get a new one, and they come up with the difference quotient. It's beautiful, and it gives them a sense of ownership of this awkward (and seemingly redundant) formula.

    As for the Power Rule, I've decided I don't really have time for Pascal's Triangle (as nifty as it is), so this semester I created this handout:
    It actually went really well, except for those who had seen the Power Rule before, and they had a really hard time articulating any thoughts whatsoever.

    It's so often the case that the ones that had calculus in high school are the most procedurally-committed (and therefore weakest) students in my class.

  5. Sorry I never answered this question, MD.

    I waited until I was done with derivatives and beginning to explore anti-derivatives - then I got into the formal definition of a limit and continuity. My students did very well.

    "I just don't think it's possible to "get" calculus without really understanding limits."

    Newton and Leibniz didn't use limits. It took mathematicians 150 years to come up with a good definition of the limit. I think understanding of limits is likely to come slowly for students, while understanding derivatives can be much easier.

    I love how you have the students build the definition. And that's a nice handout. (Learning LaTex is on my agenda for this summer.)


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