[WTF] Understanding student thinking

I debated whether to begin the year on a positive note. I had fun with Oreos (more coming in that department, by the way). But now it’s time to get serious.

I posted this last May:

Alexs Pate, the author of Amistad, visited my college last year to speak about rap, writing and a whole mess of stuff that was on his mind. I ran across a one-sentence note I made during his talk:

Writers need an empathetic imagination with their characters.

Rephrase this as:

Teachers need an empathetic imagination with their students.

Of course I would add that textbook authors need this also.

So I’m using Stewart’s Calculus, Early Transcendentals (7th Edition; don’t get me started on editions). It’s my first time teaching Calculus 2 so I’m reading the text extra closely-in an attempt to model critical reading skills for my students and to imagine what will be easy, helpful, challenging and unhelpful as my students engage with the text.

On the surface, I imagine that about 90% of Calc texts are more or less identical. The curriculum has certainly converged. (heh) Limits, then derivatives, then integrals.

So it’s really only at the level of minute detail that most texts differ. Do these details matter? I’m gonna say yes. We are trying to get students to read the text. In order to convince them that this is a reasonable thing to do, we have to convince them that the text’s author gets them as the audience.

So here’s a thought experiment.

Consider the following problem (an example in the text): A tank of water in the shape of an inverted cone is filled with water to a depth of 8 m. The tank has a base of radius 4m and a height of 10 m. How much work is required to empty the water by pumping it to the top of the tank?

Do you have a picture of the cone in your mind? Have you oriented it in space so that you can begin to label variables? If so, then you’re ready…

  1. Where did you put the origin?
  2. What did you label the axis along which you’ll integrate?
  3. Which direction is positive on this axis?

Stewart gets it wrong on all three questions:

  1. The origin is at the top of the cone.
  2. The vertical axis is labeled x.
  3. The positive direction on the x-axis is down.

Yes, I understand that we’ll get the same answer either way. But if we are trying to teach students, we need to choose examples from which they can learn. This involves understanding something about how they think and then using this knowledge to make good instructional choices.

Nitpicking, you say? But did I mention that this is the SEVENTH EDITION?!? And have you seen the guy’s house?

"I had no idea you could make any money writing books. That was not a motivation at all. It was a surprise, but it enabled me to build this house. And I’ve got to continue to work to pay for the house. The house’s cost ($24 million) is double the original estimates."

I don’t begrudge the man a nice home. But that cone is sloppy writing. When the most commonly adopted Calculus text does such a poor job of meeting students where they are, I get angry.

Maybe we’ll get a better one in the eighth edition.

Now back to work.


12 responses to “[WTF] Understanding student thinking

  1. For all the shade you throw at my #anyqs exercise, it attempts to nurture in teachers exactly the kind of empathy you describe here. (ie. “Here’s a thing worth asking questions about. What questions do you anticipate students will have about it?”)

  2. Wow. I should write a textbook, if it brings in this kind of money. One can almost look past the garbage calculus for a house like this.

  3. This is what happens when you get a guy who made his bones studying harmonics (and plays the violin) writing mathematical physics problems instead of physical mathematics problems for Calc students. It’s like a point of pride among physicists to call the direction of any one-dimensional problem “x,” despite Euclidean sensibilities, and math books don’t always shake it off.

    Also, it’s something of a minor sport to watch physicists talk about the arbitrary placement of axes, origins, &c., but there ought to be some sort of sensible limit to the arbitrariness we tolerate in practice. We get it: the work done is the work done, is the work done, but do I really gain anything by choosing a horizontal axis that, e.g., requires me to consider the philosophical strangeness of negative potential energy? (If you want an interesting perspective, watch Walter Lewin here: http://bit.ly/w1mUay)

    As a halfhearted defense, I will say that the (3) things Stewart gets wrong are really more like (2), or possibly even (1), if you’re generous. The axis-naming business is annoying, but not a real roadblock, one would hope, to a college Calc student. The other two, I think, should be lumped together. I’ll at least entertain the possibility that there is an advantage to thinking of “down” as “positive” w/r/t gravity and work and signs of same, and once you do that, you’re pretty much required to put the origin at the top of the tank, because it’s the new bottom.

    • Sadly, ctlusto, puppies die in college courses too.

      What I mean is this: Students are intimidated by Calculus textbooks. They are struggling to read dense mathematics text and they have an imperfect grasp of algebraic and spatial skills. When they read a solution to an example in the text, they tend to approach it with the slimmest of hopes. So while I agree that “to a college Calc student” naming axes in arbitrary ways ought not be problematic (by which I mean I wish that it weren’t) it is. When teachers (in the larger sense that includes textbook authors) choose examples, we need to be aware of the mindset of our students and meet them where they are at. This means either (1) setting up the solution in a way that follows their likely intuitions, or (2) being explicit about the ways our solution might not do so and helping them to learn from it.

      And, really, you want down to be positive because of gravity? You don’t think that’s a setup for disaster in physics later on?

  4. You’re absolutely right about meeting students where they’re at. I was trying to agree with you in my previous comment, but maybe didn’t do it particularly well. The only point I was trying to make in the last ¶ was that my physics background isn’t strong enough to make the assertion that there is absolutely no good reason to point the positive axis in the direction of gravity. Maybe there is, and I’m not smart enough to realize it. I was just attempting to give the benefit of the doubt to a problem that seems, on its surface, to be completely ridiculous.

    • Yeah. This is how I know you’re smart (besides the fact that Karim said so)-you argue even when you agree. It’s a characteristic I admire (and that I share-see Dan Meyer’s earlier comment on this post).

  5. It’s not just Stewart. I’ve seen very similar solutions in Thomas’ Calculus and other texts. I teach at a C.C. and we’ve not adopted Stewart because it may be better suited for engineering/physics/math majors and a lot of our Calc 1/2 students are in the life sciences.

    When teaching Calc 2, I like to do that kind of problem with the students two or three ways (the two you reference and perhaps one where the positive direction is up, but the origin is at the top) and show they all come to the same solution.

  6. Yeah, I know, rmf. It’s a systemic battle. I get that. It depresses me sometimes, but I get it.

    That doesn’t excuse the author’s behavior, though.

    Now your approach-that can be empowering for students. There are multiple correct ways of approaching things, and they all get us to the same place. In that context, it’s fabulous to put the origin at the top. But as the single approach from which students are to learn some other technique? No way.

  7. Pingback: What is the chain rule? (heh) | Overthinking my teaching

  8. Christopher

    Have you seen Rogawski’s text? I adopted that at my new school for our Calc AB kids and was pleased. Not perfect, but good. I have to use Stewart next year for a number of annoying reasons and I am a little concerned about how much supplementing I’ll need to do.

  9. Pingback: Course design question | Overthinking my teaching

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