Tag Archives: student thinking

Anticipating student struggles…

UPDATE: Results from the class described have been added at the end of the post.

Putting my money where my mouth is tonight.

I have had some saucy things to say about our adopted Calc text. In particular, that it has not been written from a perspective that is empathetic with students’ states of mind.

So tonight we’re proving the area of a circle formula as an introduction to trigonometric substitution. There’s a rabbit-out-of-the-hat moment in which the substitution in question gets introduced:

x=r•cos (theta)

I anticipate that this will be a challenging point for students. So we’ll stop there and collect responses to the question, What does this make you wonder, question or think about?

Here’s what I anticipate they’ll say:

  • Why r?
  • Why cosine (instead of, say, sine)?
  • Why theta?
  • Is this like u-substitution?

As always, I am prepared to be schooled.

I know they’ll have ideas I never thought of. Can’t wait to see what they are.

Post mortem

Sue writes in the comments:

I don’t start with the strange substitution. I start with a triangle. For me, the triangle I draw explains the substitution I decide to use. Then it’s all about reasoning, instead of memorizing (?) 3 very strange substitutions.

This is spot on.

Students asked about the theta, they asked about the r. They didn’t make any connection to u-substitution. Lots of them worked on triangles. Students thought about vectors in physics and they thought about the unit circle.

In contrast with Sue’s suggestion, and in contrast with where my students’ minds were, our text talks about this:

In general, we can make a substitution of the form x=g(t) by using the Substitution Rule in reverse. To make our calculations simpler, we assume that g has an inverse function; that is, g is one-to-one…This kind of substitution is called inverse substitution.

And the triangle Sue suggests? That comes in at the end of the technique as a tool for changing variables back from theta to x. So we finish with where students’ minds are, but we start with an abstract, backwards version of something they’re not even thinking about to begin with.

And we wonder why students don’t become better problem solvers in their college courses, and why they don’t develop the kinds of critical thinking skills we would like them to. And why students don’t read the textbook. They don’t read the textbook because it’s not speaking to them.

[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.

Application of a framework

Dan Meyer busted out a new #anyqs video the other day. If you haven’t already, watch it before you read further.

In the spirit of #anyqs, I replied with my first question, together with some others (via Twitter)…

FIRST? Which is it? Top or bottom? SECOND? Why 10 seconds of outro? THIRD? What thinking is behind the wrong answer?

And then FOURTH…How did @ddmeyer identify the best misconception to include in his multiple choice #anyqs?

…and then I did a little work and noticed that they were both wrong answers.

So now we have an interesting question of task design. If we take seriously the framework of five practices for creating productive mathematical discussions in classrooms, then my third and fourth questions aren’t just throwaways, they’re really important.


  1. Anticipating
  2. Monitoring
  3. Selecting
  4. Sequencing
  5. Connecting

Dan has anticipated and selected two student ideas for the first act of his lesson. On what basis did he choose them, and how might he have chosen differently (either to better or worse effect)?

Here are three arguments that occur to me.

1. These are the best choices

Students are likely to hold one of two conceptions. (i) Light travels instantaneously and (ii) Light takes time to travel across the solar system. Both of the possibilities presented in the video call (i) into question. Students who hold (ii) are likely to identify with the bottom scenario in the video.

Thus discussion is generated by the absence of conception (i) as a choice. Kids will introduce that one, and many students will gravitate to (ii). Perhaps these students recall that light takes eight minutes to get here from the sun, and they know the moon is closer to us, so the bottom choice looks appealing.

And even those rare students who know that light takes time to move, but who think the moon must be much, much closer than suggested by the bottom choice will choose the top answer. And that top answer is also wrong, which they will discover in the process of justifying their work.

2. these choices are not ideal

Maybe there need to be more choices.

Or maybe the instantaneous transmission of light conception should be one of the two choices (instead of the top one).

Or maybe we have missed some important student conceptions about light that should be represented among the choices.

In any case, being explicit about what we anticipate students will think about this scenario makes it possible to make purposeful choices of which ones to present in act one.

3. there should be no choices

The discussion so far has assumed that the two choices exist to provoke student discussion. But they also serve up the question. Without those choices, there’s just some guy shining his flashlight into the sky. Without those choices, I’m never gonna wonder how long the light takes to get to the moon.

Maybe we need a different way to get at that question-a way that offers students a wider array of entry points.

This isn’t it (via Scott Farrar), but I’m not sure I know what is.


We can all get a lot better at designing tasks (for curriculum or our own classrooms) by thinking through technical and challenging questions of this nature. As always, I’m thankful for Dan serving up such rich food for thought.

Throwing darts at a target

From a really interesting article on action research in the classroom. (via SupTeach?)

My analogy is that I’m throwing darts at a target trying to hit it but I have no idea where the target is. Because unless you help the student to figure out how to make their thinking visible to you, you just keep giving reading lessons aimlessly throwing these darts out hoping you meet the target. So this research is an attempt to figure out where the target is.