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 posted this last May:


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 here’s a thought experiment.



  1. Where did you put the origin?
  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.





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

Now back to work.

Application of a framework








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


Here are three arguments that occur to me.

1. These are the best choices




2. these choices are not ideal

Maybe there need to be more choices.




3. there should be no choices






Throwing darts at a target

