Tag Archives: discussion

Questions as evidence of learning

I have argued that learning is having new questions to ask.

Here are a few questions that have surfaced in the early weeks of the semester. These are all student questions in College Algebra.

(1) Can it still be a variable if it only has one value? 

This was asked by a student as we were sorting out whether y=2 counts as a function, and whether it counts as a one-to-one function.

(2) How do you solve x=|y| for y?

This was asked by a student as were considering the relationships among functionsinverses and inverse functions.

(3) Is the inverse of a circle an inside-out circle?

See, we were using a set of equations, considering x as the domain and y as the range. We were asking whether each equation—so viewed—is a function and whether it is one-to-one.

Then we were switching domain and range (i.e. swapping x and y) and asking the same questions about this new equation. Bonus question was to solve each of the new equations for y.

One of our equations was x^{2}+y^{2}=1. Swap x and y and get back the same thing. Thus, a circle (as a relation) is its own inverse. Which fact I had never considered.

But my purpose here is to check in on the progress I am making in fostering and noticing student questions as evidence of learning.


Teaching for deeper learning

A good reminder for those of us starting our teaching week. From KQED’s Mind/Shift.

[F]or educators to engage in deeper learning with students, researchers say they must begin with clear goals and let students know what’s expected of them. They must provide multiple and different kinds of ideas and tasks. They must encourage questioning and discussion, challenge them and offer support and guidance. They must use carefully selected curriculum and use formative assessments to measure and support students’ progress.

That bold part is important to me and is so often neglected in mathematics classrooms and curriculum (see also: Khan Academy, ALEKS, Saxon, Explicit Direct Instruction, etc.)

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.