One thing Malke Rosenfeld and I agreed on over breakfast the other day is that the question, *What did you learn?* makes us uncomfortable. Weird, right? We are teachers and find both answering and asking this question makes us uncomfortable.

I have many reasons for not liking the question: that it implies the process has ended; that when I ask it of my students, they may be inclined to say what they think I want to hear; that it doesn’t invite further questions; on and on.

Being asked this question in Malke’s (fabulous) workshop* led me to something new, though.

New to me, anyway.

This coming school year, I will characterize learning—for myself and for students—in the following way.

*Learning is having new questions to ask.*

If I have learned something, it is because I can ask questions that I previously could not. Some examples…

### example 1: Algebra II

Reading Nicholson Baker’s article on Algebra II in Harper’s [behind pay wall; also available at your local library. And seriously, a Harper’s subscription is like $15 a year.] recently, I didn’t learn anything. Much of what he had to say about the course and the way students experience it is pretty familiar and the tone resonated with many of my feelings. But when I read Jose Vilson’s response to it, I had questions. Jose writes,

If someone said, “Let’s end compulsory higher-order math tomorrow,” and the fallout happens across racial, gender, class lines, then I could be convinced that this was a step towards reform.

I wondered whether I would view Algebra II differently if I were a man (or woman) of color. I wondered yet again about the place and effect of developmental math and College Algebra on the economically and culturally diverse population of my community college. I have new questions to ask, so I learned something from my colleague Mr. Vilson that I didn’t learn from Mr. Baker.

And you are reading Jose Vilson’s blog on a regular basis, right? If not, now would be a good time to start.

### Example 2: Percussive Dance

At Malke’s workshop this week, I began asking about:

- the relationship between
*variable*and*attribute*, - the importance of decomposing things by their attributes and paying attention to one of these attributes at a time, and whether that is a fundamental characteristic of mathematical activity,
- whether a characteristic of a novice is an inability to distinguish noise from pattern,
- how children’s experiences with sameness in their non-mathematical lives informs and constrains their ability to work with sameness in mathematics,
- whether I was taking seriously my responsibility and opportunity to use physical classroom space for student learning, and
- what kinds of
*equivalence relations*we could use in Malke’s percussive dance work, and whether we can form a group from the resulting elements, together with composition (my hunch is*yes*and that the resulting group is non-Abelian, but I haven’t worked out the details).

Now you should watch Malke in action. I’ll be surprised if this 3-minute video doesn’t give you some new questions to ask.

### Conclusion

See, in math classes asking questions is usually a sign that **you** **have not learned**.

“Any questions?” is a signal to students to speak up if they *don’t *get what has just been explained.

We have it all backwards.

It shouldn’t be, “*What questions do you have?” [I hope you have none so that I can tell myself you learned* *something.] *

It should be, “*What new questions can you ask?” [I hope you have some because otherwise our work is having no effect on your mind.]** *

—

*Asked by someone who is not Malke, for the record.