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.

I really like your point about how we have it all backwards in math class. The ability to ask a question is a great sign of learning.

It seems to me that asking questions is a skill like any other. Some people find that life has equipped them with that skill without any conscious effort. Others find themselves without this ability, and they have to learn it.

I love the idea that a sign of learning is the ability to ask questions, but I wonder if you’ll find yourself having to teach some of your students how to ask questions.

(If learning is having new questions to ask, and you’re learning how to ask new questions would that mean that you’re having new questions to ask about how to ask oww that hurts.)

Yes,

Michael, I do think I will need to teach them to ask questions. I think I do a lot of this already and instinctively. But my new focus and your reminder will help me do this intentionally. Which means that I’ll get better at it. Thanks.I’d read a post that detailed how you help people learn to ask questions.

I’m thrilled that you had so many questions during and after your experience in my workshop but you do know that I’m going to stay on you for the answers, right? I have a (strong) feeling that those answers (including the process of me working to understand them) will bring us all to some really interesting and exciting new understanding about what can be gained by learning at the intersection of math and percussive dance.

And, although I know you know this, I do want to point out that you came to these questions in a situation that required you to be an active participant in a hands-on learning experience.

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Thank you for all this. One thing to add: there is such a thing as a stupid question, though I won’t say it aloud. I try to get kids to ask the question that matters and hopefully work it amongst themselves. Otherwise, we’re wasting our time constantly being the ones asking questions. Then again, forgive me because I’m writing this comment at 948pm.

In any case, I’ll have to expound on this soon.

I appreciate your conclusion (mostly because I’d never thought of it before and it makes a ton of sense).

Meaningful question-asking, in addition to being a sign of learning, is a sign of engagement and a sign of student self-ownership of the learning process. All reasons to hope it happens when we ask “What questions do you have?”

I have focused on helping students take ownership of their learning while working in class on problems etc. Often when they have questions or express that they don’t get it, I follow with some good questioning to get them to think through what they know and what they are working to figure out…all the while encouraging them to form questions and work at developing mathematical reasoning. Your blog, especially the conclusion, has pushed me to think even further. I love the expectation that if you have truly learned then you should have new questions. Brilliant!

This simple statement “Learning is having new questions to ask” coupled with Jo Boaler’s online course “How to learn math” have inspired me to think more about mathematics learning and teaching than I have in a long time. The way you both explain your ideas, provide support for your views, ask questions, and inspire with your words leaves me in awe. This post is what I have needed. I have struggled with how to break from my old beliefs about mathematics teaching and learning in moment-to-moment interactions in the classroom. Sometimes when teaching, I find that autopilot kicks in and I teach in ways I was taught (e.g., focusing on procedures, covering material, lecturing, asking for questions in insincere ways as opposed to providing opportunities that encourage questioning naturally). This way of viewing learning and the way you have operationalized it will be a guiding theme for my teaching practice this semester. I will let you know how it goes!

Relevant: Alan Kay’s “Similarities over differences” non-universal. Creating/noticing (a philosophical rather than action-based distinction) attributes means seeing similarities in different things. http://learningevolves.wikispaces.com/nonUniversals#simDiff

I’m pleased to discover your blog via one of Malke’s recent posts. I think perhaps you might enjoy the paper “Teaching Students to Ask Questions Instead of Answering Them”(http://www.nea.org/assets/img/PubThoughtAndAction/Bowkershort.pdf) by an interdisciplinary college professor, Matthew H. Bowker. It’s all about helping students to think independently by asking insightful questions themselves. Although Bowker’s examples come from the humanities, it would be interesting to consider how his recommendations might apply to maths teaching.

The rediscovery of enjoyment, meaning and value in questions is something that really appeals to me. I think about it often in the course of my philosophical enquiries with children.

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