*Learning To Think Mathematically*:

As teachers, we want to value every question that comes out of a kid's mouth. But not all questions are equally valued in math -- asking mathematically productive questions is a skill that successful math students learn.

Take this enormous badminton puck or whatever you call it:

You ask most students: "What do you wonder when looking at this picture?" Tell me if I'm wrong here, but I think that they'd respond like this:

- What is that?
- What's that thing in the back?
- Why did someone put it there?
- Is that a sculpture?
- What's it made out of?
- Where is this?

And so on. Compare that with the list of questions that a bunch of teachers came up with:

- How big would the racket need to be?
- How big would the person who could hit this thing be?
- What time is it?

With time and practice, the students will more closely resemble their teachers. They'll notice what questions get picked up in class for lengthy discussions. They'll compare their questions to the questions their teacher asks. Slowly, they'll get a nose for the sorts of questions that animate mathematicians.

Asking a good question in math isn't natural. It's something that people learn how to do.

Which leaves me with a big question: how important is it to play to students' natural curiosity? Should we follow Annie Fetter's lead and ask students what they wonder? Should we follow Dan and ask kids what the first question that pops into their head is?

On one hand, the case for natural curiosity seems strong. If we care about what engages kids, it's important to know what kids are interested in. If we care about assessment, we shouldn't want kids to hold back. And if we care about including everyone, there should be a low barrier to participation.

On the other hand, the ability to ask an interesting mathematical question is something that is learned, and it is important. And if something is important and capable of being learned, shouldn't we teach it?

**Questions:**

- Should we also be worried about teaching productive question-asking? Can we teach it effectively if we always play for our students' natural curiosity?
- From Max Ray's
*Powerful Problem Solving*: "Noticing and wondering is something students get better at over time." How do kids get better at being naturally curious? Are we changing what they're naturally curious about? - Are there times when we want kids to be naturally curious and times when we want them to be unnaturally curious? When would those times be?

Another possibility, that occurs to me after I published the post, is that you ask more productive mathematical questions over time, but that this isn't an independent skill. Rather, asking interesting questions is dependent on what you know about math.

ReplyDeletePreviously on Rational Expressions, we looked at a Paul Silvia, who thinks that what people find interesting is one part newness, one part ability-to-answer-ness. ("Interest -- The Curious Emotion")

Maybe you can't really skip ahead much in your question-asking. The things that you're capable of being naturally interested in depends on what you know how to answer. Question-asking is something that just happens, over time.

I don't think that this can be the whole story. First, those teachers cited at the top of the post knew enough math to ask that question, but they didn't. Second, I personally find that I'm capable of asking questions that don't naturally occur to me. I also find this to be really productive, and it pushes my learning ahead a bit. There's probably a stretch-zone for the questions that we're capable of asking?

I think you can do both, right? Have a brainstorming session where all questions are valid and interesting, but only pursue the ones in math class that are mathematically interesting.

ReplyDeleteSome questions are "trivia." They can only be answered by looking up the answers or already knowing the facts. Other questions can be figured out or solved based on the information you have. There's not much to be done with the first type other than look up the answers. The second type is much richer and possibly worth time considering.

"I think you can do both, right?"

DeleteI guess. The thing I worry about sometimes is that we're teaching kids how to ask mathematically interesting questions under the guise of just taking any interesting question. But I'm probably stressing out over nothing.

now attempting to distil questions in my head!

ReplyDeleteOne of my most memorable teaching moments was when a struggler actually commented, "but I don't ask the questions you do" as I modelled mathematical thinking. Since then, I've been on a lookout for ways that such thinking could be thought.

I think you're right that it is a skill and also that curiosity plays a huge part. So does context and a whole other variables like interest, mindset, etc.

Anyway, I'm now studying Master in Special Ed and one thing that comes across strongly is the need for explicit instruction. This applies to content as well as strategies, of which mathematical thinking is an example of. It would help some (many? all?) students if we can teach them a strategy for looking and generating questions. There are tons of resources on that but you can probably make these up yourself - creating frames of reference to scaffold the inquiry.

I feel like I'm rambling a bit but I've been on this journey a while and coming up with something cohesive eludes me but I'll attempt to summarise my points:

- asking questions can (should) be taught, explicitly

- you can decode your own processes and write them up as strategies to teach

- there are plenty of resources out there as well

All of this sounds like good advice, Malyn. Like I said on twitter, thanks for all this.

DeleteI want to say something meaningful soon but I am on dorm duty now. I do want to echo the last comment with my own anecdote. Early in my teaching career I worked with a small grow of kids for four years straight.I taught them from Alg II through Calc BC. The week before the Calc BC test I spoke with one of my anxious students. She told me that she was confident about the upcoming test because, in her words, if I get confused on a problem I know I will just hear the questions you keep asking us. Now, this was a girl who had engaged in conversations with me for four years so that's an unusual situation. But I think it offers some semblance of evidence that we can successfully model question asking behavior.

ReplyDeleteThe quote from Max makes me think about how in art classes students learn to notice things in a different way. They train their eyes to look for line and shadow and probably a thousand other things that I don't know about (not having had said training). Math questioning is similar--we are training our mathematical "eye" to look for things that are intriguing *and* mathematically interesting.

ReplyDeleteYeah!

DeleteI'm going to have to say even though I'm a math teacher, if I look at the picture honestly I do not care whether it is 20/30/40 feet tall or whatever. That has no interest to me intrinsically.

ReplyDeleteThe most mathematical question that occurs to me (that I'd actually care about the answer to) is "how much did it cost to make?"

Are we training students to ask the most interesting questions, or just the ones that can be easily examined in a math class?

We can't have things that are both really interesting questions and also be examined in math?

DeleteAfter all, you eventually landed on a question that you find mathematically interesting, right?

This is such a great question! When I did Dan Meyer, etc. tasks in my classroom last year, I would often get questions similar to the ones you listed. Then I would get the kid(s) that kind of "knew better" and would start asking mathematical questions, and I eventually led them to a couple of choice questions. But they weren't really asking questions at that point, they were playing a game of 'What is in Mrs. Longworth's head?' I just finished my first year, be kind :)

ReplyDeleteI agree that students need to be taught how to ask these types of questions, but I also agree with Jason and even though I'm a math teacher, some mathy questions aren't going to interest me. If they don't interest me, they are not likely to interest my students.

Just brainstorming, but I might have the students generate a list of their questions, show them the teacher questions, and in a Venn Diagram format come up with the most interesting or "best" questions. That way it's kind of a win-win. Because what good is a totally engaging image/clip if I am going to eventually lead them to MY question that I had prepared for? I'd rather explore an interesting and mathematical problem and learn along with the students.

"at that point, they were playing a game of 'What is in Mrs. Longworth's head?'"

DeleteI wonder whether we can avoid playing that game by being explicit about our goals. If we're asking "what do you find interesting?" then when the questions are unproductive we set the stage for kids to swoop in and pretend that something they have interest in something that they don't.

But if we are explicit that finding an interesting question is a sometimes hard but always worthwhile task, then we can make finding an interesting but mathematically worthwhile question our goal.

Michael, I recently read Kate's blog and picked up this nugget:

ReplyDelete"We got some fun wonderings like, "Does Angela have an official diagnosis of OCD, or... I mean, Miss Nowak, who counts grapes?" but focusing on questions we have the power to explore mathematically, quickly settled on "How many grapes did she eat on Monday?"

There are two things here: 1) we should explicitly direct kids to focus on things that can be answered using math, but 2) we shouldn't limit them when they want to focus on other things sometimes. In other words, without #1, they will wonder why 99% of their questions are viewed as "off-topic." But without #2, they may feel they have nothing to contribute that fits #1.

Here's a link to the rest of Kate's post:

http://function-of-time.blogspot.com/2014/08/day-1-sooooo-school.html?utm_source=feedburner&utm_medium=email&utm_campaign=Feed%3A+blogspot%2FRrDwm+%28f%28t%29%29

Let's jump ahead to the next time that Kate asks for some wonderings.

DeleteWon't the kids remember that their wacky but off-topic question was ignored in class last time? And won't they just come to believe that their wackiness isn't valued in class, all on their own?

Or am I overthinking all this?

I don't think you're overthinking this. I don't think this process needs to be as tense and needle-threading as people make it either.

DeleteI believe there isn't a wrong way to be curious about something. There aren't better curiosities than others. That isn't to say that students aren't sometimes INSINCERE about their curiosity, asking questions they don't really wonder but which they think will provoke a class to disorder. A sincere question is a good question is a question for which we ought to be grateful. I ask for student questions not because they'll advance my curricular goals but because they let me know my students better. Those are my theoretical biases.

So when we ask for questions, we honor all the questions. We're grateful for all the questions.

Then we model how mathematicians think, how mathematicians ask questions, and we ask our OWN questions that we know lead towards productive mathematical goals. That's the requirements of our class. And once we've satisfied those goals, it's awfully nice to loop back around and try to tackle the other questions also.

And the way I measure the success of this process is whether or not students are more or less likely to reveal their sincere curiosity the next time I ask for it.

Thanks for this. I love your emphasis on honoring all questions, and I think you're right that if this is going well that it should play out in the sorts of questions that we get next time.

DeleteIf we're modeling mathematical question-asking, I take it that it's something that we're OK teaching our kids. So is this just a matter of technique? Are you describing the best way to teach kids how to ask mathematically productive questions?

In other words, are you saying that modeling is the best way to teach mathematical question-asking? Or are you saying that you'd rather not teach mathematical question-asking, instead you'd rather just honor all sincere curiosities?

I'm late to this, but I always feel obligated to point out that lots of people (raises hand) don't come up with *any* questions. That is, extreme Jason Dyer.

ReplyDeleteAnd yes, I think insincerity is an obvious problem. The kids are just learning to play the game you want them to.

I'd ask something like "What mathematical questions do you see in this picture that we could research and answer, assuming we cared enough?"

I personally also like the way Kate Nowak did it -- giving space for some non-mathematical discussion but honing in on the math. I think kids think it is reasonable to focus on math in math class, especially if they believe us when we tell them it's useful and lets us find out cool things. Also, some kids will get very impatient with other kids' questions if you let the whole thing extend too long.

ReplyDeleteBut that's not really why I'm posting a comment. As a former high school and college badminton player, I just wanted to express my horror and disappointment at the phrase "enormous badminton puck". The proper word is either birdie (informally) or shuttlecock (more formally), either of which properly convey how utterly cool and savage the sport is. (Although the phrase "enormous shuttlecock" would perhaps generate more curiosity and engagement than you really want in the classroom.)

By the way, I was a dreadful player. But it had its moments, and it was character-building to keep doing something I kind of sucked at.

Michael, great post! Thank you everyone who has already added some great comments. I fear that the scenario of a teacher tossing a picture like this up and asking for student questions is far below the norm. That said, any math teacher doing something remotely close to this type of activity is on the path to considering student curiosity. Yes, there could be a teacher agenda, but that's where sincerity comes in. Don't patronize kids. If the teacher is sincere, the kids will see it and that sincerity will resonate.

ReplyDeleteI've learned from experience to take all the questions, no matter what. I treat or react to them all the same. I've made the mistake of celebrating when a student asked a question I thought of. Rookie move.

The student questions you listed are curious questions, ones that might tell nuances of a story. Many of them are fun questions, too. Some we can answer and some we can speculate. I think the art of teaching is to honor their curiosity. I think the art of teaching math is to coach their mathematical curiosity. Think of the differentiation involved with just questioning (hence your post). As the teacher, it’s important to acknowledge questions like "Why did someone put it there?”. However, the art in coaching is to appropriately challenge them to think of questions where mathematical thinking (modeling) could help predict an outcome, quantity, or property.

Let's call out the elephant in the room: we're in math class. Chances are extremely high that mathematical questions are preferred. I think this should be discussed with students AND modeled by the teacher. Part of the discussion is to extend our mathematical curiosity past the classroom door and out into the world. Part of the discussion is keeping track of things your curious about. Students might not share your curiosity in this picture, just like you might not share their curiosity in something part of their life. I share with my students my curiosity of life experiences or events, captured or in a story form, because it’s important to me. Curiosity can be contagious, it can be mathematical, it can be non-mathematical, it can be sincere, it can be absurd, it can be simple, etc. The point is: it can be and must be. I model my curiosity while I applaud and encourage students to take on curiosity too.

There are no easy answers here. Your questions are legit.

1- Yes we should consider (not worry) teaching productive questioning. Effectiveness I think is measured in your next questioning session like Dan mentioned.

2- Students can get better, but how? I think the teacher must do everything in their power to create an atmosphere where questioning (mathematical or not) is safe. The teacher must model questioning. (see above) This goes back to listening to the students is greater than for the right answer (question).

3- The timing of naturally curious questions seems too contrived anymore to me. I’ve been there. There’s no point in gaming students for math questions or scorning them for non-math questions. I say we place a greater emphasis on time in a different format. Create time for questioning sessions in class, good or bad. Collectively laugh at silly questions. Collectively be awed at math questions. Give time for students to ask a non-math questions and what they think could be a math question.

Follow-up questions:

I was at a training recently and received some great advice from Jeff Zweir: Formulate the mathematical conversation you’d like students to have. Write it down and teach toward it.

My takeaway: it might appear this is scripted teaching, but in essence I think the backwards planning is a great way to help teacher plan how they intend to coach/teach students to articulate their reasoning in math. This means the teacher can provide the students with tools and strategies to support them. I’d love to carry this over to questioning. If this last thought makes sense, what do you think that would look like for both he teacher and student?

Michael, This post, together with the comment thread, is so very timely for me and for the teachers I work with. For the first time, we are making a concerted effort to encourage our students to ask questions instead of just answering them. There is so much to work with and think about here as we begin that process. Thanks to you and all those who took the time to weigh in with their thoughts.

ReplyDelete