The teacher put up this photo at the start of class and asked her students, “Where’s the maths in this picture?” She asked them to discuss the question in their groups while she took attendance. After four minutes, she pulled them back together and asked a student from each group to tell her where they found the maths in Wembley Stadium.
“The amount of seats,” one student said.
“Area and perimeter,” said another student.
“The number of seats,” said a third student.
“We already have the number of seats,” said the teacher.
The student tried again. “The, uh, perimeter … of the lines.”
The teacher pressed a little – what lines? what perimeter? – and then accepted it.
After two more groups, a girl named Sarah said something I can’t quite make out on the video, but the teacher was visibly floored.
“Have you read my lesson plan?!” she said. “Because that is very, very spooky. Because what we’re actually going to look at today is based on what Sarah just said. I’m a little bit … that’s very odd … but good stuff.”
What I’m suggesting with #anyqs and my last post is that:
- If you give students some photo of their world and tell them, “We’re definitely applying math to this –Â you figure out how,” you’re confusing the master and the servant in the relationship between math and their world.
- If a majority of your students are interested in a single question (eg. the number of seats) then use it. That’s a gift. Can math help your students resolve that curiosity?
- Conversely, if you are shocked when your students’ questions zero in on the point of your lesson, you’re designing your curriculum for the only person in the room you shouldn’t care about.
2011 May 24. Bowen Kerins picks up some of my slack:
There’s a big fault with this sort of question that you didn’t mention: students start skipping the mathematics altogether and try to determine what it is the teacher wants them to say. It’s the equivalent of the teacher asking “What number am I thinking of?” then waiting for a bunch of answers. It’s a totally different game, decidedly not math, and not even close to good inquiry-based teaching. Such questions should either be clarified or just not asked in the first place.
MikeMay 23, 2011 - 1:57 pm -
I must say that I am intrigued by the topic the girl and teacher were both thinking of, but perhaps that’s because I’m more inclined like them. It hits me that this is a very familiar difference between myself and most people I meet and socialize with, where I’ll ponder some thought in public, staring at a logo or piece of architecture which when published as a problem in a book would be clear-cut pseudo-context.
Most people I knew didn’t care about the relative motion of trains or the strange physics one assumes for the length of time of a single math textbook problem.
Looking at that photo, I think that at that resolution and lack of a unit of scale, seat count and field/stadium dimensions are rather difficult to parse. I think the choice of research based on the relative difficulty and/or interestingness is an worthy question for class discussion. However that generally might require instructors be experienced researchers themselves.
Do you do what you love, or do what you get paid to do?
If anyone cares, my thought went quickly to “How long does it take to mow that lawn and what’s the pathing?” After that “What’s the average/best seat in the stadium (considering price and/or view)?”
MattMay 23, 2011 - 1:58 pm -
I’m not sure I understand. Is this hypothetical? I can’t see the video anywhere.
Dan MeyerMay 23, 2011 - 2:13 pm -
This is a video I was watching for a research project. I’m summarizing what I saw. Names have been changed, etc.
Mr. KMay 23, 2011 - 2:13 pm -
I’m assuming the use of comic sans is intentional.
BrianMay 23, 2011 - 2:18 pm -
I agree with your points, and I was thinking about #3 before I got to it, but now I disagree because of one exception:
if the question your lesson is answering is one of those questions where the students say something like “ooohhh. . .yeah, that’s something I’d like to know, I just didn’t think of it myself” (not that I have any students like that), or “wait, that’s MATH?!? How?” That could still be a good curriculum item with a good intro, even if only one other person thought of the same question on his/her own.
But otherwise, I’m right with you.
BenMay 23, 2011 - 6:15 pm -
in an ideal world, the teacher would care about delivering the curricula in a way that cares enough about what every learner needs. Perhaps you meant to say, the only person in the room you don’t need to WORRY about?
louiseMay 23, 2011 - 6:44 pm -
So put up all the questions, and then let them have at it. They can pick one, then put up the solutions together. Remind them that we don’t really know the answers, we’re going for reasoning. Have them give a list of possible things that could make their estimate wrong (e.g. seats removed for wheelchair access). Oops! suddenly we got science.
Dan MeyerMay 23, 2011 - 6:47 pm -
@Mr. K, same photo and same font as the teacher used, believe it or not.
@Brian, can you favor me with an example?
@Ben, what I’m saying is that the teacher is designing curricula for herself. I’ve clarified the text. Thanks.
Dan MeyerMay 23, 2011 - 6:52 pm -
Not all questions are answerable given the information available. Not all questions require the same mathematical competencies. Problems?
Bowen KerinsMay 23, 2011 - 7:15 pm -
When we were writing our materials, we called this sort of thing a “what’s in my head” question. There’s a big fault with this sort of question that you didn’t mention: students start skipping the mathematics altogether and try to determine what it is the teacher wants them to say. It’s the equivalent of the teacher asking “What number am I thinking of?” then waiting for a bunch of answers.
It’s a totally different game, decidedly not math, and not even close to good inquiry-based teaching. Such questions should either be clarified or just not asked in the first place.
It’s interesting to compare this to #anyqs, where the goal seems to be to make as certain as possible that the question the kids come up with is the same as the one in the lesson plan — taking the path of not asking the question, letting students construct it. (Can we ever be certain it will happen the “right” way, though?)
Timon PicciniMay 23, 2011 - 8:18 pm -
So in this case Dan (let’s get rid of the fact the teacher was surprised), how many students does it take to make our anyqs legit? I know there is not a magic number, and not every student is going to come up with the same question, but how do we gauge if our anyqs has generated a natural question that the whole class can buy into.
I know we can post them to twitter, but I feel like a group of math teachers are naturally going to gravitate more toward the “mathy” questions.
Dan MeyerMay 24, 2011 - 1:47 am -
I’m probably playing a bunch of semantic games here, but I don’t want to suggest there’s a “right” question. I do have a question I’d like students to consider, a question I think is important and worthy of their time. That question comes from a certain context. How do I represent that context to my students so that the question I’m wondering seems compelling to them also?
I’m curious, for instance, how the classroom would have looked if the teacher had shown some video of some kind of stadium stampede, something to trigger the thought that, “my word, we need to do something about this.”
A majority, maybe? But consider that students may have lots of questions about a given context. Maybe “how many fire exits are there?” is the predominant question for only 40% of the students but it’s somewhere in the top three for the rest. I can live with that. I can’t live with instructional design that disregards the likely questions students might have about a given context, which is what we’re working against with #anyqs.
BenMay 24, 2011 - 2:05 am -
And Dan’s just hit on the most broken piece of instruction today. It was mentioned earlier in the comments as “guess what’s in my head” and teachers do it everytime. The concept of #anyqs is powerful because you’re going up the “power” of directing what the class should be learning to your students.
Yes, you should probably be strategic with the use of your #anyqs to at least craft them in such a way that your image or video is more likely to draw out the questions that are pertinent to what you need to cover, but when you ask a room full of students something as broad and unengaging as “where’s the math” you better be prepared to follow the student’s lead on that one, otherwise you’re going to disenfranchise their curiosity.
Joshua SchmidtMay 24, 2011 - 6:28 am -
I think that it’s also going to be different for each classroom. In a classroom with leaders who are going to eat up mathematical context, the questions usually will be more to what I would have “wanted”. However, in the lower level classrooms where leaders are harder to come by, it’s important that the question that you as a class are pursuing is important to the students. I don’t like the idea of having to put a number on it, as long as your students truly value the question that you are solving. Most of the time, I don’t think the question really matters as long as the kids are learning something valuable.
My point is that each classroom setting is going to be different, and we should embrace that. Taking a “one size” fits all approach seems counterintuitive to inquiry based teaching in general. If you want a specific question answered, ask that question, the students will learn the material that is valuable to that teacher. If you create a good enough visual or manipulative, then the question will be created for you (or at least one that is worthwhile).
MattMay 24, 2011 - 7:11 am -
Dan, I feel like you need a “hashtag” definition in your sidebar or something. I’m getting very frustrated trying to figure out what these stand for. Like does anyqs stand for any question? And that one that starts with a w…no clue… Being late to the game and reading your blog gets frustrating sometimes as I feel out of the loop.
R. WrightMay 24, 2011 - 7:34 am -
Matt, WCYDWT stands for “what can you do with this,” I think, but to be honest, I’m not entirely sure that I understand what that means.
cheesemonkeysfMay 24, 2011 - 10:06 am -
I vehemently agree with Bowen and Ben about the need to avoid asking “Guess what I’m thinking questions.”
For this reason, I wonder if it might not be better to separate the generative part of the process (having students pose questions) from the “settling on a specific question for the group to investigate” part of the process.
In teaching writing, we try to emphasize separating the generative process from the editing process. Why not do the same thing with mathematical question-posing?
Starting from that assumption, I find myself wondering why would we not pose the #anyqs question to the students as, How many mathematical questions can you come up with? And then capture them ALL on the board.
Or set a timer for 5-7 minutes and have pairs or groups generate lists of as many mathematical questions as they can come up with before the timer goes off. THEN start gathering or sorting them on the board as a whole class.
– Elizabeth (aka @cheesemonkeysf on Twitter)