By The Numbers
27 people posted 45 photos and videos tagged #anyqs in two weeks. @colintgraham posted four, more than anybody else. The balance, so far, is 60/40 in favor of photos over video. The median length of an #anyqs video is 31 seconds.
So what are you doing with the feedback to your #anyqs entry? If you intended to represent a perplexing application of math to the world around you but the responses were mixed and the enthusiasm was low, what do you do? If you’re Lisa Henry, you revise and resubmit.
I asked her if she could clarify the context without weakening the task she had in mind. She came back with gold:
The problem space was clear to me. Four different navigation sites returned different durations for the same trip. A question now gripped me – why? – whereas earlier I was mostly confused.
Teacher-Centered Curriculum Design
See if you can spot a recurring theme in the discussion around #anyqs:
Colin Graham, describing #anyqs:
… viewers should respond with the first (mathematical) question that springs to mind.
What is the question?
Bryan Battaglia, responding to Christopher Danielson’s killer #anyqs entry:
That one’s easy! How many times will Griffy make it around?
what is the question?
These quotes indicate a belief that there is a right way to be curious, that students should seek out the question the teacher wants them to ask, that the question should be mathematical. I’m not suggesting that math isn’t the point of math class or that student interest should exclusively determine how you spend your class time. I’m suggesting that, given an infinite number of ways to represent a problem space, you represent it as skillfully as possible, in such a way that you can anticipate the questions your students will have about it. Conversely, if you can anticipate they won’t have any questions about it, consnider a different problem space.
Lisa Henry could have stopped with her first draft and asked her students to meet her more than halfway. She could have stood at the front of class and played the “guess what’s in the teacher’s head” game, waiting for a student to ask the “right” question. Instead, she put the burden on herself to make a stronger representation of the problem space. Her curriculum design was centered around her students, not their teacher.
Let’s Push Things Forward
Let’s say you’ve managed to anticipate the question your students will wonder about your photo or video. (Plenty difficult on its own.) How can you help them answer it? Have you gathered the information your students will need for the second act? Have you recorded an answer to the question, something you can reveal in the third act to pay off on all their hard work from the second? If you’re looking for a harder challenge than #anyqs, that’s it right there.
Huge Open Question
To what extent is the response of math teachers on Twitter to these photos and videos a useful proxy for the responses of our students? If a bunch of math teachers wonder, “how many dolls are inside?” does that mean that students will also? If teachers don’t wonder that question does that mean that students won’t? Is there a better way to test out curriculum design this quickly and easily?
Pam Grossman, my adviser, at a panel discussing teacher education:
Classrooms are somewhat unforgiving places to learn to teach.
Problem posing is a core practice of math teaching but the classroom is an unforgiving place to learn it. When you pose a problem in class, you’re betting a lot of time and motivation from a lot of students against the possibility you totally misjudged the task. When you pose a problem on Twitter to your teacher buddies, that risk drops to zero. I hope #anyqs proves itself a useful exercise of classroom practice that doesn’t require a classroom. There aren’t a lot of those. We’ll see. I only know that this exercise is most productive when we submit each new photo or video with the perspective that “this is just a first draft – I will be revising this.”
I’m doing some work in Singapore this week. I have a couple of items set to auto-post but my commenting will be light. Real talk, though: if I come home and there isn’t a pile of Graphing Stories waiting for me to edit, you are all in big, big trouble.
Telannia NorfarMay 20, 2011 - 2:55 pm -
Dan, I have to admit this is probably a rough time of year for most teachers. I have at least two things I want to submit for anyqs and I am thinking about what to submit for graphing stories. However, it is the closing of the school year! I am knee deep in end of year activities. I am starting to see light so I may be able to do some things this weekend. However, graduation is not until next Wednesday so I will probably be crazy until then :). Is this the case for anyone else out there?
monika hardyMay 21, 2011 - 6:00 am -
hey Dan – on this:
How can you help them answer it? Have you gathered the information your students will need for the second act? Have you recorded an answer to the question, something you can reveal in the third act to pay off on all their hard work from the second?
coming from an intense year of reflection on this very thing, and knowing i have much to learn, i’m dying to know what you and others think of this:
once you yourself have
gathered info for students
recorded an answer
planned a revealing
aren’t you (we) compromising curiosity?
aren’t we blocking adjacent possibilities?
possibilities that could potentially provide a more useful answer?
but more important, and what i’m really seeking insight about,
don’t the kids see the pattern, don’t they become aware you’ll be having those answers.. i’m wondering if that knowledge of where this is headed whether or not they fully engage, encourages them to be mindless.
i’m wondering if our focus on being good teachers, even less helpful teachers, is still keeping our students from becoming fully engaged. i’m wondering if we need to deliberately not teach.
blaw0013May 21, 2011 - 11:01 am -
Well stated. This sure is hard for me to ACT on in my classroom though. I guess I do plan for class seeking to draw out a question I wish for students to ask. To what extent can I let go, yet still create a mathematically productive environment?
And if not obvious, by selecting one question to pursue, the community of learners in my classroom are not each pursuing their own questions.
Which seems to beg the question, why not offer the question you wish for the students to engage (rather than feign the interest in your students’ pursuit of their own question), and allow them to pursue the question in whichever manner they see fit.
This seems to be the method observed by Jo Boaler in the “open” classroom described in her dissertation work, Experiencing School Mathematics.
AmyMay 21, 2011 - 6:19 pm -
What an interesting thought that as teachers we expect students to figure out the “right” questions to ask. We are limiting their critical thinking and questioning skills. So, instead of the teacher preparing “the answer” I like how you suggest to prepare as many questions and information you can gather regarding the initial problem. What a great skill for students to learn… problem solving.
Dan MeyerMay 21, 2011 - 11:44 pm -
This goes too far for me. Teachers bring a great deal of value to a classroom. Some of that is bound up in their ability to explain concepts, but students can access plenty of explanations from other sources. I also add value to a classroom when I notice different solution strategies around a classroom for the same problem, when I help students fix the incorrect strategies, and when I pull those different strategies together in a whole-class discussion at the end of the problem.
I can’t add that value in an unfocused problem space. The problem space needs to be rich but tightly focused if we’re all going to benefit from the different work that our different classroom colleagues are doing.
Multimedia gives us the opportunity to create problem spaces that are, in many cases, richer than what’s possible on paper. (Certainly that isn’t always the case.) I come to class with multimedia that’s been captured in such a way that I can anticipate what questions students will have about it. I gather as much information about that space as possible (measurements, etc) that will enable students to answer those questions. And while I’m capturing the space, I capture the answer to the question I anticipate most of them will want to answer.
Certainly, we need to keep a loose grip on the goals of our lesson. If the majority of the class wants to work on a different question (that’s still mathematically productive) I have no problem setting aside the answer video I captured. But the existence of the answer video is only adding value to the class.
Sure. I can’t say I have a problem with this. At some point, though, I need to ask a group of people: “is the question I’m going to ask students to pursue here the question that most of them will want to answer?” I can get that feedback by asking students, “what question interests you here?” Or, increasingly, by asking a group of teachers: “#anyqs.” At a certain point, I may get good enough to trust my instincts.
On the other hand, just because we don’t have the resources to take up every student’s question, it isn’t a priori insincere to ask students what they’re wondering. And sometimes we can take up multiple questions. Ten students wonder how many tickets are on the ticket roll. Three wonder how long the ticket roll would be stretched end to end. It’s a simple matter to take up both of those questions. But if I had only asked students how many tickets are on the roll, I would have missed an easy opportunity to validate student curiosity in my math classroom.
monika hardyMay 23, 2011 - 4:59 am -
thank you Dan..
nickMay 29, 2011 - 11:59 am -
I liked Lisa Henry’s poster, but felt that for my English pupils a trip in a place they might not recognise would be less interesting. Here’s my more UK-friendly version: