I have been experimenting recently with different ways of having student curiosities drive our work together. In some ways it has been successful. In some ways it feels like I don't know how to do this well. I thought I'd blog about a few examples from the past two weeks and hopefully you all can help me sort this out a little more.

EXAMPLE #1

I put this up on the white board:

x/2 + 5

Me:         "Someone give us a number."
Student:  "7!"
Me:         "Ok. I heard 7. We are going to put 7 in for x in the expression on the board. Then, whatever we                 get as the result, we are going to put in for x. And then again...and again....and again. But we                     aren't EVER going to stop. What do you think is going to happen? Tell your partner."

They had various ideas, we tested them out, and made some cool observations. After that, I encouraged them to experiment with anything they were curious about. What happens if we change the expression? What happens if we change the starting value? What happens if we use two rules instead of one and alternate? There were lots of options.
PROS
  • Most students found something to pursue on their own
  • Most gravitated toward something that was appropriately challenging
CONS
  • The initial task was relatively narrow and was defined by me, not them
  • It was difficult to have students understand, respond to, and challenge each others' work because they were all working on something different


EXAMPLE #2

I posted about this problem before, but this is an extension of my thinking about the launch of that problem. For my first two classes, I gave students the following problem:
They seemed to rely on me to define the task for them and set parameters. So, for the third class, I put this up on the board:
I said, "For the next five minutes, everyone experiment with something you find interesting." They experimented for a bit and then I had them compare their activity with the rest of their group. There were a few different ideas. Most students hovered somewhere around the question of "which numbers can you create" but there was a lot of discrepancy about parameters.
PROS
  • Students were the ones engaged in "finding the task" and setting the parameters
  • It opened up the possibility for students to pursue questions outside of the one that might have been intended or suggested
CONS
  • It took a lot longer and was more difficult to facilitate
  • It felt like I was tricking students into asking THE question versus actually giving them freedom to explore their own questions
  • Students had a difficult time accepting the suggested parameters of a different group if they were not the parameters THEY saw as fitting (there might be some implications here about the questions we pose as teachers seeming unnatural to students?)




EXAMPLE #3

I put the following images up on the projector:
I gave students some time to examine the images and then asked them to brainstorm a list of questions that were raised by the images. We put a list of them up on the board. It was interesting that a lot of the questions were clarifying questions rather than problems to be investigated (Is the first one just a zoomed in portion of the second? What is the dot? Does the line always have to cross diagonally through the small squares?) It seemed like they were so used to asking about parameters, rather than setting them, that it didn't occur to them to just set the parameters and ask a solvable question based on them.

Eventually, we got a few questions with potential. Will it always hit the corner? How many times will it hit the sides before it hits a corner? Does it matter if the side lengths are odd or even? Is it possible to hit every grid line on the side BEFORE it hits the corner? Is it possible to end up in the same corner that you started in?

I suggested that each group: 1) pick a question they were interested in, 2) set their own rules/parameters and 3) get to work. It was interesting. Groups worked for a couple days and then things really stalled out. Because there were only four people (or so) working on a problem, there wasn't the same opportunity for them to bounce ideas off of other groups, for us to work through difficult things together as a class, or the same chance that someone might have an insight that led to progress for the whole class. We eventually proved that 1) it would ALWAYS end in a corner and 2) that corner would NEVER be the starting corner (assuming you launch at a 45 degree angle from a corner). So, from there, I suggested we all work towards finding a way to predict exactly which corner it would land in based on the rectangle size.
PROS
  • This was the closest I have come to having student curiosities drive the work; felt like students had a genuine opportunity to follow their own question
  • Students were engaged in questioning, setting parameters, exploring, and then adjusting their question or parameters if they needed to
  • Many students really enjoyed the freedom and creativity involved
  • Almost all students were engaged in mathematical activity
CONS
  • Students seemed to, initially, search for a more shallow level of depth than we usually accomplish as a class
  • Some students were very turned off by the ambiguity and openness
  • I didn't know how to bring things together or take it further when students were all over the place
  • As a result, I eventually defined a question for the whole class. Even though it evolved out of their work, it was still defined by me




Help Me Out...

I'm really interested by idea of using student generated questions but I feel like I need help on how to make it work. Things that I'm thinking about:
  1. I don't love the idea of tricking students into asking the question you want them to ask, but I also have trouble when students are all working on different things.
  2. I'm curious about the "initial event" that prompts student questions. Should I start small and well-defined and then move to open exploration (example #1) or should I start wide open and leave it wide open (example #3)?
  3. In most cases, I found it hard to facilitate student work. How can I get students to share work, challenge each other, and challenge themselves?
Mostly, I would love to hear about your experiences, questions, advice, or thoughts.
 


Comments

12/18/2012 10:35pm

First, I cannot help but think of <a href="http://blog.mrmeyer.com/?p=5789">this</a> when I read the post. Some really good discussion in the comments about the use of this. How much to give students, how much to direct it, etc.

I think you have to remember that there is a good balance of student led problems and teacher initiated problems. Not every problem that you deal with in life or anywhere is going to be instigated by your curiosities. There are somethings that will make you curious that you never expected. Let students experience that.

When doing investigations I notice that it is difficult to let them explore so much, because if they are learning a NEW topic, then I pretty much have to take the reins. If they are applying a topic, I take hands off.

With the Barbie Bungee I showed kids how to take the data, but I let them solve on their own. I had kids solving graphically (never taught them to do that) solving by guess and test (I often encourage that), and I challenged some to come up with more succinct ways (for example using their derive formula, and doing standard algebraic manipulation). It was fun to see students dig, and to challenge them. I think that is fine to say, "All right, make that more elegant."

In exploration we shouldn't be afraid of saying, that is a great way of doing it, can we make it even easier? more efficient? more succinct?

I feel I am the rambling stage, so I am done...

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12/19/2012 8:39am

I did a similar version of your Example #3 with math and science teachers at an (optional) evening PD. I used almost the exact same prompts and set-up, but I also added the task of choosing one problem to work on as a group, by popular vote. The obvious pro was that we could all (theoretically) talk together about our work so far. The con (which surprised me at the time but in retrospect was "duh!") was that it was really, really hard for the groups to define the parameters and the task when it wasn't their question -- as you saw in Example 2.

One thing I've noticed in your blog is that you're very open with your students about the kinds of learning and doing math experiences you want them to be having and why. I wonder if they might help you think about the tradeoffs between having lots of minds at work together on a single task, and having a task you came up with for yourself that you full understand and are invested in.

When is it worth doing a lot of up-front work to define a task so that everyone is doing the same thing, and what makes that easier? Is it better for those tasks to have a narrow definition or goal (e.g. set up the Example 3 task as a challenge -- you pick the grid to get the ball into the left corner pocket, like calling your shot when you sink the 8-ball at the end)?

My hunch is that there will be some times when you want students to do exploration and pose challenges of interest to them and then live with the con that they're one of the few people in the world invested in this challenge at the moment -- and then have them work together after solving the problem to try to map out what they, collectively, know about the whole domain and what questions are left to explore.

Other times, you'll want them to be doing a particular kind of math thinking together, and so it will be worth posing the task (in a game or challenge or cliffhanger kind of way) so that the question you would ask feels very natural -- like how students almost always want to know with games, "how do I win?" or "who is going to win/who is best?"

Hope that's helpful -- it's basically just a long-winded way of saying, "I too feel that conundrum and wonder if we just have to live with it and choose which pros and which cons for which tasks."

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melanie
12/19/2012 2:13pm

#2 is giving me a lot to think about. I think there is a decision we as math teachers have to make, which is, what do we want our students to come out of the class with? And that answer can change on any given day. So the day when I want them to be investigating a problem in a straightforward day, the question is provided. The day when I want them to be doing something much more open-ended, the question isn't provided. I see a real connection in the second way you presented the problem to scientific inquiry, where a scientist gathers some data and designs an experiment that could look completely different from the next scientist's experiment.

In conclusion: I don't think I'm adding much to the conversation but I do really appreciate the conversation that's happening! :)

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12/19/2012 5:29pm

Timon, Max, and Melanie...

I'm really interested by this common thread in your comments. It seems like everyone is noting the "conundrum," as Max points out, of living in the tension between direction and indirection. I'm sitting here thinking about how to sort out my thoughts in a nice, elegant comment but my thinking isn't quite there yet....so excuse the messiness. I have some questions about how/when/why we, as teachers, decide to control the flow of thinking in the classroom. How do students benefit from this? In what ways? How do students suffer when we do this? In what ways? Why do we decide it is ok to direct in some instances but not others? How does student work differ in each instance?

Timon... Why would you say we have to take the reins when students are learning a new topic?

Max... You mentioned it was difficult for participants to set parameters when the question was not theirs. I wonder what the implications might be here about students working with teacher (or textbook) posed problems. Do the parameters seem natural to students? If they don't, what are the implications for the student?

Melanie... in #2, which task would you use? Why?

I hope you will continue to drop your thoughts here when, or if, they keep unfolding. I'm curious.

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12/20/2012 7:04am

My students rebel so much when I make things wide open, that I've almost always backed off some, and directed more.

In math circles, though, I like to make things as open-ended as I can, and then I feel that tension more between having them each work on their own and end up going in different directions, or leading a group discussion of what we might do next on a problem.

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Bryan Meyer
12/20/2012 8:14am

Hi Sue...

How do you usually respond to that tension in your math circles?

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12/20/2012 8:53am

I think my solution to #1 tends to be similar to Max's, in that we explore things more openly for a while, then generate a list of questions on the board, and then collectively choose a question or two or three to focus on (sometimes based on what more of them are interested in, and sometimes based on the direction I want to go with the exploration).

A lot of times I'll ask a deliberately vague question, like in your billiard ball example, I'll ask "What happens with other rectangles?" and then get suggestions to help clarify what that should mean -- how many bounces, whether it ends up in a corner, which corner, what if it's not integer-sided, or whatever it might be.

I think both starting small and starting open can be good. In either case, you need some kind of "what if", I think. In the first example, it's "what if we have other functions? what if we alternate between two functions?" and so on. In the third example, it's "what if we have a different rectangle?"

Getting students to understand and challenge each others' work is always tough. I tend to get collections of data that may form patterns of various sorts. Then at first they start correcting each others' data, and then I collect a bunch of conjectures about the patterns, and then they either refute the conjectures with counterexamples, investigate why some of the conjectures are true, or ideally do both by refuting a conjecture but then fixing it up a bit to make it true.

I sometimes instead will give a problem that's way too hard, and have the class collectively work on the strategy of finding an easier problem that they can do and that relates in some way to part of the original problem. I'm not sure quite how that fits in with all of the above, though.

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12/21/2012 11:32am

Hi Josh...


Thanks for the thoughtful comments. I have a couple short questions based on what you wrote:
1. How do you "finish" a problem with your students?
2. What are the organizing principles you use for your curriculum?

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Dave
12/20/2012 10:53am

Hi Bryan,

Have you asked your students about this? I don't mean what they prefer and why. I mean, what they think your intentions are when you pose a small, well-defined problem versus a wide-open problem, and what they think the outcome will be.

Ideally, all of your students should respond positively to both types of problems. You don't want students who will only engage in problems that they generate themselves or students who will only engage in problems that have a well-defined path and clear solution at the end.

Students come into your classroom with a lot of baggage. Some students are instantly turned off by wide-open problems because, when they have done them in the past, they've been completely lost and forced to observe while some of their smarter peers have a blast. Or they do something that is fun, but does not seem to have any tangible benefits to them or their future, so it feels like a waste of time. Some students are instantly turned off by small, well-defined problems because they feel like you are manipulating them... and something that starts out interesting is just going to turn into boring math class at some point.

Personally, I don't think that you can avoid feeling like you have ulterior motives when posing problems to students. You do. It is your job to set the goals for your students and the class. I have found that the key is building a culture where students trust that most problems, whether small and well-defined or wide-open will lead to a good and rewarding experience in the end. That takes time.

If we are doing our jobs effectively as teachers and helping students develop agency, then I would expect the way that my students respond to the problems I pose them to change drastically between the beginning of the year and the end of the year.

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05/17/2013 11:00am

If you're curious to learn more about the billiards problem, you might enjoy a look at my handout (first page) http://sanjosemathcircle.org/handouts/2007-2008/20071003.pdf
and perhaps if you want more details, at
http://sigmaa.maa.org/mcst/PosterActivitySessions/documents/BilliardsPoster.pdf

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