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:
 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.
 I'm curious about the "initial event" that prompts student questions. Should I start small and welldefined and then move to open exploration (example #1) or should I start wide open and leave it wide open (example #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.