Something happened in my class last week that really got me thinking and reflecting. My class was working on this problem...
...when one of the students came up to the board and drew this...
If you haven't already, it might be helpful to think about the prompt and the student's proposal for a bit. Essentially, the question they were considering was "if we started with a perfect square, are all five of the divisions also perfect squares?" As you might have noticed, it is possible to draw it so that it in a way that makes you stop and think. There are a host of interesting things to discuss here including measurement, drawing versus abstraction, and others, but here is what interests me most at the moment...

Another student in the class responded to this student's drawing with a friendly counterargument. In essence, he provided the student with a logical argument about why these five divisions were not squares. The student who had drawn the proposal on the board seemed to recognize the fault in his argument and we kept discussing other cases. Then, on his way out when class was over, I overheard him say, "I wonder what size starting square would make my five case possible?" It seemed to me that this student was still searching for ways to preserve his mental equilibrium; hesitant to make a (perhaps?) necessary accommodation.

All of this got me thinking about the ways our minds work and the role of the teacher in working with students. It's fascinating to me how people work with their ideas. I'm convinced that any lasting change must come from a student. I use the example to make a larger point about our goals in working with students. In the context of this puzzle-like problem, it's pretty easy to suggest that we just keep letting this student play with other size starting squares; to keep letting him play with his own ideas. I know that in my own teaching, it isn't always as easy to preserve this play-like quality with things that we are "supposed" to teach. But to hold students accountable to those things (or what we perceive to be their mastery of that Mathematics) is an oversimplification of the workings of the human mind and a coercive, unjust position to hold in education. Because if we were to force this student to abandon his ideas about that square, to insist that his thinking is wrong, he would inevitably start to reject his own thinking as mathematical. He would inevitably start to surrender his own agency.

I think it brings up some interesting questions about the goals of education. I've been reading The Having of Wonderful Ideas recently (which I highly recommend) and I think Eleanor Duckworth puts it quite nicely:
"Many people subscribe to the goals of the it's-fun (interest) and I-can (confidence) types, but when it comes to detail, almost never does one see a concern with anything other than the-way-things-are beliefs. Lesson-by-lesson objectives are almost without exception of this type, despite the fact that general goals very often mention things like interest, confidence, and resourcefulness. This is, of course, because it is difficult to produce a noticeable change in any of these in the course of one 50-minute lesson. But notice that as a result all the effort is put into attaining the objectives stated for the lesson."
I'm not sure where all of this leaves me other than the realization that when we listen to kids and their ideas, working with students is challenging, perplexing, and a ton of fun.


12/09/2012 2:14pm

We have chatted about this some already. The thought that jumps to mind right now is how excited many of my math leadership friends are (math teacher trainers, for example) because of the opportunities for teaching that the CCSS-M Standards for Math'l Practice (SMP) seem to provide. I too am excited.
However, it is my sense that for the teacher to really work to develop the SMPs, it seems they may have to loosen their grip on the foregrounding of the particular mathematical ways of thinking and knowing as they are defined by the discipline, and evidenced in the CCSS-M Content Standards.
This tension will hopefully make EVERY math teacher wince, but become embraced as what defines the professionalism of our field.

Bryan Meyer
12/14/2012 11:30am

"it is my sense that for the teacher to really work to develop the SMPs, it seems they may have to loosen their grip on the foregrounding of the particular mathematical ways of thinking and knowing as they are defined by the discipline"

I couldn't agree more. Will the assessments value students engaging in "mathematical practice" to the same extent they value the discipline of mathematics? As in, how will they evaluate the student who demonstrates incredible perseverance but arrives at an incorrect answer or who makes incredible attempt to make sense of the problem in a way that works for them but not in the way that the question intended? Will the value be distributed equally, or will it still reside with the discipline? Not sure what will (or should) happen....but I'm interested and curious.

12/10/2012 11:25pm

This is one of my favorite in-class group PS (problem solving) activities. We started this in geometry, and already there's been great conversations within small groups and across the groups. We'll continue tomorrow. My guiding questions: "What number of squares are possible? Which ones are not? How do we know if we can find them all?..." Great stuff.

Bryan Meyer
12/14/2012 11:34am

Pretty great problem. I like that there are a few different ways to think about a solution and that the problem offers multiple opportunities to engage students in proving or justifying things. My only critique with it is that it seems a bit closed...the problem is narrowly defined and not easily extendable.

I have often wondered (sometimes publicly here on this blog) about how we can make every day a problem solving capture the puzzle-like quality of mathematics in all of our work with students. It's a challenge, but a fun one. I would love to hear your thoughts.


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