How can we communicate our thinking?
The math teachers I work with often express frustration with their learners' inability to communicate their thinking when it comes to solving problems. If learners are stuck, they often struggle to articulate what they have tried. If the problem has been solved, learners have trouble explaining their efforts. Too often, learners respond to a teacher's question about their thinking with, "I don't know." This does not present teachers with the assessment data necessary to evaluate what learners can do or are trying to do, which makes it difficult to plan what comes next.
I encountered this same problem while working with fifth-graders on fraction computation. They were practiced in giving answers and even showing work but when I asked them to share their thinking they often said, "I don't know." This provided me with an opportunity to try a response suggested by Ellin Oliver Keene in a session at the 2009 MRA Conference: "Pretend that you did know - what would you say?" The fifth-graders found that this framing supported their communication efforts by freeing them to take a risk because they were "pretending."
While this got them talking about their thinking, they still needed more support to organize their efforts. I thought it would help to demonstrate what a reasoning recount might look like. I was introduced to the recount text form through Margaret Mooney's book, Text Forms and Features. Here is the model reasoning recount I wrote based on the prior efforts of the class to think about adding fractions.
Recently, I have been collaborating with Jennifer Brokofsky via Twitter and email about ways to connect reading, math, and writing. The figure below represents our current thinking. I hope this vignette further demonstrates the link. We recognize that this is work in progress, and your support would be appreciated. Please share your thinking in the comments.
This is really interesting, David. Hope you will keep writing about it. I've been thinking a lot about the connection between reading and math, and it seems to me that "questions" and "inferring" and especially "monitoring" belong in both boxes. Should we be distinguishing between "Mental images" and "visualizing"? Or should we be emphasizing the similarities? I can think of reasons for both. Is "estimation" really a kind of "inferring"? I've been influenced here by the Foundation for Critical Thinking, which helped me make my thoughts more concrete. They use a framework that applies one set of mental habits to a wide variety of disciplines, and I found it persuasive.
ReplyDeleteI like your point about "pretending." I'm starting to realize that my students tend to avoid inferring because the consequences of being wrong seem too high. An interesting corollary is that, if I push them to infer, they seem to treat their inference as if they now have to defend it to the death. Sort of "in for a penny, in for a pound." This makes it very hard on them when they find evidence against their idea -- almost a personal affront. It also seems to dissuade them from looking for contradictory evidence. I'm looking for ways to encourage the concept of "tentativeness" -- the idea that formulating an idea is not the same as committing to it. That you can formulate an idea and hold it at arm's length, gathering evidence both for and against. That it is not an expression of your identity, so that if you decide to abandon it, it doesn't feel like you're carving a piece out of your self-worth! I don't think that "pretend" would work for my group of adult learners, but I need to find something in the same vein.
I appreciate what you're saying about modelling your thinking. I'm continually underestimating how much modelling my students need, and how much they benefit from it. I don't think I would use the word "recount" though, since every time I read it I have to reorient my brain, which has started thinking about election recounts.
Mylène,
ReplyDeleteThank you for your thoughtful comments.
I agree that the figure needs more work/thought. There are overlaps and distinctions that need to be made. I began working with a venn Diagram but had some of the same questions as you. Let's keep thinking about this.
Your point about using "pretending" is interesting. I thought the same thing until my colleagues started having success using it with their students in college math classes. Maybe we all need permission to pretend.
I hadn't though about the "recount" terminology in that way {smiling}.
Peace,
Dave