October 14, 2012 by ray_emily
Although it’s been a good long while since I’ve written, I’ve been doing lots of thinking. Today, I want to share with you some of the thoughts I’ve been thinking about manipulatives and mathematical modeling. (Bear with me; I need to provide a bit of background information, first…)
In case you haven’t read the post, the activity goes like this: One kid draws draws a slip of paper with an equation on it. That kid illustrates (on a mini whiteboard) the appropriate amount of positive and negative integer chips to represent said equation, being sure to cross out any zero pairs (of one positive chip and one negative chip). Kid shares his whiteboard illustration with a buddy (or buddies). Buddy tries to figure out what equation is being represented, and jots that down on his whiteboard. (It’s Pictionary, but with integers. Awesome, right?)
In Timon’s post on this activity, he explains how he has previously observed that students struggle to draw out mathematical models, due in part to their perception that “memorizing mnemonics is true math.” The benefit of the activity, in Timon’s words, is that “it gets students thinking laterally about mathematics, and translating something concrete into something abstract.”
Naturally, I was excited to incorporate Integer-nary. I made a mental note to reserve some class time for it when my unit on integer operations rolled around.
A week or so before my Integer Operations unit, I read an article for a class about Zoltan Paul Dienes (developer of Base Ten Blocks). (I have no idea who the author of this article is. Sorry!) Most notably, I read about (and was fascinated by) Dienes’ “Multiple Embodiment Principle,” which states that:
Mathematical abstractions occur when students recognize structural similarities shared by several related models… It is not enough for students to work with a single model; they must also investigate ‘mappings’ to other models.
And, it occurred to me that I do not, in my classroom, embrace the Multiple Embodiment Principle. In the instances when I do expose students to multiple models, I have (eek!) told kids that it is okay to stick with the model that works for them. (I’ve noted that some kids prefer number lines, others prefer chips. No biggie, right? Go with what you know—maybe?) I wondered how testing out this Multiple Embodiment stuff might help my kids meaningfully come to grips with the concepts I hope for them to master…
And then, I had my “ah ha!” moment—when I thought, “Hey, maybe I can modify Timon’s activity to help my kids think about the mappings that exist between different mathematical models!” It seemed easy enough…
And so, here is the adjustment I made: I required students to draw a number line model (with an accompanying arrow to illustrate the distance ‘jumped’ to the left or to the right of the starting value), rather than simply writing down the equation being represented. My hope was that the process of creating, viewing, and analyzing these two models, concurrently, would spur analysis and investigation of the similarities (and differences) between the models.
After discussing the activity with colleagues, I have recognized that I missed the boat in a lot of ways. In other words, I could have made this activity way more meaningful and worthwhile than I did!
Here are some modifications we thought up, which I’ll definitely try next year:
What I did: Kids began each round by analyzing the numeric equation, first. (For instance, they looked at a slip of paper that said “-5 + 2 = -3.”)
What I should have done: At the start of the round, I could have provided slips of paper with number line representations, or even with chips representations! I could have / should have varied the starting point of the activity. This adjustment would have upped the rigor, and inspired kids to more meaningfully make the connections (in multiple directions) between the models.
What I did: After the kid who started the round analyzed the equation and then illustrated the chips representation, his partner always wrote out the equation (with numbers)—and then provided the number line representation.
What I should have done: I think that omitting the ‘write the equation’ step would have forced kids to do more pondering of the models and their similarities.
What I did: There was no thoughtful introduction to the activity, nor was there an awesome wrap-up, afterwards.
What I should have done: Opening questions—to guide kids’ thinking—might have been valuable. I am also wondering what questions I might have asked after the activity, to help students verbalize the similarities and differences between the models. (I’d love help in brainstorming these questions, as this task overwhelms me a bit.)
What do y’all think? Do you have ways in which you help kids to make these cognitive connections? I want to do more of this! Inspire me, please–as you always do. =)