In 20 years of teaching I have seen standards and curriculum come and go… or evolve. And in the flurry of activity now, as people continue to refine and innovate, and move to implement the standards in the Common Core, there is always something in each for me to learn, or an activity to take away. But I have always felt there was something missing, something in the core philosophies that wasn’t serving my students, especially the students who struggle with math.
I think there is a key perspective that math education in the U.S. cannot or will not adopt. In fact I think sometimes we actively avoid it. We want our students to learn to think mathematically, and we focus on designing tasks where there is the opportunity to do so, and we talk a lot about how to structure discussions so that students can share their thinking and learn from each other: guide on the side, not sage on the stage. But there is a lot of “sink or swim” hidden in that scenario. What about the students who really struggle with mathematical thinking? What tools are they missing? Can we teach them to think mathematically through direct instruction? Should we do that?
We adopted Primary Math at my school, one of the curricula that originated in Singapore. I try to stick as much to the original design as I can when I use it, and I love using bar models. Or I should say, I love teaching bar models. Because not only am I posing problems that give my students a chance to independently and flexibly apply mathematical thinking, I am also training them to use a tool that increases their ability to do so. As teachers we sometimes see visual models as a way to make explanations clearer. In Singapore, they serve a slightly, but significantly different role. The educators in Singapore recognize that visualizing mathematical relationships is a key skill, perhaps the most essential skill, to being able to independently solve problems in mathematics. And they also recognize that certain students seem to figure out on their own how to visualize mathematical relationships in useful ways, while others don’t. So they have very carefully looked for visual tools that are effective, and have designed a curriculum largely around developing those tools, and helping the students to internalize them, over a time frame of many years.
There is something else I have seen teachers from Singapore do, or at least one of them. Yeap Bar Har is an educator from Singapore who speaks frequently at conferences. During one workshop, both in front of teachers and while leading students, I watched him solve bar model problems. I was surprised how little wait time there was after each question, and how in fact he seemed to be giving us the answer relatively quickly. And then I realized he was modeling his thinking. He was in fact solving the problem in front of us, telling us what he was thinking at each step… and more specifically, how he was using the information to build his visual model, where he was looking, what he was noticing, what conclusions that led him to. Again, he knows there are students in the room who can do that themselves, maybe without the bar model. He is talking to the students who can’t: the ones who, even when the other students explain things, just still don’t see it. Literally. He is helping them to develop their ability to do more mathematical thinking for themselves.
Singapore Bar Model Method
In a later blog, I'll share an insight I had recently for my pre-algebra students about functions, to help them see better how functions and tables are related.
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