I believe that number talks are changing the face of mathematics instruction. They put more emphasis on the sense making and less on computation. The results of this shift are remarkable. I have seen so many students (and teachers!) excited by the conversations surrounding mathematical ideas/concepts. To put it in the words of a 6th grade student when asked how he liked his first number talk experience, he said, “It’s like math…only better!”
But what about the kids who are still not accessing the content that other students are sharing? How do we provide that access to all the students? Attaching models (e.g. number lines, arrays) to student thinking is a great way to provide a representational element to an abstract student thought.
This is an example of a number string presented to students who shared a great deal of thinking, including place value and counting up strategies. None of these students, however, suggested the use of a model to represent their thinking initially. As the facilitator of the number talk, I saw this as an opportunity to provide a model FOR their thinking. Through the use of number lines, number bonds, tallies, and base ten block representations, many students who were not following the abstract reasoning of the their classmates were more able to make sense of the thinking. By the time that we had considered the final problem in the string, 14 + 15, a student suggested a model OF her thinking. She had been exposed to the number line as a representation of this type of problem a sufficient number of times to influence the way in which she thought about it.
In this example, note that student thinking for the multiplication expression 4×10 was represented with a number line and several small arrays (to support repeated addition reasoning) and with a large array and area model (to support multiplicative reasoning). Once again, I made the decision to attach a model to the more abstract thinking of student solutions to provide greater access to the entire class.
The use of models can provide students with a tool that reduces the cognitive demand of the work by providing a representation to support their thinking. “Mathematizing demands the development of mathematical models. In order to mathematize, children must learn to see, organize, and interpret the world through and with mathematical models…Models that are developed well can become powerful tools for thinking.” (Fosnot, et. all)