The Best Writing on Mathematics 2011 by Pitici Mircea; Dyson Freeman;

Author:Pitici, Mircea; Dyson, Freeman;
Language: eng
Format: epub
Publisher: Princeton University Press

Figure 1. Sample note page.

With a few adjustments to the structure of the classroom, students would likely develop a deeper understanding of mathematics and begin to see the relevance of this content in their lives. The remainder of this chapter focuses on three areas that we know to be effective ways to engage students in thinking about mathematics (e.g., Fisher and Frey, 2007): modeling, vocabulary development, and productive group work.

Engaging Students in Thinking about Mathematics

MODELING

There exist decades of evidence that teacher modeling positively impacts student performance and achievement (Afflerbach and Johnston, 1984; Duffy, 2003; Olson and Land, 2007). Modeling provides students with examples of the thinking required, as well as the language demands, of the task at hand. In essence, the student gets to peer inside the mind of an expert to see how that person thinks about, processes, and solves a problem.

Unfortunately, there is significant confusion between modeling and explaining. Think of a lecture you've attended. It was probably full of explanations. And explanations aren't all bad. We all need things explained to us sometimes. But we also need modeling, which personalizes the experience for the learner as the teacher uses “I” statements to share his or her thinking. Modeling also provides information about the cognitive process that went on in the mind of the expert; it's the why that we're after here. But as Duffy (2003) pointed out, “The only way to model thinking is to talk about how to do it. That is, we provide a verbal description of the thinking one does or, more accurately, an approximation of the thinking involved” (p. 11).

Accordingly, mathematics teachers must model their thinking by talking and thinking aloud for their students. Some of the common areas of thought that math teachers model include:

• Background knowledge (e.g., “When I see a triangle, I remember that the angles have to add up to 180°.”).

• Relevant versus irrelevant information (e.g., “I've read this problem twice, and I know that there is information included that I don't need.”).

• Selecting a function (e.g., “The problem says 'increased by,' so I know that I'll have to add.”).

• Setting up the problem (e.g., “The first thing that I will do is…because…”).

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