I have been enjoying reading Suzanne Chapin's Classroom discussions: Using math talk to help students learn. One thing that has stood out for me so far is her discussion about our over emphasis on correct answers. She makes the point that helping students develop the ability to talk about and through their reasoning in a given problem is more important to achieve a deeper understanding of mathematical concepts. She does not dismiss correct answers of course, only postpones them. She also talks about when it is important to provide "the answer" right away and that is based on the idea of knowledge that is "external" to a student rather than knowledge that can be discovered through reasoning. Such knowledge includes the social conventions of math such as the meanings of symbols etc.
A good example of focusing on reasoning is Case 1(page 22). In this classroom example, the teacher is introducing problems which involve two equations and two unknowns. The unknowns are indicated by a square and a triangle. In each set of equations the values of the squares and triangles must be the same. This type of problem helps begin preparing students for algebra. In Case 1, the teacher circulates as students try and solve the first set. She notices that a number of students do not understand that the values for the squares and triangles must be the same in both equations. She notices one student that has the "right answer." She brings the students back together and purposefully calls on a student who did not understand the problem to demonstrate their "solution." She asks her to explain her answer. Then she asks if everyone agrees. The girl with the right answer raises her hand and says she disagrees. The teacher asks her to explain why. A class conversation ensued in which the girl who had provided the initial answer comes to understand her error and the girl who had the right answer develops a deeper understanding of her reasoning by having to explain it. The right answer is never actually shown to the class. Instead, after the discussion, she gives them another problem to work on with a partner.
I admit, at first I was surprised that the right answer was not presented. As I thought about it, I realized it was a specific strategy of the teacher to de-emphasize the right answer. I can see how this could be beneficial for students to help them realize that the process is as important (maybe more so in some cases) than the product. It might, over time, decrease the student's desire to just be given the answer.
Very thoughtful entry, Lauren. I'm thrilled to read about the ways you're continuing to read from, and think about, the important pedagogical ideas in the Classroom Discussion book. The way you've honed in on the teachers work around reframing what is valued in mathematical discussions is insightful. This is artful. Especially when you consider the question, what are her students coming to learn is important in mathematics? From your excerpt, I can imagine the ways her students are receiving the message mathematics is about forming a conjecture, discussing ideas, agreement and disagreement, reasoning and explaining..... much more than the correct answer. In subtle but powerful ways this teacher is teaching her students how to be mathematicians in school that think like mathematicians in the field of mathematics. If you want to experiment with this in your spring placement, which I can imagine you doing, let me know how I can support you.
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