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The visions of teaching set forth in the Professional Standdrds for Teaching Mathematics (NCTM 1991) do not, in themselves, ensure that we can successfully teach all students mathematics. To reach all students, we must understand how students think and then develop instruction compatible with their thinking. To begin to understand, we must observe, listen, and gather a variety of evidence of what and how students are learning. Although we might view a mathematical concept or algorithm as simple, it is "a mysterious, almost inexplicable phenomenon from the point of view of the outsider" (Davis and Hersh 1981, 43). Examining students' thinking through their interactions with mathematical tasks can help unlock a bit of the mystery. This process of analysis is the amalgamating item in the "Standards for Teaching Mathematics" section of the Professional Standards for Teaching Mathematics and emphasizes the need to listen carefully to our students.
Earlier in this volume year, issues of the Mathematics Teacher addressed various standards for teaching by considering (a) worthwhile mathematical tasks; (b) empowered discourse; and (c) challenging, but comfortable, learning environments. This article focuses on how purposefully listening to students can help us realize the visions of the Professional Teaching Standards. We invite readers to "listen in" on students' dialogue and to consider possible insights into their thinking processes. We then propose a few strategies for analyzing teaching.
CAREFUL LISTENING
To highlight students' reasoning processes, we gathered data from middle school through university students. We used actual interview data about the concpts of area and perimeter to create a fictitious but familiar classroom scenario centered on the task in figure 1. (Figure 1 omitted) We offer this dialogue for your analysis of the thinking of three students as they work in a small group in an inquiry-based classroom. We invite readers to listen for the following in the dialogues:
* Evidence of mathematical understanding
* Indications of mathematical communication
* Empowering or inhibiting students' attributes
Al: To find the perimeter, we count the nails. Start in the lower left and follow the path to the upper right and then back down and around.
Pause
I get eleven because there are eleven nails on the rubber band.
Viv: But, another nail could fit between the last nail and...