Our answers to students' questions about the relevance of what we teach might paint mathematics into an undesirable corner.

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### Beth Cory and Ken W. Smith

Through these calculus activities, students reach an understanding of the formal limit concept in a way that enables them to construct the formal symbolic definition on their own.

### Colin Foster

Exploring even something as simple as a straight-line graph leads to various mathematical possibilities that students can uncover through their own questions.

### Carol J. Bell

Reasoning and Proof is one of the process standards set forth in NCTM's *principles and standards for school mathematics* (2000).

### Mark W. Ellis and Janet L. Bryson

Sitting in the back of Ms. Corey's sixthgrade mathematics class, I enjoyed seeing students enthusiastically demonstrate their understanding of absolute value. On the giant number line on the classroom floor, they counted the steps that they needed to take to get back to zero. The old definition of absolute value of a number as its distance from zero—learned by students and teachers of the previous generation—has long ago been replaced with this algebraic statement: |x| = x if x ≤ 0 or − x if x < 0. The absolute value learning objective in high school mathematics requires students to solve far more complex absolute value equations and inequalities. However, I cannot remember students attacking the task with enthusiasm or having any understanding beyond “make the inside positive.”

### Randall E. Groth

The lesson study model of professional development that originated in Japan is becoming increasingly popular in the United States (Lesson Study Research Group 2009; Stigler and Hiebert 1999). At its core, lesson study is a means of bringing teachers together to carry out the process of planning a lesson, implementing and observing it, and then examining it during a debriefing session (Yoshida 2008). The debriefing component is one of the most distinctive characteristics of this type of professional development. It provides a means–discussion–for reflecting on the strengths and weaknesses of the collaboratively planned lesson. As such, the debriefing component merits special attention from those currently engaged in lesson study as well as those considering using it.

### Rob Wieman

After many years of teaching mathematics, I still fall into the trap of assuming that my students think as I do. Indeed, this failure to recognize my own assumptions and to acknowledge that others may not share them is at the root of most of my teaching problems.