Teachers pose a variety of questions to their students every day. As teachers, we recognize that some questions promote deeper mathematical thinking than others (for more information about levels of questions, see Martens 1999, Rowan and Robles 1998, and Vacc 1993). For example, when asking, “Is there another way to represent or explain what you are saying?” students are given the chance to justify their thinking in multiple ways. The question “What did you do next?” focuses only on the procedures that students followed to obtain an answer. Thinking about the questions we ask is important, but equally important is thinking about the patterns of questions that are asked.

To analyze students' geometric thinking, use both formative and summative assessments and move students along the van Hiele model of thought.

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This study investigated changes in 22 secondary school mathematics teachers' self-perceived and observed teaching practices during their involvement in a three-year professional development program. Data were collected over four years and analyzed for changes over time. Sources included a survey given at the beginning and end of the program that addressed the teachers' perception of their practice, a classroom observation instrument, interviews, and participants' written reflections. Both teachers' self-perceived and observed practices were found to move towards the targeted goals, although not necessarily at the same rate. As the teachers' practice approached exemplary, the discrepancy between observed and self-perceptions decreased. When discrepancies did exist, the teachers tended to perceive themselves as being closer to exemplary than was supported by observation.