In addition, he believed (from conversations with colleagues) that posing challenging problems
generated frustration and negative student reactions. He also thought it was inappropriate to “put
students on the spot” by asking for explanations since that could be interpreted as confrontational. He
believed his role was to present rules and solve simple problems and that other activities were likely to
be criticized by faculty and students. Alex was generally disappointed with his teaching, but believed he
just needed to “make clearer presentations” to improve. Since what Alex reported were not the course
supervisor’s intentions, I encouraged him to have students work on problems collaboratively and present
their solutions. We discussed lesson plans and strategies for engaging students. Since researchers have
documented significant difficulties that even strong and experienced teachers have implementing
reform-oriented practices, it seemed unlikely that a relatively inexperienced teacher who was struggling
as much as Alex would succeed in transforming his practices in a short period of time.
Second observation: It was, therefore, quite surprising that during my observation several weeks
later, I watched Alex orchestrate dramatically different learning opportunities for his students. The
classroom atmosphere was different, as were the ways Alex engaged students. The topic was volume
and surface areas of cylinders and cones. Instead of presenting rules and examples, Alex began by
discussing shape characteristics and asking for definitions of various figures. Next, he showed paper
models and then derived formulas with ideas volunteered by students. For the remaining hour, students
worked collaboratively on problems, presented solutions, and engaged in whole-class discussions.
Problems involved finding and comparing areas and volumes. All problems entailed use of formulas
discussed earlier, but each also required reasoning about shapes both before and after using the formula.
Alex’s learning experiences: From our discussions, I discovered that Alex’s teaching practices
during my first observation were not ones he had experienced as a learner. He had active learning
environments throughout his mathematics schooling and believed he and others had learned well from
working on challenging problems together, discussing and presenting their thinking. He believed such
environments were “more like how learning math really happens” and that the traditional practices he
had tried seemed straightforward, but said he did not actually understand their role in students’ learning.
Conclusions
Findings suggest that ability to “teach as one was taught” is not limited to those who experienced
traditional instruction. Reform-oriented experiences and beliefs appear to be extremely resilient–
resilient enough to withstand the influences of several years of teaching in a context that promoted
traditional instruction. This resilience is also evidenced by how easily Alex transformed his unsuccessful
traditional practices into the more reform-oriented ones with which he was familiar. These findings are
encouraging: if the mathematics education community succeeds in helping current teachers develop
reform-oriented practices, their students may be powerful levers to successful implementation of reform.
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Convention