Impact of dualistic rhetoric, lesson planning, and analyzing student work on elementary teachers' conceptions of mathematical understanding

Teaching for conceptual understanding is an enduring challenge for mathematics education reform. Though widely valued in policy documents, it remains a largely unachieved goal as the robustness of traditional practices attests. In the district where this study was conducted, pervasive dualistic rhetoric expressed as skills acquisition versus problem solving appeared in policies, textbooks, and mandatory testing. All contributed to a complex matrix of issues that impede improvement of instructional practices by inadvertently posing antagonistic views of the mathematics to be taught, leaving the teachers to resolve the dualism. The district offered little assistance in helping teachers grasp the conceptual underpinnings of the mathematics they taught. As a result, for the 4 elementary teachers in this study, conceptual understanding remained problematic as they tried to make sense of their district's contradictory policies.; Within this context this case study describes the teachers' evolving conceptions of mathematical understanding as they participated in a professional development effort. With a focus on teaching for conceptual understanding, this effort took the form of weekly teacher discussion groups. Thirty sessions conducted over 11 months engaged these teachers in their principle professional activities: planning lessons, teaching lessons, and assessing student work products.; The effect of the district's dualistic rhetoric was a bifurcation of mathematical content that manifested in teachers' talk and during this study's interventions. Initially, what constituted mathematical understanding was procedural understanding, though the teachers thought they were expressing what conceptual understanding means. Analyses of their efforts to plan and teach a fractions lesson revealed both traditional and reformed instruction, but in bifurcated fashion. While analyzing their students' work on an open-ended problem, the teachers struggled to integrate its procedural and conceptual components. In and through these practical activities, teachers began to construct richer notions of mathematical understanding, making strides toward integrating procedural and conceptual aspects of the activities they engaged. By the end of the study, teachers no longer expressed understanding as steps or procedures. The results suggest that teacher educators need to recognize how the bifurcation of mathematical understanding manifests in teachers' professional culture and instructional activities and how it can be resolved.