UC San Diego

Though it is widely acknowledged that teachers' knowledge of mathematics is a cornerstone on which their instructional practices are based, little research exists documenting the impact of changes in teachers' mathematical content knowledge on their teaching practices. As proving is a central activity in the study of mathematics, a teacher's proof schemes (in the sense of Harel and Sowder, 1998) enable and constrain instructional approaches. For professional developers hoping to better understand the impact of teachers' proof schemes on their instructional practices, examinations of specific cases, with special attention to the nature and mechanisms of change in proof schemes and teaching practice, provide insight into the organization of effective professional development (PD) for teachers in the domain of proving. The case study reported here examines the development of proof schemes and teaching practices of one in-service secondary mathematics teacher who participated in an off- site PD for two years. Two data sources were examined: video of the participant doing mathematics at the PD and footage of her own teaching. The analysis of proof schemes focuses on proof production during the PD. Development of the teacher's practice was also investigated during the two academic years following each summer. The study includes theoretical connections (using Harel's DNR Theoretical Framework) between developments in the teacher's proof schemes and teaching practices. Specifically, this study asked : (1) What changes were observed in one participant's proving and proof schemes as she participated in the PD? (2) What connections can be found between her experiences at the PD (including changes in her proof schemes) and the evolution of her teaching practices in a whole class setting? It was found that she became increasingly able to identify pivotal statements in her own proofs that had previously been left unattended. The participant showed evidence of a transition from empirical to deductive proof schemes. The greatest developments in teaching practices were observed in the practices of handling students' solutions by encouraging student-to-student talk, asking students to prove conjectures, soliciting alternative solutions in the presence of correct solutions, and attending to mathematical detail in correct solutions