| Geometry and measurement represent topics of great significance in mathematics; however, efforts to teach this content in the middle grades have been formulaic, with students memorizing formulas and definitions without conceptual understanding. Moreover, students and teachers demonstrate gaps and misconceptions in their knowledge of geometry and measurement, particularly with respect to relationships between measurable quantities of geometric figures and proof. This study investigated changes in knowledge needed for teaching geometry and measurement through engagement in a practice-based course for preservice and practicing teachers.; Pre- and post-course measures showed significant teacher growth along all three aspects of knowledge needed for teaching. Teachers grew in their ability to attack non-routine problems relating dimension, perimeter, and area and dimension, surface area, and volume; and in their use of multiple solution methods, multiple representations, and production of mathematically sophisticated solutions. Teachers also grew in content knowledge for teaching, becoming more representationally fluent and increasingly able to modify tasks to target key geometry ideas and about the affordances of different formulas for area and volume, and in knowledge of proof, including identification of the key aspects of the definition of proof, the role of proof in the classroom, and creation of proofs and proof-like arguments.; Teachers grew in knowledge of mathematics for student learning as conceptualized by the five practices for productive use of student thinking: anticipating student solutions to a mathematical task, the use of high-level questions to assess and advance student thinking, selecting and sequencing student work to share, and connecting that work in ways that targeted the big mathematical ideas. Teachers also grew in their identification of routines, an example of practices that support teaching. Qualitative analysis of the course tied these results to opportunities to learn in the course.; The results suggest that teachers can grow in their knowledge of content and pedagogy through practice-based teacher education experiences. The results suggest a value for focusing methods courses on particular slices of mathematical content. The design principles articulated in the analysis predicted teacher learning, and generalize to the design of teacher education experiences that enhance knowledge needed for teaching mathematics. |
