This paper reports an ongoing effort to address the problem of instructional capacity for high school geometry from a systems improvement perspective. In an effort to understand the system that contains the high school geometry instructional capacity problem, we identified key stakeholders and conducted preliminary interviews to learn about the…

Researchers have identified students' difficulties reasoning about inverse functions. Through our review of this literature, three meanings stand out: a formal, 'undoing', and quantitative meaning. Using these meanings as a guide, we analyzed student work collected from a lesson on the topic of inverse functions taught by an experienced high…

We show how we used a national distribution of responses from 416 practicing teachers to items of a test of mathematical knowledge for teaching geometry to estimate changes in knowledge by a group of 11 practicing teachers who participated in a 2-year practice-based professional development programme. To draw a reliable interpretation of the…

How can basic research on mathematics instruction contribute to instructional improvement? In our research on the practical rationality of geometry teaching we describe existing instruction and examine how existing instruction responds to perturbations. In this talk I consider the proposal that geometry instruction could be improved by infusing it…

This study presents an analysis of whether geometry teachers create opportunities for student discussion when engaging students in proving as revealed through the use of a multimedia survey instrument. We presented 42 secondary mathematics teachers with 8 multimedia narratives set in the situation of doing proofs in high school geometry and asked…

In this paper, we argue that to prepare pre-service teachers for doing complex work of teaching like leading classroom mathematics discussions requires an implementation of different pedagogies of teacher education in deliberate ways. In supporting our argument, we use two frameworks: one curricular and one pedagogical. The curricular framework is…

This paper reports on a study of the instructional situation in high school Geometry that Hsu (2010) called Geometric Calculation in Algebra (GCA). In particular, we conducted a virtual breaching experiment in order to examine the extent to which high school teachers recognized breaches of two norms that we conjectured to describe geometry…

This study compared conversations among groups of teachers of high school geometry that had been elicited by a representation of instruction (either a video or an animation) and facilitated with an open-ended agenda. All artifacts used represented instruction scenarios that departed from what, according to prior work, had been hypothesized as…

This paper builds on Grossman's notion of approximations of practice as scaled-down opportunities for preservice teachers to learn to teach by doing. The authors propose the use of media rich, collaborative web-authoring tools for preservice teachers to create, complete, or edit scenarios in which they practice particular activities of teaching,…

We describe the conceptualization, development, and piloting of two instruments--a survey and a scenario-based assessment--designed to assess, teachers' recognition of an obligation to the discipline of mathematics and the extent to which teachers justify actions that deviate from what is normative on account of this obligation. We show how we…

We inspect the hypothesis that geometry students may be oriented toward how they expect that the teacher will evaluate them as students or otherwise oriented to how they expect that their work will give them opportunities to do mathematics. The results reported here are based on a mixed-methods analysis of twenty-two interviews with high school…

This study examined High School Geometry standards from recent individual states' and the Common Core Standards (CCSSM) featuring reasoning and proof. We examined specificity of content and use of proof or reasoning as a process in the language of standards. Results indicated standards with proof as its primary process are more likely to have…

This paper documents efforts to develop an instrument to measure mathematical knowledge for teaching high school geometry (MKT-G). We report on the process of developing and piloting questions that purported to measure various domains of MKT-G. Scores on the final set of items had no statistical relationship with total years of experience…

This article reports on an investigation of how teachers of geometry perceived an episode of instruction presented to them as a case of engaging students in proving. Confirming what was hypothesized, participants found it remarkable that a teacher would allow a student to make an assumption while proving. But they perceived this episode in various…

This article discusses how a teacher can prepare the terrain for students to understand what it means to define a figure. Drawing on writings from mathematicians and mathematics educators on the role of definitions in mathematics, the authors argue that students develop a greater appreciation for the conciseness of a mathematical definition if…