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In this paper we examine how facilitators' prior experiences as mathematics teachers frame their work when facilitating a practice-based professional development (PD) for the first time. We focus on the experiences of a novice facilitator of StoryCircles, a professional learning process in which teachers collectively script and visualize a problem-based lesson, arguing about their rationales for different decisions connected to discussions of students' work. We situate the challenges the facilitator encountered by identifying the expectations that were not met while she was facilitating and offer possible connections to the PD design. We close by suggesting a perspective to account for facilitation challenges. [For the complete proceedings, see ED630210.]

A historical review of mathematics textbooks suggests a canonical method to solving equations that teachers often see as "the" way to solve equations. In this paper, we examine data from a nationally-distributed sample of 524 secondary mathematics teachers who responded to scenario-based survey items that represent the instructional situation of solving equations. The items featured scenarios in which students presented non-canonical solution methods and asked participants to share how they would respond. Using a framework that draws on systemic functional linguistics, we describe the linguistic resources teachers used. While closed moves are frequently used to avoid discussion of non-canonical solutions, our results suggest that teachers find ways to make regular use of: (1) closed moves for accommodating non-canonical solutions; and (2) open moves when steering the conversation back to the canonical method. [For the complete proceedings, see ED630060.]

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 problem from their perspective. We use these interview data to describe the system in more detail and to identify six major factors contributing to the high school geometry instructional capacity problem. [For the complete proceedings, see ED629884.]

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 school mathematics teacher, using a novel task. In analyzing the data, we noticed tensions between students' understanding of the context and of inverse function as treated in curricula. In this paper, we illustrate these tensions and describe potential implications for students' productive construction of the meanings of inverse function. [For the complete proceedings, see ED629884.]

This paper reports on an ongoing project aimed at developing an inter-institutional system of professional support for the improvement of the Geometry for Teachers (GeT) courses that mathematics departments teach to preservice secondary teachers. In alignment with the literature on improvement science (see Bryk et al., 2015; Lewis, 2015), it is essential to develop and deploy practical measurement tools to inform improvement. We describe three key forms of measurement our team has been using to drive this work as well as some preliminary findings. [For the complete proceedings, see ED606556.]

Using mixed-effects regression, we analyzed teachers' responses to a multimedia survey of instructional practices in posing proof problems in geometry. Teachers described and rated for appropriateness three different ways of involving students in deciding what to prove, including one in which the teacher chooses the givens and the conclusion to prove, and two others that expand the students' role in different degrees. While teachers recognized the former as normative, their ratings identified an alternative as more appropriate, having more positive value, and less negative value than the normative one. This alternative has students propose the givens or the conclusion to prove, and it allows the teacher to control the complexity of instruction by endorsing one proposal before students are to write the proof. [For the complete proceedings, see ED606531.]

This paper reports on a project aimed at developing a system of professional support for the improvement of the Geometry for Teachers course that mathematics departments teach to preservice secondary teachers. We share data from interviews with 20 instructors to report on how they perceive their position of geometry instructors and the work they do in the course. To inspect this set of interviews, we use the framework of professional obligations to the discipline, to individual students, to the institution, and to the classroom community. We share how references to these professional obligations emerged in the interview data. [For the complete proceedings, see ED606531.]

We discuss affordances and liabilities of using a storyboard to depict a written case of a teacher's dilemma that involves race, opportunity to learn, and student community. We rely on reflections by the teacher educator who authored the written case and later depicted it as a storyboard to use it with his preservice teachers (PSTs). The analysis involved, first, organizing the signifiers in each of the two representations of practice into what we call concentric spheres of stratification, and secondly, contrasting the various meanings attributed to signifiers by both the author and his PSTs. We suggest that the resources of storyboard allow for more inquiry and alternative narratives than is available from the single modality of text in the written case. [For complete proceedings, see ED581294.]

This paper is part of a three-year inquiry that supports and investigates the work of groups of mathematics teacher educators using technological tools to design and implement multimedia practice-based teacher education curriculum materials. This paper describes the kinds of activities, interactions, and tools used by mathematics teacher educators to engage in such work. Using Engeström's Activity Theory as a framework, we organize our observations of the groups' work sessions, noting differences across the groups' objectives and ways of organizing the division of labor and tools for engaging in the work. Our results suggest the activity of collective curriculum development amongst teacher educators can take on at least three distinct types of interactions. We present these types of interactions as "caricatures" (Lambdin & Preston, 1995), using data from all of the groups to represent composite descriptions. [For complete proceedings, see ED581294.]

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 with activities where students use representations of figures to model their experiences with shape and space and I show how our basic research on high school geometry instruction informs the implementing and monitoring of such modeling perspective. I argue that for mathematics education research on instruction to contribute to improvements that teachers can use in their daily work our theories of teaching need to be mathematics-specific.

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 teachers' expectations of this work context. The results of our analysis of the data (using z-tests and mixed effect regression models) provide evidence that, in the situation of GCA, (1) teachers appear not to take issue with giving students tasks that require them to set-up and solve equations whose solutions have no geometric meaning (e.g., the length of a side of the figure is zero), and (2) teachers do not appear to expect students to document the geometric theorem or property that justify the setup of those equations (highlighting the contrast between the situation GCA and that of doing proofs). [For the complete proceedings, see ED583989.]

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 have used classical test theory and item response theory to select items for the instruments and we provide information on their reliability, using a sample of 88 high school mathematics teachers. [For the complete proceedings, see ED597799.]

We report on the development and piloting of a method that provides a sufficient condition for confirming that an observed regularity in a classroom is a norm. The method we describe is a refinement of the breaching experiment technique (Garfinkel, 1963; Mehan, 1979) that uses random assignment to experimental conditions as a means to facilitate controlled comparisons between participants' reactions to different episodes of instruction. We use this method to confirm the existence of normative ways that teachers scrutinize the details of proofs in geometry. [For the complete proceedings, see ED597799.]

We describe the development of measures of teachers' recognition of an instructional norm--that proof problems in high school geometry are presented in a diagrammatic register. A first instrument required participants to openly respond to depictions of classroom scenarios in which the norm was breached. A second instrument was a survey that required participants to rate the extent to which they agreed with various explicit statements about instruction. A third instrument capitalized on pros of the other two. We demonstrate how this instrument development process improved our conceptualization of the components included in the diagrammatic register norm. [For the complete proceedings, see ED584443.]