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Abstract The complexity of mathematics teaching is especially evident in lessons where teachers build on students’ genuine ideas, such as problem-based lessons. To enhance teachers’ capacity for rich discussions in problem-based instruction, we have developed a unique approximation of practice: digital asynchronous simulations where teachers make subject-specific decisions for a virtual teacher avatar. The simulations are based on materials and principles from a practice-based professional development (PD) program, implemented with small groups of teachers. The self-paced simulation model offers flexibility and scalability, allowing more teachers to participate on their own schedules, but it lacks key affordances of collaborative PD. To examine how to leverage the affordances of collaborative, practice-based PD, this paper uses a design-based research approach to explicate the mechanisms in which digital simulations can support mathematics teachers’ learning about problem-based lessons. We focus on two cycles of design, implementation, analysis, and revisions of the simulation model, drawing on data from focus groups with mathematics teacher educators, prospective teachers’ performance, and teachers’ reflective assignments. The analysis illustrates how two design principles –Authenticity to the teacher’s work, andNuanced feedback– were transformed to better reflect aspects of practice-based teacher learning. We argue that self-paced, asynchronous simulations with indirect feedback can effectively emulate aspects of collaborative, practice-based PD in supporting teachers’ growth. The paper also contributes to the literature on mathematics teachers’ noticing and decision-making, examining how the two interact in simulated environments. We suggest implications for designing practice-based asynchronous digital simulations, drawing on emerging technologies. 

This paper examines the learning process of facilitators engaging in a practice-based professional development (PD) called StoryCircles facilitator training. We observed the language used by facilitators when engaging in a demonstration1) of practice in which another facilitator leads a group of secondary mathematics teachers engaged in StoryCircles PD to consensus. Subsequently, facilitators simulate the role of teachers in StoryCircles PD to anticipate challenges they, as facilitators, may face. By analyzing facilitators' collective dialogue across the different practice-based approaches within the StoryCircles facilitator training, we highlight the adoption of others' roles within the PD context and facilitators' understanding of their own roles as a result. Findings indicate that the demonstration of a facilitator's practice was a context in which facilitators developed an understanding of their own responsibilities. Meanwhile, simulating the participants' practices allowed facilitators to better understand the role of the facilitator from the demonstration of practice, as they navigated the potential tensions arising from teachers adopting practices that facilitators deemed less desirable. This paper highlights how such situated practice-based learning experiences may be associated with facilitators' perceptions of their roles within the StoryCircles framework, offering insights into the intricacies of PD facilitator training and practice. 

We illustrate how concepts from systemic functional linguistics are adapted for the analysis of multimodal representations of practice used in activities where teachers and teacher educators transact meanings about practice. We focus on the transactive register used to project practice meanings to the audience of these representations. We showcase the systems called visibility (how much of the classroom experience happening is made visible to the viewer), temporality (how sequence and duration of events are represented), and theme (how semiotic resources maintain and develop themes). We apply these systems to examine the differences between two storyboards of algebra lessons that were used in a professional development context and the different kinds of reactions teachers offered to the different storyboards. 

This paper contributes to understanding the work of teaching the university geometry courses that are taken by prospective secondary teachers. We ask what are the tensions that instructors need to manage as they plan and teach these courses. And we use these tensions to argue that mathematics instruction in geometry courses for secondary teachers includes complexities that go beyond those of other undergraduate mathematics courses–an argument that possibly applies to other mathematics courses for teachers. Building on the notion that the work of teaching involves managing tensions, and relying on interviews of 32 instructors, we characterize 5 tensions (content, experiences, students, instructor, and institutions) that instructors of geometry for teachers manage in their work. We interpret these tensions as emerging from a dialectic between two normative understandings of instruction in these courses, using the instructional triangle to represent these. 

A central goal of lesson-centered professional development programs (PD) for mathematics teachers is to learn by constructing an artifact, for example, by designing and improving a lesson plan together. That leads to the questions, what does it mean, for mathematics teachers, to improve a lesson? And how can improvements be accounted for in the analysis of the resulting artifacts, especially when these are multimodal? This paper lays the groundwork for answering such questions by drawing on empirical data from a lesson-centered PD program for secondary geometry teachers. We show how semiotic choices were made to convey that the teacher would need to support students when geometry instruction moves from a construction task to a proof, by (1) addressing students’ confusion; and (2) creating a shared language to discuss diagrams. We relate these findings to teachers’ professional growth and the conference theme. 

This research article contributes to the growing literature highlighting the potential for innovation in mathematics education through design cycles that involve creative risk-taking and failure-based learning. Specifically, we explore how “failed” cycles of StoryCircles—a practice-based professional development approach that centers on teacher collaboration—have been productive in fostering innovations within the program. Our focus is on the challenges that arose in our efforts to enable feedback mechanisms within the StoryCircles system that support teachers’ interrogation of their own instructional practice, as they collaboratively develop lessons and expand their collective knowledge base for teaching mathematics. Through examples of three challenges, we illustrate how various lesson artifacts, including those constructed by teachers in anticipation of implementation and those extracted from actual implementations, failed to serve as the sole source of feedback for supporting teachers’ growth. 

To leverage pedagogies of approximation to improve teacher education, educational researchers must first tackle the methodological question, “How can we determine the potential of pedagogies of approximation for supporting the growth of prospective teachers’ knowledge and practices for teaching?” In this paper, we offer a model called the Dual Action Cycles of Approximations of Practice for identifying the pedagogical practices made available to pre-service teachers within various approximations of practice and describe its use to empirically investigate the potential of StoryCircles—a facilitated process of collaboratively representing a lesson using a multimedia storyboarding tool. We illustrate ways the model was used to make observations of pre-service teachers during StoryCircles. A key feature of the model is that it provides a bi-focal perspective on both the preactive and interactive phases of teaching, which help bring into focus the interdependent nature of these two phases. We close by suggesting that the development of new forms of approximation needs to be accompanied by research frameworks capable of investigating the potential of these various innovations. 

In this paper, we examine the relationships between teachers’ subject matter preparation and experience in teaching and their performance on an instrument measuring mathematical knowledge for teaching Algebra 1. We administered the same instrument to two different samples of teachers−high school practicing teachers and community college faculty − who teach the same algebra content in different levels of institutions, and we compared the performance of the two different samples and the relationships between the measured knowledge and their educational and teaching background across the samples. The comparison suggested that the community college faculty possess a higher level of mathematical knowledge for teaching Algebra 1 than high school teachers. The subsequent analyses using the Multiple Indicator Multiple Causes (MIMIC) models based on our hypothesis on the factors contributing to the differences in the knowledge between the two teacher samples suggest that experience teaching advanced algebra courses has positive effects on the mathematical knowledge for teaching Algebra 1 in both groups. Highlighting the positive effect of algebra-based teaching experience on test performance, we discuss the implications of the impact of subject specific experience in teaching on teachers’ mathematical content knowledge for teaching.