Gil Schwarts Lab

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 teachers work, and Nuanced 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.

Critical elements of the expertise of mathematics teacher educators (MTE) can be identified in the artifacts they design for working with prospective teachers (PT), specifically for engaging PT in the double role of practitioners and students of practice. While MTE are increasingly utilizing designed multimodal representations of practice (such as storyboards), theoretical frameworks and methods for analyzing these pedagogical artifacts and the meanings they support are still in early development. We utilize a semiotic framework, expanding systemic functional linguistics to encompass non-linguistic elements, to identify aspects of what we call navigational expertisewhich supports PTs in engaging both as practitioners and students of the practice. We view this expertise as tacit knowledge-in-use, enacted through artifact production. The paper demonstrates MTEs navigational expertise by showing how MTE design storyboards of practice to engage PT as fellow teachers experiencing the lesson taught by a peer and as university students who consider the artifact (including the represented lesson) as an object of study. The framework allows us to identify how MTEs navigational expertise lies in seamlessly interweaving these two purposes and navigating between them, knowing when, how, and for what purposes to invoke each. Through analysis of storyboards from four MTE in diverse US settings, we uncover some of the nuanced tacit components of navigational expertise, highlighting linguistic and non-linguistic design choices and their potential meanings for transactions with PT.

Mathematically Responsive Teaching (MRT) is widely recognized as a cornerstone of high-quality instruction. However, its common definition is narrowly focused on specific teacher moves (e.g., eliciting student thinking). This theoretical paper redefines MRT to consider student experiences and needs alongside their mathematical ideas. Methodologically, it advocates moving beyond teaching moves to recognize a responsive stance toward students and emphasize authentic relationships with them, rooted in Buberian principles. The identification of an MRT stance is demonstrated through an analysis of a 10th-grade low-track statistics lesson.