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Facilitators of professional development (PD) for mathematics teachers currently gain increasing attention, as their practices are crucial for the success of spreading mathematics educational ideas and innovations into schools and strengthening the professional expertise of teachers. So far, mainly two components of facilitation have been empirically researched: facilitators' knowledge resources and facilitation moves. In this paper, we propose to develop a more comprehensive conceptual framework in order to explain facilitators' practices and underlying decisions. For this purpose, we lift Schoenfeld's ROG framework for teachers' decision-making (comprising resources, orientations, and goals) to the facilitator level, and add identity as a fourth component to obtain what we call the ROGI framework. We draw on data collected simultaneously in two PD projects for mathematics teachers in Germany and in Israel. Through three cases of facilitation, we illustrate how the adapted ROGI framework helps to understand processes of decision-making made by facilitators during mathematics professional development sessions.

Professional development (PD) courses are the main context for mathematics teachers' lifelong learning. Leaders with expertise are needed to facilitate these courses; thus, there is a growing interest in understanding the nature of this profession, its core practices, and the challenges it entails. This paper focuses on a specific group of facilitators: experienced mathematics teachers who have just begun facilitating PD courses in addition to their classroom teaching. To better understand these novice facilitators' practices, a commognitive approach was implemented to examine how they draw on their mathematics teaching experience when leading PD courses. In commognitive terms, these practitioners draw on multiple professional discourses, but mostly on the discourses of teaching and facilitation. The analysis of the challenges and affordances associated with participating in these two professional discourses showed that novice facilitators bring into play their teaching practices in four distinct ways: enacting a familiar practice, negating a familiar practice, questioning the relevance of a familiar practice, and generating a new practice based on their teaching experience. We claim that novice facilitators' well-established identity as teachers is both a challenge and an asset in grounding successful facilitation practices. Overall, facilitators modify their teaching experience through the adoption, adaptation, and retraction of their teaching practices. Implications for the preparation and support of facilitators, within processes of upscaling PD programs, are discussed.

In this paper I explore the question of what counts as successful implementation of a mathematics professional development (PD) project at scale. Two kinds of upscale are presented and differentiated from one another: upscale of settings and upscale of values, and the link between this differentiation and how success may be defined is discussed. As part of the discussion, three models of taking a PD project to scale are described, namely replication, adaptation and reproduction, and these are tied to the degree of professional autonomy of teachers within the PD. The examination of these ideas draws on data from two PD projects for secondary mathematics teachers in Israel, that are now years after their pilot phase and still being implemented. The first project is aimed at supporting mathematics teachers in teaching mathematically at-risk students and in helping these students to succeed in high-stakes exams. The second project is aimed at enhancing mathematics teachers' reflection on their teaching practices, using classroom video as a tool. Both projects were implemented at scale across Israel, and reviewing results from these implementations enables some general insights to be gained regarding the success of scaled-up PD projects. The crucial role of PD facilitators in successful implementations is highlighted.

This paper reports on a study conducted within VIDEO-LM (Viewing, Investigating and Discussing Environments of Learning Mathematics), a video-based professional development project for secondary mathematics teachers that aims to enhance reflection on practice. The study explored VIDEO-LM sessions where Israeli teachers watched an 8th grade geometry lesson from Japan. We analyzed teachers' discussions around this lesson and characterized main themes and reflection types. Findings suggest that watching the Japanese lesson, using the VIDEO-LM framework, led to productive discussions among teachers. The fact that the videotaped lesson was culturally "far" from the teachers' reality provoked them to re-see everyday aspects of their practice, revealing some dissonances which promoted reflective thoughts and actions.

While proof is often presented to mathematics undergraduates as a well-defined mathematical object, the proofs students encounter in different pedagogical contexts may bear salient differences. In this paper we draw on the work of Dawkins and Weber (2017) to explore variations in norms and values underlying a proof across different pedagogical contexts, including the lecture, the textbook, home assignments and received feedback. We conclude that shifts in norms of proof within instruction and assessment, and the consequent difficulties in student proof learning, may be the result of an inherent tension between mathematicians' values and pedagogical goals.

Is it possible that a meeting of mathematicians and primary school teachers will be productive? This question became intriguing when one professor of mathematics initiated a professional development course for practicing primary school teachers, which he taught alongside a group of mathematics Ph.D. students. This report scrutinizes the uncommon meeting of these two communities, who have very different perspectives on mathematics and its teaching. The instructors had no experience in primary school teaching, and their professed goal was to deepen the teachers' understanding of the mathematics they teach, while teachers were expecting the course to be pedagogically relevant for their teaching. Surprisingly, despite this mismatch in expectations, the course was considered a success by teachers and instructors alike. In our study, we analyzed a lesson on division with remainder for teachers of grades 3-6, taught by the professor. The framework used for the data analysis was "mathematical discourse for teaching," a discursive adaptation of the well-known mathematical knowledge for teaching framework. Our analysis focuses on the nature of the interactions between the parties and the learning opportunities they afforded. We show how different concerns, which might have hindered communication, in fact fueled discussions, leading to understandings of the topic and its teaching that were new to all the parties involved. The findings point to a feasible model for professional development where mathematicians may contribute to the education of practicing teachers, while they are gaining new insights themselves.

The VIDEO-LM project (Viewing, Investigating and Discussing Environments of Learning Mathematics) is aimed at enhancing mathematics teachers' reflection on their professional practice, through watching and discussing videotaped lessons of unknown teachers. The discussions are guided by the use of an analytic framework, comprised of six detailed "viewing lenses." In this article, we first provide a brief historical context of practicing teacher education, followed by the description of the framework we developed. We then present selected data from 17 implementation sites that illustrate the impact of VIDEO-LM courses on the professional growth of participants. Findings show that using the six-lens framework in video-based peer conversations supports the development of a reflective language, with which mathematics teachers engage in deep discussions about core issues of their profession.

Teaching mathematically gifted students is a challenging pursuit, especially at the secondary level. However, characterizing teachers who can successfully meet this challenge has so far been scarcely addressed directly. In this paper, elements of the desired profile of such teachers are extracted from the general literature on mathematical giftedness. The difficulty to recruit and train teachers to satisfy this profile, resulting from the reality of severe shortages of mathematics teachers in many countries, is described. Findings are presented from a study on a unique 2-year training course for teachers of mathematically gifted and advanced students, including analyses of the prerequisites of the course, the syllabus and the profile of graduates. The findings point to a problem where compromises in entry requirements, made in light of shortage in suitable candidates, may limit the number of successful graduates even in comprehensive and carefully designed programs. Implications of these results for decision makers are discussed.

This article discusses the possibility of using nonprofessional tutoring as means for advancing low achievers in secondary school mathematics. In comparison with professional, paraprofessional, and peer tutoring, nonprofessional tutoring may seem less beneficial and, at first glance, inadequate. The described case study shows that nonprofessional tutors may contribute to students' understanding and achievements, and thus, they can serve as an important assisting resource for mathematics teachers, especially in disadvantaged communities. In the study, young adults volunteered to tutor low-achieving students in an urban secondary school. Results showed a considerable mean gain in students' grades. It is suggested that affective factors, as well as the instruction given to tutors by a specialized counselor, have played a major role in maintaining successful tutoring.

This article examines the notion of informal mathematical products, in the specific context of teaching mathematics to low achieving students at the secondary school level. The complex and relative nature of this notion is illustrated and some of its characteristics are suggested. These include the use of ad-hoc strategies, mental calculations, idiosyncratic ideas, everyday rather than mathematical language, non-symbolic explanations, visual justifications and common-sense based reasoning. The main argument raised in the article concerns the challenge of valuing informal mathematical products, created by low achievers, and using them within the mathematics classroom as means for advancing such students. The data draws from several research and design projects conducted in Israel since 1991. Selected examples of students' products, gathered from low-track mathematics classrooms involved in these projects, are presented and analyzed. The analyses highlight various features of such products, and portray the possible gains of teaching approaches that legitimize, and build onwards from, informal products of low achievers.