Publications

The goal of this chapter is to showcase and characterize the diversity of interactions between policy and implementation of digital resources for teaching and learning mathematics. Based on a review of the professional literature on educational policy and implementation, we put forward a set of conceptual distinctions, including the following: types of policy (e.g., policy as governing text vs. policy as negotiation of power), typology of objects of implementation (e.g., material-centered vs. material-interactive), and roles of stakeholders (e.g., original vs. secondary proponents of an innovation, government-affiliated policymakers vs. researchers and teachers as policymakers). These distinctions are first illustrated by means of several past studies on mathematics education digital resources and then are systematically put to use in the context of three ongoing R&D projects from Greece, Israel, and the USA. These projects, as different as they are in respect to their goals, operation, and policies involved, are comparable with respect to their digital components, namely digital platforms containing collections of digitalized resources for teaching and professional learning of mathematics teachers. The chapter concludes with remarks about the emerging patterns of policy implementation relationships and suggestions for future research.

Professors in proof-based mathematics courses often intend that the feedback they provide on students' flawed proofs will promote proof comprehension. In this theoretical article, we investigate how such feedback can be formulated. Drawing on Lakatos's process of proof and refutation, we propose the notion of heuristic refutation feedback for feedback on a flawed proof that contains a mathematical argument, possibly incomplete, that logically implies that the student's proof is invalid. Such feedback is heuristic in the sense that interpreting and utilizing it invites mathematical reasoning that can contribute to development of proof comprehension. We build on Toulmin's model of argumentation to analyze possible variations in the formulation of feedback. Based on data from a Real-Analysis course, we highlight two key decisions entailed in formulating refutation feedback: deciding what flawed (possibly implicit) claim in the student's proof to refute, and deciding how explicitly to present the various elements of the refutation argument. We exemplify these decisions in two particular types of refutation feedback that we have identified: refutation by counter-example and refutation by false implication. We show how different formulations of refutation feedback may afford different opportunities for student-engagement with particular facets of proof comprehension. Our findings suggest how professors can purposefully tailor feedback in order to achieve particular didactic goals.

The importance of mathematical problem solving has long been recognized, yet its implementation in classrooms remains a challenge. In this paper we put forth the notion of problem-solving implementation chain as a dynamic sequence of intended, planned, enacted and experienced activity, shaped by researchers, teachers and students, where the nature of the activity and its aims may change at the links of the chain. We propose this notion as an analytical framework for investigating implementation of problem-solving resources. We then illustrate this framework by a series of narratives from a project, in which the team of task-designers develops problem-solving resources aimed at reaching middle-school students via their teachers, who encounter these resources in professional development communities. We show how the problem-solving activity evolves along the implementation chain and then identify opportunities for mutual learning that emerge from tensions in perspectives on PS held by the different parties involved.

In their enactment of the curriculum, teachers have a substantial role as instructional designers. Accordingly, any evaluation of the progression of students learning should first be concerned with the pedagogical intentions of the teacher. In this article we present a method for reconstructing teachers implicit and tacit considerations in their selection, sequencing and enactment of tasks. Two 11^th grade teachers tagged all of the tasks that comprised a 5-week learning progression. Tagging consisted of assigning values to prescribed categories of metadata. Visual representations of the metadata revealed patterns in the tagged progressions, and allowed the teachers to reflect upon these patterns. Both teachers, though guided by very different didactical considerations, validated that many of their explicit and implicit intentions were revealed in the representations of the progressions. Furthermore, both teachers had the opportunity to reflect on tacit aspects of their instructional design that they were not previously aware of.

With the emergence of e-textbooks, along with expectations to integrate technology in instruction, teachers are becoming significant co-designers of the curriculum. Informed selection and sequencing of learning resources requires sensitivity to didactic nuance, and tools to support development and application of such sensitivity. Teachers practices are constrained by the aspects of resources that are \u201csearchable,\u201d which are limited using standard search engines, and the selection of tasks is influenced by the engines ranking of search results, reflecting their popularity. We are developing and researching a coupled pair of tools to support mathematics teachers in making informed curricular decisionsa tool for tagging learning resources with prescribed categories of didactic metadata and a dashboard for browsing collections of resources according to this tagged metadata. In this article, we investigate affordances of these tools for the professional development of mathematics teachersboth practicing and pre-service teacher candidates. Viewing the dashboard, along with the metadata that it encodes, as a boundary object between the teachers and the researchers perspectives on curricular design, we show how teachers learned through acts of boundary crossing, conceived as transitions and interactions between the two communities curricular discourses. We show how using the dashboard in a task-selection assignment encouraged teachers to reflect on their practicemaking explicit the tacit considerations that they apply to curricular decisions and articulating them from the researchers perspective. We also describe the emergence of \u201chybrid\u201d search strategies, integrating multiple perspectives to create practices that are both didactically informed and practically relevant for instruction.

Research mathematicians often play a central role in determining educational policies, yet the relevance of their mathematical expertise may be (and indeed often is) questioned. A mathematician who developed a game based on non-Euclidean geometry, and a high school teacher, participated in a group discussion on the game, and were interviewed separately to elicit their perspectives on enriching advanced-track students through inquiry. Though their perspectives appeared in many ways incompatible or incommensurable, we suggest ways in which many of their conflicting concerns can be reconciled. In this we are proposing a model of cooperation among communities, where mathematicians and teachers contribution to mathematics education is mediated by mathematics education researchers.

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.

Jason Cooper is a research fellow at the University of Haifas Faculty of Education and at the Weizmann Institutes Department of Science Teaching. His research concerns various aspects of teacher knowledge, including roles of advanced mathematical knowledge in teaching and contributions of research mathematicians to the professional development of teachers. He has been a board member of the European Society for Research in Mathematics Education (ERME) since 2015. Abraham Arcavi is the incumbent of the Lester B. Pearson Professorial Chair at the Department of Science Teaching of the Weizmann Institute of Science, Israel. He works on the teaching and learning of mathematics, curriculum development and professional development of mathematics teachers. At present (2016-2020), he is the secretary general of the International Commission on Mathematical Instruction (ICMI).

In this paper we investigate task design in an unusual professional development course for elementary school teachers, conceived and taught by research mathematicians. Prior analysis singled out relevance for teaching as a critical design issue for engaging teachers in effective learning. The aim of the current research is to uncover how relevance for teaching was achieved without compromising mathematical rigor and depth. Findings are based on an analysis of three representative cases of task designing in which the authors where involved one as instructor and task designer, the other as participant observer. Our analysis reveals a designing model that first addresses purely mathematical concerns and then refines tasks, taking into consideration a series of constraints imposed by the requirement of relevance for teaching. Using Schoenfelds Resources-Orientations-Goals framework for decision-making, we show how the mathematicians drew on their special knowledge of mathematical content to achieve such relevance in ingenious ways. We find that tasks were best aligned with Knowless principles of Adult Learning in cases where the designers appropriated the teachers point of view, no longer seeing the need for relevance as a constraining imposition, but rather as an opportunity to combine and merge knowledge specific for teaching and purely mathematical knowledge.

The TWG14, University Mathematics Education (UME), was launched in CERME7 (Nardi, González-Martín, Gueudet, Iannone, & Winsløw, 2011) showcasing the fast growth of research in UME and also the specificity of the research context. In particular, the abstract, formal nature of a significant portion of the mathematical content; the absence of national curriculum guidelines, and therefore the great variations in organization and practices across institutions; the general lack of professional development or of systematic preparation for teaching; and, the volume of content to learn in a short period of time, and the degree of autonomy expected from the students and faculty.