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This study explores the diversity of university mathematics teachers' perspectives and experiences in relation to the (STT). Data for this study consist of responses of 310 university mathematics teachers from 30 countries to a survey. The survey design and data analysis are informed by an adaptation of a theory of improvement for education and by four idealized approaches for improving STT as identified in the literature: jump-oriented, enculturation-oriented, diversity-oriented, and cross-community-oriented. We characterize the diversity of perspectives and experiences in the data and examine ideas for future projects in light of these four approaches. We conclude that while jump-oriented approaches for improving STT are still the most prevalent, in part due to a persistent perspective of STT as a "deficit," university mathematics teachers' discourse on STT is becoming more diverse and nuanced. We find it particularly promising that participants' suggestions for future action are aligned with the four approaches, and most notably, with the cross-community approach expressing the emergent need for greater communication between the secondary and tertiary mathematics education communities.

This paper explores student emotion and learning experiences fostered by lecturing-style instruction in Real-Analysis problem-centered lessons. We focus on two lessons that were taught by two reputable instructors and involved challenging, mathematically-related problems the students did not understand. Nonetheless, one lesson evoked negative emotional reactions, while the other positive emotional reactions -- a phenomenon we aimed at explaining. The main data comprise the filmed lessons and subsequent stimulated-recall interviews with nine students. The analysis draws on conceptual tools from three perspectives: mathematical discourse, variation theory, and a recently developed construct of "key memorable events" (KMEs) that offers an affective-cognitive lens for investigating the interrelation between teaching and learning. The findings indicate that the positively-perceived lesson contained instances of what we call "heuristic-didactic discourse": a meta-level discourse that presents heuristics monitored from an expert's perspective, yet derived from a student's perspective. Implications for research and practice are drawn.

Inquiry into the teaching and learning of mathematics is a central aspect of the work of both the mathematics teacher and the mathematics education researcher, yet there are profound differences between the practices and processes underlying educational inquiries within each community. These differences are known to hinder implementation of research, yet they can also become a valuable resource. This paper explores a particular model of implementation of research: mathematics teachers adopt and adapt practices and processes of disciplined educational inquiry in a co-learning partnership with mathematics education researchers. We present a narrative inquiry of a teacher-researcher community that designed and studied classroom activities aimed at encouraging students to ask meaningful mathematical questions. The data analysis, informed by the literature on boundary objects and boundary crossing, highlights how the teachers and researchers leveraged their different processes and practices of educational inquiry as a resource in their collaborative inquiry. We conclude by suggesting that informed and thoughtful attention to the differences between teacher inquiry and disciplined inquiry may support and enhance mathematics education research implementation.

Implementation has always been a paramount concern of mathematics education, but only recently has the conceptualizing and theorizing work on implementation as a phenomenon begun in our field. In this survey paper, we conduct a hermeneutic review of mathematics education research identified as related to the implementation problematics. The first cycle of the review is based on examples of studies published in mathematics education journals during the last 40 years. It is organized according to five reasons for developing implementation research. The second cycle concerns 15 papers included in this special issue and is organized by four themes, as follows: objects of implementation, stakeholders in implementation, implementation vs. scaling up, and implementability of mathematics education research. The paper is concluded with a refined glossary of implementation-related terms and suggestions for future research.

The looking-back stage is rarely observed in students' problem solving in spite of its recognized importance. The importance of this stage is attributed to practices of engagement with queries on verification of the obtained solution(s), comparative consideration of alternative solutions, and formulation of implications for future problem solving. We refer to such practices as "looking-back practices." In the present study we explored the hypothesis that the looking-back practices can be evoked in small-group classroom discussions of controversial worked-out solutions to word problems. Such tasks are known as Who-Is-Right tasks. The data consisted of audio- and videotapes of six small groups of high-school students working on a Who-Is-Right task in the context of percentage. The data analysis, informed by a discursively-oriented perspective on problem solving, attended to strategies, dialogical moves and mathematical resources enacted by the students towards attempted agreement as to which of the solutions should be endorsed and why. The findings imply that Who-Is-Right tasks have undeniable potential for supporting collective looking-back practices. In addition, the study contributes to the literature on enactment of mathematical resources in problem-solving discourse and on patterns of students' dialogic participation in small-group problem solving.

The research question studied in this paper is how teachers' task designing can be enhanced in professional development. We show that this can be done by scaffolding. Scaffolding is used in problem solving and we apply it by considering designing a task as a problem. In the paper, we describe a professional development of four teachers led by a mathematics educator. The professional development consisted of 10 meetings. The teachers were first exposed to a well-designed task in geometry as learners and then were asked to design new geometry tasks for their students. The mathematics educator helped them by supplying scaffolds such as modelling, reflection, articulation and worked examples. The data consisted of video-recordings of the workshops and a semi-structured interview with each of the teachers at the end of the project. The data analysis shows how the use of particular scaffolds was helpful while preserving the teachers' autonomy as task designers. [ABSTRACT FROM AUTHOR]

An iterative unpacking strategy consists of sequencing empirically-based theoretical developments so that at each step of theorizing one theory serves as an overarching conceptual framework, in which another theory, either existing or emerging, is embedded in order to elaborate on the chosen element(s) of the overarching theory. The strategy is presented in this paper by means of reflections on how it was used in several empirical studies and by means of a non-example. The article concludes with a discussion of affordances and limitations of the strategy.

Aiming to enhance understanding of visual obstacles inherent in two-dimensional (2-D) sketches used in high school spatial geometry instruction, we propose a measure of visual difficulty based on two attributes of the sketches: potentially misleading geometrical information (PMI) and potentially helpful geometrical information (PHI). The difficulty of 12 normatively oriented cube-related sketches was theoretically ranked according to their ratios, #PHI/#PMI. The ranking was compared to the actual visual difficulty as measured by the percentage of correct or desired comprehension, individual spatial ability, and study-time allocation. This procedure was repeated for unnormatively oriented sketches, obtained by vertically flipping the original sketches. In both cases, the findings substantiate #PHI/#PMI as an a priori measure of visual difficulty. Practical, theoretical, and methodological implications are inspected and discussed.

This study aims to explore a phenomenon of a one-off manifestation of mathematical creativity on the part of a student, against the background of her normative and not especially creative behavior--a flash of creativity. We seek to elaborate on this phenomenon in terms of the 4P (person, press, process and product) model of creativity. Namely, most of the time the student having particular personality traits (person) studies mathematics in a regular to her learning environment (press), in which she manifests problem-solving behaviors (process) and produces solutions (product) that are considered, by the student, her classmates and her mathematics teacher, as normative or "usual" to that student. However, the same individual may at times become unusually and unexpectedly creative when coping with a challenging problem. We first discuss this phenomenon based on an historical example and theoretically, and then illustrate it through a case of a high-school student while focusing on a nexus of intellectual and emotional triggers that led the student to drastically change, for one time only, her normative mathematical behavior. Implications for research and practice are drawn.

This theoretical article explores an issue of developing education research competencies in mathematics teachers through their involvement in mathematics education research. We first argue that the development of education research competencies is beneficial for the teachers' professional growth. We then identify opportunities for mathematics teachers to develop education research competencies through different modes of research-practice partnerships. In the main part of the paper, we present a particular theoretical-organizational framework for large-scope teacher-researcher collaborations in educational research. The framework is called Teacher-Researcher Alliance for Investigating Learning (TRAIL), and consists of a set of theoretically laden premises, design heuristics, and provisional partnerships.

This article describes the following phenomenon: Gifted high school students trained in solving Olympiad-style mathematics problems experienced conflict between their conceptions of "effectiveness" and "elegance" (the EEC). This phenomenon was observed while analyzing clinical task-based interviews that were conducted with three members of the Israeli team participating in the International Mathematics Olympiad. We illustrate how the conflict between the students' conceptions of effectiveness and elegance is reflected in their geometrical problem solving, and analyze didactical and epistemological roots of the phenomenon. (Contains 5 figures and 2 endnotes.)

This article concerns student sense making in the context of algebraic activities. We present a case in which a pair of middle-school students attempts to make sense of a previously obtained by them position formula for a particular numerical sequence. The exploration of the sequence occurred in the context of two-month-long student research project. The data were collected from the students' drafts, audiotaped meetings of the students with the teacher and a follow-up interview. The data analysis was aimed at identification and characterization of the algebraic activities in which the students were engaged and the processes involved in the students' sense-making quest. We found that sense-making process consisted of a sequence of generational and transformational algebraic activities in the overarching context of a global, meta-level activity, long-term problem solving. In this sense-making process, the students: (1) formulated and justified claims; (2) made generalizations, (3) found the mechanisms behind the algebraic objects (i.e., answered why-questions); and (4) established coherence among the explored objects. The findings are summarized as a suggestion for a four component decomposition of algebraic sense making.

This paper presents a part of a larger study, in which we asked "How are learning and teaching of mathematics at high level linked to students' general giftedness?" We consider asking questions, especially student-generated questions, as indicators of quality of instructional interactions. In the part of the study presented in this paper, we explore instructional interactions in two high-school classes for mathematically promising students with specific focus on questions that students ask. The first class included generally gifted students (IQ =130) who were motivated to study mathematics at a high level (hereafter, a gifted class), and the second class included students characterized by high motivation regardless of their IQs (hereafter, motivation class). We analysed questions asked by the students during algebra and geometry lessons. Two types of questions are considered: elaboration and clarification. We found that students in a gifted class mostly asked elaboration questions, whereas students in a motivation class mostly asked clarification questions. We connect the revealed inclination to ask elaboration questions with intellectual curiosity that characterizes generally gifted students. Accordingly, we suggest that in classes of students who are motivated to study mathematics at high level, students who are generally gifted may create mathematical discourse of higher quality. We also argue that the identified differences in students' questions observed in classes of different types are not only student-dependent (i.e. depend on the students' levels of general giftedness) but can also be teacher-related and content-related.