Publications

In this paper, we aim to present an overarching picture of Mathematical Problem Posing (MPP) with a focus on the impact of task variables on MPP products and processes. Admittedly, there are many different settings with which to approach research related to task variables and their associated products and processes in MPP among mathematics students and teachers. In this paper, we approached related research in three kinds of settings: (1) the individual setting, (2) the group setting, and (3) the classroom setting.

Arguably, all educational research is conducted with the goal of improving teaching and learning. However, research may not explicitly suggest implications for practice. Furthermore, teachers may not be convinced by findings based on academic theories and methods of analysis that are foreign to them. This suggests that findings of such research may have a low level of implementability. We posit that replication of research by teachers can lead them to findings that are more implementable for them. In this article we elaborate the notion of implementability of educational research findings, make a theoretical case for teachers' qualitative replication of basic research, and offer the notion of pedagogical sensemaking as the activity by which research findings may become implementable for teachers. We investigate this practice based on empirical data gathered in a graduate course for practicing teachers, who were required to replicate a well-known piece of basic research in mathematics education.

Researchpractice partnerships around educational research may have beneficial outcomes but also present tensions. By considering the dynamics of manifested tensions, our study aims to understand how teachers engage with the various stages of the research in an inquiry-based professional development community consisting of eleven in-service teachers and three mathematics education researchers. In light of Heider's Balance Theory, we identify and analyze tensions expressed by teachers in the community discourse. Findings indicate that epistemic tensions related to teachers' and researchers' different cultural orientations act as powerful generators of inclusionary and exclusionary actions shaping community members' participation paths. While downplaying epistemic tensions can evoke individual actions detrimental to learning and destructive to the community's existence, awareness of and well-timed coping with tensions can become a springboard for community development.

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.

We substantiate the following claim: multi-variable narrative in qualitative research on problem posing bears promise for a better understanding of causality relationships between ways in which problem-posing activities are organized on the one hand, and characteristics of processes, products, and effects of problem posing on the other hand. Our notion of multi-variable narrative is first introduced by means of a hypothetical scenario. We then discuss relationships between different types of variables while adapting the terminology developed in mediation analysis literature to problem-posing situations and suggest heuristics for choosing problem-posing variables in research that aspires to inform practice. This is followed by an illustration in the context of a problem-posing activity by mathematics teachers. The illustration shows how features of the posed problems can be related to the problem-posing task organization, and how these relations may be mediated or moderated by particular features of the problem-posers, and by choices they make.

Background: Influential philosophers have suggested that interdisciplinarity is crucial for ecosystem management and scientific practice, and for education to democracy. However, a historical review of the rise of disciplines points at their compartmentalization in schools. An analysis of core construct categories of three disciplines, shows that this compartmentalization may decrease when dialogic argumentation is enacted. This background led us to launch an interdisciplinary program in schools. In previous publications, we identified multiple constraints in its implementation and listed design principles for affording interdisciplinary dialogic argumentation. Method: We adopt a narrative approach to analyze classroom talk, and ask whether and how interdisciplinary processes emerge in this talk. Findings: Students maintain dialogic argumentation around Interdisciplinary Social Dilemmas, but guidance is necessary for integrating knowledge from different disciplines. When the teacher is attentive to students unarticulated references to disciplinary ideas, she may subtly guide the emergence of interdisciplinary dialogic argumentation. Often, the teacher misses those opportunities and declaims the integration of knowledge in a non-dialogic talk. Contribution: Dialogic Education is crucial for the success of interdisciplinary programs in schools, but the actual emergence of interdisciplinary processes depends on the handling of organizational and institutional constraints, on huge design efforts, and on subtle guidance.

In February 2024, part of the editorial team of IRME (Uffe Thomas Jankvist, Mario Sánchez Aguilar and Morten Misfeldt) had the pleasure of discussing the development of implementation and replication studies in the context of South American STEAM (Science, Technology, Engineering, Arts, and Mathematics) education. We participated in the 2nd Latin American STEAM Education Research Conference and Summer School in Buenos Aires, Argentina. We also were invited to host a dedicated meeting at the Uruguayan Society of Mathematics Education in Montevideo, Uruguay. These academic events were indeed an interesting experience in several ways. Firstly, they made us aware of the international resonance and relevance of implementation and replication studies in mathematics education. Secondly, they made us reflect on the European origins from which this perspective has developed under the CERME (Congress of the European Society for Research in Mathematics Education) congress tradition. In this editorial, we will describe the work we did at these events and the reflections they led to.

Theorising is one of the most endorsed practices in contemporary mathematics education research. At the same time, such questions as \u201cwhat counts for useful theorising?\u201d and \u201chow can theory inform practice?\u201d are still subjects of debate. Arguably, this is in part because of the theoretical diversity characterising our field (e.g. Sriraman & English, 2010), and in part because of the diversity of approaches to handling the research-practice relationship (e.g. Schoenfeld, 2020).

In this plenary talk, I address two longstanding challenges for mathematical problem solving as a teaching and learning practice for all: (1) the many potential (and divergent) roles of a teacher in problem-solving instruction; (2) the vast diversity in intentions, goals, and meanings of tasks intended to be problems in different classrooms. In relation to the first challenge, I develop a metaphor of a problem as a living (discursive) creature whose \u201clive\u201d depends on who and how attends to it. This metaphor posits teachers as \u201crevivers\u201d of problems in their classes. In relation to the second challenge, I show how mathematical tasks are transformed in a chain of intended, planned, enacted, and experienced activity, and argue for research-practice partnerships as a useful perspective for making problem-solving instruction feasible.

We devote this editorial to implementability rather than implementation of research products and research-based innovations in mathematics education. This includes implementability of research findings, theoretical frameworks, professional development tools, instructional materials, interventions in regular practice and other innovations. The rationale for choosing this topic is as follows.

Theory in mathematics education can play different roles, among them the role of guiding the action of different stakeholders (e.g., teachers, teacher educators, curriculum designers, researchers, and policymakers). Researchers and practitioners alike have argued that the theories typically provided by researchers are often not sufficient for guiding action and have called for research-practice partnerships as, in part, a means of generating theory to guide action. So far, we have only partially recognized which theories can guide action, and how researchers and practitioners can engage in research-practice partnerships in order to generate theories which can better guide practitioners actions. In this chapter, we disentangle different kinds of theory elements and their interplay, in service of guiding practice, and we discuss typical challenges in reaching the goal. We draw from existing literature and our own experiences to suggest how discourses of researchers and practitioners can be transformed so that theories are better positioned to guide practitioners actions. We suggest acts of joint theorizing and supportive conditions which can increase the chance that a developed theory can guide stakeholders actions. Finally, we reflect on the implications for the culture and policies within the academic field of mathematics education.

The aims of this chapter are: (i) to showcase the state-of-the-art for theoretical perspectives for studying mathematics teacher collaboration; (ii) to identify promising theoretical and methodological perspectives for future studies. The ideas that are synthesised in this chapter are based on the presentations, papers, and discussions that occurred as part of the ICMI Study 25 Conference. In this introductory section, we briefly review the results from the previous ICME-13 research survey on mathematics teacher collaboration (Robutti et al., 2016; Jaworski et al., 2017). These results, which functioned as a starting place for the Theme A Working Group, not only informed the organisation of our group but were also reified in the ICMI Study 25 Conference discussion document (IPC, 2019).

s previously mentioned in this journal (Jankvist et al., 2021), the work carried out in the Thematic Working Group 23, \u201cImplementation of research findings in mathematics education\u201d (TWG23) of the Congress of the European Society for Research in Mathematics Education has found a natural continuation in the pages of Implementation and Replication Studies in Mathematics Education (IRME). The direct reason for this natural continuation is the active involvement of IRMEs editors in developing the TWG23 over the past six years. Nevertheless, the TWG23 has been important for shaping an overt and precise discussion about implementation in mathematics education and a hub for the newest ideas in implementation research in mathematics education.As a result of the nature of the TWG23 and our involvement in it, we will, in this editorial, present a report from this working groups latest meeting at CERME13, the 13th Congress of the European Society for Research in Mathematics Education, held in Budapest in July 2023. Due to her role as co-leader of the TWG23 at CERME13, Linda Marie Ahl has been invited to co-author this editorial. The editorial provides an overview of the papers presented in the TWG23 and the thematic discussions in this working group. It also provides a synopsis of the papers included in this issue of IRME.

When we launched Implementation and Replication Studies in Mathematics Education (IRME) two years ago, it was in a sense a culmination of several years of work trying to increase the focus on implementation (and replications) in mathematics education research (Jankvist et al., 2021). Our feeling at the time was that improvement of mathematics teaching and learning based on various insights accumulated in research had been the omnipresent concern of mathematics education research to such a large extent that it had been overlooked as an independent research interest.We of course recognized that attention to implementation was in no way novel. This attention has for instance been expressed by Bruckheimers (1979) focus on inhibitors of implementation, in Wittmans (1995, 2021) characterization of mathematics education as a design science, in Cobbs (2007) description of the philosophical foundations for mathematics education research, and in Artigues connected paradigms of design-based research and didactical engineering (e.g., Artigue, 2021). One can argue that the question of implementation is framing the mathematics education enterprise (e.g., Niss, 1999). In the introductory article of the special issue in ZDM on \u201cImplementation and implementability of mathematics education research\u201d (Koichu et al., 2021), we argued that even though the literature on implementation exists and can be surveyed, the literature that explores this perspective overtly is sparse. Asking mathematics educators about the quality and nature of the implementation they engage in is like asking a fish about the quality and nature of the water it swims in. The answer in both cases can very well be \u201cwhat are you talking about?\u201d.1 We as mathematics education researchers are immersed in a large and systemic attempt to improve the teaching and learning of mathematics, to a degree where we might overlook the influence of dedicated implementation initiatives on our research and practice. So in that sense, we have taken on the task of making implementation an overt phenomenon with clear boundaries. To stay in the fish metaphor, the water is everywhere and any attempt to distinguish it from the rest of the fish-surroundings would be flawed. This leads to a tricky situation. Of course, having overt and analytical attention towards implementation is an improvement over using naive unconscious images of what it is. Yet, because it is rather difficult to capture and describe the full implementation space, any model of implementation will lead to problems and need to be criticized.In this editorial, we discuss the concept of post-implementation as a way to organize the needed cautiousness and skepticism. We build on what we interpret as a current post movement in educational research, which questions traditional categories and myths about teaching and learning in order to conceptualize the fragile and unstable constructions that education rests upon. This movement uses concepts such as post-human (Harari, 2016), post-digital (Jandric et al., 2018), and post-colonial.2 A clear and prominent example of this is the post-digital attention to the digitalization of life and education and how this occurs at an increasing speed. Still, other areas are under scrutiny as well, e.g., climate crisis, political instability, and the increased focus on the development of identity, minority status, and sexual orientation. All this is challenging our image of educating for a stable future and justifies a critique of the myths and categories that drives and maintains the standard image of teaching, learning, and formation of the youth.Hence, in this editorial, we will discuss the increasing use of the prefix post in educational research, and exemplify it with the current discourse around post-digital education. We then go on to define and describe what meaning we could attach to post-implementation in mathematics education research, and discuss what this shows us about future directions for implementation (and replication) studies in mathematics education research.

This study explores the diversity of university mathematics teachers perspectives and experiences in relation to the secondary-tertiary transition (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 \u201cdeficit,\u201d 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 reactionsa 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 experts perspective, yet derived from a students perspective. Implications for research and practice are drawn.

This study aims to examine the types of mathematical tasks that teachers select or modify to elicit argumentative dialogue among students, and to identify the dialogical characteristics attributed to these tasks by teachers. The data were collected in a task-based PD program aimed at fostering dialogical teaching in mathematics classrooms. The results indicate that when it comes to the independent selection or modification of a task, many types are considered, but most of the teachers prefer tasks that based on explication of students' mistakes and misconceptions. Additionally, we identified three features of tasks that make them dialogical from the teacher's perspective: Triggering cognitive conflict, mathematical complexity, and encouraging multiple solutions or approaches to the same problem.

Past research has shown various benefits of combining the argumentative and the dialogic in cognitive development. However, it has also shown that attempts to implement dialogic argumentation in school fail to leave a sustainable impact. One reason for this situation is related to the lack of explicit knowledge about how to design and organize dialogical activities in realistic school settings. The present study addresses this lacuna while putting forward interdisciplinarity as a promising venue for promoting student dialogue and argumentation. We first elaborate on the connection between dialogic argumentation and interdisciplinarity, review the relevant literature on instructional and task design and then suggest a configuration of the design principles needed to be developed: these are content-related principles, pedagogy-related principles and organization-related principles. This is followed by an illustration of how specific design principles of these types emerged from and are reflected in task-design practices of project team of researchers in science, mathematics and philosophy education. Finally, we show that while the focus on interdisciplinarity seems to narrow the issue of the implementation of dialogue and argumentation in schools, it in fact opens it and provides instructional design principles in complex educational projects relevant to the society.

The word fidelity regularly pops up in mathematics education research and practice. This is in particular when dealing with matters of digital technologies, instructional interventions and replications. The experience shows that it is not always clear what is meant by fidelity. That it may have slightly different meanings in different situations. In which meaning, for example, can we talk about fidelity of teachers implementation of curriculum reforms or about the use of digital technologies in the classroom? The word may also have different connotations for different people. As pointed out by Ahl et al. (2022, p. 70), \u201cfidelity is an elusive concept to capture in analysis.\u201dThe Oxford Advanced Learners Dictionary (2022) asserts that fidelity is associated with loyalty to somebody or something, as well as to the quality of being accurate in relation to something, as in the following examples: (i) \u201cfidelity to your principles,\u201d (ii) \u201cfidelity to your partner,\u201d (iii) \u201cthe fidelity of the translation to the original text,\u201d or (iv) \u201cthe story is told with great fidelity to the original.\u201dFor our purposes, both loyalty and accuracy are of relevance. For example, one can talk about the quality of or level to which a teacher, a textbook author or an institution is loyal to a given instructional approach (cf. example i). Or to what extent researchers are loyal or accurate in relation to a previous study being replicated (cf. examples iii and iv). On some occasions, the term faithfulness also enters the picture, for example, when addressing the extent to which one is truthful to the original intent of an innovation. Surely, words such as loyal, faithful and truthful are more emotionally laden than accuracy. This may of course be related to example (ii) above, since here fidelity implies the opposite, namely infidelity. Hence, if stating that the fidelity of a given implementation/replication is low, then this may insinuate some, or even a high, degree of unfaithfulness, disloyalty or even inaccuracy certainly not positively-perceived qualities as such.To be clear, the purpose of this editorial is not to suggest that we now use another word than fidelity because of the above mentioned potential connotations and associations people may or may not have. Rather, we find it to be important to be aware of these when we have a look at how the notion of fidelity is used in our research field. When looking into the literature on fidelity, both inside and outside of mathematics education research, it becomes clear that the notion of fidelity is diverse, both in terms of definition and measurement aspects. Our editorial is not meant to be a review of fidelity studies. Still, we find it worth the endeavour to address the notion of fidelity. Not to remedy the discrepancies in definition and use, but rather to draw any such to the attention of the readers of Implementation and Replication Studies in Mathematics Education (IRME) as well as the researchers who publish in this journal.In the following, we address the notion of fidelity from three perspectives. Firstly, the meaning(s) of fidelity in mathematics education research. Secondly, fidelity in implementation research. And thirdly, fidelity in replication studies. Along the way we introduce the four papers in the current issue, and relate these to our discussion of fidelity.

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.

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.

The primary concern of Implementation and Replication Studies in Mathematics Education (IRME), is the development of a vibrant environment around research and development in implementation and replication studies in mathematics education. The Thematic Working Group 23 (TWG23) entitled \u201cImplementation of research findings in mathematics education\u201d at the three last Congresses of ERME (European Society for Research in Mathematics Education) has had an important impact on developing implementation research (IR) in mathematics education research (MER). In this editorial, we provide an extended report from TWG23 at CERME12, the twelfth Congress of ERME. TWG23 was led by IRMEs three associate editors, Mario Sánchez Aguilar (chair), Boris Koichu and Morten Misfeldt, along with two younger researchers, Rikke Maagaard Gregersen (until August 2021) and Linda Marie Ahl (from September 2021). Linda Marie Ahl was invited to co-author this editorial, which is an extensive report from this years TWG23.Due to the Covid-19 situation, CERME12 was held virtually. TWG23 consisted of eight sessions involving presentations of papers and posters as well as three thematic discussions among the groups 21 participants from 12 different countries. This was the third time that TWG23 took place, and the group has now proven somewhat stable in terms of participant number and contributions. We will say more about the evolution of the group in the next section of the editorial, but it is important to be aware of the value of the work of this TWG for the existence and evolution of IRME. (This was already acknowledged by Michèle Artigue (2021) in her overview of theoretical resources for implementation research in mathematics education in the very first issue of IRME.)In the present editorial, we will first describe the evolution of the work on IR in the context of CERME, and then we will describe the contributions at CERME12s TWG23. Afterwards we will describe three overarching themes that were discussed at this years meeting in TWG23: (1) the role of change and theory of change; (2) matters of scaling; and (3) the conception of stakeholders. These three themes are also used to situate the four papers in this issue of IRME, since they all are extensions of studies presented at this years CERME12 (TWG23). The editorial concludes with several observations and considerations about the next steps for IR in mathematics education in general and for the TWG23 at CERME in particular.

Autonomous student problem solving is still rare in mathematics classes, and its incorporation in lessons requires profound knowledge and readiness of the teacher. In this paper we present a study conducted in the context of a professional development course aimed to enhance 12 teachers readiness to create opportunities for students autonomous problem solving. We characterize processes that the teacher-participants went through and report the extent to which the course achieved its goals. Based on qualitative and quantitative data, we offer a conceptual framework for characterizing the adoption processes and argue for the feasibility of the desirable change in the teacher readiness to adopt problem-solving instruction.

Mathematical problem posing (MPP) has been at the forefront of discussion for the past few decades, and a wide range of problem-posing topics have been studied. However, problem posing is still not a widespread activity in mathematics classrooms, and there is not yet a general problem-posing analogue to well-establishedframeworks for problem solving. This paper presents the state of the art on the effort to understand the cognitive and affective processes of problem posing as well as task variables of problem posing at the individual, group, and classroom levels. We end this paper by proposing a number of research questions for future studies related to task variables and processes ofproblem posing.

We present a study of a model for professional development of mathematics teachers, based on their participation in a collaborative problem solving in online discussion forums, in two roles. At the first stage of the study, 47 high-school mathematics teachers participated in the forums as students. At the second stage, they mediated forums as mentors. The first stage of the study showed gradual development of group synergy among the teachers-as-students. The second stage showed that the experience of group synergy gained by the teachers at the first stage has supported the development of their mathematical fluency in teaching.

In this second editorial of Implementation and Replication Studies in Mathematics Education (IRME), we address the question of what to look for in research results and findings from mathematics education research in terms of sources for replication studies. Surely, a joke in the mathematics education community goes that mathematics education researchers do not want to replicate, they want to get replicated! Probably this is true for many research fields, not only mathematics education. Still, we deem this not to be the major reason for the relatively small number of replication studies in our field (Aguilar, 2020; Jankvist et al., 2021). Rather we believe this has to do with the research culture. As some of us recently pointed out in the context of studying the use of digital technologies in mathematics education (Jankvist & Misfeldt, 2021), researchers in mathematics education seem more likely to prefer introducing new theoretical constructs for \u201cnew problems\u201d rather than looking into the back catalogue of our research field in order to spot already well-developed constructs that may readily provide insights into these new problems (Mason, 2016). This is of course in line with the (mis)interpretation of the old saying \u201cIf you want to get ahead, get a theory\u201d (Karmiloff-Smith & Inhelder, 1975) to mean that if you want to get ahead, develop a new theory (and have others replicate it). Still, as we know from Schoenfeld (2014), \u201cIf you really want to get ahead, get a bunch of theories , and data to test them\u201d (p. 7). This quote is at the heart of replication in mathematics education, we find. Namely, to gather (different types of) data to \u201ctest\u201d the theoretical constructs that we apply and adhere to in our research. Still, this does not answer the questions of what to replicate and why.Now, another reason for not resorting to constructs from the back catalogue of our research field may of course be that new coming researchers are not necessarily well familiar with the back catalogue. Surely, we have some collections of \u201cclassical papers\u201d (e.g. Bishop, 2010), but maybe some \u201cpre-digestion\u201d of our fifty years of research results is needed in order to direct and guide (young) mathematics education researchers in a pursuit of research findings to potentially replicate. We find that the listing by the Editorial Committee of the European Mathematical Society of so-called \u201csolid findings\u201d (in mathematics education research) to some extent illustrates this.

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.

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 teacherresearcher 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.

This book explores the idea that mathematics educators and teachers are also problem solvers and learners, and as such they constantly experience mathematical and pedagogical disturbances. Accordingly, many original tasks and learning activities are results of personal mathematical and pedagogical disturbances of their designers, who then transpose these disturbances into learning opportunities for their students. This learning-transposition process is a cornerstone of mathematics teacher education as a lived, developing enterprise. Mathematical Encounters and Pedagogical Detours unfold the process and illustrate it by various examples. The book engages readers in original tasks, shares the results of task implementation and describes how these results inform the development of new tasks, which often intertwine mathematics and pedagogy. Most importantly, the book includes a dialogue between the authors based on the stories of their own learning, which triggers continuous exploration of learning opportunities for their students.

While the affordances of problem-based learning are broadly recognized, implementation of this innovative approach is not common, particularly in tertiary mathematics education. This study investigates early stages of an implementation of problem-based instruction in 1^st year mathematics courses for engineering students, within a project encompassing 12 universities and colleges across Europe. Twenty-three lecturers from participating institutions took part in a preparatory workshop. Framing the project as a case of diffusion of innovations, we analyze post-workshop questionnaires to reveal the participants' conception-of and attitudes-toward the innovation. We highlight some challenges that the innovation entails, and how they relate to participants general attitude toward implementing the innovation.

The rise of citizen science in the past decade has brought many opportunities for scientists and publics alongside many challenges and questions regarding best practices. These include questions regarding public engagement, project design and measures of success. The aim of this study is to better understand what makes citizen science projects scientifically successful, and highlight what can be learned and implemented in future project design. We focus on scientifically productive projects as a success measure that can encourage greater scientists' involvement in citizen science, and analyze five of these projects for factors that contribute to their success. We found that although all projects have strong scientific goals, they all have additional strong emphases on communication and social practices, providing a good user experience and generating motivation and empowerment. We provide five heuristics for the future design of citizen science projects, which focus on engagement and communication features which we believe are important for citizen science project success.

Three teams of pre-service mathematics teachers were engaged in an assignment of designing teaching sequences aimed at empowering high-school students to understand solutions to International Mathematics Olympiad problems. The assignment was rich with problem-posing opportunities. The data the sequences, posed problems and reflections were analyzed in order to reveal the underlying structure of the participants' problem-posing experiences. The findings support two conclusions. First, problem posing in mathematics teacher education can be promoted a-didactically, as an implicit objective of an activity having teaching for problem solving as an explicit goal. Second, problem posing that requires pre-service teachers to cope with mathematics that is challenging for them, provides rich learning opportunities and can result in worthwhile posed problems.

Recent research in problem solving has shifted its focus to actual classroom implementation and what is really going on during problem solving when it is used regularly in classroom. This book seeks to stay on top of that trend by approaching diverse aspects of current problem solving research, covering three broad themes. Firstly, it explores the role of teachers in problem-solving classrooms and their professional development, moving ontosecondlythe role of students when solving problems, with particular consideration of factors like group work, discussion, role of students in discussions and the effect of students engagement on their self-perception and their view of mathematics. Finally, the book considers the question of problem solving in mathematics instruction as it overlaps with problem design, problem-solving situations, and actual classroom implementation. The volume brings together diverse contributors from a variety of countries and with wide and varied experiences, combining the voices of leading and developing researchers. The book will be of interest to any reader keeping on the frontiers of research in problem solving, more specifically researchers and graduate students in mathematics education, researchers in problem solving, as well as teachers and practitioners.

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.

Student transition from school-level mathematics to university-level mathematics, often referred to as the secondary-tertiary transition (hereafter STT) is an enduring, complicated and multi-faceted process. STT is a long-standing issue of concern, which has merited significant attention in mathematics education research and practice. In particular, STT was discussed on the pages of this Newsletter several years ago (Gueudet on behalf of the Education Committee of the EMS, 2013) At its 2018 meeting in Cyprus, the EMS Education Committee recognized that our knowledge about successful ways of dealing with STT is still insufficient and that moving forward requires a large scope effort on the part of all parties involved, including mathematics lecturers, school teachers, education researchers, policymakers and students in transition. As part of this effort, the Committee is conducting a survey among mathematicians. The goal of the survey is to collect and report to the mathematics community information needed in order to devise national and international actions that can essentially improve the state of the art with respect to STT.

This paper explores a particular model of implementation of research: teacher adaption of research procedures and ideas in their classrooms as part of participation in community educational research. The TRAIL (Teacher-Researcher Alliance for Investigating Learning) project seeks to guide the design and conduct of co-learning partnerships between mathematics teachers and mathematics education researchers. In TRAIL, mathematics teachers actively participate in formulating research goals and designing research tools, and then collect data in their classrooms and analyze together the shared data corpus. In the first part of the paper, we present theoretical underpinnings of implementation through participation in TRAIL. In the second part, we examine implementation through participation in an illustrative case, in which a group of teachers designed and explored classroom situations aimed at promoting student questions in the classrooms.

The goal of this chapter is to present and theorize our more successful and less successful attempts to enhance long-term collaborative problem solving in high school, by means of online problem-solving forums. We focus on two classroom communities and their interactions, during two school years, with an additional community, a research group that initiated the use of the forums. In one of the classroom communities, online problem solving has eventually become a routine practice and a valuable addition to classroom problem solving. In another classroom community, the forum did not become active despite considerable effort made, but enduring attempts to activate it led to enhancement of student-student interactions in the classroom. All three communities (i.e., two classroom communities and the research group) gradually developed. Taking the Diffusion of Innovations perspective, we characterize stages of the development and identify its main agents. Taking the Communities of Practice perspective, we characterize each community and illustrate boundary interactions between them as a driving force for their development.

A theoretical discussion of possible connections between some non-conventional external representations (i.e., textual tasks of a particular format) and their corresponding internal representations (i.e., perceptions of these tasks by mathematics learners) is presented in this paper. Our goal is to analyze the interplay between external and internal representations in relation to nonconventional textual tasks of "who-is-right?" format. Such tasks involve a relatively long textual story introducing a situation that can be interpreted in (at least) two contradictory ways, which are explicitly given. The solvers of the task are required deciding which interpretation is correct and support their decision by an argument that would convince their peers. The presented theoretical analysis in terms of representations can serve as a tool for teachers and teacher educators in designing tasks specifically tailored to their students' needs.

Professional learning communities (PLCs) are considered an effective vehicle for teacher professional development, yet their emphasis on discussions-based learning practices may create tension with the expectation for growth of content knowledge. We have been leading a PLC of practicing and prospective heads of school mathematics departments, in which this tension was particularly salient. We investigate ways in which lectures and workshops conducted by content-experts can support the development of desirable PLC characteristics, rather than being at odds with them. Findings suggest that the tension can be reconciled by means of ongoing debriefings with a focus group comprised of the PLC participants, contributing to the careful design of community activity surrounding the expert-provided lectures.

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 behaviora 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 chapter presents a proposal for an exploratory confluence model of mathematical problem solving in different instructional contexts. The proposed model aims at bridging the knowledge of how problem solving occurs and the knowledge of how to enhance problem solving. The model relies of the premise that a key solution idea to a problem is constructed as a result of shifts of attention stipulated by the solver's individual resources, interaction with peers, or with a source of knowledge about the solution. The exposition converges to the conclusion that successful problem solving is likely to occur in choice-affluent learning environments, in which the solvers are empowered to make informed choices of a challenge to cope with, problem-solving schemata, a mode of interaction, an extent of collaboration, and an agent to learn from. The theoretical argument is supported by an example from an empirical study.

In September 2012, the Center for Educational Technology together with the Israeli Ministry of Education and the Trump Foundation, launched the first Israeli Virtual High School (VHS). The goal was to increase the number of students who study advanced-level mathematics and physics by addressing a two-tiered problem in Israel's periphery: The shortage of qualified teachers who can teach advanced-level mathematics or physics, and the low numbers of students who opt for these studies - too low to justify funding dedicated classes for them at their home schools. This chapter describes the design, instruction, and support mechanisms of the VHS and presents its growth since its inception in 2012. It then presents preliminary findings gathered from a questionnaire disseminated in 2015 among the first graduating cohort of the VHS and their teachers about their experience.

We present three examples of online problem-solving forums that accompany the learning of mathematics in Israeli high schools. The first example introduces a Facebook forum intended to prepare students from different schools for the secondary school graduation exam in mathematics. The second example concerns a WhatsApp asynchronous forum opened by a novice mathematics teacher for her 10th grade. The forum has gradually developed into a multifunctional space for students seeking assistance and sharing solutions to homework problems. The third example is a synchronous Google+ forum in an 11th grade initiated by an experienced mathematics teacher. This forum functioned as a platform for collaborative problem solving after school hours. Trends in development of self-organized online activity are discussed by the end of the chapter.

This was my first reaction to the Shinno et al. (2018) article in 38(1). Indeed, as Mason (2010) has argued, brief-andvivid recollections of observations that would resonate or trigger the others own recollections has special value in mathematics education research. For me, the article was definitely resonating and triggering. My second thought was about Shinno and colleagues approach to situating their observations in international context, by means of a proposal for a particular reference epistemological model (REM). I asked myself: how can the proposed three-layer REM account for the complexity of teaching and learning proof in school mathematics instructions in different countries? I briefly discuss these two questions in this commentary.

The paper focuses on student learning experiences during large-group undergraduate Calculus tutorials. We identify eight types of Key Memorable Events emotionally loaded events that are meaningful for the learning process in class from a student perspective. The findings are predominantly based on stimulated-recall interviews with 36 students, corresponding to 7 filmed lessons. Implications are drawn in relation to both the learning and teaching in the undergraduate mathematics classroom.

This study is aimed at characterization of action strategies in DGE (Cabri 3D),by using a measure of visual difficulty of 2-D sketches depicting 3-Dgeometric situations (specifically, of cubes with auxiliary structures), as a toolallowing dynamic monitoring of the problem-solving processes. Twenty-onestudents, seven of high, seven of medium and seven of low spatial ability level,were engaged in DGE-supported solving spatial geometry problems ofdifferent visual difficulty levels, during individual work-sessions, immediatelyfollowed by semi-structured interviews. The data analysis consisted ofidentification of changes in the visual difficulty of the sketches undertaken bythe students on the computer screen and of their problem-solving moves, asexpressed in the interview verbatim. Findings show that learners work withDGS to reduce visual difficulty in a nonlinear process, that is influenced by theindividual spatial skills, the initial visual difficulty of the problem, as well asthe solution-stage in which the software is introduced. Significant differencesin the strategies employed by students of different spatial ability levels, forsolving spatial geometry problems, can be explained by differences in thestructure of knowledge of experts versus novices. Consequently, these findingshave important implications for teaching spatial geometry with DGS.

We illustrate how the classical dialogues  Galileos Dialogue on Infinity from Dialogues Concerning Two New Sciences, Platos Meno, and Lakatos Proofs and Refutations  can be used in teacher education. By re-capturing our conversation, we demonstrate the use of the classical dialogues to revisit mathematical notions, such as infinity, or to highlight meta-mathematical issues, such as definitions and proofs. We share several scripting assignments used with teachers and several student-written scripts produced in response to such assignments. We elaborate on the benefits of bringing classical dialogues for discussion in classes of mathematics teachers. These include, but are not limited to, enculturation by exposure to historical context, reinforcement of mathematical ideas and concepts, introduction to subsequent readings and assignments, and extended variety of tasks for the use in mathematics teacher education.

The pursuit of understanding in communication between interlocutors of different knowledge is one of the most distinct characteristics of teaching as a professional occupation. Consequently, mathematics teacher preparation programs should provide opportunities for the prospective teachers to put themselves in the shoes of more and less knowledgeable individuals who interact on mathematical matters. Our study is on one of such opportunities: a task of writing a script for a dialogue between fictional characters of historically different backgrounds, who discuss a particular proof from Euclids Elements, in the context of number theory. Twenty-three prospective secondary teachers responded to the task. We enquired what do the script writers identify as presumed points of difficulty in the given proof and what explanations are constructed by the participants to result in \u201cI understand\u201d claims. Three types of potential difficulty were attended to: number theory concepts and terms, claims within the proof, and the central idea of the proof. The points of difficulty were explained in various ways, such as producing procedural definitions and translating Euclidian terms into the modern terms. Implications for research on script writing and on the use of historical sources in mathematics education are drawn.

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.

We investigated aesthetic responses of 60 middle school students as they engaged in a pair of similar looking geometry problems in one-on-one semi-structured interviews. The investigation was driven by three predictions. The first two predictions were about the association between the evaluative aesthetic response and surprise stemming from the solution to each problem. The third, main, prediction was that the problem with more surprising solution would be evaluated as more beautiful. The extent of surprise was manipulated by the order in which two problems were given. The third prediction came to be true in 90% of the cases, in which the first two predictions were fulfilled. The findings suggest that school students' evaluative aesthetic response to mathematical problems can be stimulated in instructional settings. Implications for research and practice are drawn. 

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 studentgenerated 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.

The goal of this article is to present and theorize our more successful and less successful attempts to create and sustain problem-solving forums, in which exploratory discourse takes place. The main argument is that many implementation-related phenomena that we have encountered when working with seven high-school classes for one or two school years can be characterized and explained with the aid of conceptual tools provided by Rogers' Theory of Diffusion of Innovation. The most successful process of forming an online forum in one of the classes is presented in some detail, and the parallel processes in the rest of the classes are presented in the form of an aggregated summary. Implications for future design-based implementation research are drawn.

This article presents 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 events and algebraic activities in which the students were engaged while making sense of the formula. We found that the students' conviction, by the end of the project, that the formula "makes sense" emerged when they justified the formula, checked its generality, discovered a geometry mechanism behind it, and found that it came to cohere with additional formulas. The findings are summarized as a suggestion for a four-component decomposition of algebraic sense making.

This article presents 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 occurred in the context of two-month-long student research project. The data analysis was aimed at identification and characterization of the activities in which the students were engaged and the processes involved in the students self-imposed sense-making quest. We found that the sensemaking process consisted of a sequence of generational and transformational algebraic activities in the context of a global, meta-level activity, long-term problem solving. In this process, the students: (1) formulated and justified claims; (2) made generalizations, (3) found the mechanisms behind the algebraic objects; and (4) established coherence among the explored objects. The article is concluded by a proposal for a four-component decomposition of algebraic sense making.

The main argument of this article is that "challenging mathematics for all" can be more than just a nice slogan, on condition that all students are empowered to make informed choices of: a challenge to be dealt with, a way of dealing with the challenge, a mode of interaction, an extent of collaboration, and an agent to learn from. Pedagogies supporting such choices are called choice-based pedagogies. The article begins from a theoretical discussion of the relationships between the notions 'mathematics with distinction,' 'giftedness', 'challenge' and 'choice'. As a result, the learner-centered conceptualization of mathematical challenge is proposed. Then two examples of choice-based pedagogies enabling students with different background and abilities to be engaged in mathematics with distinction are presented. Implications are drawn.

The presentation is based on the results of two studies aimed at exploring opportunities for enhancing students active learning through problem solving in frontal lectures and tutorials of linear algebra and calculus university courses. In the first study, traditional linear algebra lectures and lectures involving Classroom Response Systems were explored. The study resulted in an identification of a set of practices by which experienced lecturers created opportunities for interactive problem solving. The second study explored students learning strategies in calculus tutorials. We found that while some students followed the exposition, other students periodically stopped listening, engaged in independent problem solving, and then attempted to catch up with the exposition. Implications are drawn.

This paper reports on part of a study regarding student learning-experiences and affective pathways in undergraduate calculus tutorials. The following question is pursued in this paper: How do the students' key affective states relate to the type of mathematical discourse conducted in class? We present and discuss two lessons where two similar problems were considered. The lessons were filmed and followed by stimulated recall interviews with nine students. Though the students in both lessons did not understand the solution to the challenging problem, they evaluated the lessons and subsequent learning experiences very differently. We suggest the difference was related to the type of discourse employed by the instructor. The lesson that evoked a negative reaction utilized only an object-level discourse. The lesson that evoked a positive reaction additionally utilized a meta-level discourse. We will call this heuristic-didactic discourse. Implications are drawn.

This paper discusses the process of proving from a novel theoretical perspective, imported from cognitive psychology research. This perspective highlights the role of hypothetical thinking, mental representations and working memory capacity in proving, in particular the effortful mechanism of cognitive decoupling: problem solvers need to form in their working memory two closely related models of the problem situation- the so-called primary and secondary representations- and to keep the two models decoupled, that is, keep the first fixed while performing various transformations on the second, while constantly struggling to protect the primary representation from being "contaminated" by the secondary one. We first illustrate the framework by analyzing a common scenario of introducing complex numbers to college-level students. The main part of the paper consists of re-analyzing, from the perspective of cognitive decoupling, previously published data of students searching for a non-trivial proof of a theorem in geometry. We suggest alternative (or additional) explanations for some well-documented phenomena, such as the appearance of cycles in repeated proving attempts, and the use of multiple drawings. 

הישגים גבוהים בגאומטריה מרחבית בבית הספר התיכון, מיוחסים לעתים קרובות ליכולתו של הלומד "לראות במרחב", ולדמות בעיני רוחו אובייקטים גאומטריים תלת-ממדיים מתוך הסרטוטים הדו-ממדיים המוצגים בפניו (1996, Gutiérrez) .אולם, הדבר אינו תמיד טבעי וקל. כיצד נוכל למשל לזהות בוודאות את האובייקט באיור 1? האם מדובר במשושה המצוי במישור הדף? האם ישנה כאן פירמידה ישרה עם בסיס משושה במבט מלמעלה או מלמטה? ייתכן ונוכל לראות כאן קובייה? כל האינטרפרטציות הללו הן כמובן אפשריות ונכונות. על מנת לזהות אובייקטים במרחב, המוח האנושי משלים אינפורמציה החסרה בסרטוט. כך למשל, כאשר אנחנו רואים תמונה של שולחן (ראו איור 2) ,המוח מעבד את המידע ואנחנו מפרשים את התמונה כתמונה של שולחן מלבני, למרות שלוח השולחן שבתמונה הוא בצורת מקבילית.

The paper presents and analyses a sequence of events that preceded an insight solution to a challenging problem in the context of numerical sequences. A three­week long solution process by a pair of ninth­-grade students is analysed by means of the theory of shifts of attention. The goal for this article is to reveal the potential of this theory as an analytical tool that can explain the course of the exploration towards the insight solution. The explanation is provided by inferring from the data what, how and why the students attended to when working on the problem.

We introduce virtual duoethnography as a novel research approach in mathematics education, in which researchers produce a text of a dialogic format in the voices of fictional characters, who present and contrast different perspectives on the nature of a particular mathematical phenomenon. We use fiction as a form of research linked to narrative inquiry and exemplify our approach in a dialogue related to various proofs of infinitude of primes. We view Lakatos' (1976) dialogue in the seminal Proofs and Refutations as an example of virtual duoethnography. We discuss the affordances of this approach as an alternative to the formal ways of presenting research in mathematics education.

An exploratory confluence framework for analysing mathematical problem solving in socially different educational contexts is introduced. The central premise of the framework is that a key solution idea to a problem can be constructed by a solver as a result of shifts of attention that come from individual effort, interaction with peer problem solvers or interaction with a source of knowledge about the solution. The framework consolidates some existing theoretical developments and aims at addressing the perennial educational challenge of helping students become more effective problem solvers.

An important premise of creative thinking is that a problem solver is in position to choose. The literature on open-ended problem situations focuses on the mechanisms underlying learner choices of problem-solving moves. This article argues that there is room for empowering students to make choices of additional types: choosing an extent of collaboration, choosing a mode of interaction, and choosing an agent to learn from while solving a challenging problem. Theoretical argument is supported by an empirical example drawn from an on-going study on the interaction between cognition and affect in high-school students long-term geometry problem solving.

The aim of this chapter was to identify mathematics teachers conceptions of the notion of \u201cproblem posing.\u201d The data were collected from a web-based survey, from about 150 high school mathematics teachers, followed by eight semi-structured interviews. An unexpected finding shows that more than 50% of the teachers see themselves as problem posers for their teaching. This finding is not in line with the literature, which gives the impression that not many mathematics teachers are active problem posers. In addition, we identified four types of teachers conceptions for \u201cproblem posing.\u201d We found that the teachers tended to explain what problem posing meant to them in ways that would embrace their own practices. Our findings imply that most of the mathematics teachers are result-orientedas opposed to being process-orientedwhen they talk about problem posing. Moreover, many teachers who pose problems doubt the ability of their students to do so and consider problem-posing tasks inappropriate for their classrooms.

The goal of the case study presented in this paper was to examine a student's perspective on creative products in project-based learning. In this paper we dismantle, by means of the theory of shifts of attention, a two-month long sequence of events that preceded an unexpected invention made by a ninth-grade student: the student invented a new mathematical symbol, and valued this invention higher than his solution to a complex mathematical problem. INTRODUCTION This article is part of a series of reports, in progress, on the results of a research project " Open-ended problems in mathematics " (Palatnik, in progress). At the beginning of a yearly cycle of the project, a 9 th grade class of one of schools in Israel is exposed to a set of about 10 challenging problems. The students choose a particular problem to pursue and then work on it in teams of two or three for several weeks. The initial problem serves as a basis for follow-up inquiries, which last for additional 2-3 months. At the end, the teams present results of their work in front of their classmates and academic audience at the workshop organized at the Technion. The project is designed as a venue for fostering mathematical creativity through mathematical problem solving and problem posing () theory of shifts of attention is chosen as a theoretical lens for dismantling and explaining the appearance of the creative products produced by the students in the framework of the project (Palatnik & Koichu, submitted; in prep.)

The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insight solution to a challenging problem by a ninth-grade student. A three-week long solution process was analysed by means of the theory of shifts of attention. We argue that concurrent focusing on what, how and why the student attends to when working on the problem can adequately explain his insight.

המאמר מציג מסגרת תאורטית המאפשרת אבחון וקידום של ידע מורים בנושא האלגוריתמים הסטנדרטיים של ארבע פעולות החשבון. המסגרת התאורטית פותחה על בסיס ניתוח מתמטי של האלגוריתמים, ועל בסיס אנלוגיה למחקרים העוסקים בידע מתמטי ייחודי להוראה של מספרים רציונליים. בהתאם למסגרת המוצעת, נבנו משימות מתמטיות, זאת לצורך אבחון וקידום ידע המורים בנושא. במאמר מוצגות דוגמאות למשימות הקשורות לידע העקרונות המתמטיים שבבסיס האלגוריתמים, אשר נוסו עם 64 מורים בבית ספר יסודי במסגרת התמקצעות המורים.

The chapter includes four contributions on different aspects of the relationship between problem solving in mathematics and in mathematics education. Gerald Goldin points out that besides the importance of teaching students how to solve certain classes of problems, problem solving is a means of achieving some more general purposes pertaining to mathematics learning. Israel Weinzweig develops the claim that certain sequences of mathematical questions can provide students with problem-solving experiences similar to those of research mathematicians, and that such experiences are beneficial for promoting students conceptual understanding. Shlomo Vinner discusses the role of schemata and creativity in mathematical problem solving, and argues that the notions \u201cproblem solving in mathematics\u201d and \u201cproblem solving in exam-oriented mathematics instruction\u201d are incompatible. Roza Leikin presents a study aimed at identifying unique cognitive traits of intellectually gifted students who have the potential to become research mathematicians in the future. The chapter concludes with a reflective summary, in which the points made by the contributors are considered as parts of a longer-term debate on the relationships between problem solving in mathematics and in mathematics education, a conversation that has developed over the years according to a certain spiral pattern.

This paper is situated within the ongoing enterprise to understand the interplay of students empirical and deductive reasoning while using Dynamic Geometry (DG) software. Our focus is on the relationships between students reasoning and their ways of constructing DG drawings in connection to directionality (i.e., \u201cif\u201d and \u201conly if\u201d directions) of geometry statements. We present a case study of a middle-school student engaged in discovering and justifying \u201cif\u201d and \u201conly if\u201d statements in the context of quadrilaterals. The activity took place in an online asynchronous forum supported by GeoGebra. We found that student's reasoning was associated with the logical structure of the statement. Particularly, the student deductively proved the \u201cif\u201d claims, but stayed on empirical grounds when exploring the \u201conly if\u201d claims. We explain, in terms of a hierarchy of dependencies and DG invariants, how the construction of DG drawings supported the exploration and deductive proof of the \u201cif\u201d claims but not of the \u201conly if\u201d claims.

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.

Twelve participants were asked to decode that is, interpret and make sense of a given proof of Fermat's Little Theorem, and present it in a form of a script for a dialog between two characters of their choice. Our analysis of these scripts focuses on issues that the participants identified as problematic in the proof and on how these issues were addressed. Affordances and limitations of this dialogic method of presenting proofs are exposed, by means of analyzing how the students correct, partial or incorrect understanding of the elements of the proof are reflected in the dialogs. The difficulties identified by the participants are discussed in relation to past research on undergraduate students difficulties in proving and in understanding number theory concepts.

Twenty-four mathematics teachers were asked to think aloud when posing a word problem whose solution could be found by computing 4/5 divided by 2/3. The data consisted of verbal protocols along with the written notes made by the subjects. The qualitative analysis of the data was focused on identifying the structures of the problems produced and the associated ways of thinking involved in constructing the problems. The results suggest that success in doing the interview task was associated with perception the given fractions as operands for the division operation and, at the same time, the divisor 2/3 as an operator acting over 4/5. The lack of success was associated with perception of division of fractions as division of divisions of whole numbers and using the result of division of fractions as the only reference point. The study sheds new light on the teachers' difficulties with conceptualization of fractions.

"Success stories," i.e., cases in which mathematical problems posed in a controlled setting are perceived by the problem posers or other individuals as interesting, cognitively demanding, or surprising, are essential for understanding the nature of problem posing. This paper analyzes two success stories that occurred with individuals of different mathematical backgrounds and experience in the context of a problem-posing task known from past research as the Billiard Task. The analysis focuses on understanding the ways the participants develop their initial ideas into problems they evaluate as interesting ones. Three common traits were inferred from the participants' problem-posing actions, despite individual differences. First, the participants relied on particular sets of prototypical problems, but strived to make new problems not too similar to the prototypes. Second, exploration and problem solving were involved in posing the most interesting problems. Third, the participants' problem posing involved similar stages: warming-up, searching for an interesting mathematical phenomenon, hiding the problem-posing process in the problem's formulation, and reviewing. The paper concludes with remarks about possible implications of the findings for research and practice.

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.

Twelve participants were asked to decode a proof of Fermats Little Theorem and present it in a form of a script for a dialogue between two characters of their choice. Our analysis of these scripts focuses on issues that the participants identified as problematic in the proof and on how these issues were addressed. Affordances and limitations of this dialogic method of presenting proofs are exposed, by means of analyzing how the students correct, partial or incorrect understanding of the elements of the proof are reflected in the dialogues. The difficulties identified by the participants are discussed in relation to past research on undergraduate students difficulties in proving and in understanding number theory concepts.

The study presented in this paper offers a reference framework for characterizing primary school teachers mathematical knowledge of the standard algorithms of the four basic arithmetic operations. The framework was first devised theoretically, from mathematical analysis of the algorithms and from past research on knowledge for teaching arithmetic operations with rational numbers. It was then applied to designing tasks for charactering and deepening the teachers' conceptual understanding of the algorithms. The paper contains examples of tasks related to the knowledge of mathematical principles underlying the algorithms, which were tested with a group of 46 primary school mathematics teachers.

The goal of the study presented in this article was to examine how variations in task design may affect mathematics teachers learning experiences. The study focuses on sorting tasks, i.e., learning tasks that require grouping a given set of mathematical items, in as many ways as possible, according to different criteria suggested by the learners. We present an example of a sorting task for which the items to be grouped are related to basic concepts of analytical geometry that are connected to the notion of loci of points. Based on a design experiment of three iterations with practicing secondary school mathematics teachers, we report on intended and enacted objects of learning inherent in three versions of the task. Empirically based suggestions are drawn about design of sorting tasks that potentially evoke desirable learning experiences.

Identifying mathematical and didactical conditions under which mathematics learners can encounter an intellectual need for defining and proving is recognized as a challenging research enterprise. This paper presents a particular configuration of conditions under which a group of pre-service mathematics teachers successfully constructed a definition of an eminent artistic object, the Penrose tribar, and a proof of its impossibility in 3-D space. The paper focuses on the instructors pedagogical choices and interventions, which made the exploration feasible for the students and, at the same time, preserved their autonomous learning. The interventions in the form of auxiliary problems are put forward.

The paper introduces an exploratory framework for handling the complexity of students mathematical problem posing in small groups. The framework integrates four facets known from past research: task organization, students knowledge base, problem-posing heuristics and schemes, and group dynamics and interactions. In addition, it contains a new facet, individual considerations of aptness, which accounts for the posers comprehensions of implicit requirements of a problem-posing task and reflects their assumptions about the relative importance of these requirements. The framework is first argued theoretically. The framework at work is illustrated by its application to a situation, in which two groups of high-school students with similar background were given the same problem-posing task, but acted very differently. The novelty and usefulness of the framework is attributed to its three main features: it supports fine-grained analysis of directly observed problem-posing processes, it has a confluence nature, it attempts to account for hidden mechanisms involved in students decision making while posing problems.

This chapter traces the long-term cognitive development of mathematical proof from the young child to the frontiers of research. It uses a framework building from perception and action, through proof by embodied actions and classifications, geometric proof and operational proof in arithmetic and algebra, to the formal set-theoretic definition and formal deduction. In each context, proof develops over the long-term from the recognition and description of observed properties and the links between them, the selection of specific properties that can be used as definitions from which other properties may be deduced, to the construction of crystalline concepts whose properties are a consequence of the context. These include Platonic objects in geometry, symbols having relationships in arithmetic and algebra and formal axiomatic systems whose properties are determined by their definitions.

This paper is one of the reports on a multiple-case study concerned with the intertwining between affect and cognition in the mechanisms governing experts when posing new mathematical problems. Based on inductive analysis of a single case of an expert poser for mathematics competitions, we suggest that the desire to experience the feeling of innovation may be one of such mechanisms. In the case of interest, the feeling was realized through experts reflections on the problems he created in the past, by systematically emphasizing how a new problem was innovative in comparison with other familiar problems based on the same nesting idea. The findings are discussed in light of past research on expert problem posers and expert problem solvers.

Posing an original mathematical problem is frequently regarded as an instantiation of one's mathematical creativity. However, one of the most robust observations from the research literature on mathematical problem posing is that diagnostic and learning opportunities associated with this activity are, as a rule, much more impressive than problems produced as a direct result of the effort. This is hardly surprising, given that the main research effort has been put into studying students and teachers, who did not have substantial problem-posing experience, but posed problems upon request in different instructional settings. There are also a few reflective publications by mathematics practitioners and educators on what they do when posing mathematical problems. These reflections are helpful for identifying various sources for posing new problems, as well as for mapping expert problem-posing strategies. Still, reflection-based publications, as fruitful as they are, cannot be a substitute for systematic observation-based studies aimed at revealing the nature of problem posing by those who successfully and systematically invent new problems for different educational needs. In particular, surprisingly little empirically-based knowledge exists about posing problems for the actual use at mathematical competitions, exams or mathematics classrooms. This research lacuna may be considered as one of the reasons for which a comprehensive theoretical framework for investigating problem posing as an authentic mathematical activity is still not established (e.g., Christou et al., 2005; Singer et al., 2009). This essay outlines some of the effort of the Technion mathematics education research group towards constructing such a framework. An additional goal of this essay is to invite interested colleagues from different countries to collaborate on pursuing the research questions presented below.

A considerable portion of research on mathematical giftedness seeks to compare between problem-solving experiences of gifted and average-ability schoolchildren. In some comparative studies, either quantitative or qualitative, some of the identified differences can be (implicitly) embedded in the study design. In light of the evaluation criteria adopted from research on general intellectual abilities and problem-solving competences, such studies bear a danger of falling into the pitfall of circularity. The goal of this article is to discuss three ways of overcoming this pitfall. The discussion converges to methodological implications for evaluating past research and conducting further research on problem solving by mathematically gifted schoolchildren.

An operational definition offered in this paper posits learning as a multi-dimensional and multi-phase phenomenon occurring when individuals attempt to solve what they view as a problem. To model someone's learning accordingly to the definition, it suffices to characterize a particular sequence of that person's disequilibriumequilibrium phases in terms of products of a particular mental act, the characteristics of the mental act inferred from the products, and intellectual and psychological needs that instigate or result from these phases. The definition is illustrated by analysis of change occurring in three thinking-aloud interviews with one middle-school teacher. The interviews were about the same task: \u201cMake up a word problem whose solution may be found by computing 4/5 divided by 2/3.\u201d

This article discusses an issue of inserting mathematical knowledge within the problem-solving processes. Relatively advanced mathematical knowledge is defined in terms of three mathematical worlds; relatively advanced problem-solving behaviours are defined in terms of taxonomies of proof schemes and heuristic behaviours. The relationships between mathematical knowledge and problem-solving behaviours are analysed in the contexts of solving an insight geometry problem, posing algebraic problems and calculus exploration. A particularly knowledgeable and skilled university student was involved in all the episodes. The presented examples substantiate the claim that advanced mathematical knowledge and advanced problem-solving behaviours do not always support each other. More advanced behaviours were observed when the student worked within her conceptual-embodied mathematical world, and less advanced ones when she worked within her symbolic and formal-axiomatic worlds. Alternative explanations of the findings are discussed. It seems that the most comprehensive explanation is in terms of the Principle of Intellectual Parsimony. Implications for further research are drawn.

We report an on-going design experiment in the context of a compulsory calculus course for engineering students. The purpose of the experiment was to explore the feasibility of incorporating ideas of active learning in the course and evaluate its effects on the students' knowledge and attitudes. Two one-semester long iterations of the experiment involved comparison between the experimental group and two control groups. The data were collected from observations, research diary, course exams, attitude questionnaire and two additional questionnaires designed to explore patterns of students' learning behaviors. The (preliminary) results show that active learning can have a positive effect on the students' grades on condition that the students are urged to invest considerable time in independent study.

This theoretical essay presents a consolidated theoretical framework for analysing mathematical problem posing. The main feature of the suggested framework is that it builds upon broadly accepted problem solving models, and, simultaneously, includes theoretical constructs that are identified as specific for problem posing. The framework consists of four facets: resources, problem posing heuristics, aptness and social context in which problem posing occurs. The framework contributes to the existing research literature by consolidating findings on particular aspects of problem posing that have been explored so far and suggests a research agenda for further advancing the field.

This chapter describes the development of out-of-school activities designed to foster creativity and giftedness in the area of mathematics, and analyses the socialand historical circumstances that have affected these activities. The discussionextends to current problems, acute tasks, and possible future activities.

The goal of the presented study is to characterize the process of creating conditions for adopting alternative assessment in a pre-university institution adhering to traditional assessment in mathematics. In accordance with Roger's model of decision-making during diffusion of innovations, we identified the needs of such an institution that can be addressed using alternative assessment. We then explored which types of research evidence can convince the academic staff to implement alternative assessment. The most convincing evidence came from the statistical analysis. It showed that some of the traditional exams used at the institution had low predictive validity, and thus, could be substituted with alternative assessment tools with no risk of decreasing the predictive validity of overall grades.

This book breaks through in the field of mathematical creativity and giftedness. It suggests directions for closing the gap between research in the field of mathematics education and research in the field of creativity and giftedness. It also outlines a research agenda for further research and development in the field. The book consists of a balanced set of chapters by mathematicians, mathematics educators, educational psychologists and educational researchers. The authors of different chapters accept dynamic conception of creativity and giftedness. The book provides analysis of cognitive, affective and social factors associated with the development of creativity in all students and with the realisation of mathematical talent in gifted students. It contains theoretical essays, research reports, historical overviews, recommendations for curricular design, and insights about promotion of mathematical creativity and giftedness at different levels. The readers will find many examples of challenging mathematical problems intended at developing or examining mathematical creativity and giftedness as well as ideas for direct implementation in school and tertiary mathematics courses. They will also find theoretical models that can be used in researching students' creativity and giftedness.

This article presents an instructional approach to constructing discovery-oriented activities. The cornerstone of the approach is a systematically asked question If a mathematical statement under consideration is plausible, but wrong anyway, how can one fix it? or, in brief, If not, what yes? The approach is illustrated with examples from calculus and geometry. It is argued that the If not, what yes? approach facilitates conjecturing and proving, constructing meaningful examples and counterexamples and has a potential for creating learning situations, in which responsibility for achieving desirable mathematical results is devolved from an instructor to the learners.

In this theoretical essay I suggest that considerations of intellectual parsimony, in general, and balancing between different kinds of parsimony, in particular, is a mechanism explaining many well-documented phenomena in mathematical problem-solving. This suggestion is supported by re-analysis of data taken from three recently published research papers. Further, an attempt to incorporate the considerations of parsimony in selected theoretical models of problem solving is undertaken; some implications are drawn.

A clinical task-based interview can be seen as a situation where the interviewerinterviewee interaction on a task is regulated by a system of explicit and implicit norms, values, and rules. This paper describes how documenting and mapping triadic interaction among the interviewer, the interviewee, and the knowledge negotiated can be used to increase procedural replicability of the interview and accuracy of drawn conclusions about the interviewees thinking process. Excerpts from interviews with 25 inservice mathematics teachers working on a task to make up a problem whose solution requires division of two fractions are discussed. The excerpts illustrate the relationship between methodological decisions taken by the interviewer during the interview and the applicability of the interview output to the research questions. A divergent analysis of the interviews with these teachers, which spanned over two years and were conducted by four interviewers, is used to offer a framework for analyzing data collected in clinical task-based interviews.

The relationships between heuristic literacy development and mathematical achievements of middle school students were explored during a 5-month classroom experiment in two 8th grade classes (N = 37). By heuristic literacy we refer to an individual's capacity to use heuristic vocabulary in problem-solving discourse and to approach scholastic mathematical problems by using a variety of heuristics. During the experiment the heuristic constituent of curriculum-determined topics in algebra and geometry was gradually revealed and promoted by means of incorporating heuristic vocabulary in classroom discourse and seizing opportunities to use the same heuristics in different mathematical contexts. Students' heuristic literacy development was indicated by means of individual thinking-aloud interviews and their mathematical achievements - by means of the Scholastic Aptitude Test. We found that heuristic literacy development and changes in mathematical achievements are correlated yet distributed unequally among the students. In particular, the same students, who progressed with respect to SAT scores, progressed also with respect to their heuristic literacy. Those students, who were weaker with respect to SAT scores at the beginning of the intervention, demonstrated more significant progress regarding both measures.

Individual discourse concurrent with solving a mathematical problemthinking aloud discourseis considered a valuable source of information about thought processes in doing mathematics. Analysis of thinking-aloud discourse is associated with various methodological issues, such as validity or viability estimation and qualitative comparison of thinking-aloud protocols. The paper presents ways of dealing with these issues in an empirical study concerning middle-school students problem-solving behaviors. Based on examples from that study, we discuss research implications of particular analytical decisions and describe a \u201csafety net\u201d of a multistep process of drawing conclusions about the students problem-solving behaviors.

In this paper, we present an approach to exploring students' aesthetical preferences in mathematics. Based on analysis of 9 undergraduate students' responses and behaviors in two problem solving workshops, we report essential elements of a preliminary student-centred model of the notion "beautiful mathematical problem." The preliminary model includes cognitive, metacognitive and social factors and seeks to appreciate the complexity of the students' aesthetical judgements.

Heuristic literacy an individual's capacity to use heuristic vocabulary in discourse and to apply the selected heuristics to solution of routine and non-routine mathematical tasks was indirectly promoted in a controlled five-month classroom experiment with Israeli 8th grade students (N=92). The experiment achieved a moderate mean effect size, which is in line with some previous research on heuristics. The novel result of the study is that those students of the experimental group who were below sample average at the beginning of the experiment benefited from the heuristically-oriented intervention significantly more than the rest of the students. It is argued here that this is, in part, due to communicational aspects of the intervention.

We report repeated clinical interviews with 12 Israeli middle school students. The 8(th) graders of different aptitudes were instructed to think aloud while solving word problems and geometry problems that looked like familiar tasks, but, in fact, were not. Based on principles of constant comparison method, four patterns of the students' heuristic behaviors-naive, progressive, circular and spiral are distinguished. We also found that the interviewees demonstrated multiple heuristic behaviors both in algebra and geometry contexts and that the weight of naive heuristic behavior decreased from the first to the last interview.

This article deals with characterizing the mental act of problem posing carried out by middle-school algebra teachers. We report clinical task-based interviews, in which 24 teacher-participants were asked to think aloud while making up a story problem whose solution may be found by computing 4/5 divided by 2/3. The interview protocols were analysed in accordance with principles of grounded theory. Two types of ways of thinking emerged from the analysis and were validated: coordinating and utilizing reference points. The results show that success in doing the interview task is associated with coordinated approach and utilizing a particular reference point.

In this paper, we present SciTech, the summer international research program for talented high school students, organized in the Technion ? Israel Institute of Technology [1][1]The first author of this paper is proud of founding SciTech in 1992 when he was the head of the Harry and Lou Stern Family Science and Technology Youth Center at the Technion. Information on the various activities of the Youth Center aimed at promotion of excellence and interest in mathematics, science and technology of high school students can be found in the website of the Technion (http://www.mechina.technion.ac.il/en/index.html). Our major thesis is that while taking part in scientific projects under supervision by the Technion research stuff, the high school participants in SciTech are given the opportunity to work as researchers, and not just as learners. We begin with a brief discussion of the term ?good research.? This is followed by a description of SciTech: we outline how students are accepted to the program and what they do in it. The main part of this paper is devoted to discussion of some, not necessarily the most successful, mathematical projects, which we had the pleasure to supervise. We conclude the paper with general observations concerning the projects, based on our experience as SciTech mentors.

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.

The discovery of a three-dimensional (3D) extension of the classical Ceva's theorem by a student is discussed. Ceva's theorem provides the concurrency conditions f Cevians which states the segments connecting vertices of triangle with points on the opposite side. Ceva's theorem in space can solve many 3-D puzzles which form a source of learning activities either for extracurricular projects or in a regular classroom. It was also proved that the dihedral angle bisectors of a tetrahedron meet at one point, which is the center of the sphere inscribed in the tetrahedron.

We report on transformations of teachers beliefs about students heuristic strategies in problem solving. Twenty in-service teachers responded to questions on their and their students heuristic experience. Then two of them took part in a sixmonth teaching experiment focused on heuristic training of their 8-graders. We found that the teachers considerations on usefulness of particular heuristics and their pedagogical applicability were changed while learning through teaching. They also developed and used in their instructional practices an empirical scheme of students problem solving behavior.