Boris Koichu

Weizmann Institute of Science | weizmann · Department of Science Teaching

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...

Research–practice 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 teacher...

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 occurre...

Theorising is one of the most endorsed practices in contemporary mathematics education research. At the same time, such questions as “what counts for useful theorising?” and “how can theory inform practice?” are still subjects of debate. Arguably, this is in part because of the theoretical diversity characterising our field (e.g. Sriraman & English...

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 A...

This introduction outlines the seventeen contributions presented in the TWG23 at CERME13. The content of the contributions gravitates around five pivotal themes: large-scale implementations, theoretical underpinnings, problem-solving research, computational thinking paradigms, and practical teaching model implementations. Furthermore, two thematic...

Have you ever designed a challenging mathematical problem that you absolutely love, just to find that teachers hate it, that students don’t understand it, or that mathematicians find fault with it? The authors of this chapter comprise a diverse team of problem posers, including mathematics education researchers, teacher educators, and practicing te...

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: type...

Thius document includes a collection of the position documents presented at the International Workshop of Israel Science foundation. https://curiosity-isf.edu.haifa.ac.il/

PME confrence proceedings Plenary Lectures Working Groups Seminar Colloquium National Presentation Oral Communications Posters

Volume 4 Research Reports P – Z

Volume 2 https://pme46.edu.haifa.ac.il/ Research Reports A-G

Volume 3 https://pme46.edu.haifa.ac.il/ Research Reports H-O

Over five decades, the field of mathematics education research has generated a wealth of innovations aimed at improving the teaching and learning of mathematics. However, while mathematics education research has produced solid findings related to fundamental phenomena in teaching and learning mathematics, has constructed elaborate theoretical frame...

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 teach...

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 emoti...

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 imp...

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 dial...

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....

This introduction offers an overview of the eighteen contributions (15 papers and three posters) to the TWG23 at CERME12. The three thematic discussions that took place in this Thematic Working Group are addressed, namely: the role of change, matters of scaling, and the conception of stakeholders.

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 mat...

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...

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 CERME12’s TWG23. Afterwards we will describe three overarching themes that were discussed at this year’s meeting in TWG23: (1) the role of change and theory of change; (2) matters of scaling; and (...

This volume contains the papers presented at the International Conference Building on the Past to Prepare for the Future held from August 8-13, 2022, in King’s College, Cambridge, UK. It was the 16th conference organised by The Mathematics Education for the Future Project - an international educational and philanthropic project founded in 1986 and...

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.

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 c...

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 researc...

We are launching a new international research journal in mathematics education. “What?”, you might ask, “Doesn’t mathematics education already have a myriad of journals?” Yes, this is indisputable. A quick count will easily reveal more than a hundred, counting also regional and national journals. Yet, as the title of this journal reveals, Impleme...

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....

Anyone interested in mathematics has heard about Euclid’s Elements. Some scholars (e.g., Boyer 1991) refer to Elements as one of the most influential mathematical texts ever written. That Elements, written at about 300 BC, consists of 13 books that present a collection of definitions, propositions, and proofs related to geometry, and number theory...

Like many of our colleagues, we are fond of problem-solving instruction. As such, we adhere to both teaching for problem solving and teaching via problem solving in our courses for prospective and in-service teachers. This distinction – for problem solving vs. via problem solving – is referred to in many literature sources (e.g., Schroeder and Lest...

Epigraph: “People are eager for stories. Not dissertations. Not lectures. Not informative essays. For stories.” (Haven, 2007, p.8)

Many people, including ourselves, have been intrigued when first introduced to a story of inventing a formula for solving cubic equations. We invite the reader to recall a sixteenth-century story, in which characters possessing sonorous Italian names – del Ferro, Tartaglia, Cardano, and Bombelli – were on the stage. Different authors have depicted...

We stated previously that the tasks described in this book are triggered by our previous interactions with our students, many of whom are prospective teachers. There is a repeated “experience of disturbance” when students draw erroneous conclusions or offer suggestions which lead to erroneous conclusions. However, at times, ideas that appear at fir...

Disturbance is a multifaceted word whose use in the language overlaps but does not coincide with the use of such words as interference and perturbation. A Google search for “disturbance vs. perturbation” first led us to the WikiDiff website, where we read that disturbance refers to the act of disturbing, whereas perturbation refers to the state of...

Can we imagine a thrill experienced by a mathematician who sees an unexpected structure in a string of numbers (consider 43, 47, 53, 61, 71, 83, 97, 113, 131…) or by a mathematics learner who has a sudden illumination that leads her to formulating a statement about an interesting mathematical property, which she could not find in her textbook (cons...

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...

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 1st year mathematics courses for engineering students, within a project encompassi...

This chapter suggests a discursively oriented conceptualization of mathematical problem solving and then illustrates the conceptualization by putting it in use for rethinking results of two past studies on problem solving. In the first study, a practice of developing and using heuristic discourse was facilitated in two eighth-grade classes during a...

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 scie...

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...

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 wh...

This study aims to explore a phenomenon of a one-off manifestation of mathematical creativity on the part of a student, against the background of her normative and not especially creative behavior—a flash of creativity. We seek to elaborate on this phenomenon in terms of the 4P (person, press, process and product) model of creativity. Namely, most...

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 anal...

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...

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...

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, co...

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 inmathematics. The second example concerns aWhatsApp asynchronous forum opened by a...

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 shorta...

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 teach...

We illustrate how the classical dialogues – Galileo’s Dialogue on Infinity from Dialogues Concerning Two New Sciences, Plato’s 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 high...

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 knowledgeabl...

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 prob...

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 projec...

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 sol...

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 e...

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 suppor...

This paper presents a part of a larger study, in which we asked “How are learning and teaching of mathematics at high level linked to students’ general giftedness?” We consider asking questions, especially student-generated questions, as indicators of quality of instructional interactions. In the part of the study presented in this paper, we explor...

Little is known about instructional means by which the aesthetic experience of mathematics can be enhanced for undergraduate learners. This paper presents and discusses an iterative lesson design process towards creating an opportunity for students to appreciate the beauty of an unexpected solution to a challenging calculus problem. The lesson desi...

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operation...

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 three-week 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...

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 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 inq...

The aim of this chapter was to identify mathematics teachers' conceptions of the notion of “problem posing.” The data were collected from a web-based survey, from about 150 high school mathematics teachers, followed by eight semistructured interviews. An unexpected finding shows that more than 50% of the teachers see themselves as problem posers fo...

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 mathe...

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...

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 pre...

This paper is concerned with organizational principles of a pool of familiar problems of expert problem posers and the ways by which they are utilized for creating new problems. The presented case of Leo is part of a multiple-case study with expert problem posers for mathematics competitions. We present and inductively analyze the data collected in...

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 mat...

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 in...

“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 m...

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 innova...

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...

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 fac...

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...

Gilah Leder thoroughly reviews, within the constraints imposed by the structure of this volume, what seems to be a very rich but not fully institutionalized area of research enterprise. An important (and, probably, inevitable) decision about the reviewing strategy was “[not to attempt] to provide a unique definition of mathematically gifted student...

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 ado...

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 f...

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 disequilibrium–equilibrium phases in...

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...

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...

This chapter is aimed at bridging the gap between research on general giftedness and research in mathematics education. This is accomplished by designing a model of mathematical giftedness that qualifies problem-solving acts in relation to components of general giftedness. The problem-solving acts considered in the model are (a) solving a problem i...

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 balan...