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Teacher beliefs and practices related to them could provide varying opportunities for students to develop their agency related to learning math. The aim of the current paper is to explore how different math teacher beliefs (concerning the nature of mathematics, constructivist beliefs, and self-efficacy) relate to different dimensions of student agency in primary education. A two-level multilevel model was analyzed to find out to what extent selected teacher beliefs are associated with student math competence and agency dimensions. The study was conducted in Estonia and data from 3rd and 4th grade students (N = 1557) and their teachers (N = 121) was included in the analysis. The results showed that teacher self-efficacy beliefs, constructivist beliefs, and static views on the nature of mathematics were related to students' agency dimensions. Possible explanations and directions for further studies are discussed. [ABSTRACT FROM AUTHOR]

What do ethical relations look like in the context of the many injustices that pervade mathematics education? In this paper, I argue, first, that violence is the relation that characterizes much of contemporary mathematics education and, second, that understanding ethical relations requires considering mathematics as an equal actor in creating possible relations rather than simply treating it as a context for human relations. I examine how literature in care theory, emancipatory pedagogies, and mathematics education have framed ethical relationality and suggest that the feminist new materialist conceptualization of response-ability offers several contributions for rethinking agency, justice, and praxis for mathematics teachers concerned with addressing mathematical violence. [ABSTRACT FROM AUTHOR]

In Sweden, as in many other countries, programming has been added to the curriculum in recent years. This paper explores how and under what conditions programming is transformed into teaching in technology education. Teachers in grade 7–9 were interviewed about what and how they teach about programming in the context of technology education, and what influenced the choices they made. Data was analysed using content analysis. The results suggest that teachers mainly see programming as a medium to explore and understand technological systems and construction work. The results further implies that teachers are uncertain of what programming means in terms of practices and concepts, and how programming knowledge can be assessed. The process of transformation presents teachers with several challenges, related to their own knowledge of programming, quality of teaching materials, and knowing pupils' prior knowledge about programming. In addition, the results indicate that technology teachers work in isolation. In consequence, the intended interdisciplinary collaboration between technology and mathematics teachers on programming is largely absent. [ABSTRACT FROM AUTHOR]

This paper provides an empirical exploration of mathematics teachers' planned practices. Specifically, it explores the practice of foreshadowing, which was one of Wasserman's (2015) four mathematical teaching practices. The study analyzed n = 16 lessons that were planned by pairs of highly qualified and experienced secondary mathematics teachers, as well as the dialogue that transpired, to identify the considerations the teachers made during this planning process. The paper provides empirical evidence that teachers engage in foreshadowing as they plan lessons, and it exemplifies four ways teachers engaged in this practice: foreshadowing concepts, foreshadowing techniques, foregrounding concepts, and foregrounding techniques. Implications for mathematics teacher education are discussed. [ABSTRACT FROM AUTHOR]

Teachers' questioning plays an essential role in shaping collective argumentative discourse. This paper demonstrated that rationality dimensions in teacher questions can be assessed by adapting Habermas' three components of rationality. By coordinating Habermas' construct with Toulmin's model for argumentation, this paper investigated how two secondary mathematics teachers used rational questioning to support student participation in collective argumentation. This paper identified various ways in which two participating teachers used rational questioning to support student participation in argumentation via contributions of argument components. The results establish a theoretical connection between the use of rational questions and students' contributions of components of arguments. The results indicated that not all rational questions were associated with a component of argument, and rational questions may additionally support argumentation in general for the development of a culture of rationality. The study has implications in terms of theory and professional development of teachers. [ABSTRACT FROM AUTHOR]

In India, a curriculum reform inspired by critical perspectives has sought to transform primary mathematics teaching and learning. It is aimed at strengthening socio-cultural-political connections between school mathematics and students' life experiences, thereby challenging traditional textbook culture. At the same time, this initiative has retained the textbook as a vehicle of reform while seeking to subvert many of its established conventions. Guided by Remillard's idea of modes of engagement, this paper analyses the innovative Math-Magic textbooks associated with the Indian National Curriculum Framework. It investigates how these textbooks represent and communicate the framework ideas, focusing on key curricular elements and on the teacher as reader. Analysing the 'voice' and 'structure' of the textbooks as well as the 'contexts' used, it is revealed that they use a radically unique voice to introduce school mathematics while also attempting to use authentic and socially relevant contexts within their tasks. However, they have limited structural support to communicate these ideas clearly to the teacher-reader. The paper has implications for studying reformed textbooks in primary school mathematics in the Global South, where they remain the main teaching resource for teachers. Further, by focusing on 'context', the notion of modes of engagement within textbooks is extended through socio-cultural perspectives. [ABSTRACT FROM AUTHOR]

Proving and refuting are fundamental aspects of mathematical practice that are intertwined in mathematical activity in which conjectures and proofs are often produced and improved through the back-and-forth transition between attempts to prove and disprove. One aspect underexplored in the education literature is the connection between this activity and the construction by students of knowledge, such as mathematical concepts and theorems, that is new to them. This issue is significant to seeking a better integration of mathematical practice and content, emphasised in curricula in several countries. In this paper, we address this issue by exploring how students generate mathematical knowledge through discovering and handling refutations. We first explicate a model depicting the generation of mathematical knowledge through heuristic refutation (revising conjectures/proofs through discovering and addressing counterexamples) and draw on a model representing different types of abductive reasoning. We employed both models, together with the literature on the teachers' role in orchestrating whole-class discussion, to analyse a series of classroom lessons involving secondary school students (aged 14–15 years, Grade 9). Our analysis uncovers the process by which the students discovered a counterexample invalidating their proof and then worked via creative abduction where a certain theorem was produced to cope with the counterexample. The paper highlights the roles played by the teacher in supporting the students' work and the importance of careful task design. One implication is better insight into the form of activity in which students learn mathematical content while engaging in mathematical practice. [ABSTRACT FROM AUTHOR]

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

This paper reports on concept images of 38 secondary school mathematics prospective teachers, regarding the evenness of numbers. Written assignments, individual interviews, and lesson transcripts uncover salient, erroneous concept images of even numbers as numbers that are two times "something" (i.e., 2i is an even number), or to reject the evenness of zero. The notion of concept image serves in the analysis of the findings, and the findings serve in offering two refinement notions: mis-in concept images that mistakenly grant non-examples the status of examples (e.g., 2i is an even number), and mis-out concept images that mistakenly regard examples as non-examples (e.g., zero is not an even number). We discuss possible benefits in distinguishing between these two refinement notions in mathematics education. [ABSTRACT FROM AUTHOR]

This paper focuses on the importance of covariational reasoning within the processes of mathematics teaching and learning. Despite the internationally recognized relevance of covariation, research shows that only a small percentage of students and teachers is able to adopt covariational reasoning and the majority of mathematics curricula do not contain explicit references to covariational skills. In particular, when covariational reasoning manifests as conceptual knowledge, it becomes challenging to assess, and the need for innovative methods of assessment emerges; there is a need for suitable assessment to highlight the characteristics of covariation and capture the various features that characterize conceptual understanding. Comparative judgement (CJ) is an innovative assessment method based on collective expert judgements of students' work rather than requiring scoring rubrics. Due to its holistic approach, CJ is particularly suitable for assessing complex mathematical competencies, and, as we shall see in this study, it proved to be appropriate for the covariation's assessment. In details, our study aims to investigate the perception and relevance attributed by mathematics teachers to covariation as a theoretical construct and the way CJ can help in the assessment of covariational reasoning skills underlying a less structured modelling task. [ABSTRACT FROM AUTHOR]

Offering explanations is a central part of teaching mathematics, and understanding those explanations is a vital activity for learners. Given this, it is natural to ask what makes a good mathematical explanation. This question has received surprisingly little attention in the mathematics education literature, perhaps because the field has no agreed method by which explanation quality can be reliably assessed. In this paper, we explore this issue by asking whether mathematicians and undergraduates agree with each other about explanation quality. A corpus of 10 explanations produced by 10 mathematicians was used. Using a comparative judgement method, we analysed 320 paired comparisons from 16 mathematicians and 320 from 32 undergraduate students. We found that both mathematicians and undergraduates were able to reliably assess the quality of a set of mathematical explanations. Furthermore, the assessments were largely consistent across the two groups. Implications for theories of mathematical explanation are discussed. We conclude by arguing that comparative judgement is a promising technique for exploring explanation quality. [ABSTRACT FROM AUTHOR]

Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to students' self-efficacy across large time intervals (months or years), and paid less attention to changes at the level of lesson sequences. Knowledge of how self-efficacy changes during a sequence of lessons is important as it can help teachers better support students' self-efficacy in their everyday work. In this paper, we expanded previous studies by investigating changes in students' self-efficacy across a sequence of 3–4 lessons when students were learning a new topic in mathematics (nStudents = 170, nTime-points = 596). Nine classes of Norwegian grade 6 (n = 77) and grade 10 students (n = 93) reported their self-efficacy for easy, medium difficulty, and hard tasks. Using multilevel models for change, we found (a) change of students' self-efficacy across lesson sequences, (b) differences in the starting point and change of students' self-efficacy according to perceived task difficulty and grade, (c) more individual variation of self-efficacy starting point and change in association with harder tasks, and (d) students in classes who were taught a new topic in geometry had stronger self-efficacy at the beginning of the first lesson as compared to those who were taught a new topic in algebra (grade 10), and students in classes who were taught a new topic in fractions had steeper growth across the lesson sequence as compared to those who were taught a new topic in measurement (grade 6). Implications for both research and practice on how new mathematics topics are introduced to students are discussed. [ABSTRACT FROM AUTHOR]

This paper explores how different schools of thought in mathematics education think and speak about preparing mathematics for teaching by introducing and proposing certain metaphors. Among the metaphors under consideration here are the unpacking metaphor, which finds its origin in the Anglo-American school of thought of pedagogical reduction of mathematics; the elementarization metaphor, which has its origin in the German school of thought of didactic reconstruction of mathematics; and the recontextualization metaphor, which originates in the French school of thought of didactic transposition. The metaphorical language used in these schools of thought is based on different theoretical positions, orientations, and images of preparing mathematics for teaching. Although these metaphors are powerful and allow for different ways of thinking and speaking about preparing mathematics for teaching, they suggest that preparing mathematics for teaching is largely a one-sided process in the sense of an adaptation of the knowledge in question. To promote a more holistic understanding, an alternative metaphor is offered: preparing mathematics for teaching as ecological engineering. By using the ecological engineering metaphor, the preparation of mathematics for teaching is presented as a two-sided process that involves both the adaptation of knowledge and the modification of its environment. [ABSTRACT FROM AUTHOR]

Although there is much research exploring students' covariational reasoning, there is less research exploring the ways students can leverage such reasoning to coordinate more than two quantities. In this paper, we describe a system of covariational relationships as a comprehensive image of how two varying quantities, having the same attribute across different objects, each covary with respect to a third quantity and in relation to each other. We first describe relevant theoretical constructs, including reasoning covariationally to construct relationships between quantities and reasoning covariationally to compare quantities. We then present a conceptual analysis entailing three interrelated activities we conjectured could support students in reasoning covariationally to conceive of a system of covariational relationships and represent the system graphically. We provide results from two teaching experiments with four middle school students engaging in tasks designed with our conceptual analysis in mind. We highlight two different ways students reasoned covariationally compatible with our conceptual analysis. We discuss the implications of our results and provide areas for future research. [ABSTRACT FROM AUTHOR]

This paper reports from a case study which explores kindergarten children's mathematical abstraction in a teaching–learning activity about reflection symmetry. From a dialectical perspective, abstraction is here conceived as a process, as a genuine part of human activity, where the learner establishes "a point of view from which the concrete can be seen as meaningfully related" (van Oers & Poland Mathematics Education Research Journal, 19(2), 10–22, 2007, p. 13–14). A cultural-historical semiotic perspective to embodiment is used to explore the characteristics of kindergarten children's mathematical abstraction. In the selected segment, two 5-year-old boys explore the concept of reflection symmetry using a doll pram. In the activity, the two boys first point to concrete features of the sensory manifold, then one of the boys' awareness gradually moves to the imagined and finally to grasping a general and establishing a new point of view. The findings illustrate the essential role of gestures, bodily actions, and rhythm, in conjunction with spoken words, in the two boys' gradual process of grasping a general. The study advances our knowledge about the nature of mathematical abstraction and challenges the traditional view on abstraction as a sort of decontextualised higher order thinking. This study argues that abstraction is not a matter of going from the concrete to the abstract, rather it is an emergent and context-bound process, as a genuine part of children's concrete embodied activities. [ABSTRACT FROM AUTHOR]

We are interested in understanding how university students learn to use programming as a tool for "authentic" mathematical investigations (i.e., similar to how some mathematicians use programming in their research work). The theoretical perspective of the instrumental approach offers a way of interpreting this learning in terms of development of schemes by students; these development processes are called instrumental geneses. Nevertheless, how these schemes evolve has not been fully explained. In this paper, we propose to enrich the theoretical frame of the instrumental approach by a model of scheme evolution and to use this new approach to investigate learning to use programming for pure and applied mathematics investigation projects at the university level. We examine the case of one student completing four investigation projects as part of a course workload. We analyze the productive and constructive aspects of the student's activity and the dynamic aspect of the instrumental geneses by identifying the mobilization and evolution of schemes. We argue that our approach constitutes a new theoretical and methodological contribution deepening the understanding of students' instrumented learning processes. Identifying instrumented schemes illuminates in particular how mathematical knowledge and programming knowledge are combined. The analysis in terms of scheme evolutions reveals which characteristics of the situations lead to such evolutions and can thus inform the design of teaching. [ABSTRACT FROM AUTHOR]

Understanding the intricate quantifier relations in the formal definitions of both convergence and continuity is highly relevant for students to use these definitions for mathematical reasoning. However, there has been limited research about how students relearn previous school mathematics for understanding multiply quantified statements. This issue was investigated in a case study in a 5-week teaching unit, located in a year-long transition course, in which students were engaged in defining and proving sequence convergence and local continuity. The paper reports on four substantial changes in the ways students relearn school mathematics for constructing quantified statements: (1) endorse predicate as formal property by replacing metaphors of epsilon strips with narratives about the objects ε, Nε, and ∣an − a∣; (2) acknowledge that statements have truth values; (3) recognize that multiply quantified statements are deductively ordered and that the order of its quantifications is relevant; and (4) assemble multiply quantified statements from partial statements that can be investigated separately. These four changes highlight how school mathematics enables student to semantically and pragmatically parse multiply quantified statements and how syntactic considerations emerge from such semantic and pragmatic foundations. Future research should further investigate how to design learning activities that facilitate students' syntactical engagement with quantified statements, for instance, in activities of using formal definitions of limits during proving. [ABSTRACT FROM AUTHOR]

Visuospatial representations of numbers and their relationships are widely used in mathematics education. These include drawn images, models constructed with concrete manipulatives, enactive/embodied forms, computer graphics, and more. This paper addresses the analytical limitations and ethical implications of methodologies that use broad categorizations of representations and argues the benefits of dynamic qualitative analysis of arithmetical-representational strategy across multiple semi-independent aspects of display, calculation, and interaction. It proposes an alternative methodological approach combining the structured organization of classification with the detailed nuance of description and describes a systematic but flexible framework for analysing nonstandard visuospatial representations of early arithmetic. This approach is intended for use by researchers or practitioners, for interpretation of multimodal and nonstandard visuospatial representations, and for identification of small differences in learners' developing arithmetical-representational strategies, including changes over time. Application is illustrated using selected data from a microanalytic study of struggling students' multiplication and division in scenario tasks. [ABSTRACT FROM AUTHOR]

This study is part of the international survey studies on teachers' beliefs concerning the integration of children's literature in mathematics teaching and learning, and this paper reports the findings of the thematic analysis of open-ended survey responses elicited from 287 primary school teachers and teacher trainees in Taiwan. Using the seminal social psychology theory, the Theory of Planned Behaviour (Ajzen in Organizational Behavior and Human Decision Processes, 50, 179–211, 1991) to frame the findings, this study highlights 11 perceived barriers and 11 perceived enablers that are thought to influence the teachers' intention to integrate children's literature in their mathematics teaching. More specifically, we identified time constraint, lack of pedagogical knowledge and confidence, and resource constraint as being the most-cited perceived barriers, while pedagogical benefits, desire to improve teaching, and enabling social norms were identified as the top perceived enablers. Ultimately, this article offers several recommendations to address some of these key perceived barriers. [ABSTRACT FROM AUTHOR]

Rational number knowledge is a crucial feature of primary school mathematics that predicts students' later mathematics achievement. Many students struggle with the transition from natural number to rational number reasoning, so novel pedagogical approaches to support the development of rational number knowledge are valuable to mathematics educators worldwide. Digital game-based learning environments may support a wide range of mathematics skills. NanoRoboMath, a digital game-based learning environment, was developed to enhance students' conceptual and adaptive rational number knowledge. In this paper, we tested the effectiveness of a preliminary version of the game with fifth and sixth grade primary school students (N = 195) using a quasi-experimental design. A small positive effect of playing the NanoRoboMath game on students' rational number conceptual knowledge was observed. Students' overall game performance was related to learning outcomes concerning their adaptive rational number knowledge and understanding of rational number representations and operations. [ABSTRACT FROM AUTHOR]

We come together, five mathematics teacher educators from varied cultural backgrounds and diverse academic pathways, interested in teacher professional learning, and interested in exploring our understanding of colonial practices in mathematics education specifically and in education more generally. In this paper, we share our stories, drawing upon our own experiences in conversation with each other and in dialogue with academic readings. Our paper studies tensions encountered as we explore decolonizing educational practices within colonial structures, paying close attention to place/land-based pedagogies. In recognizing education as a colonial act, we examine some of the literature in decolonizing education and research before introducing arguments for the role of mathematics education in colonizing educational practices. As a collaborative research group, we met regularly over a period of 3 months, collecting audio recordings of our meetings, transcriptions, shared individual writings, and written responses to each other's writing and academic readings. We draw on narrative inquiry to structure our experiences of decolonizing mathematics teacher professional learning through living and telling stories and then through re-telling and re-living stories. We argue that this process is rewarding and challenging, and requires individual and collective ongoing dialogue. With our stories, we challenge the places where mathematics is performed, for example from classrooms to learning gardens, and what counts as mathematics. We conclude with questions to frame future directions and dialogues with an invitation to others to respond through critical dialogue and practice. [ABSTRACT FROM AUTHOR]

In this paper, we reflect on our experiences teaching and working with mathematics teachers in Canada, Ghana, India and Swaziland to explore challenges and opportunities for creating mathematics tasks for peace and sustainability. Our exploration of these experiences is oriented around our interest in embedding peace and sustainability into mathematics education following the Sustainable Development Goals identified by the United Nations. We claim that attention to local contexts affords mathematics educators a medium for engaging in authentic, meaningful and context-driven mathematics tasks that address issues of local environmental, cultural and societal concerns. However, we argue that globalisation has already colonised local communities. The associated "technoscientificity", along with the conservative nature of textbooks, time constraints and the dominant force of poverty remain hindrances in creating mathematics tasks that are issue centric and socially relevant to address concerns of the local community. We end this paper by suggesting a development of a mathematics task to illustrate the possibility of creating what we call a "situated mathematics task" for students that respond to issues of local community using rural Ghana as a context. [ABSTRACT FROM AUTHOR]

Teacher education programs should have as one of their purposes the promotion of self-regulatory skills for learning among students who aspire to be teachers so that they can take a leading role in their learning and foster these skills in their future students. Considering the importance of knowing what students in teacher education programs do to study and learn, as well as how efficacious they feel to deal with academic demands, this study is part of a larger research and aims to investigate the learning and study strategies and self-efficacy for learning beliefs of 220 students enrolled in teacher education programs in Biological Sciences, Chemistry, Physics and Mathematics of a Higher Education Institution in the state of Piauí, and examine them in relation to age, gender, licentiate area, and course semester. Brazilian translations of the Learning and Study Strategies Inventory (LASSI — Third Edition) and the Self-efficacy for Learning Form were used for data collection. Scales were administered in the classrooms both through online platforms and in paper and pencil. Nonparametric inferential statistical approaches were used to test hypotheses regarding group differences. Statistically significant differences were found in LASSI in relation to gender, licentiate area, and course semester. Overall, students in Physics dealt better with anxiety; in Mathematics showed more favorable attitudes towards learning; in Chemistry reported managing their time better; in Biological Science showed significantly lower scores on many scales than did other students. Findings from this study could help inform curricular design decisions regarding teacher education programs and inform the design of interventions to strengthen the learning and study strategies and the self-efficacy for learning beliefs of future teachers. [ABSTRACT FROM AUTHOR]

In 2020, the emergency due to the COVID-19 pandemic brought a drastic and sudden change in teaching practices, from the physical space of the classrooms to the virtual space of an e-environment. In this paper, through a qualitative analysis of 44 collected essays composed by Italian mathematics teachers from primary school to undergraduate level during the spring of 2020, we investigate how the Italian teachers perceived the changes due to the unexpected transition from a face-to-face setting to distance education. The analysis is carried out through a double theoretical lens, one concerning the whole didactic system where the knowledge at stake is mathematics and the other regarding affective aspects. The integration of the two theoretical perspectives allows us to identify key elements and their relations in the teachers' narratives and to analyze how teachers have experienced and perceived the dramatic, drastic, and sudden change. The analysis shows the process going from the disruption of the educational setting to the teachers' discovery of key aspects of the didactic system including the teacher's roles, a reflection on mathematics and its teaching, and the attempt to reconstruct the didactic system in a new way. [ABSTRACT FROM AUTHOR]

Wenger's theory of learning proposes the concepts of participation and reification as dimensions of educational design. Based on this proposal, this position paper looks at the design of one Mathematics Professional Development Program (MPDP) through the lenses of these concepts. We argue that the reification concept allows professional designers to keep in mind the resources they should use to organize teachers' learning, while the participation concept allows them to ponder the types of activities that should be implemented in order for the teachers to adequately assimilate the meaning of those resources. The example that serves as the context in which we situate our discussion is the Data Analysis Project, a professional development project on statistics in which 5 in-service secondary school teachers participated. Due to the dialectic relation between participation and reification, we explain how these two processes intertwine to maintain the balance needed to allow teachers' learning to proceed within an MPDP. We argue that this dialectic relation is the key element for MPDP designers to connect the resources produced by research in mathematics education to the practice of mathematics teaching. Such a connection allows an MPDP to act as a bridge between two communities of practice: that of researchers in the field of mathematics education and that of mathematics teachers. Ultimately, the article shows how Wegner's concepts of participation and reification may be used to design an MPDP. [ABSTRACT FROM AUTHOR]

Video can be a powerful tool, with a long history of use in teacher education. Despite the increased popularity of using video in teacher education over the years, questions remain about effective ways to create and use video in online video-based instructional modules. Given this, in this paper we describe some lessons learned creating videos for online video-based instructional modules for secondary mathematics teacher candidates, as a part of a National Science Foundation (NSF) IUSE program (Award No. 1726543) funded project. We specifically focus on how we recorded interviews with middle and high school students, and iteratively developed the online platform used for the project. We conclude with implications for practice. [ABSTRACT FROM AUTHOR]

This paper explores how in-service teachers enrolled in a graduate proof course interpret, understand, and use generic examples as part of their proving and justification activities. Generic examples, which are capable of proving and justifying with strong explanatory power, are particularly important for teachers considering teaching proof in their classrooms. The teachers in our study used generic examples to produce three types of proof: example-based arguments enhanced with generic language; incomplete generic examples; and complete generic examples. We found that teachers conflate generic examples and visual representations, prefer visual generic examples for teaching, and consider a generic example with symbolic representation to be more convincing than a generic example without. We conclude with implications for secondary school teaching, as well as suggestions for future professional development efforts. [ABSTRACT FROM AUTHOR]

One major obstacle for integrating the history of mathematics in teaching (hereafter referred to as IHT) is how to help teachers, particularly those who lack experience in IHT, use historical materials in their teaching. Based on many IHT practices in Shanghai, this paper proposes an IHT framework including two parts: one is a triangular pyramid IHT model, and the other is a design-based IHT procedure. The pyramid model is composed of three different communities: teachers, researchers, and historians of mathematics. The design-based procedure, also called operating model, includes cyclic stages, which are investigation, design, implementation, assessment, and publication. The primary purpose of this framework is to build a bridge between theory and practice by combining the two models to address particular teaching concerns and improve teaching. In addition, this paper details the process of the dynamic pyramid model and the design-based IHT procedure, and illustrates an example of the application of this framework. Finally, this paper discusses the differences and similarities between this model and others and also discusses the challenge of applying this model in practice. [ABSTRACT FROM AUTHOR]

Existing studies have quantitatively evidenced the relatedness between problem posing and problem solving, as well as the magnitude of this relationship. However, the nature and features of this relationship need further qualitative exploration. This paper focuses on exploring the interactions, i.e., mutual effects and supports, between problem posing and problem solving. More specifically, this paper analyzes the forms of interactions that happened between these two activities, the ways that those interactions supported prospective primary teachers' conceptual understanding, and the difficulties that prospective teachers encountered while engaged in alternating problem-posing and problem-solving activities. The results indicate that problem posing contributes to problem-solving effectiveness while problem solving supports participants in posing more reasonable problems. Finally, multiple difficulties that demonstrate prospective primary teachers' misunderstanding with fractions and their operations provide insight for teacher educators to design problem-posing tasks involving fractions. [ABSTRACT FROM AUTHOR]

Previous studies of in-service teachers indicate strong links between teacher self-efficacy and factors such as instructional quality and pupils' achievement. Yet, much of this research approaches self-efficacy from the perspective of teaching, and not of subject knowledge. Furthermore, the majority of such studies employ quantitative measures of self-efficacy. Drawing on semi-structured interviews with 22 experienced elementary teachers, this paper takes a different approach. The interviews, broadly focused on teachers' mathematics-related beliefs, brought to the surface four themes around which teachers construct their mathematics teacher self-efficacy. These concern participants' perspectives on their mathematics-related past experiences, mathematical competence, ability to realise their didactical visions and resilience in the face of challenging mathematical situations. These themes, which are discussed in relation to existing literature, not only confirm the complexity of self-efficacy but also highlight the need for greater attention to its conceptualisation and measurement. [ABSTRACT FROM AUTHOR]

Girls' underrepresentation in high-level post-compulsory mathematics is a longstanding issue of concern in many Western nations, with innumerable efforts to increase their participation producing little impact. In this paper, we shed new light on girls' underrepresentation through a post-structural feminist investigation of mathematics teachers' discursive constructions of high-level senior secondary mathematics students. Our analysis of semi-structured interviews with 22 Australian mathematics teachers revealed gendered views that serve to exclude many students from the high-level mathematics student category. Most concerning was their recurring naturalised construction of successful high-level mathematics students as endowed with the right, invariably male, brain. In so doing, teachers repeatedly closed off the possibility of success to those lacking such a 'mathematics gift', effectively 'counting girls out'. We argue that increasing girls' participation in mathematics requires moving beyond current efforts to raise female interest and confidence to, more profoundly, disrupt enduring discourses of male superiority in mathematics. [ABSTRACT FROM AUTHOR]

This paper describes outcomes from a project conducted at a small US university where 23 secondary school mathematics teachers participated in the study. Based on sketches created in a dynamic geometry software (DGS) and written responses to an assignment related to developing the trapezoid area formula, we explored how teachers used DGS when visualizing and verifying geometrical conjectures. We found that teachers used a combination of four strategies—eye-balling, measurement, constructions, and written statements—in their DGS sketches. Some teachers used DGS as paint software and their DGS files did not pass the dragging test. Teachers' misconceptions became obvious while they were visualizing conjectures in the software, which were mediated by the skill for use of technology. Also, some used shortcuts that disagreed with conjectures and distorted their solutions, thus employing practices that could be damaging in teaching situations. These results and the methodological approach used in this study will be of interest to mathematics teacher educators, facilitators of professional development programs, and mathematics teachers. [ABSTRACT FROM AUTHOR]

In this paper, we offer a framework for teacher monitoring routines—a consequential yet understudied aspect of instruction when teachers oversee students' working together. Using a comparative case study design, we examine eight lessons of experienced secondary mathematics teachers, identifying common interactional routines that they take up with variation. We present a framework that illuminates the common moves teachers make while monitoring, including how they initiate conversations with students, their forms of conversational entry, the focus of their interactions, when and how they exit the interaction as well as the conversation's overall participation pattern. We illustrate the framework through our focal cases, highlighting the instructional issues the different enactments engage. By breaking down the complex work of groupwork monitoring, this study informs both researchers and teachers in understanding the teachers' role in supporting students' collaborative mathematical sensemaking. [ABSTRACT FROM AUTHOR]

The epilogue, Chapter 7, describes Guccia's death and the decay of the Circolo from the interwar period through the Second World War, when the bombing of Palermo in 1943 destroyed the Palazzo Guccia. Two mathematicians particularly assisted Guccia in improving his mathematical society and the I Rendiconti i : the French Poincaré and the well-known Italian mathematician Vito Volterra (1860-1940). The other important mathematician in Guccia's scientific life was Poincaré, who supported Guccia throughout his career by contributing important papers to the I Rendiconti i . However, the book remains captivating, sheds light on a still little-known story, and contains unpublished documents on the origins of the Guccia family, the role of Guccia's uncle Giulio Fabrizio Tomasi, and the first years of Guccia's life. [Extracted from the article]

To assess the strength of conceptual understanding of number sense, a three-tier number sense test (TTNST) for fifth-grade students was developed and validated in this study. The three-tier test includes a content tier that assesses content knowledge of number sense, a reason tier that assesses a reason given for selecting a first-tier response, and a confidence tier that assesses how confident fifth-grade students are in their responses in the first two tiers. A total of 819 fifth-grade students from elementary schools in Taiwan participated in this study. Collected data showed that the test had good reliability and validity. The results revealed that many of the sample students performed poorly on number sense but maintained extremely high confidence; this indicated that many of the students held severe misconceptions and some were lacking in number sense. In addition, this study confirmed that the inclusion of a third tier (with confidence ratings) in the number sense test can mitigate the weaknesses of a previously used two-tier test. The educational implications of the findings are discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper reports findings from a case study of the impact that teaching using guided notes has on university mathematics students' note-taking behaviour. Whereas previous research indicates that students do not appreciate the importance of lecturers' non-written comments and record in their notes only what is written on the board when taught with the traditional chalk and talk method, some students in our study recorded the non-written comments as well as some of their own links between sections of the lecture. We did not, however, find students' attitude towards those comments to be different from what previous research found. We conclude that guided notes can be an appropriate way of teaching university mathematics but on their own cannot make the pedagogical intentions of the lecturer clearer to the students. We also found that the educational environment plays a big part for all aspects of student learning, including decisions related to note-taking during lectures. [ABSTRACT FROM AUTHOR]

In this paper, we take a semiotic/dialogic approach to investigate how a group of UK 12–13-year-old students work with hierarchical defining and classifying quadrilaterals. Through qualitatively analysing students' decision-making processes, we found that the students' decision-making processes are interpreted as transforming their informal/personal semiotic representations of "parallelogram" (object) to more institutional ones. We also found that students' decision-making was influenced by their inability to see their peers' points of view dialogically, i.e., requiring a genuine inter-animation of different perspectives such that there is a dialogic switch, and individuals learn to see the problem "as if through eyes of another," in particular collectively shared definitions of geometrical shapes. [ABSTRACT FROM AUTHOR]

Are students with special educational needs excluded from the reform promise of "mathematics for all"? This paper explores the discursive production of students with special educational needs in the context of professional development (PD) for collaborative problem-solving teaching. We held interviews with Chilean primary school teachers after their participation in PD and used a post-structural analysis to examine them. We turned to policy and institutional practices to understand the disability discourses that were evident. Teachers called on medical and deficit discourses to produce these students as abnormal and problematic in their learning of mathematics. Yet teachers also blurred the lines of categorisation between and within labels of special needs, including other students in these terms. Simultaneously, the reform PD created space for a counter discourse of ability. We suggest PD should help teachers of mathematics resist deficit discourses and see the ways in which experience may run contrary to them. [ABSTRACT FROM AUTHOR]

School discursive practices produce and reproduce acceptable notions of the good mathematics teacher, thereby shaping identity and agency in becoming a teacher. In this paper, I draw on key aspects of Bourdieu's social field theory-his conceptual 'thinking tools' and his reflexive sociology-to explore the relations and discourses of school mathematics classrooms as experienced by two novice secondary mathematics teachers. Presentation and analysis of interview transcript data, juxtaposed with fictional 'dear novice teacher' letters from the field, reveal the ways in which the two novice mathematics teachers carefully negotiate space for enacting agency amid institutional school 'others.' The reflections in this paper are made relevant for mathematics teacher education through a better understanding of novice mathematics teacher agency, including an account of how these two teachers are being 'schooled' on the structures and strategies of classroom practices. An additional contribution of this paper to theory in mathematics education lies in the approach to analysis that draws on Bourdieu's reflexive sociology, specifically the concept of a field of opinion, to introduce competing discourses offered by novice teachers in mathematics classrooms and by teacher educators/researchers in teacher education programs. [ABSTRACT FROM AUTHOR]

The work that mathematics teachers do is frequently mathematical in nature and different from other professions. Understanding and describing common ways that teachers draw upon their content knowledge in the practice of teaching is important. Building on the descriptions by McCrory et al. (Journal for Research in Mathematics Education 43(5) 584-615, ) of two primary ways teachers use their content knowledge-decompressing and trimming-this paper differentiates between micro- and macro-levels of mathematical complexity as a way to extend and further conceptualize the moves that teachers make at a more nuanced grain size. This paper explains and portrays their counterparts, micro-level trimming and macro-level decompressing, which we discuss as concealing and foreshadowing complexity, respectively. Support for their use in teachers' work is provided through specific examples. Implications for the mathematical preparation and professional development of teachers are discussed. [ABSTRACT FROM AUTHOR]

Collaborative work in small groups is often a suitable context for yielding substantial individual learning outcomes. Indeed, small-group collaboration has recently become an educational goal rather than a means. Yet, this goal is difficult to attain, and students must be taught how to learn together. In this paper, we focus on how to prepare teachers to become facilitators of small-group collaboration. The current case study monitors a group of six prospective teachers and their instructor during a one-semester course. The instructor was a skilled mathematics teacher with strong beliefs about what is entailed in establishing a mini-culture of learning to learn together and about how to facilitate student group work in problem-solving situations. We describe the learning path followed by the instructor, including the digital environment. The findings show that by the end of the course, the students became more competent facilitators of learning to learn together. [ABSTRACT FROM AUTHOR]

Many mathematical concepts may have prototypical images associated with them. While prototypes can be beneficial for efficient thinking or reasoning, they may also have self-attributes that may impact reasoning about the concept. It is essential that mathematics educators understand these prototype images in order to fully recognize their benefits and limitations. In this paper, I examine prototypes in a context in which they seem to play an important role: graphical representations of the calculus concept of the definite integral. I use student data to empirically describe the makeup of the definite integral prototype image, and I report on the frequency of its appearance among student, instructor, and textbook image data. I end by discussing the possible benefits and drawbacks of this particular prototype, as well as what the results of this study may say about prototypes more generally. [ABSTRACT FROM AUTHOR]

This paper deals with the competence of Cultural (epistemological, historical and anthropological) Analysis of the Content (CAC), which is important for teachers' awareness and autonomy when dealing with educational choices in a changing cultural and institutional context. We report on an 18-hour intervention in a teacher education course at the undergraduate level attended by 12 participants during the fall of 2013. The goal of the intervention was to enhance participants' CAC in the case of Elementary Arithmetic Theorems by using the rationality construct derived from Habermas' work. Qualitative analyses of participants' discussions showed how the construct was gradually appropriated by them and contributed to their CAC. Trends emerging from quantitative comparisons with previous implementations of the same course provided some evidence regarding the effects of the intervention on the quality of participants' CAC performances at the end of the course. The method chosen for participants' assessment and their cultural background will be discussed as conditions for the success of the intervention. [ABSTRACT FROM AUTHOR]

This paper proposes a framework for identifying the mathematical knowledge teachers activate when using curriculum resources. We use the term knowledge of curriculum embedded mathematics (KCEM) to refer to the mathematics knowledge activated by teachers when reading and interpreting mathematical tasks, instructional designs, and representations in mathematics curriculum materials. The KCEM framework is situated within existing research on content-specific teacher knowledge. Through analysis of elementary mathematics teacher's guides from the USA, we identified elements of curriculum resources teachers interact with when using them to plan instruction. These findings were complemented by interviews of teachers using curriculum guides to plan lessons, uncovering how they interacted with different elements of their guides. From this analysis, we propose four overlapping dimensions of KCEM: foundational mathematical ideas, representations and connections among these ideas, problem complexity, and mathematical learning pathways. Using an excerpt from one elementary curriculum guide and one of the teacher interviews on using this guide to plan a lesson, we illustrate how these dimensions might be activated as teachers read and use their guides. [ABSTRACT FROM AUTHOR]

This paper applies and extends Valsiner's 'zone theory' (zones of free movement and promoted actions) through an examination of an intervention to establish inquiry communities in primary mathematics classrooms. Valsiner's zone theory, in a classroom setting, views students' freedom of choice of action and thought as mediated by the teacher. The extension of this theory adds that teachers' freedom of choice of action and thought is mediated by 'significant others' and also by the actions of students with new freedoms. The paper presents necessary theoretical constructs and provides extracts from lessons and from teacher-teacher collaboration to illustrate the theoretical extension. [ABSTRACT FROM AUTHOR]

This paper is a case study of the teaching of an undergraduate abstract algebra course, in particular the way the instructor presented proofs. It describes a framework for proof writing based on Selden and Selden () and the work of Alcock () on modes of thought that support proof writing. The paper offers a case study of the teaching of a traditionally-taught abstract algebra course, including showing the range of practice as larger than previously described in research literature. This study describes the aspects of proof writing and modes of thought the instructor modeled for the students. The study finds that she frequently modeled the aspects of hierarchical structure and formal-rhetorical skills, and structural, critical, and instantiation modes of thought. This study also examines the instructor's attempts to involve the students in the proof writing process during class by asking questions and expecting responses. Finally, the study describes how those questions and responses were part of her proof presentation. The funneling pattern of Steinbring () describes most of the question and answer discussions enacted in the class with most questions requiring a factual response. Yet, the instructional sequence can be also understood as modeling the way an expert in the discipline thinks and, as such, offering a different type of opportunity for student learning. [ABSTRACT FROM AUTHOR]

Technology is available and accessible in many mathematics classrooms. Adopting technology to support teaching and learning requires teachers to change their teaching practices. This paper reports the responses of a diverse cohort of 92 secondary mathematics teachers who chose to respond to an Australian state-wide survey (Mathematics with Technology Perceptions Survey) developed using a Theory of Planned Behaviour framework. The items discussed in this paper targeted mathematics teachers’ perceptions of possible barriers and enablers to their intention to use technology in their teaching. The responses are varied but, overall, strength of agreement with enablers outweighed agreement with perceived barriers. However, it is clear that despite an overall positive attitude towards the use of technology for teaching mathematics, some perceived barriers to change are notable. It is, therefore, helpful if those responsible for professional development, promoting the use of technology, recognise and address these barriers as well as working to strengthening enablers. [ABSTRACT FROM AUTHOR]

This paper investigates the construction of systems of competence in two middle school mathematics classrooms. Drawing on analyses of discourse from videotaped classroom sessions, this paper documents the ways that agency and accountability were distributed in the classrooms through interactions between the teachers and students as they worked on mathematical content. In doing so, we problematize the assumption that competencies are simply attributes of individuals that can be externally defined. Instead, we propose a concept of individual competence as an attribute of a person's participation in an activity system such as a classroom. In this perspective, what counts as “competent” gets constructed in particular classrooms, and can therefore look very different from setting to setting. The implications of the ways that competence can be defined are discussed in terms of future research and equitable learning outcomes. [ABSTRACT FROM AUTHOR]

Building on the papers in this special issue as well as on our own experience and research, we try to shed light on the construct of example spaces and on how it can inform research and practice in the teaching and learning of mathematical concepts. Consistent with our way of working, we delay definition until after appropriate reader experience has been brought to the surface and several ‘examples’ have been discussed. Of special interest is the notion of accessibility of examples: an individual’s access to example spaces depends on conditions and is a valuable window on a deep, personal, situated structure. Through the notions of dimensions of possible variation and range of permissible change, we consider ways in which examples exemplify and how attention needs to be directed so as to emphasise examplehood (generality) rather than particularity of mathematical objects. The paper ends with some remarks about example spaces in mathematics education itself. [ABSTRACT FROM AUTHOR]

In this paper I shall discuss data from a study on Colombian mathematics teachers’ conceptions of their own teaching practices of beginning algebra, which led to the development of a theoretical model of teachers’ thought structures designed as a thinking tool at the initial stage of the study. With a focus on the perspectives of teachers, the study investigated the relationship between the teachers’ conceptions of beginning algebra and their conceptions of their own teaching practices, with a view to unravelling their conceptions of change in their practices. Significant findings which threw light on the aforementioned relationship have been presented in Agudelo-Valderrama, Clarke and Bishop (2007), highlighting a direct association between a teacher’s conceptions of the nature of beginning algebra, the crucial determinants of her/his teaching practice, and her/his attitude to change. After an overview of the study, this paper focuses on specific evidence which clearly shows that in contrast to the strong relationship between a teacher’s conceptions of mathematics and her/his teaching practice, assumed in the theoretical model of teachers’ thought structures, the teachers see a strong relationship between their conceptions of social/institutional factors of teaching and what they do in their teaching. Implications of the findings for teacher education in Colombia are identified. [ABSTRACT FROM AUTHOR]