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This paper reports on students' conceptions of minima points. Written assignments and individual interviews uncovered salient, concept images, as well as erroneous mis-out examples that mistakenly regard examples as non-examples and mis-in examples that mistakenly grant non-examples the status of examples. We used Tall and Vinner's theoretical framework to analyze the students' errors that were rooted in mathematical and in real-life contexts. [ABSTRACT FROM AUTHOR]

Mathematics education is like many scientific disciplines witnessing an increase in scientific output. Examining and reviewing every paper in an area in detail are time-consuming, making comprehensive reviews a challenging task. Unsupervised machine learning algorithms like topic models have become increasingly popular in recent years. Their ability to summarize large amounts of unstructured text into coherent themes or topics allows researchers in any field to keep an overview of state of the art by creating automated literature reviews. In this article, we apply topic modeling in the context of mathematics education and showcase the use of this data science approach for creating literature reviews by training a model of all papers (n = 336) that have been presented in Thematic Working Groups related to technology in the first eleven Congresses of the European Society for Research in Mathematics Education (CERME). We guide the reader through the stepwise process of training a model and give recommendations for best practices and decisions that are decisive for the success of such an approach to a literature review. We find that research in this period revolved around 25 distinct topics that can be grouped into four clusters: digital tools, teachers and their resources, technology experimentation, and a diverse cluster with a strong focus on student activity. Finally, a temporal analysis of these topics reveals a correlation between technology trends and research focus, allowing for reflection on future research in the field. [ABSTRACT FROM AUTHOR]

Over the last three decades, educational researchers and policymakers have increasingly promoted instructional strategies that centralize group work in mathematics. One difficulty teachers face in implementing group-based instruction in mathematics involves facilitating meaningful group interaction amongst students. In this paper, we explore student noticing as a novel strategy for supporting college students to collaborate effectively during group work in mathematics. First, we construct a noticing framework named student noticing of collaborative practices which provides a lens for "seeing" how students notice their collaborative practices. Then, we use the framework to explore how 25 college students noticed their collaborative practices in mathematics. After working on a novel mathematics task in groups, the college students listened to audiorecordings of their group interactions and responded to reflection questions about the effectiveness of their collaboration. We identified themes regarding how and what students noticed related to their collaborative practices. The findings reveal that students attended to many aspects of their collaboration, including their talking turns and propensity to listen to others. Students demonstrated a desire to change their collaborative practices in the future. The findings imply that teachers and researchers might leverage student noticing as a tool for improvement in mathematics group-based classrooms. [ABSTRACT FROM AUTHOR]

This is a commentary on the ESM 2021 Special Issue on Innovations in Measuring and Fostering Mathematical Modelling Competencies. We have grouped the ten studies into three themes: competencies, fostering, and measuring. The first theme and the papers therein provide a platform to discuss the cognitivist backgrounds to the different conceptualizations of mathematical modelling competencies, based on the modelling cycle. We suggest theoretical widening through a competence continuum and enriching of the modelling cycle with overarching, analytic dimensions for creativity, tool use, metacognition, and so forth. The second theme and the papers therein showcase innovative ideas on fostering and on the definition and analysis thereof. These reveal the need for a social turn in modelling research in order to capture aspects of student collaboration and agency, as well as tensions in fostering when tasks are derived from real-world scenarios, but socio-mathematical norms come from the (pure) mathematics classroom. The third theme, measuring, and the papers therein offer insights into the challenges of positivist research that aims to develop innovative measurement instruments that are both reliable and valid, particularly in light of student group work, cultural background, and other socio-cultural aspects. Drawing on the three discussions, we go on to make recommendations for further research. [ABSTRACT FROM AUTHOR]

Mathematical modelling competencies have become a prominent construct in research on the teaching and learning of mathematical modelling and its applications in recent decades; however, current research is diverse, proposing different theoretical frameworks and a variety of research designs for the measurement and fostering of modelling competencies. The study described in this paper was a systematic literature review of the literature on modelling competencies published over the past two decades. Based on a full-text analysis of 75 peer-reviewed studies indexed in renowned databases and published in English, the study revealed the dominance of an analytical, bottom-up approach for conceptualizing modelling competencies and distinguishing a variety of sub-competencies. Furthermore, the analysis showed the great richness of methods for measuring modelling competencies, although a focus on (non-standardized) tests prevailed. Concerning design and offering for fostering modelling competencies, the majority of the papers reported training strategies for modelling courses. Overall, the current literature review pointed out the necessity for further theoretical work on conceptualizing mathematical modelling competencies while highlighting the richness of developed empirical approaches and their implementation at various educational levels. [ABSTRACT FROM AUTHOR]

Mathematicians frequently attend their peers' lectures to learn new mathematical content. The goal of this paper is to investigate what mathematicians learned from the lectures. Our research took place at a 2-week workshop on inner model theory, a topic of set theory, which was largely comprised of a series of lectures. We asked the six workshop organizers and seven conference attendees what could be learned from the lectures in the workshop, and from mathematics lectures in general. A key finding was that participants felt the motivation and road maps that were provided by the lecturers could facilitate the attendees' future individual studying of the material. We conclude by discussing how our findings inform the development of theory on how individuals can learn from lectures and suggest interesting directions for future research. [ABSTRACT FROM AUTHOR]

Mathematical Internet memes are examples of how the creative thrust characterising the Web 2.0 environment reaches the field of mathematics, translating mathematical statements into a new digital form endowed with an epistemic potential that is capable of initiating a process of mathematical argumentation. The research presented in this paper aims to shed light on the creative process of mathematical memes, contributing to building a body of knowledge on mathematical memes that, prospectively, could enable educators to profit from these objects in their teaching. Theoretically, this is based on a widened concept of creativity that focuses on the connection linking digital culture with mathematics, and on distinguishing and merging three perspectives to disclose the meanings of mathematical memes. Methodologically, the process of mathematical memes' creation is investigated through a reverse engineering approach on a dataset of about 2100 items collected in a 3-year-long ethnographic observation within online communities. The result is a heuristic action model of the creation process, that is validated by creating two new mathematical Internet memes that are shared online within the observed communities to explore if they retain the mathematical and epistemic characteristics of Web-found ones. [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]

Variance and invariance are two powerful mathematical ideas to support geometrical and spatial thinking, yet there is limited research about teachers' knowledge of variance and invariance. In this paper, we examined how high school teachers deal with the task of looking for invariant properties in a dynamic geometry environment (DGE) setting. Specifically, we investigated if they even attend to invariant properties; what invariant properties they discern and discuss; and how DGE can support such discernment. Our analysis found that teachers tend to discern and discuss invariant properties mainly when they were probed to consider invariance. We also found four categories of invariant properties that seem to be important for a robust and rich understanding of geometric objects in the context of invariance and DGE. The use of DGE allowed teachers to see and interact with invariant properties, thus suggesting that accessing geometry dynamically may have structural affordances especially when exploring invariance. Teachers were able to enact different DGE movements to discern and discuss invariant properties, as well as to reason with and about them. We also saw that teachers' backgrounds and past experiences can play an important role in their descriptions of invariant properties. Possible future research directions and implications to teacher education are discussed. [ABSTRACT FROM AUTHOR]

Learning to calculate with natural numbers by structuring seems promising but how this can be taught in a sustainable manner remains an open question. An eight-month-long intervention based on the idea of using a structural approach to addition and subtraction, and particularly bridging through ten, was implemented in four Swedish first-grade classes. One goal was that by the end of first grade, students would be able to solve tasks such as subtracting 8 from 15 by using part-whole number relations. In this paper, we report on learning outcomes from task-based interviews with intervention and control groups before, immediately after, and one year after the intervention, in order to investigate long-term effects and whether students used a structural approach when solving tasks in a higher number range in the second grade. In comparison to controls, students in the intervention group showed higher increases in their learning outcomes. Moreover, the intervention group used a structural approach to a larger extent when solving tasks in higher number ranges, whereas students in the control group more commonly used single-unit counting to solve such tasks. These findings have implications both for teaching and for research on students’ development of arithmetic skills. [ABSTRACT FROM AUTHOR]

As argumentation is an activity at the heart of mathematics, (not only) German school curricula request students to construct mathematical arguments, which get evaluated by teachers. However, it remains unclear which criteria teachers use to decide on a specific grade in a summative assessment setting. In this paper, we draw on two sources for these criteria: First, we present theoretically derived dimensions along which arguments can be assessed. Second, a qualitative interview study with 16 teachers from German secondary schools provides insights in their criteria developed in practice. Based on the detailed presentation of the case of one teacher, the paper then illustrates how criteria developed in practice take a variety of different aspects into account and also correspond with the theoretically identified dimensions. The findings are discussed in terms of implications for the teaching and learning about mathematical argumentation in school and university: An emphasis on more pedagogical criteria in high school offers one explanation to the perceived gap between school and university level mathematics. [ABSTRACT FROM AUTHOR]

The metaphors that students form and encounter have been shown to exert a powerful influence on how they think about mathematics. In this paper, we explore the linguistic metaphors about learning and doing mathematics that were prevalent in 11 advanced mathematics lectures. We present four metaphor clusters that were common in the corpus that we analyzed: Learning Mathematics is a Journey, Doing Mathematics is Work, Mathematics is Discovery, and Presenting Mathematics is a Story. We conclude the paper by discussing the implications of the metaphors that mathematicians use and students hear in their mathematics courses. [ABSTRACT FROM AUTHOR]

Teaching mathematical modeling is a demanding task. Thus, fostering teachers' competencies in this regard is an essential component of teacher education. Recent conceptualizations of teachers' competencies include situation-specific skills based on the concept of noticing, which is of particular interest for the spontaneous reactions needed when teaching mathematical modeling. The study described in this paper aims to analyze the development of a video-based instrument for measuring teachers' noticing competencies within a mathematical modeling context and obtain evidence for the validity of the instrument. Three kinds of validity are examined in three different studies: content validity, elemental validity and construct validity. Indicators for content validity could be found through different expert ratings and implementation with the target group, where participants were able to perceive all relevant aspects. The qualitative analysis of participants' reasoning, which is consistent with the coded level, indicates elemental validity. Moreover, the results of the confirmatory factor analysis suggest construct validity with one overall factor of noticing competence within a mathematical modeling context. Taken together, these studies imply a satisfactory validity of the video-based instrument. [ABSTRACT FROM AUTHOR]

This paper reports on empirical results about the influence of two different teaching designs on the development of tertiary students' modelling competency and attitudes towards modelling. A total of 144 first year engineering students were exposed to a diagnostic entrance test, a modelling unit consisting of five lessons with ten tasks, enframed by a pre- and a post-test, and at the end a questionnaire on attitudes towards mathematical modelling. Similar to the German DISUM study, in the modelling unit, one group of participants followed an independence-oriented teaching style, aiming at a balance between students' independent work and teacher's guidance, while two other groups were taught according to a more traditional teacher-guided style. Linear mixed regression models were used to compare pre- and post-test results. The results show that all groups had significant learning progress, although there is much room for further improvement, and that the group taught according to the independence-oriented design had the biggest competency growth. In addition, this group exhibited more positive attitudes than the other groups in five of six attitudinal aspects. [ABSTRACT FROM AUTHOR]

Although mathematical modeling plays an important role in many curricula worldwide, significant discrepancies persist in the importance of mathematical modeling in ordinary mathematics classrooms and teacher education. This paper compares pre-service mathematics teachers' professional mathematical modeling competencies in three different regions—Germany, Mainland China, and Hong Kong—where educational and cultural traditions differ, including the role of mathematical modeling. In total, 232 pre-service mathematics teachers from the three regions completed a modeling task covering mathematics content knowledge (MCK) of modeling and mathematical pedagogical content knowledge (MPCK) of modeling. The results show that pre-service teachers from Germany demonstrated the strongest MCK and MPCK of mathematical modeling; by contrast, pre-service mathematics teachers from Mainland China and Hong Kong demonstrated relatively weaker MCK and MPCK of mathematical modeling. MCK and MPCK of mathematical modeling were also found to be unevenly developed at different competence levels for the three regions. These differences may be attributed to the history of mathematical modeling in mathematics curricula, teacher education, and teaching culture in these three regions. [ABSTRACT FROM AUTHOR]

Modelling competencies are currently included in numerous curricula worldwide and are generally accepted as a complex, process-oriented construct. Therefore, effective measurement should include multiple dimensions, like the sub-competencies required throughout the modelling process. Departing from the characteristics of modelling problems as open and often underdetermined real-world problems, we propose to enrich the current conceptualisation of mathematical modelling competencies by including creativity, which plays an important role in numerous phases of the mathematical modelling process but has scarcely been considered in modelling discourse. In the study described in this paper, a new instrument for the evaluation of this enriched construct has been developed and implemented. The modelling competencies incorporating creativity of the students were evaluated based on the adequacy of the models and the modelling processes proposed, and the appropriateness and completeness of the approaches were evaluated in detail. Adapting measurement approaches for creativity that have been developed in the problem-solving discourse, certain criteria of creativity were selected to evaluate the creativity of the students' approaches in tackling modelling problems—namely, usefulness, fluency, and originality. The empirical study was conducted among 107 Chinese students at the upper secondary school level, who attended a modelling camp and independently solved three complex modelling problems. The results reveal significant correlations between fluency and originality in students' performances across all tasks; however, the relationships between usefulness and the other two creativity aspects were not consistent. Overall, the results of the study support the importance of the inclusion of creativity in the construct of modelling competencies. [ABSTRACT FROM AUTHOR]

This paper discusses the place of precision in mathematics education by exploring its role in curricular guidelines and in classroom life. By means of a joke on precision delivered by a school student in South Sweden, our study focuses on student participation in mathematical tasks that require precision in processes of measuring and reasoning. The paper uses theories on humour and inferentialism to revisit the normative place of "precision" in mathematics classroom discourse. [ABSTRACT FROM AUTHOR]

This paper focuses on the emergence of abstraction through the use of a new kind of motion detector—WiiGraph—with 11-year-old children. In the selected episodes, the children used this motion detector to create three simultaneous graphs of position vs. time: two graphs for the motion of each hand and a third one corresponding to their difference. They explored relationships that can be ascribed to an equation of the type A – B = C. We examine the notion of abstraction on its own, without assuming a dualism abstract-concrete according to which more of one is less of the other. We propose a distinct path for the attainment of abstraction, which involves navigating a surplus of sensible qualities. The work described in this paper belongs to early algebra, we suggest, because it involves the elementary symbolic treatment of unknowns and generals. More broadly, it advances a perspective on the nature of mathematical abstraction. [ABSTRACT FROM AUTHOR]

Assessment is a crucial factor in the implementation of curriculum reform. Little is known, however, on how curriculum changes can be reflected adequately in assessment, particularly if the reform concerns process skills. This issue was investigated for the case of assessing mathematical thinking in a mathematics curriculum reform for 15–18-year-old students in the Netherlands. From 2011 until 2017, these reform curricula were field tested in pilot schools, while other schools used the regular curricula. The research question is how this reform was reflected in national examination papers and in student performance on corresponding assignments. To address this question, we developed a theory-based model for mathematical thinking, analyzed pilot and regular examination papers, and carried out a quantitative and qualitative analysis of students' work on assignments that invite mathematical thinking. The results were that the pilot examination papers did address mathematical thinking to a greater degree than the regular papers, but that there was a decrease over time. Pilot school students outperformed their peers in regular schools on assignments that invite mathematical thinking by 4–5% on average and showed more diversity in problem-solving strategies. To explain the decreasing presence of mathematical thinking in examination papers, we conjecture that conservative forces within the assessment construction process may push back change. [ABSTRACT FROM AUTHOR]

In this paper, motivated by the desire to understand which forms of parent-initiated activity are productively implicated in young children's mathematics learning, we present a methodological critique of recent research. Many such studies, based on assumptions that parent-initiated activities can be categorised as formal or informal, direct or indirect, or advanced or basic, exploit surveys to elicit how frequently parents engage their children in various predetermined activities. While such survey data have the potential to yield important insights, the analytical procedures typically employed prevent them. Studies involving factor analyses yield uninterpretable factors, which are then used to create summative variables based on the scores of individual activities. Other studies, drawing on untested preconceptions, simply create summative variables. In all cases, these summative variables are based on such a wide range of qualitatively different activities that labels like formal or informal become arbitrary and the potential of individual activities to support learning gets lost beneath colleagues' desires for statistical significance. In closing, we ask colleagues, albeit somewhat rhetorically, what is the purpose of such research? Is it to identify those activities that support learning or to offer statistically robust factors, which, due to the diversity of activities embedded within them, offer few useful insights? [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]

This paper explores peer interactions in an elementary mathematics classroom (ages 9–10) where the teacher intentionally shared authority with her students and supported them in learning to share authority with one another. Authors examine how students shifted between shared, concentrated, and contested social and intellectual authority relations in partner and small group work during a three-week unit on place value. Findings show that (a) students were able to share both social and intellectual authority, and did so often; (b) the distribution of social authority was more dynamic than that of intellectual authority; and (c) when groups shifted into shared intellectual authority, shifts were usually preceded by a student making some aspect of the collaborative task public. We connect these findings to research on authority in mathematics classrooms that serve racially and linguistically minoritized students and offer directions for future work. [ABSTRACT FROM AUTHOR]

This paper focuses on the mathematical sensemaking by women of color in the USA as part of the global effort of dismantling deficit narratives about historically marginalized groups of students. Following Adiredja's anti-deficit framework for sensemaking, this cognitive study invited a group of women of color to share their understanding of basis from linear algebra to construct a sensemaking counter-story. Extending the framework, this study examines a task that explores the boundaries and nuances of a concept to support the effort of going beyond students' deficits. Eight women extended the concept of basis (and vector spaces) to 22 distinct everyday contexts, drawing from their everyday lives as well as topics from their academic experiences. Their explanations revealed analytical codes describing roles and characteristics of a basis. These codes suggest ways that students can mobilize the concept of basis beyond its logical underpinnings. Contrasting interpretations using a deficit and an anti-deficit perspective construct a counter-story that showcases these women's creativity and flexibility in understanding the concept, and potential resources for the teaching and learning of linear algebra. [ABSTRACT FROM AUTHOR]

This paper reports changes of the quality of instruction and students' attitudes towards mathematics at one high school during 3 years while teachers participated in a teacher community. The setting for this research was a small-sized high school in South Korea with 154 students and three mathematics teachers. I analyzed the teachers' mathematics classes based on types of interaction, types of questions, and mathematical quality in instruction. Their students' responses for the National Assessment of Educational Achievement study about attitudes towards mathematics were also analyzed. The findings show that this school, over all, improved the mathematical quality and changed the types of questions and the types of interactions towards students' participation in discussing, exploring, and probing mathematical meanings. The students' attitudes towards mathematics became more positive throughout the 3-year period. However, the changes in each teacher's mathematics instruction were different. This study discusses the challenges of having an effective teacher community and researching teachers as learners. [ABSTRACT FROM AUTHOR]

This paper describes elementary students' awareness and representation of the aggregate properties and variability of data sets when engaged in predictive reasoning. In a design study, 46 third-graders interpreted a table of historical temperature data to predict and represent future monthly maximum temperatures. The task enabled students to interpret numbers in context and apply their understanding of inherent natural variation to create a generalised data set. Student predictions, representations, and written and verbal descriptions were analysed using two frameworks—Awareness of Mathematical Pattern and Structure (AMPS), and Data Lenses. While 54% of students used the variability of the given data table to predict temperatures that were within the historical range for each month, only 20% described the table by focusing on aggregate properties. Student representations varied from highly structured line and bar graphs to idiosyncratic drawings on weather-related themes. In total, 83% of student representations were either idiosyncratic or direct copies of the data table. These findings suggest a progression in students' predictive reasoning, with an awareness of range and seasonal patterns emerging before a multifaceted aggregate view. [ABSTRACT FROM AUTHOR]

A great deal of progress has been made in dealing with the multiplicity and diversity of theories in mathematics education. However, relatively little attention has been paid to the opportunities offered by conflicts, tensions, and paradoxes among accepted yet opposing theoretical perspectives for theory building and theory advancement. In this paper, four modes of dealing with opposing perspectives are outlined: (1) taking contrasting theoretical perspectives as incommensurable; (2) holding opposites not as conflicting but as complementary; (3) dissolving or surpassing oppositions by blending perspectives; and (4) preserving paradoxes by recognizing the interdependence of constitutive oppositions. These four modes are illustrated by application to the long-standing debate of knowledge-in-structures versus knowledge-in-pieces and further exemplified by turning to the research literature on students' understanding of limit. [ABSTRACT FROM AUTHOR]

The networking of theories is an increasingly common and powerful approach to analyzing complex phenomena such as learning processes in classrooms. In this paper, we aim to advance the theoretical coordination of two approaches that we have previously combined to analyze individual, small group, and whole class mathematical progress. The theoretical advances we make are twofold. First, we identify and illuminate environmental and internal-theoretical commonalities across the two approaches, commonalities that contribute to the productivity of networking. Second, we propose an argumentative grammar for the networking, thus elevating the methodological logic and rationale of networking in this case and potentially in general. [ABSTRACT FROM AUTHOR]

This paper presents the findings from a survey used to investigate how mathematicians perceive the genre of mathematical proof writing at the undergraduate level. Mathematicians were asked whether various proof excerpts highlighted in four partial proofs were unconventional in each one of three pedagogical contexts: undergraduate mathematics textbooks, what instructors write on the blackboard in undergraduate mathematics courses, and how students write mathematics in these courses. There are four main findings. First, there are some potential breaches of mathematical language that participants found unconventional regardless of the context in which they occur. Second, there are some differences in how mathematicians perceived the linguistic conventions in blackboard proofs and student-produced proofs. Third, textbook authors are expected to adhere to stricter writing norms than mathematics instructors and undergraduate students when writing proofs. Fourth, there were some potential breaches of mathematical language that the literature suggests were unconventional, which were not evaluated as unconventional by the mathematicians. We argue that this diversity of expectations regarding the language of mathematical proof writing in undergraduate classrooms, together with the potential disconnect between those expectations and the types of proofs that students see presented in those classrooms, could make it difficult for students to become proficient in this important mathematical discourse. [ABSTRACT FROM AUTHOR]

We report a new phase of research of the scaling of Cornerstone Maths (CM), technology-enhanced curriculum units for lower secondary mathematics that embed dynamic mathematical technology (DMT). These combine web-based DMT, pupil and teacher materials, and teacher professional development that focus on developing the mathematical knowledge and pedagogies for teaching with technology. This paper presents the background research with teachers using CM (111 teachers from 42 London secondary schools in the period 2014–2017) that suggests the need for a web-based "professional development toolkit" to support the sustainability of the innovation and "within-school" scaling beyond the timeline of the funded project. It concludes with the research basis of the toolkit's design principles and structure that are designed to support teachers to implement DMT in their classrooms. [ABSTRACT FROM AUTHOR]

Reversibility of concepts, a key aspect of mathematical development, is often problematic for learners. In this theoretical paper, we present a typology we have developed for categorizing the different reverse concepts that can be related to a particular initial concept and explicate the relationship among these different reverse concepts. We discuss uses of the typology and how reversibility can be understood as the result of reflective abstraction. Finally, we describe two strategies for promoting reversibility that have distinct uses in terms of the types of reverse concepts they engender. We share empirical results which led to our postulation of these strategies and discuss their theoretical basis. [ABSTRACT FROM AUTHOR]

This paper focuses on kindergarten teachers' interactions with young children during mathematical learning activities involving the use of digital tools. We aim to characterise the teachers' roles and actions in these activities and extend considerations of teachers' orchestrations current in the research literature with regard to agency and mediation. Our analysis of teacher-children-digital tool interaction reveals that the kindergarten teachers took three roles in their work with young children, which we call Assistant, Mediator and Teacher roles. These roles were used interchangeably and purposefully by the kindergarten teachers. With regard to agency and mediation, we argue that agency is distributed over the human and non-human agents in the activity and that agency and mediation are interrelated. [ABSTRACT FROM AUTHOR]

Textbooks have the potential to be powerful tools to help students develop an understanding of mathematics. However, many students are unable to use their textbooks effectively as learning tools. This paper presents a framework that can be used to analyze factors that impact the ways students read textbooks. It adapts ideas from reader-oriented theory and applies them to the domain of mathematics textbooks. In reader-oriented theory, the reader is viewed as actively constructing meaning from a text through the reading process; this endeavor is shaped and constrained by the intentions of the author, the beliefs of the reader, and the qualities the text requires the reader to possess. This paper also discusses how reading mathematics textbooks is further constrained by the authority and closed structure of these textbooks. After describing the framework, the paper discusses recommendations for future avenues of research and pedagogy, highlighting the importance of teachers' roles in mediating their students' use of textbooks. [ABSTRACT FROM AUTHOR]

This paper examines a Special Issue of Educational Studies in Mathematics comprising research reports centred on Peircian semiotics in mathematics education, written by some of the major authors in the area. The paper is targeted at inspecting how subjectivity is understood, or implied, in those reports. It seeks to delineate how the conceptions of subjectivity suggested are defined as a result of their being a function of the domain within which the authors reflexively situate themselves. The paper first considers how such understandings shape concepts of mathematics, students and teachers. It then explores how the research domain is understood by the authors as suggested through their implied positioning in relation to teachers, teacher educators, researchers and other potential readers. [ABSTRACT FROM AUTHOR]

The basic assumption of this paper is that mathematics and history of mathematics are both forms of knowledge and, therefore, represent different ways of knowing. This was also the basic assumption of Fried (2001) who maintained that these ways of knowing imply different conceptual and methodological commitments, which, in turn, lead to a conflict between the commitments of mathematics education and history of mathematics. But that conclusion was far too peremptory. The present paper, by contrast, takes the position, relying in part on Saussurean semiotics, that the historian's and working mathematician's ways of knowing are complementary. Recognizing this fact, it is argued, brings us to a deeper understanding of ourselves as creatures that do mathematics. This understanding, which is a kind of mathematical self-knowledge, is then proposed as an alternative commitment for mathematics education. In light of that commitment, history of mathematics assumes an essential role in mathematics education both as a subject and as a mediator between the aforementioned ways of knowing. [ABSTRACT FROM AUTHOR]

This paper draws on data from the development of annual national mathematics assessment materials for 7-year-old pupils in Wales for use during the period 2000–2002. The materials were developed in both English and Welsh and were designed to be matched. The paper reports on item analyses which sought items that exhibited differential performance in relation to whether the materials were English medium or Welsh medium. The items that exhibited consistent differential item functioning in relation to language during pre-testing are reviewed in order to discuss the linguistic factors that could affect such behaviour. [ABSTRACT FROM AUTHOR]

This paper presents the results of a study developed in the context of indigenous teachers education from Xingu Indian Park, Brazil. The indigenous bilingual (or multilingual in some cases) teachers that participated in this education program were from 14 ethnic groups. The research focused on a mathematics textbook production, written in indigenous language by indigenous teachers to be used at schools in the Park. The paper discusses the numeracy-literacy practices in this process and focuses on the meanings, values and ways of use that are related to numbers, writing and drawing. In particular, mathematics problems written by the indigenous teachers (in indigenous language and Portuguese) are analyzed. The analysis shows how aspects of orality influence the writing of these problems. [ABSTRACT FROM AUTHOR]

The framework for this paper is a recently developed theory of abstraction in context. The paper reports on data collected from one student working on tasks concerned with absolute value functions. It examines the relationship between mathematical constructions and abstractions. It argues that an abstraction is a consolidated construction that can be used to create new constructions. Newly formed constructions are fragile entities. In the course of consolidating a construction this student creates connections between the new construction and already established mathematical knowledge and develops a language to describe and guide mathematical actions related to the construction. The resulting abstraction is a more resilient form of the construction and the student is able to justify his assertions. The paper also examines issues in designing tasks to consolidate a construction. [ABSTRACT FROM AUTHOR]

This paper reports a survey of research in mathematics teacher education from 1999 to 2003 done by an international team of five mathematics educators and researchers. The survey included published research in international mathematics education journals, international handbooks of mathematics education and international mathematics education conference proceedings. Some regional sources from various parts of the world were also included. We investigated who was writing, from and in what settings, with what theoretical frameworks, and with what sorts of study designs for what core questions. We also examined the range of findings and conclusions produced in these studies. Our analysis presented here focuses on four themes that stood out from our initial investigation of almost 300 published papers, and systematically elaborated through a focused study of a 160 papers across key journals and conference proceedings in the field. From this vantage point, the paper offers a reflection on the current state of the field of mathematics teacher education research. Our aim is to stimulate discussion that can support the development of the field, not make final pronouncements about its nature. [ABSTRACT FROM AUTHOR]

This paper proposes a genetic development of the concept of limit of a sequence leading to a definition, through a succession of proofs rather than through a succession of sequences or a succession of εs. The major ideas on which it is based are historical and depend on Euclid, Archimedes, Fermat, Wallis and Newton. Proofs of equality by means of inequalities precede the notion of limit. For example, the determination of the volume of a pyramid precedes the definition of limit. The algebraic details given here are anachronistically modern, and the notion of a vice between two inequalities which provides the distinctive perspective of this paper would not have been recognized in this form by any of the named mathematicians since it presumes the existence of negative numbers and their comparability in a modern sense with positive numbers. [ABSTRACT FROM AUTHOR]

This paper applies APOS Theory to suggest a new explanation of how people might think about the concept of infinity. We propose cognitive explanations, and in some cases resolutions, of various dichotomies, paradoxes, and mathematical problems involving the concept of infinity. These explanations are expressed in terms of the mental mechanisms of interiorization and encapsulation. Our purpose for providing a cognitive perspective is that issues involving the infinite have been and continue to be a source of interest, of controversy, and of student difficulty. We provide a cognitive analysis of these issues as a contribution to the discussion. In this paper, Part 1, we focus on dichotomies and paradoxes and, in Part 2, we will discuss the notion of an infinite process and certain mathematical issues related to the concept of infinity. [ABSTRACT FROM AUTHOR]

This paper is part of a broader study that draws on Wenger's (Wenger, E.: 1998, Communities of Practice: Learning, Meaning, and Identity, Cambridge University Press, New Work) social practice perspective to investigate teacher learning. The study extends Wenger's complex model of interrelated components of leaning (as meaning, practice, identity and community) to describe and explain teacher learning that occurs within a mathematics senior-phase in-service program that was stimulated by curriculum change. The study uses qualitative ethnography in which the researcher performs the dual role of both coordinator and researcher of the in-service practice. In a longitudinal study the phenomenon of confidence emerged in teachers' descriptions and explanations of their learning. In this paper I explore this phenomenon both empirically and theoretically. The extension of Wenger's (1998) theory to include the overarching and interacting component of confidence is embedded in and derived from data analysis of 10 teachers' learning, over a 2-year period, during a time of radical curriculum change. Since it would be incoherent within this framework to draw on psychological explanations of confidence I set out to explore confidence from within a social practice frame in a way that is grounded in data of the teachers in this study. The paper offers a concept of confidence in relation to teacher learning as 'learning as mastery', and confidence as both a product and a process of learning. Teachers can at once state their confidence as mathematics teachers, and their confidence to admit to what they do not know and still need to learn. It is argued that this is a primary condition for ongoing learning in a profession like mathematics teaching. In addition, the paper provides a critique of the applicability of Wenger's work to the context of teacher education and in particular highlights the absence of the notion of confidence within his work. [ABSTRACT FROM AUTHOR]

The sequential organization of actions necessary to construct a figure in a dynamic geometry environment (DGE) introduces an explicit order of construction. Such sequential organization produces what is, in effect, a hierarchy of dependences, as elements of the construction depend on something created earlier. This hierarchy of dependences is one of the main factors that determine the dynamic behavior (DB) within a DGE, and the order is often explicitly stated by terms such as parent and child. This article is a part of a larger study that examines various instruments developed by users when they use dragging. It addresses one aspect of dragging: the connection between dynamic behavior and the sequential order of construction. Junior high students and graduate students in mathematics education were asked to predict the DB of points that were part of a geometric construction they had executed using a DGE according to a given procedure, and to explain their predictions. The study reveals that while hierarchy in geometric constructions in a DGE is mirrored by the DB, user actions and perceptions of DB indicate that users often grasp a reverse hierarchy in which dragging a child affects its parent. The study reveals four categories of reverse-order predictions, and suggests that these may be caused by terms and knowledge built into paper-and-pencil geometry, which are part of the resources users bring to dragging. Examining various DGEs1 in detail reveals that in some cases the DB is in fact compatible with the reverse-order predictions. The paper concludes with a brief discussion of some implications for learning activities and software design. [ABSTRACT FROM AUTHOR]

This Special Issue of Educational Studies in Mathematics honoring the life and work of Leen Streefland is based on Research Forum 5, Realistic Mathematics Education Research: Leen Streefland's Work Continues, at the 25th Annual Meeting of the International Group for the Psychology of Mathematics Education (PME), held in Utrecht, The Netherlands, July, 2001 (see PME-25 Proceedings, Vol. 1, pp. 221­253). The papers in this issue address the work of Leen Streefland at the Freudenthal Institute, and its continuation by his former colleagues and by international researchers who were influenced by his ideas. The essential problem addressed is how to develop a school mathematics curriculum that is grounded in the ex- periential reality of the learners (hence Realistic Mathematics Education ­ RME). In honoring Leen Streefland in this way, we are also honoring Hans Freudenthal, who was the first director of the Institute that after his death was named after him. Freundenthal was the originator of many of the ideas on which the continuing research is based, although (as elaborated in sev- eral of the papers) these ideas evolved and were modified and expanded in the subsequent and ongoing research. Some of these ideas include `math- ematizing', `mathematics as human activity', `anti-didactical inversion', `phenomenological didactical analyses', `guided reinvention', `long-term learning processes', `levels in learning processes', and `developmental re- search'. Freudenthal's work was an important source of inspiration for Stree- fland, as it also was for many others. In `The Legacy of Hans Freudenthal', a Special Issue of Educational Studies in Mathematics (Vol. 25, 1­2) that was published ten years ago, Streefland emphasized how Freudenthal cre- ated the context for further development. Paradoxically, he sometimes did this by putting people back on the track of their previously held ideas. For Streefland this happened for example with the issue of interweaving learning strands, which was an early part of his work. By higlighting the importance of this issue, Freudenthal revived it. Incidentally, this aspect of interweaving was by no means isolated within Streefland's work. Like no other at that time, he was focused on making connections, such as those between the different phases in long-running learning processes, between the current didactics of mathematics and the history of mathematics, between the different roles ­ student, teacher and researcher ­ which all who are involved in the learning-teaching process Educational Studies in Mathematics 54: 1­4, 2003. [ABSTRACT FROM AUTHOR]

Niels Bohr's term `complementarity' has been used by several authors to capture the essential aspects of the cognitive and epistemological development of scientific and mathematical concepts. In this paper we will conceive of complementarity in terms of the dual notions of extension and intension of mathematical terms. A complementarist approach is induced by the impossibility to define mathematical reality independently from cognitive activity itself. R. Thom, in his lecture to the Exeter International Congress on Mathematics Education in 1972, stated ``the real problem which confronts mathematics teaching is not that of rigor, but the problem of the development of `meaning', of the `existence' of mathematical objects''. Student's insistence on absolute `meaning questions', however, becomes highly counter-productive in some cases and leads to the drying up of all creativity. Mathematics is, first of all, an activity, which, since Cantor and Hilbert, has increasingly liberated itself from metaphysical and ontological agendas. Perhaps more than any other practice, mathematical practice requires a complementarist approach, if its dynamics and meaning are to be properly understood. The paper has four parts. In the first two parts we present some illustrations of the cognitive implications of complementarity. In the third part, drawing on Boutroux' profound analysis, we try to provide an historical explanation of complementarity in mathematics. In the final part we show how this phenomenon interferes with the endeavor to explain the notion of number. [ABSTRACT FROM AUTHOR]

Implication is at the very heart of mathematical reasoning. As many authors have shown, pupils and students experience serious difficulties in using it in a suitable manner. In this paper, we support the thesis that these difficulties are closely related with the complexity of this notion. In order to study this complexity, we refer to Tarski's semantic truth theory, which contributes to clarifying the different aspects of implication: propositional connective, logically valid conditional, generalized conditional, inference rules. We will show that for this purpose, it is necessary to extend the classical definition of implication as a relation between propositions to a relation between open sentences with at least one free variable. This permits to become aware of the fact that, in some cases, the truth-value of a given mathematical statement is not constrained by the situation, contrary to the common standpoint that, in mathematics, a statement is either true or false. In the present paper, the didactic relevance of this theoretical stance will be illustrated by an analysis of two problematic situations and the presentation of some experimental results from our research on first-year university students' understanding of implication. [ABSTRACT FROM AUTHOR]

This paper is a critical review of the ethnomathematics literature and classifies ethnomathematics according to where it might stand in relation to the teaching of formal, academic mathematics. This paper investigates what it sees as four possibilities: ethnomathematics should replace academic mathematics, ethnomathematics should be a supplement to the mathematics curriculum, ethnomathematics should be used as a springboard for academic mathematics and ethnomathematics should be taken into consideration when preparing learning situations. We argue that it is only through the lens of formal, academic mathematics sensitive to cultural differences that the real value of the mathematics inherent in certain cultures and societies be understood and appreciated. [ABSTRACT FROM AUTHOR]

Traditional approaches to research into mathematical thinking, such as the study of misconceptions and tacit models, have brought significant insight into the teaching and learning of mathematics, but have also left many important problems unresolved. In this paper, after taking a close look at two episodes that give rise to a number of difficult questions, I propose to base research on a metaphor of thinking-as-communicating . This conceptualization entails viewing learning mathematics as an initiation to a certain well defined discourse . Mathematical discourse is made special by two main factors: first, by its exceptional reliance on symbolic artifacts as its communication-mediating tools, and second, by the particular meta-rules that regulate this type of communication. The meta-rules are the observer's construct and they usually remain tacit for the participants of the discourse. In this paper I argue that by eliciting these special elements of mathematical communication, one has a better chance of accounting for at least some of the still puzzling phenomena. To show how it works, I revisit the episodes presented at the beginning of the paper, reformulate the ensuing questions in the language of thinking-as-communication, and re-address the old quandaries with the help of special analytic tools that help in combining analysis of mathematical content of classroom interaction with attention to meta-level concerns of the participants. [ABSTRACT FROM AUTHOR]

This paper examines perspectives on values, purpose and methodology in mathematics education in schools in the light of the Third International Mathematics and Science Survey (TIMSS) and current debates on standards. It argues that standards of attainment in school mathematics are closely connected to belief systems regarding value and purpose; that these systems do not always collectively offer a credible and coherent vision for mathematics education which can be effectively implemented in school classrooms; and that this coherence of vision is what to a large extent characterises the higher performing TIMSS countries. The paper forms part of a wider investigation into the processes of change in education, with a particular focus on mathematics. [ABSTRACT FROM AUTHOR]

This paper initially discusses recent changes in mathematics education and assessment in England and Wales against the background of research on mathematics performance and assessment. It then reports findings from qualitative research with 10–11 year olds undertaken with the object of exploring the validity of the pilot pencil and paper tests in mathematics developed as part of the assessment programme for the English and Welsh National Curriculum in 1993 and 1994. Fifteen children from across the ’ability‘ range were asked to work through a selection of items in the situation of an individual clinical interview. This enabled in-depth data to be collected about their interpretation of and performance on the items. This paper focuses in particular on items where the ambiguity of the boundary between everyday knowledge and mathematics might be expected to lead to problems for children in interpreting the tasks required of them by the test items. The results show that the nature of the items might well have prevented some of these children, in the context of a paper and pencil testing situation, from demonstrating mathematical capacities and understandings they seem to have possessed. Their initial performance does not always seem to have reflected their underlying competence as demonstrated in the extended interview. [ABSTRACT FROM AUTHOR]