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Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which underpins the construction of a cognitive structure that links both types of thinking. This paper proposes a theoretical and practical framework for a learning approach that supports the construction of a cognitive structure which fosters arithmetico-algebraic thinking at the beginning of secondary school by means of cultural and technological activities relating to polygonal numbers.

This article brings critical and postcolonial theories together with extended ethnographic research in a predominantly Latinx high school in California's "South Bay" to theorize the co-production and co-naturalization of mathematical and racial essentialization. Analysis of vignettes and interview excerpts illuminates both student uptake and resistance to homogenizing narratives of mathematics and racial personhood. Student voices from Sierra High School both evidenced the existence, stakes, and personal consequences of narrowly bounded, taken for granted, conceptions of mathematics and racial personhood and challenged these homogenizing categories. This paper contributes theory illuminating the co-construction of mathematical and racial essentialization along with examples of local critique and resistance from youth and their adult allies at one high school in the USA. Implications suggest that decolonial work in mathematics education must jointly address the narrow and essentializing frames for race and racialization and for mathematics itself as co-producing and co-naturalizing each other. This study contributes insight into mechanisms that perpetuate and also resist or disrupt these processes.

The use of graph paper is traced from its origins in research reports in the early 19th century to massive use in the early 1900s. Reflected in this report is the impact of "squared paper" on instructional practices and courses in technical, scientific, and mathematical areas. (MP)

Social institutions function not only by reproducing specific practices but also by reproducing discourses endowing such practices with meaning. The latter in turn is related to the development of the identities or subjectivities of those who live and thrive within such institutions. Meaning and subjectivity are therefore significant sociological categories involved in the functioning of complex social phenomena such as that of mathematical instruction. The present paper provides a discursive analysis centered on these categories of the influential OECD's PISA mathematics frameworks. As we shall see, meaning as articulated by the OECD primarily stresses the utilitarian value of mathematics to individuals and to society at large. Furthermore, molding students' subjectivities towards endorsing such articulation of meaning is emphasized as an educational objective, either explicitly or implicitly, as connected to the OECD's definition of mathematical literacy. Therefore, the OECD's discourses do not only serve to reproduce the type of mathematical instruction implied in the organization's services concerning education, but also concomitantly provide a potentially most effective educational technology through which the demand of these very services may be reproduced.

Investigating the transition between educational levels is one of the main themes for the future of mathematics education. In particular, the transition from secondary school to STEM degrees is problematic for the widespread students' difficulties and significant for the implications that it has on students' futures. Knowing and understanding the past is key to imagine the future of a research field. For this reason, this paper reports a systematic review of the literature on the secondary-tertiary transition in Mathematics Education from 2008 to 2021. We constructed two corpuses: one from the proceedings of three international conferences in mathematics education (PME, ICME, and INDRUM) and the other from peer reviewed research papers and book chapters returned by the databases ERIC and Google Scholar. A clear evolution in perspectives since 2008 emerges from the analysis of the two corpuses: the research focus changed from a purely cognitive to a more holistic one, including socio-cultural and -- to a lesser extent -- affective issues. To this end, a variety of research methods were used, and specific theoretical models were developed in the considered papers. The analysis also highlights a worrisome trend of underrepresentation: very little research comes from large geographical areas such as South America or Africa. We argue that this gap in representation is problematic as research on secondary tertiary transition concerns also consideration of socio-cultural and contextual factors.

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.

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.

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.

Little is known about the cognitive effort associated with algebraic activity in the elementary and middle school grades. However, this investigation is significant for sensitizing teachers and researchers to the mental demands of algebra learning. In this paper, we focus on the relationship between algebraic thinking and domain-general cognitive abilities. The sample of the study comprised 591 students from grades 4 to 7. The students' abilities in algebraic thinking were assessed through a test that involved three task categories: generalized arithmetic, functional thinking, and modelling languages. Test batteries were used to assess students' domain-general cognitive abilities in terms of analogical, serial, spatial, and deductive reasoning. The results of structural equation modelling analysis indicated that: (1) analogical reasoning predicts students' abilities only in generalized arithmetic; (2) serial reasoning predicts students' abilities only in generalized arithmetic; (3) spatial reasoning predicts students' abilities in functional thinking and modelling languages; and (4) deductive reasoning predicts students' abilities in all three categories. Differences between students across grades are also discussed.

The absence of discussions of identity that consider both subjective/personal and social aspects of identity is an important concern in mathematics education research. This paper proposes a framework to analyse the mathematical identities offered to and constructed by learners during peer interactions, by considering personal and social identities. To exemplify the framework, we discuss how two pairs of secondary school learners, who were engaged with similarly by their teachers and who performed similarly academically, interacted with each other and offered and constructed micro-identities during moments of a lesson, and macro-identities across lessons. We show that when learners were offered identities by a peer as higher, lower or equal status, they constructed their identities by affiliating with or resisting the identities offered by their peer. Their interactions and the identities that they constructed were informed by their personal identities, including their emotions when learning mathematics, and their social identities, including their choices to occupy certain roles in the classroom community.

International comparative studies on mathematics teaching and learning often provide unitary and harmonious images of classroom practices. This paper aims to complement those studies by describing the complex aspects of those practices. Adopting activity theory as a framework, this paper considers classroom teaching and learning of mathematics as an activity system embedded in socio-historical contexts, analyzes its contradictions, and makes a comparative analysis of different practices of mathematics teaching and learning in the same country. As a case in point, mathematics lessons in Japanese secondary schools are considered. The initial activity system of mathematics lessons in Japanese modern schools came from the whole-class instruction system and then gradually evolved into the problem-solving style lesson. The latter has been facing challenges, and then a new activity system has emerged from a new paradigm of school education. It is argued that activity systems analysis enables us to understand the contradictions prevailing in mathematics education and that comparative analysis of more than one system in the same culture can provide valuable insights.

Mathematical modelling emphasizes the connection between mathematics and reality -- still, tasks are often exclusively introduced inside the classroom. The paper examines the potential of different task settings for mathematical modelling with real objects: outdoors at the real object itself, with photographs and with a 3D model representation. It is the aim of the study to analyze how far the mathematical modelling steps of students solving the tasks differ in comparison to the settings and representations. In a qualitative study, 19 lower secondary school students worked on tasks of all three settings in a Latin square design. Their working processes in the settings are compared with a special focus on the modelling steps Simplifying and Structuring, as well as Mathematizing. The analysis by means of activity diagrams and a qualitative content analysis shows that both steps are particularly relevant when students work with real objects -- independent from the three settings. Still, differences in the actual activities could be observed in the students' discussion on the appropriateness of a model and in dealing with inaccuracies at the real object. In addition, the process of data collection shows different procedures depending on the setting which presents each of them as an enrichment for the acquisition of modelling skills.

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.

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.

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.

This paper focuses on pattern generalisation as a way to introduce young students to early algebra. We build on research on patterning activities that feature, in their work with algebraic thinking, both looking for sameness recursively in a pattern (especially figural patterns, but also numerical ones) and conjecturing about function-based relationships that relate variables. We propose a new approach to pattern generalisation that seeks to help children (grades 2 and 3) work both recursively and functionally, and to see how these two modes are connected through the notion of variable. We argue that a crucial change must occur in order for young learners to develop a flexible algebraic discourse. We draw on Sfard's () communication approach and on Châtelet's () notion of the virtual in order to pursue this argument. We also root our analyses within a new materialist perspective that seeks to describe phenomena in terms of material entanglement, which include, in our classroom research context, not just the children and the teacher, but also words, gestures, physical objects and arrangements, as well as numbers, operations and variables. [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 (n[subscript Students] = 170, n[subscript Time-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.

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.

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.

In this paper, we analyse a large, opportunistic dataset of responses (N = 219,826) to online, diagnostic multiple-choice mathematics questions, provided by 6-16-year-old UK school mathematics students (N = 7302). For each response, students were invited to indicate on a 5-point Likert-type scale how confident they were that their response was correct. Using demographic data available from the online platform, we examine the relationships between confidence and facility (the proportion of questions correct), as well as gender, age and socioeconomic disadvantage. We found a positive correlation between student confidence and mean facility, higher confidence for boys than for girls and lower confidence for students classified as socioeconomically disadvantaged, even after accounting for facility. We found that confidence was lower for older students, and this was particularly marked across the primary to secondary school transition. An important feature of the online platform used is that, when students answer a question incorrectly, they are presented with an analogous question about 3 weeks later. We exploited this feature to obtain the first evidence in an authentic school mathematics context for the "hypercorrection effect" (Butterfield & Metcalfe J EXP PSYCHOL 27:1491-1494, 2001), which is the observation that errors made with higher confidence are more likely to be corrected. These findings have implications for classroom practices that have the potential to support more effective and efficient learning of mathematics.

Formulas are involved in most parts of the mathematical curriculum in upper secondary education and in everyday mathematics classrooms, but research shows that students have difficulties using formulas adequately. When students are presented with a task, the task activates a conceptual frame in the students, making them perceive formulas in a specific way, thereby affecting their mathematical behaviour. In this paper, building ideal types of patterns of mathematical behaviours is used to conceptualise 'view on formula' specified by several specific views. The concept of 'view on formula' is applied in an analysis of a classroom episode pointing to reasons for the difficulties students have with handling formulas. When views are either missing or not used in a flexible way this can lead to an unsuccessful handling of formulas. Also, students' views on formula may indicate what knowledge is missing for solving a task involving formulas. Taken together, this points to the importance of paying attention to views on formulas in the everyday mathematics classroom.

This paper describes the difficulties faced by a group of middle school students (13- to 15-year-olds) attempting to translate algebraic statements written in verbal language into symbolic language and vice versa. The data used were drawn from their replies to a written quiz and semi-structured interviews. In the former, students were confronted with a series of algebraic statements and asked to choose the sole translation, of four proposed for each, that was semantically congruent with the original. The results show that most of the errors detected were due to arithmetic issues, especially around the distinction between product and exponent or sum and product in connection with the notions of perimeter and area. As a rule, the error distribution by type varied depending on the type of task involved.

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.

Students entering sixth grade operate with three different multiplicative concepts that influence their reasoning in many domains important for middle school. For example, students who are operating with the second multiplicative concept (MC2 students) can begin to construct fractions as lengths but do not construct improper fractions as numbers. Students who are operating with the third multiplicative concept (MC3 students) can construct both proper and improper fractions as multiples of unit fractions. This paper is a case study of one seventh grade MC2 student, Milo, who demonstrated the most advanced reasoning of all MC2 students in a large project with 13 MC2 and 9 MC3 students. In working on problems involving fractional relationships between two unknowns, most MC3 students constructed reciprocal reasoning. In contrast, the MC2 students struggled with these problems. Similar to the MC3 students, Milo showed some evidence of reciprocal reasoning, and he used proper fractions as operators on unknowns with a rationale. However, Milo did not construct reciprocal reasoning. We account for his reasoning by showing how he used length meanings for proper fractions and how he coordinated two different two-levels-of-units structures. The study expands the mathematics for MC2 students, showing what learning may be possible for students like Milo, and it suggests a change to the theory of students' multiplicative concepts. Specifically, advanced MC2 students are those who have constructed length meanings for fractions and can coordinate two different two-levels-of-units structures.

This paper responds to the burgeoning literature on mathematics teacher noticing, arguing that its cognitive orientation misses the cultural and ideological dimensions of what and how teachers notice. The author highlights Goodwin's concept of "professional vision" as a way of bringing analyses of culture and power into studies of teacher noticing. The case of a high school algebra teacher who learned to notice the mathematical strengths of students from marginalized groups is used to illustrate how this might be done. The teacher's noticing involved not only cognitive processes like attending to, interpreting, and deciding how to respond to students' thinking, but also managing dominant ideologies that position students--especially students from non-dominant communities--as mathematically deficient rather than as sense-makers whose ideas should form the basis for further learning. The paper advances the field's capacity for understanding the challenges that teachers face as they attempt to notice in ways that are ambitious as well as equitable.

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.

While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students' understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.

This paper reports the results of an international comparative study on the nature of proof to be taught in geometry. Proofs in French and Japanese lower secondary schools were explored by analyzing curricular documents: mathematics textbooks and national curricula. Analyses on the three aspects of proof--statement, proof, and theory--suggested by the notion of Mathematical Theorem showed differences in these aspects and also differences in the three functions of proof--justification, systematization, and communication--that are seemingly commonly performed in these countries. The results of analyses imply two major elements that form the nature of proof: (a) the nature of the geometrical theory that is chosen to teach and (b) the principal function of proof related to that theory. This paper suggests alternative approaches to teach proof and proving and shows that these approaches are deeply related to the way geometry is taught.

The importance of integrating history into mathematics education is widely recognised in the literature and advocated in curricula worldwide, including in China. However, under the influence of the long-standing centrally designed curricula, teachers in China are accustomed to content- and teacher-centred examination-driven teaching practices. Adopting a life story approach, this paper reports the case of a mathematics teacher who integrated history into her mathematics teaching during the initial two years of her teaching in a Shanghai high school. The agentic perspective adopted in the study allows us to focus on how the teacher's agency was enacted and achieved when engaging in teaching practices. Our findings reveal the roles played by personal qualities, prior experiences, and the structure and culture of schooling in the teacher's agency in integrating history into teaching under a dominant performance-driven context. Implications of the results for integrating history into teaching in restricted contexts are then discussed.

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.

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.

The process of proofs and refutations described by Lakatos is essential in school mathematics to provide students with an opportunity to experience how mathematical knowledge develops dynamically within the discipline of mathematics. In this paper, a framework for describing student processes of proofs and refutations is constructed using a set of heuristic rules formulated by Lakatos. A salient feature of this framework resides in the notion of increasing content by deductive guessing, which Lakatos considered to be one of the cores of proofs and refutations, where deductive reasoning is used to create a general conjecture that is true even for counterexamples of an earlier conjecture. Two case studies involving a pair of fifth graders and a pair of ninth graders are presented to illustrate that the notion of increasing content by deductive guessing is useful in examining student processes of generalisation of conjectures. The framework shown in this paper contributes to the current knowledge of mathematics education researchers about proof and proving by providing a tool to investigate how a proof can be used not only to establish the truth of a given statement but also to generate new mathematical knowledge.

Proportional reasoning, the ability to use ratios in situations involving comparison of quantities, is essential for mathematical competence, especially in the middle school years, and is an important determinant of success beyond school. Research shows students find proportional reasoning and its foundational concepts difficult. Proportional reasoning does not always develop naturally, however some research suggests that with targeted teaching, its development can be promoted. This paper reports on a large Australian study involving over 130 teachers and their students. A major goal of the study was to investigate the efficacy of ongoing teacher professional development for promoting middle years students' proportional reasoning. A series of professional development workshops was designed to enhance the teachers' understanding of proportional reasoning and to extend their repertoire of teaching strategies to promote their students' proportional reasoning skills. The workshop design was informed by research literature on proportional reasoning teaching and learning as well as the results of a diagnostic instrument administered to over 2500 middle years students prior to the professional development. Between workshops, the teachers implemented a variety of targeted teaching activities. This paper reports on pre- and post- instrument student data collected at the beginning and end of the first year of the project (i.e., after completion of half of the workshops). The findings suggest that targeted professional development and explicit teaching can make a difference to students' proportional reasoning.

This paper discusses the importance of considering bilingual learners' non-linguistic forms of communication for understanding their mathematical thinking. In particular, I provide a detailed analysis of communication involving a pair of high school bilingual learners during an exploratory activity where a touchscreen-based dynamic geometry environment (DGE) was used. The paper focuses on the "word-use," "gestures" and "dragging actions" in student-pair communication about calculus concepts as they interacted with a touchscreen-based DGE. Findings suggest that the students relied on gestures and dragging as non-linguistic features of the mathematical discourse to communicate dynamic aspects of calculus. Moreover, by examining the interplay between language, gestures, dragging and diagrams, it was possible to identify bilingual learners' competence in mathematical communications. This paper raises questions about new forms of communication mobilised in dynamic, touchscreen environments, particularly for bilingual learners.

In this conceptual paper, the author argues that equity research in mathematics education is a genre that operates according to certain implicit ideological and rhetorical rules and assumptions—or discursive formations—that form how one can think about equity and inequity. One such rule that forms the basis of this paper is the axiom of racialized deviance, a logical tool developed by whiteness to establish its dominance and to justify physical, psychic, and epistemic violence against blackness. The author takes up whiteness and blackness related to global systems of racialization beyond the reference to white and Black people that is more typical in the United States. The author proposes three ways that the deviance axiom shows up in equity research in mathematics education: ethnomathematics, repair orientations, and success counternarratives. This issue of racialized deviance unveils equity research in mathematics education as a project whose logical foundations undermine its stated aims. The logic of global white supremacy under which school mathematics operates creates a situation where it is impossible for equity in mathematics education to exist outside because the genre requires that anyone who elects to participate accepts the axiom of racialized deviance on some level. [ABSTRACT FROM AUTHOR]

Educational literature indicates that solving open mathematical tasks (OTs) is a powerful creativity-directed activity. However, the use of these tasks with school students on an everyday basis is extremely limited. To promote implementation of OTs in middle school, we manage a large-scale R&D project, Math-Key, which makes open mathematical tasks available to teachers. Additionally, we encourage teachers to pose OTs by transforming textbook mathematical problems. In this paper, we analyze teachers' skills and affective conceptions related to posing OTs and using them in teaching. Forty-four teachers with different teaching experience (years of experience--YoE) and different levels of expertise participated in a 4-h workshop that introduced them to OTs and their categorization. They were also given a homework assignment: pose OTs and solve them to demonstrate their openness. This assignment was accompanied by a 5-point Likert scale questionnaire that examined teachers' affective conceptions about engaging and teaching with OTs. We drew distinctions between different types of OTs (TOTs) posed by the teachers and the problem posing strategies they used. We found that the types of tasks and strategies that teachers use are a function of teachers' experience in terms of both the level of mathematics taught and years of teaching. In the affective dimension, we found interesting connections between conceptions regarding the difficulty of posing OTs, conceptions regarding the suitability of OTs for teaching and learning, teachers' readiness to implement OTs in their classes, and their predictions regarding teachers' and students' problem-solving behaviors.

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.

School mathematics examination papers are typically dominated by short, structured items that fail to assess sustained reasoning or problem solving. A contributory factor to this situation is the need for student work to be marked reliably by a large number of markers of varied experience and competence. We report a study that tested an alternative approach to assessment, called comparative judgement, which may represent a superior method for assessing open-ended questions that encourage a range of unpredictable responses. An innovative problem solving examination paper was specially designed by examiners, evaluated by mathematics teachers, and administered to 750 secondary school students of varied mathematical achievement. The students' work was then assessed by mathematics education experts using comparative judgement as well as a specially designed, resource-intensive marking procedure. We report two main findings from the research. First, the examination paper writers, when freed from the traditional constraint of producing a mark scheme, designed questions that were less structured and more problem-based than is typical in current school mathematics examination papers. Second, the comparative judgement approach to assessing the student work proved successful by our measures of inter-rater reliability and validity. These findings open new avenues for how school mathematics, and indeed other areas of the curriculum, might be assessed in the future. [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.

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.

This paper examines mathematical meaning-making from a phenomenological perspective and considers how a specific dynamic digital tool can prompt students to disclose the relationships between a function and its antiderivatives. Drawing on case study methodology, we focus on a pair of grade 11 students and analyze how the tool's affordances and the students' engagement in the interrogative processes of sequential questioning and answering allow them to make sense of the mathematical objects and their relationships and, lastly, of the mathematical activity in which they are engaged. A three-layer model of meaning of the students' disclosure process emerges, namely, (a) disclosing objects, (b) disclosing relationships, and (c) disclosing functional relationships. The model sheds light on how the students' "interrogative processes" help them make sense of mathematical concepts as they work on tasks with a digital tool, an issue that has rarely been explored. The study's implications and limitations are discussed.

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.

School mathematics examination papers are typically dominated by short, structured items that fail to assess sustained reasoning or problem solving. A contributory factor to this situation is the need for student work to be marked reliably by a large number of markers of varied experience and competence. We report a study that tested an alternative approach to assessment, called comparative judgement, which may represent a superior method for assessing open-ended questions that encourage a range of unpredictable responses. An innovative problem solving examination paper was specially designed by examiners, evaluated by mathematics teachers, and administered to 750 secondary school students of varied mathematical achievement. The students' work was then assessed by mathematics education experts using comparative judgement as well as a specially designed, resource-intensive marking procedure. We report two main findings from the research. First, the examination paper writers, when freed from the traditional constraint of producing a mark scheme, designed questions that were less structured and more problem-based than is typical in current school mathematics examination papers. Second, the comparative judgement approach to assessing the student work proved successful by our measures of inter-rater reliability and validity. These findings open new avenues for how school mathematics, and indeed other areas of the curriculum, might be assessed in the future.

This paper reports a case study from a mathematics teaching developmental research project. The theoretical foundation for the research comprises "communities of inquiry" and "critical alignment," with which the developmental methodology has a particular synergy. This synergy is the main focus of the paper. The paper elaborates theoretical and methodological antecedents of the project and traces these through a case study of developments in the practices of one upper secondary school team and a group of university didacticians (mathematics teacher educators and researchers) during the first year of the project. The case study reveals that critical alignment and inquiry (necessarily) bring uncertainty and risk, and foster tensions within the teachers' practice and between the practices of teachers and didacticians. In exposing these uncertainties, risks and tensions, the paper points to their value for the learning and knowledge gained by participants.

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.

In 1968, Hans Freudenthal launched the journal "Educational Studies in Mathematics." This paper describes the events leading to the foundation of this journal. It turns out that his wife, Suus Freudenthal, deserves more credit for her husband's interest and achievements in mathematics education. The couple was interested in education from a social democratic perspective. The events leading to the foundation of the journal show that lack of success for Dutch national educational reforms, combined with problems noted by his colleagues abroad on their respective reforms, triggered Hans Freudenthal to embark on this enterprise, and contributed to its success.

The paper focuses on four upper secondary students' collaborative small-group mathematical reasoning (MR) with respect to a sinusoidal function. The students were collaboratively engaged in a process of MR regarding the relationships between mathematical theoretical descriptions of parameters in the algebraic expression of the sinusoidal function and their counterparts when identifying these parameters together with their numerical value in a problem situation. Students' reasoning on the mathematical tool of this sinusoidal function is manifold and multicoloured. Adopting a dialogical approach to the analyses of student interaction, the study reveals that the students' MR, viewed as comprising both process aspects and structural aspects, is characterised by three features. Firstly, the analyses show that the students' MR is characterised by tensions between the mathematical tool and its use in situ, called resistance. Secondly, the students' MR is characterised by their use of semiotic means of objectification such as deictics, gestures, linguistic devices and the text book. Thirdly, the analyses reveal that MR is characterised by the students' efforts to interthink and collaborate in order to achieve the goal of their collective task solving. All four students contributed to the collective MR, however, in various ways and to a varying degree.

This paper builds on previous work (Black et al., "Educational Studies in Mathematics" 73(1):55-72, 2010) which developed the notion of a "leading identity" (derived from Leont'ev's concept of "leading activity") which, we argued, defined students' motive for studying during late adolescence. We presented two case studies of students in post-compulsory education (Mary and Lee) and highlighted how the concept of a leading identity might be relevant to understanding motivation in mathematics education and particularly the "exchange value" or "use value" of mathematics for these students. (Lee's identity was mediated by mathematics' potential exchange value in becoming a university student, and Mary's more by its perceived use value to her leading identity as an engineer.) In this paper, we follow up Mary's story as she progresses to university, and we see how she is now "led" by contradictory motives and identities: Mary's aspirations and decisions seem to be now as much related to her identity as a Muslim woman as to her identity as an engineer. Therefore, we argue that more than one identity/activity may be considered as "leading" at this point in time--e.g. work versus motherhood/parenting, for instance--and this raises conflicts and tensions. We conclude with a more reflexive account of leading identity which recognises the adolescent's developing awareness of self--an ongoing process of organisation as they experience contradictions in managing their education, work, domestic, community and other lives.

Mathematics teacher educators are confronted with numerous challenges and complexities as they work to inspire prospective teachers to embrace inquiry-based pedagogies. The research study described in this paper asks what a teacher educator and faculty advisor can learn from prospective secondary mathematics teachers as they construct (and are constructed by) official pedagogical discourses embedded in mathematics classrooms. Drawing on the theoretical constructs of Bourdieu, I present several pervasive discourses, or dispositions, as storied by prospective mathematics teachers. These discourses highlight prospective teachers' negotiations of conflicting habitus-field fits during their teacher education field experience. The reflections put forth in this paper offer insights into the roles of mathematics teacher educators and teacher education programs in general.