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

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

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

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]

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

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

Students' opportunities to persevere in making sense of mathematical ideas have long been considered significant to learning. Building on existing literature and a case study of video-based teacher collaborative sensemaking, we propose a conceptual framework for bridging perseverance and sensemaking. This framework synthesizes dispositional, metacognitive, and contextual-interactive theoretical perspectives on perseverance. Informed by these three research perspectives, the conceptual framework brings forth three interrelated mediators for bolstering perseverance practices and dispositions towards mathematical sensemaking: students' positions as capable sensemakers, explicit problem-solving heuristics, and facilitation of student participation within their collective Zone of Proximal Development (ZPD). We argue that the three mediators, when brought together, provide a holistic and generative lens for teaching and teacher learning. To illustrate the framework and its utility, we build on a case study featuring a veteran middle-school mathematics teacher across his classroom facilitation of students' engagement with a classical mathematical task, the Tower of Hanoi, and a subsequent video-based debrief about the lesson with his colleague and our research team. We first frame the analysis around classroom events, and then investigate teacher learning opportunities in the lesson debrief. By making explicit the complex work of directing perseverance towards sensemaking, this study provides a more nuanced understanding of perseverance for teaching and teacher learning. Moreover, developing clarity around notions of perseverance in mathematics classrooms helps mitigate the potential dangers of the term being taken up in ineffective or even harmful ways. [ABSTRACT FROM AUTHOR]

One typical challenge in algebra education is that many students justify the equivalence of expressions only by referring to transformation rules that they perceive as arbitrary without being able to justify these rules. A good algebraic understanding involves connecting the transformation rules to other characterizations of equivalence of expressions (e.g., description equivalence that both expressions describe the same situation or figure). In order to overcome this disconnection even before variables are introduced, a design research study was conducted in Grade 5 to design and investigate an early algebra learning environment to establish stronger connections between different mental models and representations of equivalence of expressions. The qualitative analysis of design experiments with 14 fifth graders revealed deep insights into complexities of connecting representations. It confirmed that many students first relate the representations in ways that are too superficial without establishing deep connections. Analyzing successful students' processes helped to identify an additional characterization that can support students in bridging the connection between other characterizations, which we call restructuring equivalence. By including learning opportunities for restructuring equivalence, students can be supported to compare expressions in graphical and symbolic representation simultaneously and dynamically. The design research study disentangles the complex requirements for realizing the design principle of connecting multiple representations, which should be of relevance beyond the specific concept of equivalence and applicable to other mathematical topics. [ABSTRACT FROM AUTHOR]

The aim of the current study was to investigate whether exposing teachers to theoretical and empirical background information regarding pedagogical aspects of geometrical thinking would affect their noticing abilities. Forty-one in-service Mathematics teachers participated in this study. The research tools used in the study included a lesson taken from the VIDEO-LM project and articles that were all presented to the teachers. Our work with an open-ended questionnaire and mixed-methods analysis revealed that exposing in-service teachers to theoretical and empirical articles may affect the three components of their professional noticing performance: attending, interpreting, and responding. After the exposure to the articles, the teachers exhibited a greater focus on the specific difficulties inherent in the specific task at hand and their subsequent interpretations of those difficulties and responses to those difficulties were also more specific than they had been before the intervention. [ABSTRACT FROM AUTHOR]

The success or failure of education systems in promoting student problem-solving skills depends on attitudinal, political, and pedagogical variables. Among these variables, the design of mathematics textbooks is thought to partially explain why students from high-achieving countries show better problem-solving ability in international assessments. In the current study, we delved into this question and compared the frequency and characteristics of arithmetic word problems (AWPs) contained in primary school math textbooks in two countries with different levels of performance in international assessments—Singapore and Spain. In our analyses, we focused on (1) the quantity of arithmetic word problems, (2) the variety of problems in terms of their additive or multiplicative structures and semantic-mathematical substructures, and (3) the quantity and nature of illustrations that were presented together with arithmetic word problems. Although a larger proportion of AWP activities was found in Singaporean textbooks, the results showed a similar variety of AWPs in both Singaporean and Spanish math textbooks. Furthermore, in both countries, math textbooks emphasized the structures classified as (additive) combine 1 and (multiplication) simple rate in AWPs. Notably, the Singaporean textbook contained a larger percentage of illustrations that reflected the semantic-mathematical structures of the problems and helped students learn how to solve AWPs (e.g., bar models). The findings are discussed in light of theories that posit that textbooks constitute a fundamental part of the teaching–learning process in the classroom. [ABSTRACT FROM AUTHOR]

In this article, we document the support provided by highly recognised journals in Mathematics Education in response to the challenges faced by English non-dominant language (EnDL) authors when they attempt to publish in English. To address this issue, we first conducted a synthesis of research literature related to those influences that direct EnDL authors' publication efforts towards English language journals and the associated challenges that result. Second, we gathered survey and interview data from participant editors-in-chief of leading journals in Mathematics Education about the support, enacted and planned, provided by their journal for EnDL authors and associated challenges. Finally, we discuss the findings of the study from the perspective of heteroglossia. Findings indicate that while a range of initiatives have been employed to support EnDL authors, current journal policies lag somewhat behind the plans of editors-in-chief, which have been limited, to date, by available resources. [ABSTRACT FROM AUTHOR]

In this article, we direct attention to what becomes critical in teaching activities for toddlers (1–3-year-olds) to learn the meaning of numbers. One activity we thoroughly explore is interactive book reading, based on previous research indicating positive learning outcomes from this type of mathematical activity, as it has shown to simultaneously embrace the child's perspective and encourage interaction and 'number talk.' A specially designed picture book presenting small quantities was developed, and variation theory principles were embedded in both the book design and the teaching acts. Through qualitative analyses, we aim to identify what is critical in the interactive book reading sessions for toddlers to discern essential aspects of numbers, with a specific focus on the conditions for making modes of representations into resources for learning. Preschool teachers frequently read the book to 27 toddlers over the course of a year. Video documentation of their reading sessions was analyzed, and exposed the significance of addressing the child's perspective when choosing what representation to emphasize and in what ways connections within and between representations can be made. Thus, the study contributes knowledge on the teaching of numbers with toddlers, and problematizes as well as extends the potential of interactive book reading as a quality-enhancing educational tool. [ABSTRACT FROM AUTHOR]

This study aimed to investigate seventh-grade students' visuospatial thinking processes in an art studio environment, where students were engaged with geometrically rich artworks. The students were asked to observe minimalist artworks, then create and critique their own and others' artworks based on the Studio Thinking Framework. Data were collected through interviews conducted with students, video recordings in the studio, and students' documents (sketches, artworks, and notes). The data were analyzed based on previous studies on spatial thinking and emergent data. The study's findings indicate that the Studio Thinking-based environment has the potential to elicit students' visuospatial thinking processes, mainly in recognizing shapes, decomposing and composing shapes, patterning, and transforming shapes rigidly and non-rigidly (scaling). The present study, which includes accounts of three studio works, suggests an emergent framework for the characterization of visuospatial thinking within a particular art-math-related environment. The findings of the study shed light on other studies on visual arts and mathematics education and on mathematical thinking and learning in informal learning settings. [ABSTRACT FROM AUTHOR]

Emotions play an essential role in pre-service teachers' competence development, particularly in mathematics. However, the emotion of shame in mathematics has been largely neglected so far. This article deals with shameful experiences of pre-service primary school teachers during their mathematical education at school and the various effects of shame on their decision to study mathematics as a subject at university. The research consists of a qualitative and a quantitative study with 311 prospective primary school teachers who responded to a survey about their experiences of shame in mathematics at school when they were students. Results of the qualitative study emphasize the different experiences in mathematics during the school years and reveal the characteristics of these situations, for example, social exposure or competition games. In the quantitative study, pre-service primary teachers' subject choice was analyzed in relation to their experienced shame in mathematics at school. Results reveal that shame experienced at school has effects on the initial choice in favor of mathematics at university. Implications for primary teacher education are finally discussed. [ABSTRACT FROM AUTHOR]

Several studies have shown that children do not only erroneously use additive reasoning in proportional word problems, but also erroneously use proportional reasoning in additive word problems. Traditionally, these errors were contributed to a lack of calculation and discrimination skills. Recent research evidence puts forward an additional explanation, namely, children's relational preference (i.e., in tasks where both, additive and multiplicative reasoning, are appropriate, some children have a preference for additive relations, while others have a preference for multiplicative relations). Children's relational preference offers a unique explanation for erroneous word-problem solving, after taking into account computation and discrimination skills in 8- to 12-year-olds. However, it is still unclear whether relational preference is also associated with word-problem solving at an earlier age, before the start of formal instruction in word-problem solving. A task measuring children's relational preference as well as three additive and three proportional word problems was administered to a large group (n = 343) of 6- to 7-year-olds. Results show that relational preference is also associated with word-problem solving behavior at this young age: an additive preference is related with better performance on additive word problems but also with more erroneous additive reasoning in proportional word problems. Similarly, a multiplicative preference is related with better performance on proportional word problems but not yet with more erroneous proportional reasoning in additive word problems. The latter is possibly due to the low number of proportional errors that were made in the additive word problems at this young age. The implications of these findings for further research and educational practice are discussed. [ABSTRACT FROM AUTHOR]

Affect (e.g., beliefs, attitudes, emotions) plays a crucial role in mathematics learning, but reliance on verbal and written responses (from surveys, interviews, etc.) limits students' expression of their affective states. As a complement to existing methods that rely on verbal reports, we explore how graphing can be used to study affect during mathematical experiences. We analyze three studies that used graphing to represent, stimulate recall, and reflect on affect. In each, students were asked to draw their perception of an affective construct, such as confidence or intensity of emotion, against time. The studies differed in participant populations, target affect, timescales of participant experience, and structural features of the graphs. The affordances of graphing include reduced dependence on verbal data, temporal ordering of participants' recollections, explicit representation of change over time, and the creation of objects (the graph) for discussion. These studies as examples show that well-structured graphing can productively supplement existing methods for studying affect in mathematics education, as a different medium through which students can communicate their experience. [ABSTRACT FROM AUTHOR]

In this study, the authors examined a group of related literatures that used the same large-scale nationally representative dataset with algebra achievement as the outcome to explore why results from the same dataset may differ across studies. More specially, the authors synthesized the extant research literature that utilized data from the High School Longitudinal Study of 2009 (HSLS:09) to investigate the student-, teacher-, school-, and parent-level characteristics associated with algebra achievement and whether these relationships were consistent across students' backgrounds (i.e., race/ethnicity, gender, socio-economic status, and prior achievement). The authors have summarized the results of 21 studies across 2 outcome variables (ninth- and eleventh-grade algebra achievement) using optimal resource theory as a framework to summarize evidence-based practices for improving student outcomes. [ABSTRACT FROM AUTHOR]

When it comes to choosing tasks, values can have a significant impact. This study explores teachers' values as they choose one task from among three that they believe will have the most potential to occasion mathematical creativity in the classroom. Participants' analyses of each task, as well as their reasons for choosing one task as most preferred, were analyzed in terms of task features and cognitive demands, as well as their explicit reference to fluency, flexibility, and originality. Two values shared by most participants were having several solution paths and a task that would challenge students. Of particular interest are task features and cognitive demands that were associated with a task at the initial stage of analysis but were not mentioned as a reason for choosing that task as occasioning mathematical creativity. Dilemmas and challenges regarding the study of values are discussed. [ABSTRACT FROM AUTHOR]

This study examines the collective mathematical reasoning when students and teachers in grades 3, 4, and 5 explore fractions derived from length comparisons, in a task inspired by the El´konin and Davydov curriculum. The analysis showed that the mathematical reasoning was mainly anchored in mathematical properties related to fractional or algebraic thinking. Further analysis showed that these arguments were characterised by interplay between fractional and algebraic thinking except in the conclusion stage. In the conclusion and the evaluative arguments, these two types of thinking appeared to be intertwined. Another result is the discovery of a new type of argument, identifying arguments, which deals with the first step in task solving. Here, the different types of arguments, including the identifying arguments, were not initiated only by the teachers but also by the students. This in a multilingual classroom with a large proportion of students newly arrived. Compared to earlier research, this study offers a more detailed analysis of algebraic and fractional thinking including possible patterns within the collective mathematical reasoning. An implication of this is that algebraic and fractional thinking appear to be more intertwined than previous suggested. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic is an unprecedented situation that influenced all aspects of society, including education. Millions of students found themselves adjusting to a new medium of mathematics instruction, not to mention the teachers who had to provide instruction through remote sources. Considering students' diverse social, economic, and academic background, this study sought to examine teachers' perspectives on factors that support or hinder how equity is attended to in mathematics during remote instruction and the extent it differed from practices utilized when instruction was provided in a face-to-face setting. We also sought to document teachers' perspectives on how they attended to equity in mathematics to support students with language barriers. We interviewed nine teachers to explore their perspectives of factors that support or hinder equity in mathematics teaching and learning during remote instruction compared to face-to-face instruction and how they support the diverse needs (inclusive of language barriers) of students. There were salient factors in this study that supported or hindered equitable mathematics instruction, such as teachers' beliefs, expectations for students, access to resources, students' socioeconomic status, and language barriers. Hence, it is recommended that policymakers, school administrators, and teachers need to collaborate to systematically plan to ensure that all students have access to quality mathematics. [ABSTRACT FROM AUTHOR]