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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]

In this paper a part of the data from a larger study, the Enactment Project, is explored. The exploration is guided by two research questions. The first is specific to the instructional core of teacher practice while the second is related to student perceptions of how they learn mathematics. The project adopted the Complementary Accounts Methodology. Micro-analysis of how three teachers enacted their instructional objectives, using qualitative data analysis methods, led to the identification of three components of an instructional core, namely, Development of concepts and skills [D], Student work [S] and Review of student work [R] that pervade the lessons of the three teachers. They enacted their instructional objectives through instructional cycles comprising combinations of D, S, and R. A lesson was found to comprise one or more instructional cycles. The post-lesson video-stimulated student interview data, showed that teachers helped their students learn mathematics in their lessons specifically in the S and R activity segments. This paper also dispels the myth about stereotypical mathematics instruction in Asian classrooms whilst contributing to this issue of ZDM in memory of David Clarke. It was his work related to the Complementary Accounts Methodology that provided colleagues in Singapore with a tool to illuminate mathematics instruction in their schools. [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]

Background: There is anecdotal evidence that many elementary teachers integrate mathematics lessons and art activities by having students first make colorful, rich material that is subsequently used in an instructional activity. However, it is unclear whether such activities effectively promote learning and transfer of mathematical concepts. The goal of the present research was to examine the use and effectiveness of such "math-and-art" activities on children's ability to acquire basic fraction knowledge. We report the results of a survey of practicing elementary school teachers in the United States, their use of activities involving physical material, and the resources they use for ideas to supplement the standard curriculum. Two experiments examined first-grade students' learning, transfer, and recognition of fraction knowledge from rich, contextualized material versus simple, generic material. Results: The survey results confirm that many U.S. teachers use math-and-art activities and are often inspired by informal sources, such as Pinterest and YouTube. Experiment 1 examined the effectiveness of colorful, contextualized student-constructed material (paper pizzas) versus simple, pre-made material (monochromatic paper circles) in an instructional activity on fractions. Students who used the pre-made circles scored higher than those who used the student-made pizzas on pre-instruction tests of basic fraction knowledge, immediate tests of learning, and delayed tests of transfer. Experiment 2 tested students' ability to spontaneously write fractions to describe proportions of pizzas and circles. Students who answered generic circle questions first were markedly more accurate than those who answered pizza questions first. Conclusions: These findings suggest that rich, contextualized representations, including those made by the student, can hinder students' learning and transfer of mathematical concepts. We are not suggesting that teachers never integrate mathematics and colorful, contextualized material, and activities. We do suggest that elementary students' mathematics learning can benefit when initial instruction involves simple, generic, pre-made material and opportunities for students to make and use colorful, contextualized representations come later. [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]

In this paper, we expand our prior work on mathematics education in contexts of language diversity by elaborating on the three perspectives on language described by Ruiz (NABE J 8(2):15-34, ): language-as-right, language-as-resource, and language-as-problem. We illustrate our arguments with data taken from research contexts in Catalonia-Spain and South Africa. In these two parts of the world, the language policy in education has long been an issue, with a monolingual orientation that values one language (i.e., Catalan in Catalonia and English in South Africa) over others. Throughout the introduction of specific examples of policy documents, classroom practices, and participants' reports, our main point is that the right of using the students' languages makes sense because it is itself more than an intrinsic human right; it is an option that potentially benefits the creation of mathematics learning opportunities. Especially for the instances of classroom practices, our examples can be considered as representative in that they point to a common situation in our data: despite the fact of the language of learning and teaching being fixed, there is room for the learners and the teacher to take or react to a decision on what language to use, with whom, and how in concrete moments of the interaction. However, on the basis of our studies and drawing on the literature in mathematics education and language diversity, we argue that language rights are not sufficiently connected to language as a pedagogical resource. The enactment of these rights is still contributing in many ways to the social and political construction of problems concerning the role of certain languages in classroom interaction. We conclude the paper by discussing some possibilities for framing language as a resource that provide effective support to all students' learning of mathematics. [ABSTRACT FROM AUTHOR]

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

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

Mathematics teachers' identities profoundly influence how they interact with and position their students to learn mathematics. In this paper, we examine how a year-long teacher learning intervention supportive of student engagement in mathematics helped shift teachers' mathematics-related identities. We use an implied identity perspective as a theoretical lens to explore changes in what teachers perceive as legitimate ways of being as a result of their participation in the intervention. Data from pre- and post-intervention concept maps and focus groups with 15 grade 5–7 teachers of mathematics were integrated for this purpose. Teachers reported shifts in their identities, describing themselves as "facilitators," "learners" and "co-creators" of knowledge. We argue that such shifts in the mathematics-related identities of teachers can have practical consequences in terms of improving students' engagement, and in particular, their autonomy for learning mathematics. [ABSTRACT FROM AUTHOR]

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

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

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

The purpose of this study was to investigate the trends of GeoGebra related studies. We focused on the articles published between 2009 and 2021 and existed in Web of Science database. The bibliometric analysis of the GeoGebra related articles exposed four-clustered structure according to the degree of relationships among terms that reflect the articles. The clusters were identified as learner anatomy (the use of GeoGebra to improve students' mathematics learning), teaching analytics (the role of GeoGebra in mathematics teaching), technological focus (the technological and procedural issues), and conceptual extension (the attempts considering the features of GeoGebra, mathematics topics and skills). This study offers evidence to show trends and practices in GeoGebra related studies, as well as to clarify their relationships with other concepts. The investigation into GeoGebra's dynamics would help researchers, teachers, students, and policymakers gain insight into areas that need further research, as well as plan for future studies. [ABSTRACT FROM AUTHOR]

This paper reports on one aspect of a larger research project conducted in the United States that designed and implemented an elementary mathematics, specialist-coach preparation program and evaluated the effect of qualified specialist-coaches on student achievement. The paper discusses a conceptual framework for coaching in which a specialist-coach is to serve as a 'more knowledgeable other' for a community of practice in a school, and ultimately to impact both the knowledge and professional practice of teachers and the school's mathematics program as a whole. Specialist-coaches have unique opportunities and challenges in this daunting task, and the paper discusses one program designed to prepare well-respected teachers for the transition to the role and responsibilities of a specialist-coach. The reported analyses document changes in specialist-coaches' mathematical content knowledge, mathematical knowledge for teaching, and beliefs regarding mathematics teaching and learning over the preparation program and during the specialist-coaches' first years of service in a school. These specialist-coaches' mathematical content knowledge grew and their beliefs became more aligned with a Making Sense perspective during the preparation program, and their changed state persisted throughout 2-3 years of service as specialist-coaches. Evidence addressing the specialist-coaches' mathematical knowledge for teaching was mixed, but suggested that growth occurred both during the preparation program and in their first year of coaching, stabilizing in the years following. [ABSTRACT FROM AUTHOR]

This theoretical paper examines a process for researchers and teachers to exchange knowledge. We use the concepts of communities of practice, boundary encounters, and boundary objects to conceptualize this process within mathematics professional development (MPD). We also use the ideas from design research to discuss how mathematics professional development researchers can make professional development the focus of their research. In particular, we examine the question: How can MPD be conceptualized and designed around research-based knowledge in ways that promote knowledge exchange about students' mathematics and mathematics learning among researchers and teachers to improve the practices of both the research and the teaching communities? We propose that MPD is a premier space for researchers and teachers to exchange knowledge from their communities, impacting both researchers' and teachers' practices without reducing the importance of either. [ABSTRACT FROM AUTHOR]

Mathematics teachers' beliefs are central to mathematics teaching and student learning. Because different aspects of motivation and affect—particularly enjoyment—primarily develop within the classroom context, examining how different teachers' beliefs may affect student outcomes in mathematics is imperative. The current study examines teachers' beliefs about the nature and learning of mathematics in connection to students' motivation (i.e. intrinsic value, utility value and perceived competence) and enjoyment of mathematics across different settings by considering students' mathematics achievement, gender and classroom composition (i.e. socioeconomic and behavioural). Data were collected from 3rd- and 4th-grade mathematics teachers (N = 686) and their students (N = 11,782) in six countries (i.e. Norway, Finland, Sweden, Portugal, Estonia and Serbia). A two-level structural equation modelling technique (TSEM) (i.e. student level and classroom level) with random slopes was employed to address our research questions. The results indicate that students' intrinsic value and perceived competence positively relate to their enjoyment of mathematics in all six countries. Teachers' beliefs about the nature and learning of mathematics moderate the within-classroom relationship between boys and girls and the motivation and enjoyment of learning mathematics in Portugal and Norway. Unlike boys, girls consistently perceive themselves as less competent in mastering mathematics, even in primary school. Classroom socioeconomic composition had a more pronounced influence on teachers' beliefs in Sweden, Norway and Serbia. In relation to teachers' beliefs, classroom behavioural composition was relevant in Estonia and Sweden. In Finland and Norway, classroom composition was essential to boys' and girls' differential motivation and enjoyment of mathematics learning. [ABSTRACT FROM AUTHOR]

Expectancy-value models assume that task values play an important role in engaging successfully with the learning content. However, teacher education students in mathematics only value tasks that deal with university mathematics to a low degree. Offering students profession-related tasks that uncover the connection between university mathematics and school mathematics might help increase students' value beliefs and effort and decrease costs. Based on Eccles and Wigfield's (2020) situated expectancy-value theory and using a sample of 133 teacher education students, we investigate whether offering profession-related tasks affects task values, costs, and effort for students with different levels of interest in school mathematics. In a within-subject design, students rated their task values, costs, and effort to solve seven mathematical tasks from number theory, which were either a profession-related task or a regular task with equal demands regarding the intended mathematical activities. Students reported higher utility value for their professional life, higher intrinsic value, and lower costs regarding profession-related tasks. No main effects were found for students' effort. However, students with a higher interest in school mathematics reported higher effort concerning profession-related tasks than regular tasks. On the theoretical level, our findings contribute to the discussion of the interplay between personal (i.e. interest) and task-specific factors (i.e. relation to future profession) for task values, costs, and effort in university mathematics. One important practical implication of our study is the recommendation to include profession-related tasks in university mathematics courses for teacher education students. [ABSTRACT FROM AUTHOR]

This special issue discusses various pedagogical innovations and myriad of significant findings. This commentary is not a synthesis of these contributions, but a summary of my own reflections on selected aspects of the nine papers comprising the special issue. Four themes subsume these reflections: (1) Gestural Communication (Alibali, Nathan, Church, Wolfgram, Kim and Knuth ); (2) Development of Ways of Thinking (Jahnke and Wambach ; Lehrer, Kobiela and Weinberg ; Mariotti ; Roberts and A. Stylianides 2013; Shilling-Traina and G. Stylianides ; Tabach, Hershkowitz and Dreyfus ); (3) Learning Mathematics through Representation (Saxe, Diakow and Gearhart ); and (4) Challenges in Dialogic Teaching (Ruthven and Hofmann ). [ABSTRACT FROM AUTHOR]

Students' concerns about mathematics can significantly affect their ability to learn the subject. In particular, their anxieties and attitudes can greatly affect how they perceive their own mathematical competence, and in return, this may make them reluctant to pursue mathematical studies. Many researchers believe in the role of active learning environments in reshaping students' perception of and emotions about mathematics. This paper reports the results of an investigation on the effect of one such environment. As a result of being exposed to an active learning environment, a group of 21 cohort students showed opinion changes from a pre- to post-survey statements significantly on emotion and noticeably on confidence and perception statements. There was, however, no statistically significant change observed on the opinions of a group of 41 traditional students. [ABSTRACT FROM AUTHOR]

This paper investigates the noticing of six Chinese mathematics prospective teachers (PSTs) when looking at a procedural error and responding to three specific tasks related to that error. Using video clips of one student's procedural error consisting of exchanging the order of coordinates when applying the distance formula, some variation was found in how PSTs attended to, interpreted, and responded to this error. A more important finding is represented by the inconsistent responses that individual PSTs provided to the three related tasks. This finding suggests that, to some extent, prior learning experience, beliefs, and orientations inform what PSTs notice. But the finding also suggests the centrality of selecting tasks that provide accurate representations of PSTs' emerging professional noticing. Implications for teacher educators are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we explore the implications of adopting (and developing the capacities necessary to adopt) an empathic second-person research perspective. Such a perspective aims to mediate participants' access to their own experience, thereby providing a rich source of first-person data as well as a powerful pedagogical tool. Working within the enactivist tradition (Maturana and Varela ; Varela et al. ), we acknowledge and welcome the co-evolution and intertwining of awareness, description, and experience that such an approach necessarily entails, and we further note a blurring of the distinction between teacher and researcher that occurs as the research method prompts changes in the very aspects of experience we are observing. We begin by weaving together insights based on Varela's 'empathic coach' (Varela and Shear ) and Gendlin's (, , ) Philosophy of the Implicit and practice of Focusing. We describe how we developed and refined our own use of these methods to prompt and describe learners' evolving experiences of mathematical doubt and certainty. We then further elaborate the nature of the empathic coach and close with a discussion of implications for teaching mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we emphasize the methodological challenges of analyzing data with/in an enactivist framework in which (1) all knowing is doing is being, while focusing on, (2) how an entry by the observer transforms usual ways of analyzing data. In this sense, the association of knowing with doing suggests a shift in attention away from what students might 'know' toward attending to the active, dynamic, enacted mathematical activity as knowing, bringing to bear the local and emergent character of the mathematical activity. In addition to illustrating data analysis along those lines, we discuss the methodological significances and challenges of, and paradigm shifts required for, replacing questions of knowledge and acquisition with ones that concern mathematical doing alone. [ABSTRACT FROM AUTHOR]

Small-group discussions involving students and their teacher that focus on meanings constructed during the mathematics lessons or solutions to problems produced in these lessons offer great potential for debate and argument. An analysis of the epistemological nature of knowledge can give deeper insight, to gain a better understanding of the emerging discontinuities in argumentations, negotiations, and clarifications about contentious meaning differences that arise. In most cases mathematical interactions between students and a teacher about contentions are very fragile and seem to be handled more or less directly-by side-stepping to another topic or by resolving via the teacher's authority, for example. Therefore, the maintenance of such negotiation processes in mathematics teaching is a specific challenge for students and the teacher. The type of closure of these processes seems to be related to the emerging maintenance processes. In this paper, small-group discussions are interpretatively analyzed in the three steps 'Initiation-Maintenance-Closing' with the focus on fundamental (dialogical) learning. [ABSTRACT FROM AUTHOR]

The BSCS 5E Instructional Model, an elaboration of the well-known and accepted three-part learning cycle, is a five-part approach to incorporating inquiry-based, active learning in the classroom. Developed in 1987 by the Biological Sciences Curriculum Study, "the 5Es" is a popular K-20 instructional practice, and as such, the Biological Sciences Curriculum Study has advocated for the empirical study of its effectiveness. We present a systematic review of English-language peer-reviewed publications of empirical studies of the 5Es for K-20 mathematics and science teaching and learning. We report our search process and analyses, revealing the characteristics of the literature found that specifically question the effectiveness of the 5Es: 45 articles have been published, beginning in 2006, in 33 journals. Of those with scores available, the average SCImago Journal & Country Rank was 0.587, and the average Impact Factor was 1.151 for year of publication. From our synthesis, we find that the 5Es are more impactful than traditional instruction, yet comparisons of the 5Es to the original learning cycle predecessor have not been done, and the 5Es are not necessarily more effective than other inquiry methods. While dosage does not seem to matter in the studies in our corpus, the 5Es are particularly studied and found to be positively impactful in secondary biology contexts. The relatively few studies with teachers and students in mathematics contribute to both a nuanced understanding of some limitations of the rest of the data corpus and are areas for future research. [ABSTRACT FROM AUTHOR]

Since their appearance new technologies have raised many expectations about their potential for innovating teaching and learning practices; in particular any didactical software, such as a Dynamic Geometry System (DGS) or a Computer Algebra System (CAS), has been considered an innovative element suited to enhance mathematical learning and support teachers' classroom practice. This paper shows how the teacher can exploit the potential of a DGS to overcome crucial difficulties in moving from an intuitive to a deductive approach to geometry. A specific intervention will be presented and discussed through examples drawn from a long-term teaching experiment carried out in the 9th and 10th grades of a scientific high school. Focusing on an episode through the lens of a semiotic analysis we will see how the teacher's intervention develops, exploiting the semiotic potential offered by the DGS Cabri-Géomètre. The semiotic lens highlights specific patterns in the teacher's action that make students' personal meanings evolve towards the mathematical meanings that are the objective of the intervention. [ABSTRACT FROM AUTHOR]

Primary mathematics teachers' ( N = 521) personal goal orientation and instructional practices were examined based on questionnaire responses. The teachers (grades 2 and 3) were oriented towards mastery goals and mastery approaches to instruction, and reported high teaching efficacy. Strong positive relation between performance orientation and performance instructional practices was established, and correspondingly between mastery orientation and mastery instructional practices. Positive relations between students' ( N = 9,980) basic mathematics performance (measured by paper and pencil tests) and both teacher mastery orientation and teaching efficacy were also found. Results indicate that mastery oriented teaching strategies are crucial for fostering basic mathematical competencies in primary students. [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]

Students understand math and science lessons better when innovative teaching resources are used. To implement a variety of teaching strategies and methods, teachers must be aware of and familiar with various innovative and effective pedagogical knowledge and skills. Some students, on the other hand, believe that math and science are difficult subjects. As a result of their mindset, they avoid or refuse to participate in these subjects in their advanced education. To that end, the purpose of this study was to assess the extent to which upper primary and lower secondary school mathematics and science teachers use modernized and innovative teaching tools after nine months of professional development. The study included 959 teachers from 10 districts in Rwanda, 300 from the primary level and 659 from the lower secondary level. The study collected quantitative data using a classroom observation tool. MS Excel and SPSS were used to analyze data. The results from the analysis showed that secondary school teachers use the provided resources by Rwanda Quality Education for Human Capital Development (RQBEHCD) at a rate of 65.5% on average, while primary school teachers use them at a rate of 63.00% on average. The findings also revealed that these teachers use modernized tools at 77.89% and 80.51%, respectively. This implies that secondary school teachers use more modern tools than primary school teachers. Although, there is no statistically significant difference in modernized tools and innovative methods in terms of teaching grade level (F = 1.067, df = 957, p >.50) between upper primary and lower secondary schools. It was found that primary and secondary teachers use ICT resources at nearly the same rate as secondary school teachers, at 41.1% and 40.6%, respectively. Our findings, therefore, recommend policymakers use more modern tools and innovative teaching and learning methods to improve the educational process. [ABSTRACT FROM AUTHOR]

This study examined the extent to which boys fell into clusters comprised of different levels of motivations and costs. In turn, the antecedents of these clusters and associations with engagement and wellbeing outcomes were considered. Based on survey responses from 168 students across Years 5, 7 and 9 from an all-boys' school in Sydney, Australia, three clusters were identified: Positively Engaged, Disengaged, and Struggling Ambitious. Performance-approach and avoidance achievement goals, mastery classroom goal structure, perceived peer valuing of mathematics and teacher enthusiasm differentially predicted profile membership. Clusters were also found to differ in terms of both wellbeing and engagement, such that students within maladaptive profiles evidenced the most negative outcomes. The study reaffirms prior work, holds implications for addressing student motivation in mathematics, and adds to understanding of the interplay of individual and classroom goal structures in relation to students' mathematics expectancies, values and resultant outcomes. [ABSTRACT FROM AUTHOR]

We investigated and compared sociomathematical norms in secondary mathematics teachers' classrooms before and after using mathematical modeling activities. Participants were four secondary mathematics teachers. The data included 104 h of videotaped classroom observations and teacher interviews. The data analysis revealed notable changes in sociomathematical norms in teachers' lectures not utilizing modeling activities beyond those utilizing modeling activities. The results implied that teachers and students were convinced by and adopted what was done in the modeling environment. We discussed that modeling activities could be a powerful tool for mathematics teachers and students in challenging and transforming traditionally oriented classroom norms to those aligned with reform-oriented mathematical classrooms and their sustainability. [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]

Nurturing autonomous learners is a fundamental instructional goal, aimed at honing students' problem-solving skills for use in later educational and career experiences. The study explores the results of a yearlong program providing professional learning to mathematics student teachers with the aim of nurturing mathematically talented students (MTS) as autonomous learners in heterogeneous classes, using generic questions. Taking one program participant (Effy) as a case study, the research examined changes in her stated goals, questioning approach and reflection in the context of nurturing students' autonomous learning while problem solving (with a focus on asking generic questions). Applying a qualitative paradigm, I analyzed Effy's reflective reports, her videotaped excerpts of her teaching situations, her assignments, and her initial questionnaire. Findings suggest that she moved from using small-step teaching towards asking generic questions through a four-stage process of change: exposure, modification, incubation, and accommodation. Changes were detected in her stated goals, instructional approach, and reflection on action. The study provides empirical support for the notion that teachers can be taught to use generic questions with the aim of promoting autonomous learners. [ABSTRACT FROM AUTHOR]

Technology-enhanced Reciprocal Peer Tutoring (RPT) was carried out with Finnish, nine to ten year-old primary school students (N=40). The RPT as a method of paired mathematics placed specific emphasis on electrical mathematical writing and drawing. Little is known about the important area of implementing digital mathematical skills at the primary level. The contribution of this study is to address the research gap in implementing Technological Pedagogical Content Knowledge (TPACK) into primary school students' digital mathematics learning. Data was collected from one primary school mathematics lessons by videotaping student's peer collaboration. In this empirical educational study, the data was analysed qualitatively with content analysis and quantified based on the Contact Summary Sheet (CSS)-instrument and TPACK -framework. This work sheds a light on how students' collaboration such as instruction, thinking aloud, feedback, support and new ideas during their peer tutoring arouse students' mathematical attainment and technological discourse. The results indicate that all dimensions of TPACK emerged during primary school students Technology-enhanced RPT. Data reveals that there was a high degree of PK in all videotaped episodes. More research on RPT with an empirical electrical mathematical focus is needed. [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 paper presents the results of a study that aimed at increasing students' problem-solving skills. Polya's () heuristic for problem solving was used and students were required to articulate their thought processes through the use of a structured diary. The diary prompted students to answer questions designed to engage them in the phases of Polya's () heuristic. While it appeared as though most students did not internalise the diary questions, further analysis of students' responses indicated that most students showed improvement in their solution strategies. These results indicate that having students write about their thinking may be beneficial for developing their problem-solving skills. [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]