Research on classroom interactions in mathematics education has shown how discourse, participation, and power shape learning opportunities and the nature of mathematical activity itself. A central strand of this work examines teacher noticing as a key practice of responsive teaching. While research continues to advance embodied and ecological perspectives on teacher noticing, the auditory dimension of how teachers use their bodies, space, and movement to listen remains undertheorized. This paper examines teacher listening as a dimension of embodied teacher noticing through an exploratory case study of a mathematics teacher. We integrate conversation and interaction analysis with embodied and spatial methods to examine how teachers’ embodied movements and spatial positioning reflect listening practices. Our analysis proposes a framework of three different listening modes—observational, concentrated, and conversational—each defined by distinct spatial and attentional patterns that reflect how teachers shift and manage auditory attention within their broader noticing practice. We illustrate how this framework extends embodied and ecological perspectives on teacher noticing, offering new conceptual and methodological tools for understanding responsive teaching and how teachers make themselves available to students’ mathematical thinking.
The objective of this research was to formally articulate Action, Process, Object, and Schema (APOS) Theory and Semiotic Representation Registers (SRR) Theory through the Networking of Theories methodology. As a result, we propose an approach denominated APOS-SRR, an explicit relationship between the subject, the mathematical object, and the semiotic representation system. This approach integrates the mental mechanisms of APOS Theory and reconceptualizes its structures as semiocognitive structures. The main methodological contribution of this study is the Semiocognitive Genetic Decomposition, conceived as an analytical tool for structuring the process of conceptual construction and for identifying how semiotic transformations are associated with cognitive constructions in advanced mathematical thinking.
In this paper, we report on a fine-grained analysis of social interaction among a group of four Chinese Year 7 students during collaborative problem solving in a mathematics classroom. The data were collected as part of the Social Essentials of Learning project, which investigates student learning in collaborative contexts. We examined the coconstitutive between meaning negotiation and authority relations during collaborative activities. Through a dual analytical lens, we first identify the group’s negotiative foci during the discussion—specifically, its focus on mathematical facts and procedures, didactical norms, and social/interpersonal considerations. A parallel analysis traces the dynamic distribution of intellectual and social authority relations across four distinct states: shared, concentrated, contested, and disbanded. By synthesizing these analyses, the study illustrates the coconstitutive nature of authority relations and meaning negotiation during students’ collaborative mathematical problem solving. The findings suggest that conflict can serve as a catalytic mechanism driving authority redistribution within the group. The analysis demonstrates that moments of mathematical disagreement frequently precipitate shifts in both intellectual authority and social authority. Furthermore, we identify how different authority states mediate the group’s capacity to resolve conceptual conflicts and advance collective understanding. These empirical insights contribute to theoretical understandings of social learning mechanisms in mathematics education while offering practical implications for structuring productive collaborative environments. The study also advances methodological approaches for analyzing microinteractional processes in mathematics classrooms through its innovative framework combining meaning negotiation and authority relations.
In this article, we analyze children’s descriptions and graphic representations of a region severely affected by industrial pollution. We also examine the mathematical spatial relations used by the children in these representations of the region they inhabit. Our data come from magazines created by these children as well as mapping activities that we conducted. The magazines were part of the Atoyac River Memorial Museum—a collaborative project carried out in the state of Tlaxcala, México, between teachers, members of a community organization, and education and environmental sciences researchers. The mapping activities were done with children from three schools located in different towns along the Atoyac River Basin. We examine the magazines and maps through a theoretical framework that brings together two perspectives: (1) land-based education, developed in dialogue with critical geography, and (2) mathematics education focused on the study of physical space. On the one hand, we show that the children expressed diverse points of view on the river basin they inhabit that are strongly related to their life experiences, including conversations with their elders. On the other hand, as the children were representing their territorial views during the mapping processes, several discussions emerged about the resemblance between their towns and their maps, mobilizing a wide variety of mathematical spatial relations. This article contributes to incorporating children’s views into territorial studies, which tend to focus on adults, and to integrating the presence of people—along with their stories, relationships, and symbolic world—into studies on representing physical space with maps.
This paper reports two conceptual replications examining how gender-segregated secondary education in New Zealand relates to students’ mathematics-related outcomes. Study 1 analyses enrolment data from the University of Auckland (UoA; 2016–2020) to investigate single-sex secondary school attendance rates by gender across Engineering and comparison programmes (i.e., Computer Science, Mathematics, Statistics, Business, and Psychology), presenting descriptive results as in the original study at the University of Canterbury (UC; 2005–2017). We also use logistic regression to further examine whether the observed differences across programmes are beyond chance. At UoA, the rate of female Engineering students who had attended single-sex secondary schools (42
As mathematics educators, we frequently encounter problematic messages about mathematics. For example, we are often told by students, parents, and people we meet that they do not have a mathematics brain or are not a mathematics person. Such speech is inaccurate and harmful to learners, but its origins are complex and responding requires care. In this article, I discuss the ethical considerations that shaped my attempt to respond to this speech through fiction and essay. My approach is grounded in Levinasian ethics, which views listening as a nonviolent response to difference. I explain how my attempts to listen to the alterity of this speech influenced my response and changed me. I offer the story-essay pairing that resulted from this process as a resource for mathematics educators working with those who might be exposed to this speech or use it themselves. I argue that listening can be a methodology for using fiction as an ethical tool to access alterity, and I share my reflections on my process as a case study of ethically guided action in a contested mathematics education space.
Students’ difficulties with mathematical argumentation, conjecturing, and proving, as well as with collaborating effectively, have long been documented in the literature. While there is some research on what constitutes successful proving processes, systematic investigations of proving process quality are rare. We investigate how individual-mathematical and social-discursive process characteristics during a collaborative conjecturing-and-proving activity relate to students’ performance in conjecturing and proving, as well as their proof-related resources, such as prior proof skills. Based on a multidimensional high-inferent coding of collaborative conjecturing-and-proving processes of N=98 prospective mathematics students who worked in dyads, we test a range of hypotheses on the overall role of process characteristics for students’ performance. Moreover, we investigate which single process characteristics best predict conjecturing-and-proving performance. We find that individual-mathematical process quality mediates the effect of students’ resources on their performance, whereas the investigated social-discursive characteristics were largely unrelated to students’ proof performance. Mathematical argument structure was particularly predictive for performance. Results further indicate that process quality is predicted by the partners’ resources, but not by the partners’ contributions earlier in the process. However, students within a dyad converge in terms of process quality. We derive implications for the conceptualization, measurement, and investigation of process characteristics.
This conceptual paper contributes to conversations about the mathematics relevant for teaching and the experiences in undergraduate studies that can serve future teachers. The construct of knowledge at the mathematical horizon is refined, with attention paid toward the Husserlian conceptualization of the inner and outer horizons of an object. Inspired by Felix Klein’s approaches to mathematics education and teacher preparation, this paper considers implications for teacher education of embracing and making explicit mathematical structure and sensibilities that can foster a connected and flexible view of the discipline.
Teachers’perceptions about effective teaching shape their practices. This study examines mathematics teachers’ perceptions and their relationship with professional background and teacher motivation. The data were collected via an open survey that included professional background, motivation for being a mathematics teacher, the top three goals of a good mathematics lesson, and the criteria for evaluating a colleague’s lesson. Drawing on 632 responses from a national sample, we used latent class analysis to differentiate teacher profiles. Informed by the pedagogical triangle—student, teacher, and content—we found that teachers focused on student-learning when discussing lessons they viewed as successful, while emphasizing teacher-teaching when evaluating their colleagues’ lessons. Location, highest qualification, school level, and teacher motivation were associated with perceptions of effective teaching. The analysis yields five classes, highlighting different teacher profiles—the levels of schooling they taught, their highest qualifications, and the teacher motivations collectively influencing their perceptions of good mathematics teaching. Methodological considerations for investigating teachers’ perceptions and their implications for professional development are provided.
In this paper, we detail the outcomes of work in initial teacher education (ITE) that sought to support moves into coherent and connected early mental mathematics teaching in South Africa. Our aims in the paper are twofold. First, in revisiting an earlier framework (the Mediating Primary Mathematics framework), we elaborate a theoretically oriented and empirically sensitive continuum of teaching quality that moves from incoherence to coherence to connection to generality. Second, we present an analysis of the instruction of a cross-attainment sample of nine pre-service teachers from the ITE cohort at one university that indicates very few instances of the incoherence identified in earlier research, and substantial evidence of episodes that are, not only coherent, but also connected. The findings suggest that a combination of good quality structured teaching materials and course mediation of these materials can contribute to supporting good quality mental mathematics instruction.
This study investigates the cognitive difficulties encountered by upper secondary students when solving mathematical modelling tasks, drawing on the five-phase modelling framework proposed by Stillman et al. (2007). Two complementary instruments were used: the Algorithmic Map of Anticipated Difficulties (AMAD) and the Coded Grid of Cognitive Difficulties in Modelling (CGCDM), both structured around specific cognitive aspects within each phase of the modelling process. The sample comprised 300 students equally distributed across three grade levels—Common Core, first-year Baccalaureate and second-year Baccalaureate—each of whom solved two tasks: a conventional mathematical application task and an authentic mathematical modeling task. Students’ written responses were classified into four performance profiles: Full Success, Partial Success, Unsuccessful with Partial Attempt, and Unsuccessful with No Response. The results indicate that difficulties were more frequent and more pronounced in mathematical modeling tasks, particularly in phases involving mathematisation, interpretation, and validation. Statistical analyses show limited cross-sectional differences among common core, first- and second-year baccalaureate students, while highlighting persistent cognitive difficulties in mathematisation/translation and validation/interpretations across all three educational levels. Key aspects such as contextualisation, hypothesis formulation, and model validation were often absent, indicating stable patterns in students’ reasoning across educational levels. These findings offer a fine-grained account of where difficulties occur within the modelling process and support the relevance of phase-level diagnostic tools for analysing students’ engagement with modelling tasks.
This article explores how the concept of risk can be meaningfully integrated into mathematics education to address pressing global challenges. As crises like climate change, pandemics, and social instability grow more complex, education must equip learners to act responsibly under uncertainty. The focus is on fostering risk literacy—the ability to understand, assess, and make informed decisions in risk situations. The paper begins by unpacking the conceptual ambiguity of “risk” and introduces a three-part typology based on vulnerability, objectivity, and subjectivity. Building on key theoretical foundations, it presents a comprehensive framework that combines classical and contemporary views. Risk literacy is conceptualised as a hybrid competence blending statistical and probability literacy and intuitive judgement. This is linked to education for sustainability, suggesting that mathematics should enable learners not only to analyse data, but also to engage ethically and socially with uncertain outcomes. The didactic model draws on the concept of argument-based mathematical inquiry developed by James Fielding (2025), which guides students in integrating context knowledge, statistical knowledge, and argumentation knowledge throughout the inquiry process. A classroom example on wildfire risk illustrates how students can work with real data, reflect on probabilities, and make socially responsible decisions. The article concludes that integrating risk into mathematics fosters critical skills for sustainable development and global citizenship. Despite challenges in aligning formal models with subjective experience, it offers a conceptual bridge between abstract risk theory and its pedagogical application.
Textbooks play a crucial role in shaping instructional practices, making it essential to examine the reasoning-and-proving (RP) opportunities that they offer. While previous research on RP opportunities in textbooks has focused primarily on the secondary mathematics level, this study applies an established analytical framework to explore RP opportunities in the “number and early algebra” domain across 12 volumes of elementary textbooks (grades 1–6) in China. Across the 12 textbook volumes analyzed, RP tasks accounted for an average of 17.18
Despite the pressing need to improve the mathematics learning conditions and resultant achievement outcomes of Black boys, it is surprising that their experiences have received relatively little attention in the literature. Furthermore, the vast majority of what has been published has not been centered on the crucial role Black families and communities play in Black boys’ mathematics education. In this conceptual paper, we draw on BlackBoyCrit Pedagogy to illuminate how Black families and communities contribute to the early mathematics success of Black boys. In doing so, we offer three principles to highlight the assets that Black families and communities contribute to Black boys’ mathematics education. We also share recommendations for early-childhood research, policy, and practice to support Black boys’ mathematics success in the early grades. This article is part of a special issue titled “Parents, caregivers, and community in mathematics education.”
Research on teacher education is increasingly concerned with how pre-service teachers learn to apply professional knowledge to instructional demands. From a cognitive perspective, this requires transforming declarative knowledge into procedural knowledge usable in teaching. Therefore, a distinction can be made between mastering the demands of lesson preparation and evaluation (slow, deliberate; reflective competence) and mastering interactive classroom demands (fast, spontaneous; action-related competence), which may involve different knowledge transformation processes. However, systematic research on and the implementation of learning opportunities that foster pre-service teachers’ ability to apply knowledge to instructional demands remain limited. Instead, initial pre-service mathematics teacher education typically focuses on the acquisition of decontextualized declarative knowledge, which often remains inert and ineffective for teaching. However, cognitive theories suggest that such transformation may be supported by learning opportunities that engage pre-service teachers with instructional demands in increasingly situated ways. To address this assumption, we developed a short-term intervention to promote the reflective and action-related competences and to investigate the underlying learning mechanisms. In a pre-post intervention study with 187 pre-service mathematics teachers, we compared three intervention formats that differed in their degree of situatedness, ranging from abstract exercises to situated text and video vignettes. Using ANCOVA analyses, results indicate a small but significant increase in reflective competence, with the video-based group outperforming the text-based group. Implications for mathematics teacher education and research on situated learning opportunities are discussed.
Generative AI makes it easy for students to obtain problem-solving solutions in mathematics, but it also raises the risk of uncritical acceptance. This critical case study develops and examines a rational questioning approach, grounded in Habermas’s (1998) theory of rational behavior, to scaffold students’ validation of AI-generated solutions in a college calculus course. Twelve STEM undergraduates engaged in AI validation activities across one semester. Data included ChatGPT chat histories and written lab justifications from an early lab (pre-rational questioning) and a later lab (post-rational questioning). A microanalysis focused on three aspects: (a) the rationality criteria students applied, (b) the follow-up questions they posed to the AI, and (c) the presence of explicit warrants in their justifications. Findings showed a shift from step-checking to criterion-guided validation guided by rationality components. Students not only attended to content and method correctness but also considered method efficiency and articulated warrants. Students also posed criterion-aligned follow-up questions that fostered rational discourse with the AI, and their written justifications more often included explicitly mathematically grounded warrants. The findings suggest that the rational questioning approach may support students’ validation across three analytic aspects while also highlighting the need for further refinement, particularly with respect to communicative rationality. Implications for instruction and directions for future research are discussed, and potential risks of using AI in mathematics teaching are highlighted.
In the context of teacher professionalization, teachers and their professional competence play a decisive role in the quality of classroom activities and students’ learning gains. Within this discourse, teacher noticing has emerged as a set of central situation-specific skills that mediate the relationship between teachers’ knowledge, beliefs, and motivation and their observable teaching performance. Due to its situational nature, teacher noticing is assumed to be highly context dependent. However, few empirical studies have systematically addressed cultural influences on teacher noticing as part of the professional context of teaching. This study addresses this research gap and investigates how far professional teacher noticing reflects culturally influenced dimensions of competence in two contrasting educational systems from Europe and South America. Data from 204 lower secondary mathematics teachers in Germany (n = 118) and Chile (n = 86) were analyzed using a standardized video-based instrument developed within the Teacher Education and Development Study in Mathematics (TEDS-M) research program. Based on differential item functioning (DIF) analyses on a mathematics-pedagogical perspective, especially in interpreting individual students’ thinking and process-oriented mathematical competences. These complementary strength patterns reflect the different educational philosophies underlying the educational systems in which mathematics teachers learn and teach. The findings highlight teacher noticing as a culturally shaped dimension of professional competence and underscore the need for context-sensitive approaches to teacher professionalization and cross-cultural competence research.
The universality of narratives that link ability in mathematics to boys and to masculinity around the globe is questionable. This study tests a conjecture that such narratives might be less common in the Palestinian/Arab Israeli context among school students. In Arabic-language schools in Israel, girls’ achievement in mathematics consistently exceeds that of boys. We surveyed 475 ninth- and eleventh-grade students using a translated version of the "Who and Mathematics" survey. Results reveal a configuration of gendered associations that diverges from Western patterns. Item-wise analysis shows that students significantly associate mathematical difficulty, disengagement, and disruptive classroom behaviors with boys but attribute enjoyment of mathematics and attention to success in mathematics to girls. These findings confirm that associations between gender and mathematics are culturally specific. Profile analysis shows different configurations of gendered constraints in mathematics, in particular, a feminization of diligence-driven success. This framing may reinforce an effort over ability logic that potentially constrains the advancement of girls and women in mathematics or pathways to mathematics-related careers.
This paper investigates how 11th-grade students’ covariational reasoning with periodic functions emerges when they choreograph function-driven animations to music in a dual digital environment. Building on new materialist perspectives, we reconceptualise covariational reasoning as an aesthetic-material practice distributed across assemblages of students, tools, graphs and sounds. We introduce the construct of aesthetically driven covariational reasoning to describe how emergent qualities of fit and misfit between motion and music participate in organising relations between the varying quantities. The study is part of a design-based research project with students working in pairs to create their function-driven “dancing animations” for given song excerpts. They initially used GeoGebra calculator to plot the periodic functions (simple trigonometric function families of sine and cosine and trigonometric polynomials) by adjusting coefficient values in order to plan the choreography and then they used the MaLT2 digital environment for simulating the animation through a dynamic square. Screen-audio recordings and researcher notes were analysed as aesthetic events, focusing on multimodal instances where reasoning shifted. The analysis identifies four recurrent aesthetic functions—composition, attunement, modulation and provisional individuation—through which reasoning unfolds. Across a series of events, we show how these functions give rise to mathematical meanings about periodic covariation, including amplitude-dependent and amplitude-independent period, relations between vertical span, monotonicity and perceived speed, and emergent derivative-like meanings of local rate of change in trigonometric polynomials. We argue that attending to aesthetic-material dynamics offers a complementary lens on covariational reasoning and suggests design approaches that treat students’ aesthetic responses as productive resources.
Open real-world problems play an important role in the teaching and learning of mathematics. Research on open problems comprises a variety of research methodologies, theoretical conceptualizations, and constructs. This systematic literature review aims to synthesize and structure these various approaches by examining the current state of research on open problems in mathematics education. With a full-text analysis of 67 peer-reviewed journal articles, we investigated the descriptive characteristics of studies in this field, the conceptual frameworks used to define the openness of real world mathematical problems, and the structure of problems in terms of their initial, intermediate, and goal states. Additionally, we analyzed the constructs employed in the studies. We identified four sets of conceptual frameworks: (1) psychological frameworks of problem-solving, (2) mathematical problem-solving frameworks, (3) mathematical modelling frameworks, and (4) suspension of sense-making frameworks. The findings demonstrated that most studies have focused on problems with open initial and intermediate states, whereas problems with open goal states have received less attention. Furthermore, cognitive constructs were addressed more frequently than affective-motivational ones. The findings of this review reveal the need for a theoretically grounded conceptualization of the notion of openness in empirical studies. Important future research directions include investigating problems with open goal states, analyzing instruction to improve student performance in solving open problems, and addressing affective-motivational constructs in studies with open problems. This review contributes to a deeper understanding of openness as an important task characteristic and provides a comprehensive synthesis of the available research on open real-world mathematical problems.
