Research indicates that partitioning provides the foundation for developing an understanding of how fractions and rational numbers more broadly can be interpreted. However, the literature examining young children’s partitioning capabilities suggests this concept is an inherently spatial activity, particularly when children distribute collections of items or partition continuous objects based on the geometric similarities and spatial arrangements of the materials. The connection between spatial reasoning and mathematics achievement is well-established in the literature. Yet the role of spatial reasoning in young children’s early partitioning experiences in relation to fractions remains implicit in the literature and seldom explored pedagogically across different early schooling contexts. This study draws on a subset of data from a Design-Based Research project involving 44 children from two, Year 1–2 classes in regional Australia. A 13-lesson teaching intervention program that utilised a spatial reasoning approach to teaching and learning fractions was developed and implemented by the researcher. The findings reveal the way in which spatial reasoning supports the development of partitioning as the foundation to early fraction understanding, beyond what children are typically expected to achieve for this age. The research provides evidence for implementing a spatial reasoning approach to early fraction learning in the early years.
Students’ understanding of the equal sign is a well-researched topic. Recently, some new insights come to emerge. One is the phenomenon of students holding simultaneous operational and relational understanding (SOR). Researcher called for more investigation to this phenomenon as it is crucial to help in supporting students’ transition toward a robust relational understanding. This study utilised Rasch analysis and student interviews to delve deeper into the SOR level, involving 1008 Chinese Grade 1 students. This study provides empirical support for distinguishing SOR as a coherent level within the continuum of students’ understanding of the equal sign. It also reveals SOR students’ nuanced reasoning and contributing factors underlying their inconsistent thinking. Based on these insights, the study offered suggestions for refining the widely used number-sentence-based activities for teaching and learning the equal sign, as well as the assessment items used to examine students’ understanding.
Interactive digital tasks have the potential to aid mathematics educators in providing equitable mathematics learning experiences. However, research indicates that mathematics educators may not be aware of how interactive digital tasks can promote equitable teaching practices. This study reports on a design-based research project exploring pre-service teachers’ perceptions of equity and interactive digital tasks. Over three cycles of design, implementation, evaluation, and revision, 57 pre-service teachers participated; this article focuses on data from the third cycle, involving 25 pre-service teachers. Using lesson analyses grounded in the Equity-centered Transformative Technology Lesson Analysis Tool (EqT-tech LAT; Suh et al., 2022), we explored their perceptions of how interactive digital tasks can promote equitable teaching. Findings suggest that while the structured framework supported pre-service teachers in identifying some equity-related features of interactive digital tasks, many struggled to articulate connections between task features and deeper equity goals. These results indicate that pre-service teachers benefit from explicit frameworks for discerning how interactive digital tasks can promote equitable teaching practices but require additional supports to develop productive perceptions of the equity constructs embedded in such tools. We discuss implications for mathematics teacher education.
Teachers’ instructional decisions reflect underlying pedagogical values, yet limited research examines how teachers recognise and interpret value differences in classroom practice. This study investigates how nine experienced Malaysian secondary mathematics teachers interpreted mathematics educational values (MEV) tensions through critical incident narratives. Data from online semi-structured interviews were analysed in two stages: first, extracting critical incidents from narratives, and second, coding underlying MEV tensions. Analysis identified six key MEV differences: application versus computation, enjoyment versus hardship, ability versus effort, exposition versus exploration, compliance versus contribution, and information communication technology (ICT) versus paper-and-pen. Findings indicate that making MEV tensions explicit through critical incident narratives can enhance teachers’ reflective practice by surfacing previously implicit value conflicts. This study advances the mathematics education literature by operationalising pedagogical values through narrative analysis and demonstrating the potential of the critical incident methodology to support values-informed professional practice. Implications for teacher preparation and professional development are discussed.
This qualitative multiple case study explores how secondary mathematics teacher leaders in Guam and Saipan leverage culturally sustaining pedagogy to advance mathematics equity for students in Micronesia. Grounded in an understanding of equity as fairness and differentiated support, this research highlights how culturally sustaining mathematics instruction can affirm students’ identities, honor community knowledge, and challenge Eurocentric curriculum traditions. Guided by frameworks of culturally sustaining pedagogy, critical mathematics education, and teacher agency, the study focuses on four award-winning secondary mathematics teachers recognized for their leadership in equitable practice. Findings reveal that participants incorporated local culture in varied ways, from embedding Micronesian histories, languages, and traditional practices into lessons, to fostering participation structures attentive to gender and cultural dynamics, and designing community-centered mathematics projects. Despite constraints from dominant math curricula, math teacher leaders enacted professional agency to adapt instruction and sustain community-rooted mathematical knowledge. These results confirm and extend prior work on culturally sustaining pedagogy by providing empirical insights from a Micronesian context, where such studies are scarce. The study further identifies teacher leadership as a key lever for sustaining equity-focused reforms, supporting calls to center Indigenous knowledge and cultural strengths in mathematics education. Implications include the need for ongoing professional development that empowers teachers to enact culturally sustaining practices, as well as policy supports that enable teacher-led innovation.
In this study, we tested a developmental progression of students’ understanding of angles involving five stages of quantification: gross, intensive, extensive, ratio, and rate. The Quantifications of Angularity Instrument was used to categorize 152 middle-school students by their stage of angle quantification. Developmental hypotheses were tested using quantitative methods. Findings from the study provide evidence that students’ understanding of angles progresses through these five developmental stages and that students develop non-circular notions of angle prior to circular notions of angle. Implications for instruction on angles and future research are discussed.
Disparities in achievement and equity associated with socio-economic status (SES) are particularly pronounced in mathematics. Research highlights that students from low-SES backgrounds often experience less effective instruction and lower teacher expectations. Factors such as ability grouping and school environment significantly impact attitudes and performance in mathematics, with low-SES students tending to experience higher anxiety and lower self-concept than their more advantaged peers. International studies show that the relationship between achievement and socio-economic status varies across countries, suggesting that with appropriate support, overcoming a disadvantaged background is possible; how this can be realised by interventions is less clear; questions remain about what forms of support effectively promote equity, and under what conditions. This review contributes by exploring evidence-based research that aims to support students’ access to mathematics. It explores how the interventions engage with complex contextual factors and reflects the forms of inequity they might disrupt. The analysis focuses on 29 empirical studies and is guided by Bronfenbrenner’s Process-Person-Context-Time (PPCT) model and Enchikova et al.’s conceptualisation of Equity. Key findings emphasise the importance of pedagogies that support low-SES students’ sense of relevance of the subject, impacting their attitudes and motivation. Adequate resources (tangible and intangible), and supportive environments reflecting high expectations are essential. Timing is key, with lower-secondary school reflecting an important inflection point. Perhaps most importantly, recognition of the centrality of the teacher, with sustained support, professional development and common planning time, is crucial to the development of the kinds of teaching and learning environments conducive to students’ access and equity in mathematics.
Writing on equity has persistently identified problems with an emphasis on outcomes and the deficit framings of marginalized communities associated with this focus. In this paper we focus on an Initial Teacher Education intervention that established a multi-institutional primary mathematics teacher educators community of practice (CoP) collaborating to develop number sense through mental mathematics. The CoP was built around shared concerns about poor mathematical outcomes and weak number sense among South African learners and primary pre-service teachers. It came together around exploring the models and materials from the Mental Starters Assessment Programme (MSAP) to strengthen the teaching and learning of efficient calculation strategies in lesson starters. Our focus in this paper is on analysis of primary mathematics teacher educators’ reflections on their experiences of participation in the CoP that pointed to shifts in orientation towards teaching for changes in learning efficient calculation strategies, rather than simply acknowledging poor outcomes. Drawing on questionnaire and interview data, and using Fullan’s (2001 and 2008) writing on multi-level change processes, we noted actions and experiences that suggested enablers for change centred around mental mathematics routines, and consistent and iterative addressing of pre-service teachers’ and learners’ early number fluencies and concepts.
Equity and marginalisation in mathematics education remains an ongoing concern. Classroom discourse has been seen to impact on student performance with dialogic practices being effective in providing for more equitable student participation. This paper investigates two examples of classroom discourse in relation to inclusive participation in early years mathematics classrooms. Drawing on data from a broader research project focusing on co-designing dialogic strategies aimed at enhancing students’ mathematical talk, we examine two classroom episodes from two low SES schools; one at the start and one towards the end of the co-design process. Using the Toolkit for Systematic Educational Dialogue Analysis (T-SEDA) framework, we analyse classroom interactions across three dimensions: openness for dialogue, invitation to contribute, and dialogic participation. Our findings suggest that classroom discourse that includes these dimensions are more indicative of inclusive learning environments through facilitating student voice and creating opportunities for dialogic participation. We argue that dialogic pedagogies, supported by systematic frameworks such as T-SEDA, hold potential for addressing inequities in mathematics learning and offer practical insights for educators committed to inclusive practice.
Ability grouping remains a widerspread and damaging practice in primary school mathematics classrooms, perpetuating deficit narratives, and replicating societal inequities. Despite increasing evidence supporting alternative approaches, some teachers encounter significant challenges in enacting these, as pedagogical decisions are shaped by deep-seated beliefs about fixed mathematical ability. This paper presents an in-depth case study of one primary school teacher, Meliame, who participated in a design-based collaborative intervention involving iterative cycles of out-of-class professional learning and in-class coaching support. Findings show that Meliame’s initial grouping approach generated rigid ability hierarchies rooted in fixed perceptions of capability. Through sustained, relational support, she was guided to identity individual student strengths, disrupting deficit narratives, and redesigning grouping practices accordingly. A pivotal classroom episode illustrated how deliberately drawing on a low-status student’s strength in visual art provided equitable access to mathematical reasoning and elevated his standing within the group and class. Three key factors supported successful implementation. High relational trust between teacher and researcher, explicit attention to status and expectations, and a strengths-based planning record. This study contributes an empirically grounded model for integrating strengths-based grouping into mathematics teaching, demonstrating that continuous, embedded professional learning is essential for disrupting entrenched inequities and positioning students as capable mathematical learners.
This study investigates how six U.S. high school mathematics teachers position themselves and their emergent bilingual students (EBs) within the classroom. eaSpecifically, it examines how translanguaging pedagogy promotes equity for EBs in mathematics learning environments. Findings indicate that teachers generally adopted inclusive and affirming storylines more frequently than exclusionary or deficit-based ones, though results varied across participants. Notably, teacher self-positioning was found to be inherently reciprocal to how the teachers position their EBs, often resulting in a dynamic distribution of power. During a year of professional development in which teachers used research-based tools and translanguaging strategies, this study demonstrates that translanguaging provides a powerful framework for challenging deficit narratives that marginalize EBs. The results suggest that teachers’ positioning with translanguaging storylines can foster more humanizing, rigorous, and linguistically inclusive mathematics classrooms.
This paper investigates how Australian teachers of mathematics, when directly asked, conceptualise success in mathematics education, drawing on responses to an open-ended survey question. While the term success features prominently in educational policy, literature, and discourse, the meanings and manifestations attached to the concept remain under-examined. Using qualitative analysis, teacher responses were categorised into six overlapping domains: Knowers of Mathematics, Users of Mathematics, Navigators of Mathematics, Humans with Mathematics, Judged for Mathematics and the Conflicted Other. Relationships between domains were then visually represented as the Agency-Time Arc, a conceptual thread that highlights how success is variably framed – from short-term academic performance to broader, agentive and long-term engagement. The apparent dominance of certain domains and the relatively limited presence of other interpretations of success raise critical implications for how students experience mathematics, particularly if success is interpreted as mastering a collection of concepts that are school-bound and temporary. The findings also demonstrate a level of tension and conflict between teachers’ personal beliefs and system-driven expectations, offering important insights for the ongoing interrogation of success in mathematics education.
All teachers mediate curriculum, even in countries with highly prescribed curricula. How this occurs equitably, despite global constraints such as accountability and performativity pressures, remains an area of investigation for both teachers and researchers internationally. Our research delves into teachers’ curriculum making practices in language diverse mathematics classrooms, an area that is currently underexplored. It examines how and why teachers mediate a prescribed maths scheme in equitable ways. We propose the ecological approach to teacher agency as a suitable theoretical framework to investigate teacher mediation of curriculum making. Drawing on interviews and observations in England, alongside relevant literature, we examine these mediation processes. In particular, we highlight the role of teachers’ beliefs, professional histories, social structures, material and relational resources which may enable or hinder teachers to achieve their agency for equitable practices. By illuminating a common international dilemma regarding how prescription can be mitigated through equitable and agentic teacher practices, this paper offers practical, methodological, and theoretical insights for teacher education, future research, and policy to better support teacher agency in ever-changing and diverse classrooms.
In this article, we report on an aspect of a study related to the implementation of challenging mathematical tasks in an initial teacher education program. Challenging mathematical tasks are complex, requiring problem solving and reasoning capabilities. Data collection consisted of a survey and a focus group interview administered after the implementation of tasks. Tasks were designed to include the challenged necessary to generate student productive struggle. Data analysis used both quantitative (descriptive statistics) and qualitative (content analysis) methods. An outcome of the study was the generation of a framework for the Design and Implementation of Challenging Online Mathematical Tasks (DICOMT). Data analysis indicated that initial teacher education students held reservations about the implementation of challenging mathematical tasks in school classrooms. These reservations appear connected to an uneven understanding of different mathematical topics and pre-existing beliefs that effective instruction should be based on problems which required familiar routine approaches to finding solutions.
Education systems worldwide are being challenged to re-examine the statistical and mathematical capabilities needed for an informed and critical citizenry, in light of disruptive phenomena such as global warming and climate change, migration, and pandemics. These phenomena are having a significant impact on the quality of life and wellbeing of citizens across the globe—and likely to continue to do so into the future. This article outlines the findings of a joint qualitative study in three countries (Brazil, Australia, South Africa) that focused on mathematics teacher autonomy during a time of disruption. The study draws on focus-group interviews with teachers regarding perceptions of enacting changes in content (what) and methods (how) they were teaching in response to the new mathematical and statistical demands associated with the COVID-19 pandemic crisis. The results contribute to new knowledge by highlighting the complex nature of teacher autonomy via a mapping of eight enabling and inhibiting influences affecting perceived situational freedom, at four levels: Classroom (curriculum demands; students’ knowledge), School (school leadership; time pressure), Profession (teacher self-efficacy; perceived responsibility for change), and External influences (from outside the school system, e.g., pandemic effect on students, parents, community; politicized sensitivities). Each influence was found to act in a dual fashion as both an enabler and inhibitor in relation to teacher autonomy. The discussion and conclusions reflect on implications for theory and practice, and point to directions for educational systems that can support the autonomy that teachers need to make changes when responding to (future) disruptions.
In this paper, we report on the challenges and opportunities of prospective and practicing teachers (PPTs) as they learn about culturally responsive pedagogy (CRP) in mathematics classrooms. Drawing on reflective journal assignments collected during several offerings of a teacher education course in Canada, we discuss how our study’s analysis was initially grounded in Ladson-Billings’ elements of CRP and then extended through our COFRI framework (Challenges, Opportunities, Fears, Resistance, Insights). We use a thematic analysis, along with a lens based in an earlier COFRI-based analysis to reveal three types of challenges and opportunities: mathematical, pedagogical, and ideological, each of which were found to consist of several sub-types. To address the challenges and opportunities expressed by PPTs, we introduce reflective- and action-based tools that mobilize PPTs to further grow their understanding of CRP in mathematics classrooms. The tools can be used as a journaling exercise, discussion prompt, or professional development activity for PPTs in mathematics teacher education programs, offering strategies for promoting equitable and culturally responsive teaching practices.
Research on integer addition often attributes students’ performance to their grade level while ignoring the role of prior whole number knowledge in influencing their learning difficulties. To explore whether these difficulties are related to their grade level or persistence in using rules based on their whole-number experiences, we first paired three second-grade students with three fifth-grade students who had similar integer addition strategies on their pretest. Then, we compared their strategy changes and performance after small-group sessions and whole-class activities around integer addition (i.e., instructional intervention). Although all pairs gained from the pretest to posttest, one fifth grader still relied on the rule of addition as getting a larger absolute value, suggesting her prior prolonged whole number addition experience impeded her progress compared to her second-grade pair. In fact, after whole-class activities, the second-grade pairs demonstrated better overall gains than the fifth-grade pairs. We extended our exploration to a larger group of 32 fifth graders and 18 second graders. Although fifth graders had higher performance on both the pretest and posttest, second graders made a significantly larger gain on two-addend integer addition problems than fifth graders. We argue that to make a more meaningful comparison across grade levels, researchers need to explore students’ strategy changes based on their prior knowledge. Our study findings suggest that some integer addition strategies, based on prolonged whole number addition experience, may hinder students’ further understanding of integer addition and contribute to learning difficulties.
Understanding functions is a cornerstone of mathematics education, yet students’ conceptions of functions often remain fragmented—especially when tasks require coordinating formal definitions with graphical, symbolic, and contextual representations. This study investigated Greek upper-secondary students’ understanding of functions across Grades 10–12 using a quantitative, curriculum-referenced design. A 10-item questionnaire assessed key strands of function knowledge (definition/uniqueness, domain, graph–rule connections, variable roles, and contextual interpretation). Participants were 185 students from three public schools in Greece (school-access sample; intact-class administration). Descriptive and inferential analyses revealed persistent conceptual difficulties across grades. The most prevalent misconceptions involved identifying functions from graphs (62
While there is widespread uptake of Number Talks (NTs) in elementary classrooms, limited research examines their effectiveness in fostering meaningful mathematical learning. Matney et al. (2020) describe this gap as a "black hole" (p. 247), noting the complexity of NTs due to five rapidly occurring, essential components: (1) creating a supportive classroom environment and community, (2) fostering productive classroom discourse, (3) the teacher's role in questioning and supporting student reasoning, (4) the role of mental math, and (5) designing purposeful computation problems. This study addresses this gap by investigating NTs enacted in Nova Scotia schools in Canada through an inquiry-based professional learning project involving 16 elementary teachers across two Regional Centres for Education. Over three full-day sessions, teachers co-planned and observed NTs, producing video data for analysis. Drawing on Maturana and Varela's (1992) enactivist perspective-which emphasizes learning as a dynamic, collective process-we explored the question: How do teachers' and students' interactions during Number Talks, viewed through an enactivist lens, reveal processes that contribute to an understanding of the efficacy of Number Talks? Our findings suggest that NTs hold potential for teachers to elicit and legitimize students' mental math strategies and for students to recognize, connect, and build on one another's ideas. These interactions illustrate how deeper mathematical thinking can emerge even within brief instructional windows. This study contributes to the emerging literature on NTs by reframing the discourse on NT efficacy and highlighting the potential of NTs, viewed through an enactivist lens, to foster collaborative and meaningful mathematical learning in elementary classrooms.
This qualitative case study investigates the interaction sequences that emerge among secondary school students engaged in mathematical modelling tasks using digital technologies. The study focuses on four groups of 14–17-year-old students in southern Norway, varying in achievement levels, as they worked on two distinct mathematical modelling tasks. Task 1 involved numerical manipulation and function development based on given data, while Task 2 required qualitative decision-making without explicit numerical constraints. Recorded interactions, comprising video and screen-capture data, were analysed using thematic analysis, focusing on pseudo, asymmetrical, reactive and mutual interaction sequences. The findings indicate that Task 1 predominantly elicited asymmetrical interaction sequences, with high-performing students taking the lead in proposing solutions, reflecting limited exchange patterns. In contrast, Task 2 fostered more reactive and mutual interaction sequences, supporting balanced and equitable exchanges within groups. Additionally, group composition shaped interaction patterns: same-achievement, high-performing upper secondary groups and mixed-achievement lower secondary groups demonstrated more mutual and reactive interaction sequences across tasks, while mixed-achievement upper secondary groups exhibited more varied patterns. The study highlights how task design, group composition and digital technology use interact to influence collaborative learning dynamics in mathematical modelling.
