Engagement has been shown to influence students’ understanding and achievement, yet little is known about how engagement with statements relates to students’ understanding of corresponding proofs. In this paper, we take a step towards addressing this gap by investigating whether engaging with statements is associated with better understanding of proofs by contradiction and contraposition. Eighty-seven Indonesian undergraduate mathematics education students, organized in four groups, participated in the study. Two groups (n = 43) were randomly selected to first engage with the statements—reflecting on their meaning, judging their truth, and attempting to prove or disprove them—before responding to an instrument designed to measure their understanding of given proofs for the same statements; the other two groups (n = 44) proceeded directly to the instrument. There were six statements overall, three proved by contradiction and three by contraposition. We found that students who engaged with the statements achieved significantly higher mean scores on measures of proof understanding than those who did not, with a medium effect size. Regression analyses confirmed engagement as a significant predictor of students’ understanding for both proof methods, and individual semi-structured interviews with 12 participants provided qualitative support for these findings. Overall, our findings identify engagement with statements as a meaningful factor in students’ proof understanding and suggests that selectively incorporating an engagement stage before introducing key proofs may enhance undergraduate mathematics instruction beyond the typical Definition-Theorem-Proof (DTP) approach.
The purpose of this study is to understand how elementary school children make sense of their mathematical experiences in the context of a playful Math Fair, and how epistemic emotions, forms of agency, and perceptions of participatory mathematical identity emerge in these interpretations. An interpretive qualitative approach was employed with 80 students aged 8–11 from urban and rural schools in the Valparaíso Region. Data collection was conducted through stimulated recall using photographs taken during the activity, interviews, and field notes. The analysis, conducted using open and axial coding, identified two central and interrelated dimensions: (a) epistemic emotions—surprise, curiosity, productive confusion, and enjoyment—associated with processes of seeking explanations, exploring alternatives, reviewing errors, and sustained participation; and (b) agency and participatory mathematical identity, expressed in perceptions of competence, legitimate participation, boldness in proposing ideas, collaboration, and peer leadership. The findings show that playful-mathematical experiences provided affective, bodily, and relational conditions for students to explore strategies, review errors, justify decisions, and construct situated mathematical meanings. Implications for the design of school experiences that integrate play, emotion, child participation, and meaningful mathematical activity are discussed.
This study focuses on the development of a hypothetical learning trajectory (HLT) for learning to solve systems of equations involving letter-symbolic unknowns. A longitudinal constructivist teaching experiment was conducted, which followed a student, Kaelyn, from the spring semester of her sixth-grade year through the fall semester of her seventh-grade year. Kaelyn could perform mental operations on a unit of units. The HLT that was developed documents instructional methods that bridged the gap between her initial operations on a unit of units and the learning goal to solve systems of equations involving letter-symbolic substitution. The phases of the HLT included tacit reasoning about unknowns, explicit reasoning about multiple, related non-letter-symbolic unknowns, explicit reasoning about non-letter-symbolic substitution, and explicit reasoning about letter-symbolic substitution. This study contributes an HLT for middle-grades students’ learning to solve systems of equations, including their critical transition from non-letter-symbolic to letter-symbolic reasoning about unknowns.
Concept image and concept definition, among the most utilised constructs in mathematics education research, emerged in a period characterised by cognitive theories of learning aimed at explaining individual students’ mathematical constructions. Acknowledging learning theories that have since developed, and in particular those that emphasise the bodily dimensions of mathematical thinking, in this article we propose concept gesture as a construct that foregrounds the role of the body in mathematical teaching and learning, while expanding on the notions of concept image and concept definition. We illustrate the construct by examining some of a lecturer’s mathematical gestures in a first-year university analysis lecture, focusing specifically on those that convey conceptual meaning in relation to the concepts of sequence and limit. We discuss how concept gestures, understood within an embodied perspective, can productively enrich the concept image/definition framework, and suggest practical, theoretical, empirical, and methodological implications.
We understand pattern problems as non-routine problems that require finding implicit regularities and mathematical structures to generalize beyond immediate evidence. Our study examines in what ways 11-and 12-year-old students' approaches to solving pattern problems are related to the types of pattern problems they pose in an unstructured task. Data were collected during four sessions in which fifty students first solved a sequence of pattern problems requiring near and far generalization and later engaged in a problem-posing task. We analyzed students' written productions to characterize the strategies they used and the extent to which they were able to generalize. We also categorized the problems they posed into null, incomplete, variation of a solved problem and new problem. Results reveal that those who did not pose any problems, or who posed null or incomplete problems, in the problem-solving task had mainly resorted to strategies that had little to do with generalization, such as extending the given drawing. In contrast, students who posed variations of previously solved problems or entirely new problems, had mainly used strategies oriented toward generalization, such as mixed strategies and the continuation of arithmetic progression. Although our findings are context-specific, they contribute to ongoing efforts to better understand how problem-solving experiences relate to students' ability to pose new problems. We conclude that research seeking to explore problem-posing processes should provide students with opportunities not only to pose problems, but also to discuss and solve them, thereby facilitating a more comprehensive understanding of their reasoning processes.
Existing research on teachers becoming teacher educators, and specifically mathematics teacher educators (MTEs), shows that identity plays a significant role in this transition. However, MTEs’ changed identities along their professional trajectories are underresearched. This study focuses on two mathematics teachers in their journey to become MTEs, in a context of a peripheral region with severe lack in educational leadership. The subjects are graduates of a unique program aiming to develop local mathematics leaders. Using a longitudinal case-study design employing in-depth interviews, held during the program and two years after its end, allowed to follow these novice MTEs’ journeys, and unpack changes in their identities. Findings show that although early on the way the dual identities of mathematics teacher and MTE were in tension, thus creating challenges for the MTE work, over time identities harmonized, with the teacher identity cultivating the MTE identity and becoming a source for credible leadership.
Structure sense is central to learning mathematics. This study analyses undergraduate students’ mathematical writing, focusing on their use of brackets to infer aspects of their structure sense. Using students’ written responses to integration and differentiation problems, we examine two nonstandard bracketing practices: mental brackets (omitted but implicitly present) and unconventional brackets (additional brackets used to emphasise structure). Drawing on 353 student scripts from Greece and Turkiye, collected through two questionnaires, the analysis shows that both forms of nonstandard brackets occur frequently. Although these uses do not align with formal conventions or curriculum expectations, they occur consistently in students’ solutions and provide insight into their structure sense. Mental brackets are mostly evident in integration tasks, where students appear to prioritise calculation over formal presentation. Unconventional brackets, meanwhile, function as structural supports, with additional notation indicating that students may use them to manage complexity and make structure visible when applying rules or substitutions. These findings suggest that nonstandard bracket use should not be viewed simply as error or redundancy, but as evidence of meaningful engagement with mathematical structure. We relate the results to research at earlier educational levels, highlighting both continuities and differences and considering implications for how instructors interpret and respond to students’ notation.
Mathematical discussions, along with other learner-centered teaching practices, are becoming more prevalent in tertiary mathematical education, yet the mathematical content of these discussions and the learning involved remain less explored. We developed the Discourse Mapping Tree (DMT), a commognitive based tool for mapping mathematical content during discussions, which allowed us to examine learning of a class as a whole in linear algebra discussions. The DMT displays the objects on which a discussion focuses, the multiple types of realizations of these objects and the narratives authoring links between the realizations. The DMT showed that the division of labor between the instructor and the students in our study was not always symmetrical. This tool provides a succinct overall image of the publicly available mathematics during a classroom discussion and allows for the mapping of complex tertiary level mathematics which include multiple objects and realizations.
This study aims to explore how university students construct the connection between Cavalieri's principle and Riemann sums through arguments generated with dynamic mathematics software within a collaborative learning methodology. Seven university students took part in the study, which lasted four weeks. Data collection included tasks, field notes, audio-video recordings, and screen recordings. Data analysis was based on the integration of the Toulmin model and the cK cent model. The specific approach helped students realize that the rectangles in the Riemann sum approach line segments as their limit approaches infinity. It also supported students in developing the concept of line segments in Cavalieri's principle based on their understanding of the rectangles in Riemann sums. It was observed that understanding line segments helped students explain the geometric rationale behind some calculus rules. In this way, students were able to move from algebraic to geometric representations within a social interaction environment. The ability to switch between representations fostered a deeper understanding of the connection between Cavalieri's principle and Riemann sums. Integrating dynamic software into a collaborative learning methodology enabled students to construct Cavalieri's principle from a more dynamic perspective.
Generalizing with algebraic expressions and variables is a challenging area of algebra content that requires conceptual understanding. Instructional videos can contribute to conceptual understanding only if students deeply engage with the video content. This deep engagement can be scaffolded by activating design features (e.g., brief prompts). However, limited research attention has been given to specifying their content-specific focus. This paper reports on a design research approach concerning generalizing with expressions, conducted in two design-experiment cycles with 16 students. Through qualitative comparative scaffolding analysis, we specified which content-specific processes needed focus at what moment through which activating feature. The first design experiment cycle, using an unscaffolded video, revealed which processes occurred spontaneously and which required scaffolding through activating features. The second cycle indicated that the designed activating features could indeed scaffold the intended processes. We discuss how this design research specification approach for content-specifically focused scaffolds could be transferred to other subject-matter content.
This study introduces and uses an analytic framework to examine how precalculus students reason about rate and point as multiplicative objects in linear equations and linear approximation. Building on prior work on ratio and rate, the analysis characterizes students’ reasoning across generalized ratio, unit rate, and interiorized ratio, foregrounding bottlenecks in coordinating rate with a reference point in linear and linear-approximation tasks, especially in graphical contexts. Clinical interviews with three precalculus students revealed bottlenecks in movement toward interiorized ratio reasoning; conceptions often remained fragile, particularly with non-integer or symbolic changes in x. Students also struggled to coordinate rate with a reference point when constructing new coordinates, exposing limited understanding of a point as a record of covariation. The findings contribute a refined theoretical lens for analyzing covariational reasoning and for supporting students’ coordination of rate and point as multiplicative structures across algebraic and geometric contexts.
This study analyzes pattern recognition in 4-year-olds by examining how they understand repeating patterns and initiate generalization in tasks with different teaching resources, with the aim of identifying the most effective resources for fostering early algebraic thinking. A teaching itinerary was designed and implemented with 24 children over one school term. This itinerary is based on an explicit pedagogical approach that promotes generalization. It starts with tasks from real-life situations, manipulatives, and games, and progresses to tasks using graphic resources, considering progressive abstraction: from informal to intermediate and formal contexts. Data were collected through systematic observation of the children's actions, verbalizations and graphical productions. Performance was assessed by analyzing the children's in-situ strategies and responses, and indicators of generalization were identified when children anticipated or transferred structural regularities across different representations. The teaching intervention emphasized both open and structured questioning to promote explanation, justification and progressive generalization. The results show that children perform better with informal resources than with intermediate and formal resources. In addition, 25 % of the participants show signs of generalization when translating patterns with different elements. We conclude, on the one hand, that the approach used is a powerful tool to promote, assess and describe generalization at early ages; and, on the other hand, that it provides a well-founded pedagogical framework for rethinking how the teaching of patterns is conceptualized, structured and implemented in early childhood, emphasizing the use of resources that support tangible, concrete and visual manipulation of structures.
Using recent ethnomathematical methods that combine mathematics and anthropology, this article explores mathematical cognition (MC) through the 4E framework (embodied, embedded, extended, and enactive) beyond classical approaches to numeration and Euclidean geometry. The research focuses on ”sand drawing” practices within the oral tradition culture of Sia Raga on Pentecost Island (Vanuatu) — an ephemeral art consisting of creating symmetrical figures on the ground using one finger, without lifting it and returning to the starting point. To address the cognitive processes at work in memorizing and creating these artifacts, the study analyzes various forms of ”external representations”. First, those of ethnomathematicians who, to analyze this practice, construct algebraic models based on group theory and group actions. Second, those mobilized by the practitioners themselves, including traces on the ground, drawing rules, and the narratives that accompany them. These representations are then examined through the lens of practitioners’ dynamic interactions with their environment, particularly with the landscape of affordances that structures it. As illustrated by the polysemous concept of cycle, our findings highlight congruencies between the different external representations and suggest that our model captures cognitive structures that reaffirm the embodied, embedded, extended, and enactive dimensions of mathematical activity. By developing a group-theoretical model of sand drawing practice, this work suggests that group actions may offer new avenues for research on mathematical cognition. The tacit nature of this type of knowledge allows for exploring educational perspectives in mathematics, particularly the potential of bodily movements, diagrammatic representations, and cultural artifacts.
In today’s elementary mathematics classroom, students are urged to construct arguments. If this is to enhance students’ learning, teachers must be able to identify and refute students’ false arguments. This requires substantial knowledge, yet little research has examined the nature of this knowledge with prospective elementary teachers. We asked 17 prospective teachers to assess the validity of students’ arguments regarding the comparison of fractions and to refute those that were false using counterexamples. Teachers did well with the mathematical aspects of this task, successfully identifying false arguments and refuting them with correct counterexamples. The pedagogical aspects of the task were more challenging, as only one counterexample explained why an argument was false and counterexamples were hampered at times by distractors. We propose that teacher educators emphasize pedagogical considerations in preparing prospective elementary teachers for such work. However, which considerations to emphasize requires additional research examining elementary students’ reactions to counterexamples.
This study aims to design, implement, and evaluate a hypothetical learning trajectory (HLT) for the concept of the kernel of a linear transformation, grounded in the heuristic of emergent models and inclusive mathematics education. A design-based research methodological approach was adopted, and the study was developed in three phases: (1) experiment preparation, in which an HLT was constructed that considered student diversity and was based on theoretical and empirical evidence; (2) teaching experiment, implemented with university students from Chile and Mexico; and (3) retrospective analysis, through which actual learning processes were reconstructed and compared to the proposed hypothetical learning process. The results show a conceptual progression from real-life contexts toward formal levels of abstraction, with differences in processes among the student groups but without compromising the understanding of the core concept. Obstacles were identified related to the notion of dimension and the generalization of the kernel concept in abstract vector spaces. The study highlights the usefulness of the emergent models approach combined with inclusive strategies to promote learning in diverse contexts. In addition, it offers an adaptable instructional proposal that can contribute to enriching the teaching of Linear Algebra in higher education.
In lectures delivered to large cohorts of students, it is challenging for a lecturer to develop a two-way communication with students. This study explores the tacit aspects of a lecturer’s communicational activity of proof teaching in mathematical analysis. We used a multi-layered commognitive analysis to discern the lecturer’s communicational actions and to connect these actions with metarules that regulate students’ participation from the lecturer's perspective. We discussed two themes of metarules: metarules about the development of proof and metarules governing proof as shaped by advanced mathematical norms. The identified metarules indicate pedagogical orientations for participation in the discourse when focusing on the introduction of proof. The study shows how lectures serve as a medium for disseminating the mathematical norms, potentially facilitating students' participation in the mathematical discourse. We conclude by calling for further research into the metalevel aspects of lecturing.
Understanding fractions as operators remains a persistent challenge in the upper grades of primary education, particularly when students are required to transfer this knowledge across different contexts and representations. This study examined how Grade 4, 5, and 6 students solve fraction-as-operator tasks involving both whole and fractional base quantities. Each task was presented in two formats—contextualised word problems and decontextualised bare-number exercises—and required either an arithmetic or a graphic solution. Using a cross-sectional, comparative ex post facto design with matched task pairs, we analysed success rates together with process-related variables to capture grade-level patterns of performance. A profile-based approach provided a fine-grained description of outcomes across tasks. The results reveal three consistent patterns. First, procedural consolidation was more visible in tasks with whole-number bases, though less stable when representational variation was required. Second, tasks with fractional bases produced high omission rates and recurrent error patterns, especially in graphic formats. Third, contextualised word problems supported performance only when linguistic and visual demands were relatively low; otherwise, outcomes tended to decline. These findings align with theoretical accounts that emphasise the verbal articulation of multiplicative relationships and representational flexibility in developing a robust understanding of the operator. The study extends this literature by documenting, at scale, the specific task conditions under which difficulties become most visible, thereby offering diagnostic value for instructional design and assessment practices.
Children experiencing difficulties learning mathematics often have a long-embedded coping mechanism of looking to others as authorities for the correctness of their solutions. In this pilot case study, we demonstrate ways in which promoting their checking of their own answers can empower their development. Specifically, we examine answer-checking schemes that a cognitively diverse 6th grader with difficulties learning mathematics used when solving additive tasks. We draw on constructivist scheme theory as a framework to analyze data from a year-long teaching experiment, demonstrating a rather rapid progress in his problem-solving schemes, from counting-all to break-apart-make-ten. Along with this rapid conceptual progress, we found that his answer-checking schemes developed in tandem with the problem-solving schemes, typically being one cognitive step behind the latter, that is, advancing from no answer-checking to a numerical count-on scheme. During problem-solving, he may have used schemes at either a participatory or anticipatory stage, whereas for answer-checking he mostly used schemes at the anticipatory stage. We discuss theoretical and practical implications of these novel findings about numerical progress in a cognitively diverse student.
Creating effective learning environments in heterogeneous mathematics classrooms remains a persistent challenge. This exploratory study examines how realistic representations can support learners across achievement levels. Through a two-part investigation, the study first conducts an a priori analysis of two representation-oriented activities to articulate the roles of realistic representations and then explores how teachers view such roles as supporting low- and high-achieving students. The initial analysis identifies four roles of realistic representations: motivating, initiating, connecting, and applying—each mediating differently between everyday and mathematical discourses. The subsequent examination through case studies of two teachers implementing these activities in their classrooms shows that one teacher recognized motivational benefits for low-achieving students, whereas the other highlighted their value in encouraging connection-making for high-achieving students. We discuss how explicating the multifaceted roles of realistic representations can help teachers and teacher educators design more inclusive mathematics instruction.
