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.
Within classroom discussions, teachers’ feedback is one way they can navigate the dual goals of sharing mathematical authority and supporting students’ progress towards mathematical goals. In this study, we consider the context of classroom reasoning-and-proving and illustrate how teachers’ feedback and other proving actions served to achieve those dual goals. Using a modified version of the Otten and colleagues (2017) framework, we coded proof actions and interactions that occurred during whole class and small group conversations in a high school classroom. We demonstrate how the teacher alternated between maintaining and attempting to disseminate authority throughout the reasoning-and-proving process in alignment with the lesson goals. Particularly, the teacher maintained authority when defining accepted statements and key vocabulary to address content goals and disseminated authority to students when establishing conjectures and when constructing and determining the validity of arguments to actively engage students in the proving process. This case study highlights one way teachers can use feedback as a way to navigate the double bind within an intellectually honest classroom to successfully attend to content goals and center students as doers of mathematics within a proof context.
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.
Understanding how students reason about the negation of logical statements is essential in supporting their mathematical development. Prior literature suggests that undergraduate students experience persistent challenges in articulating the precise negation of conditional statements, in particular. Whereas the negation of the logical implication “p implies q” is the conjunction “p and not q,” many students instead provide the logically distinct statement “p implies not q.” In this study, we investigate the challenges that introductory proofs students experience when negating implications, and the reasoning they demonstrate in addressing these challenges. Drawing on clinical interviews with three undergraduate students enrolled in an introductory proofs course, we identify four interrelated factors that contribute to and explain their treatments of negation: the role of (hidden) quantifiers, reliance on Euler diagrams, the tendency to reduce the universal set to the truth set of the hypothesis, and the effect of pragmatics and everyday language. These factors describe the challenges that students experience when reasoning about a logical implication and its negation, and, when taken together, they explain why students might treat the negation of a logical implication as “p implies not q.” Moreover, we explicate nuances in quantification that can render this statement logically equivalent to the negation. We conclude by urging instructors of introductory proofs courses to bring forth, leverage, and connect their students’ existing ways of reasoning about logical implications and negations with formal mathematical logic and language.
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.
This study investigates critical translanguaging spaces (CTSs) in a second-grade dual-language mathematics classroom. Guided by Hamman's (2018) conceptualization of CTS and the Translanguaging Structures Framework (TSF) (T & icirc;rnovan, 2023, 2024), we explore how cultivating and mobilizing critical translanguaging spaces may shape students' translanguaging practices during mathematical communication. Using a qualitative hybrid-thematic approach, data were collected from three CTS activities: student interviews, a meta-translanguaging whole-class activity, and a small-group student-written math book activity. CTSs enable students to draw upon their linguistic repertoire, engage in meaningful dialogue about translanguaging, and co-construct knowledge in mathematics classrooms. The findings illustrate the transformative potential of CTSs in shaping students' translanguaging practices toward more inclusive mathematics learning and shifting linguistic power dynamics from the teacher to the students. This research offers practical implications for curricular design, teacher professional development, and policy, providing a framework for more inclusive and empowering mathematics learning environments for multilingual learners.
This study examines how undergraduate students chunk graphs, or group complex visual information into meaningful segments, as they interpret a logistic function graph modeling a chemical titration curve. Fifteen students participated in task-based interviews where they were asked to explain the relationship between the pH of a solution and the amount of base added. Analysis revealed four chunking strategies, organized into two broader levels: superficial chunking, focused on how the graph appears, and deep-level chunking, focused on why the graph behaves as it does. Superficial chunking included (a) visual-trend chunking, where students focused predominantly on visual patterns or shapes (e.g., flat, steep) and (b) numerical/scale-based chunking, wherein students relied primarily on numerical threshold (e.g., basic, acidic) to segment the graph. Deep-level chunking included (a) mathematical interpretation, in which students chunked the graph by attending to both the magnitude and direction of change in the rate of change, and (b) chemical interpretation, in which students segmented the curve by integrating mathematical features (e.g., slope, curvature, inflection points) with chemically meaningful concepts (e.g., buffering, neutralization). All students began the graph interpretation process by using visual trend chunking strategies. Although the initial chunking was similar, students differed in how they interpreted the resulting chunks: some remained at superficial levels of sense-making, whereas others moved toward deeper-level interpretations. These findings present a novel way to analyze how students interpret graphs and highlight how attention to rate of change can anchor students’ chunking in covariational reasoning, supporting robust interpretations of logistic graphs in STEM contexts.
In this paper, we showcase how two undergraduates reinvented tests for series divergence and convergence. Through a context problem, the “Partial Sum Sequence Game,” the students argued for necessary and sufficient conditions for sequences of partial sum divergence and convergence. Ultimately, the students reinvented claims related to the test for divergence, p-series test, and comparison test. We close the paper discussing aspects of the well-designed tasks and scaffolding which supported the students in their reinvention.
In this study we employed a figure pattern task to investigate how students' mathematical justifications varied between group work and whole-class discussions during a single lesson in three Finnish Grade-7 classes (ages 13-14). Although previous researchers have typically focused on either group work or whole-class discussions, by combining both, we gained a deeper understanding of how students construct their justifications. This study comprised three lessons, each involving the same figure pattern problem. The lessons were recorded using multiple video cameras, and the students' written productions were collected. We analyzed the content and level of explicitness of the students' justifications using thematic analysis. The findings revealed significant variations in the content and explicitness of justifications between the two activities. Although group discussions often featured more detailed justifications, many of these were either simplified or not shared in whole-class discussions. In addition, reasoning was often implicit in whole-class discussions, and the teachers settled for superficial justifications in many cases. In contrast, refined or new justifications were also introduced in whole-class discussions. The teachers' selection of students' work influenced the justifications that were shared and discussed during the whole-class discussions. Although erroneous solutions were chosen for discussion, the potential to learn from the errors they contained was not maximized. The study highlights the importance of teachers monitoring group work to support richer whole-class discussions and the need to raise more diverse student justifications for discussion. Students' solutions may appear similar on the surface, but their underlying reasoning may differ significantly.
In this paper, we introduce the construct of a mathematical-pedagogical task (MP task) and analyse the responses of prospective secondary mathematics teachers as they attempted to explain a well-known relationship between the slopes of perpendicular lines. We suggest that the notion of “solution spaces”, introduced by Leikin and Levav-Waynberg (2008) in discussion of multiple solutions to a mathematical problem, is applicable in considering an MP task. Findings illustrate a variety of possible approaches and mathematical ideas that can reveal different solution spaces. We argue that MP tasks are an appropriate avenue for strengthening teachers’ knowledge.
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.
