Journal of Mathematical Behavior最新文献

A study on the different representations and performance profiles on fractions as operators in primary education 小学教育中分数作为运算符的不同表征及表现概况研究

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-06-01 Epub Date: 2025-11-27 DOI: 10.1016/j.jmathb.2025.101302

Diana Herreros-Torres , Maria T. Sanz , Carlos B. Gómez-Ferragud

{"title":"A study on the different representations and performance profiles on fractions as operators in primary education","authors":"Diana Herreros-Torres , Maria T. Sanz , Carlos B. Gómez-Ferragud","doi":"10.1016/j.jmathb.2025.101302","DOIUrl":"10.1016/j.jmathb.2025.101302","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"82 ","pages":"Article 101302"},"PeriodicalIF":1.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145610578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

From context to formal reasoning: Supporting students’ understanding of the kernel of a linear transformation through a hypothetical learning trajectory 从情境到形式推理：通过假设的学习轨迹，支持学生理解线性变换的核心

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-06-01 Epub Date: 2025-12-20 DOI: 10.1016/j.jmathb.2025.101314

Andrea Cárcamo , Juan de Dios Viramontes-Miranda , Claudio Fuentealba

{"title":"From context to formal reasoning: Supporting students’ understanding of the kernel of a linear transformation through a hypothetical learning trajectory","authors":"Andrea Cárcamo , Juan de Dios Viramontes-Miranda , Claudio Fuentealba","doi":"10.1016/j.jmathb.2025.101314","DOIUrl":"10.1016/j.jmathb.2025.101314","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"82 ","pages":"Article 101314"},"PeriodicalIF":1.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Pattern recognition and generalization in 4-year-olds: The impact of task contexts and teaching resources on the development of algebraic thinking 4岁儿童模式识别与概化：任务情境与教学资源对代数思维发展的影响

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-06-01 Epub Date: 2026-01-30 DOI: 10.1016/j.jmathb.2025.101315

Yeni Acosta , Ángel Alsina , Cristina Ayala-Altamirano , Jefferson Rodrigues-Silva

{"title":"Pattern recognition and generalization in 4-year-olds: The impact of task contexts and teaching resources on the development of algebraic thinking","authors":"Yeni Acosta , Ángel Alsina , Cristina Ayala-Altamirano , Jefferson Rodrigues-Silva","doi":"10.1016/j.jmathb.2025.101315","DOIUrl":"10.1016/j.jmathb.2025.101315","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"82 ","pages":"Article 101315"},"PeriodicalIF":1.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146076890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Group actions at the fingertip: A 4E approach to mathematical cognition in Vanuatu sand drawing 指尖的群体行动：瓦努阿图沙画中数学认知的4E方法

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-06-01 Epub Date: 2026-01-19 DOI: 10.1016/j.jmathb.2025.101313

Alban Da Silva

{"title":"Group actions at the fingertip: A 4E approach to mathematical cognition in Vanuatu sand drawing","authors":"Alban Da Silva","doi":"10.1016/j.jmathb.2025.101313","DOIUrl":"10.1016/j.jmathb.2025.101313","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"82 ","pages":"Article 101313"},"PeriodicalIF":1.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146022504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Metarules in university mathematics lectures: A commognitive analysis of proof-oriented mathematics teaching 大学数学讲座中的元规则：面向证明的数学教学的交际分析

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-06-01 Epub Date: 2025-12-04 DOI: 10.1016/j.jmathb.2025.101303

Thomais Karavi , Angeliki Mali

引用次数: 0

How can students’ engagement with instructional videos on generalizing with algebraic expressions be scaffolded? Design research for specifying the content-specific focus of scaffolds 如何让学生参与代数表达式推广的教学视频？为明确支架的内容重点而进行的设计研究

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-06-01 Epub Date: 2026-02-07 DOI: 10.1016/j.jmathb.2026.101317

Stefan Korntreff , Susanne Prediger

{"title":"How can students’ engagement with instructional videos on generalizing with algebraic expressions be scaffolded? Design research for specifying the content-specific focus of scaffolds","authors":"Stefan Korntreff , Susanne Prediger","doi":"10.1016/j.jmathb.2026.101317","DOIUrl":"10.1016/j.jmathb.2026.101317","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"82 ","pages":"Article 101317"},"PeriodicalIF":1.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146173039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

“It won’t work every time”: Prospective elementary teachers’ counterexamples for students’ false arguments about fractions “它不会每次都有效”：准小学教师对学生关于分数的错误论点的反例

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-06-01 Epub Date: 2025-12-26 DOI: 10.1016/j.jmathb.2025.101316

Michael Jarry-Shore , Marta Kobiela , Lydia Provost

{"title":"“It won’t work every time”: Prospective elementary teachers’ counterexamples for students’ false arguments about fractions","authors":"Michael Jarry-Shore , Marta Kobiela , Lydia Provost","doi":"10.1016/j.jmathb.2025.101316","DOIUrl":"10.1016/j.jmathb.2025.101316","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"82 ","pages":"Article 101316"},"PeriodicalIF":1.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Beyond the formula: Students’ reasoning about rate and point in linear equations and approximation 公式之外：学生对线性方程和近似中速率和点的推理

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-06-01 Epub Date: 2026-02-07 DOI: 10.1016/j.jmathb.2026.101318

Kyunghee Moon

引用次数: 0

Stage 3 high school students’ generalization of a cubic identity 第三阶段高中生对三次恒等式的泛化

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-03-01 Epub Date: 2025-09-06 DOI: 10.1016/j.jmathb.2025.101283

Erik S. Tillema , Andrew M. Gatza , Weverton Ataide Pinheiro

{"title":"Stage 3 high school students’ generalization of a cubic identity","authors":"Erik S. Tillema , Andrew M. Gatza , Weverton Ataide Pinheiro","doi":"10.1016/j.jmathb.2025.101283","DOIUrl":"10.1016/j.jmathb.2025.101283","url":null,"abstract":"<div><div>This paper reports on one study in a series of design research studies that have taken as a guiding design principle that combinatorial and quantitative reasoning can serve as a constructive resource for high school students to establish algebraic structure between a polynomial and its factors. Within this framing, we report on an interview study with eight 10th-12th grade students whose purpose was to investigate their progress towards generalization of the cubic identity <span><math><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mspace></mspace><msup><mrow><mn>1</mn><mo>∙</mo><mo>(</mo><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo><mo>+</mo><mn>3</mn><mo>∙</mo><mrow><mfenced><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∙</mo><mn>1</mn></mrow></mfenced></mrow><mo>+</mo><mn>3</mn><mo>∙</mo><mrow><mfenced><mrow><msup><mrow><mi>x</mi><mo>∙</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mo>+</mo><mn>1</mn><mo>∙</mo><mo>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></math></span>. The students worked on this generalization by solving cases of a 3-D combinatorics problem and representing their solutions using 3-D arrays. Findings include the identification of how differences in students’ combinatorial reasoning impacted their reasoning with 3-dimensional arrays and their progress towards a general statement of the cubic identity.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"81 ","pages":"Article 101283"},"PeriodicalIF":1.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145005201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Tandem conceptual progression (TCP): Pilot case study of a cognitively diverse student’s use of adjacent problem-solving and answer-checking schemes 串联概念进展（TCP）：一个认知差异学生使用相邻问题解决和答案检查方案的试点案例研究

IF 1.7

Journal of Mathematical Behavior Pub Date : 2026-03-01 Epub Date: 2025-11-21 DOI: 10.1016/j.jmathb.2025.101296

Helen Thouless , Cody Harrington , Ron Tzur , Alan Davis , Beyza E. Dagli

{"title":"Tandem conceptual progression (TCP): Pilot case study of a cognitively diverse student’s use of adjacent problem-solving and answer-checking schemes","authors":"Helen Thouless , Cody Harrington , Ron Tzur , Alan Davis , Beyza E. Dagli","doi":"10.1016/j.jmathb.2025.101296","DOIUrl":"10.1016/j.jmathb.2025.101296","url":null,"abstract":"<div><div>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 <em>pilot</em> 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.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"81 ","pages":"Article 101296"},"PeriodicalIF":1.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145568249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0