Journal of Mathematical Behavior最新文献

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 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2026-01-30 DOI: 10.1016/j.jmathb.2025.101315

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

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.

本研究通过考察4岁儿童在不同教学资源的任务中如何理解重复模式和启动泛化，分析了他们的模式识别能力，旨在找出培养早期代数思维的最有效资源。设计并实施了24名学生一个学期的教学日程。这个行程是基于一个明确的教学方法，促进泛化。它从现实生活中的任务开始，操作和游戏，并使用图形资源发展到任务，考虑渐进抽象：从非正式到中间和正式的上下文。通过系统地观察儿童的行为、语言表达和图形制作来收集数据。通过分析儿童的原位策略和反应来评估表现，并确定儿童在不同表征中预期或转移结构规律时的泛化指标。教学干预强调开放式和结构化的提问，以促进解释、论证和逐步概括。结果表明，使用非正式资源的儿童比使用中间资源和正式资源的儿童表现更好。此外，25% %的参与者在翻译具有不同元素的模式时表现出泛化的迹象。我们的结论是，一方面，所使用的方法是在早期促进、评估和描述概括的有力工具；另一方面，它提供了一个有充分基础的教学框架，用于重新思考如何在幼儿时期概念化、结构化和实施模式教学，强调使用支持结构的有形、具体和视觉操作的资源。

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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 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2026-01-19 DOI: 10.1016/j.jmathb.2025.101313

Alban Da Silva

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.

本文使用结合数学和人类学的最新民族数学方法，通过4E框架（具体的、嵌入的、扩展的和主动的）探索数学认知（MC），超越经典的计算和欧几里得几何方法。研究的重点是瓦努阿图Pentecost岛Sia Raga的口头传统文化中的“沙画”实践-一种短暂的艺术，包括用一个手指在地面上创造对称的数字，而不是抬起它并返回到起点。为了解决记忆和创造这些人工制品的认知过程，该研究分析了各种形式的“外部表征”。首先是民族数学家，为了分析这种做法，他们基于群体理论和群体行为构建了代数模型。二是实践者自己动员起来的，包括实地的痕迹、绘制的规则，以及伴随而来的叙述。然后，通过从业者与环境的动态互动，特别是与构成环境的能力景观的互动，来检查这些表征。正如循环的多义概念所示，我们的发现强调了不同外部表征之间的一致性，并表明我们的模型捕获了认知结构，这些结构重申了数学活动的具体、嵌入、扩展和行动维度。通过建立一种抽沙实践的群体理论模型，这项工作表明群体行为可能为数学认知的研究提供新的途径。这种知识的隐性性质允许探索数学的教育视角，特别是身体运动、图表表示和文化文物的潜力。

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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 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2025-12-26 DOI: 10.1016/j.jmathb.2025.101316

Michael Jarry-Shore , Marta Kobiela , Lydia Provost

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.

在今天的小学数学课堂上，学生们被要求建构论点。如果这是为了提高学生的学习，教师必须能够识别和反驳学生的错误论点。这需要大量的知识，但很少有研究对未来的小学教师的这种知识的性质进行调查。我们要求17名未来的教师评估学生关于分数比较的论点的有效性，并使用反例反驳那些错误的论点。老师们在这项任务的数学方面做得很好，成功地识别了错误的论点，并用正确的反例来反驳它们。这项任务的教学方面更具挑战性，因为只有一个反例可以解释为什么一个论点是错误的，而反例有时会被干扰物所阻碍。我们建议教师教育工作者在为这些工作准备未来的小学教师时强调教学方面的考虑。然而，需要强调的是，需要对小学生对反例的反应进行额外的研究。

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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 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2025-12-20 DOI: 10.1016/j.jmathb.2025.101314

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

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.

本研究旨在设计、实施和评估线性变换核概念的假设学习轨迹（HLT），以新兴模型的启发式和包容性数学教育为基础。采用基于设计的研究方法，研究分为三个阶段：(1)实验准备，构建考虑学生多样性的HLT，并以理论和经验证据为基础；(2)教学实验，由智利和墨西哥的大学生实施；(3)回顾性分析，重构实际学习过程，并与提出的假设学习过程进行对比。结果显示，从现实生活背景到正式抽象水平的概念进展，学生群体之间的过程存在差异，但不会影响对核心概念的理解。指出了维数概念和核概念在抽象向量空间中泛化的障碍。该研究强调了新兴模式方法与包容性策略相结合的有用性，以促进不同背景下的学习。此外，它还提供了一种适应性强的教学方案，有助于丰富高等教育线性代数的教学。

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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 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2025-12-04 DOI: 10.1016/j.jmathb.2025.101303

Thomais Karavi , Angeliki Mali

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.

在面向大批学生的讲座中，讲师与学生建立双向沟通是一项挑战。本研究探讨了数学分析证明教学中讲师交流活动的隐性方面。我们使用多层交际分析来辨别讲师的交际行为，并将这些行为与从讲师的角度规范学生参与的规则联系起来。我们讨论了元规则的两个主题：关于证明发展的元规则和由高级数学规范形成的管理证明的元规则。确定的元规则表明，当重点放在引入证据时，参与话语的教学方向。该研究显示了讲座如何作为传播数学规范的媒介，潜在地促进学生参与数学话语。最后，我们呼吁对讲座的金属层面进行进一步的研究。

引用次数: 0

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 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2025-11-27 DOI: 10.1016/j.jmathb.2025.101302

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

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.

理解分数作为运算符仍然是小学高年级学生面临的一个持续的挑战，特别是当学生需要在不同的背景和表示中转移这些知识时。本研究考察了四年级、五年级和六年级学生如何解决涉及整数和分数基数的分数作为算子的任务。每个任务都以两种形式呈现——语境化的文字问题和非语境化的裸数练习——并且需要算术或图形解决方案。使用横向的，比较的事后设计与匹配的任务对，我们分析了成功率与过程相关的变量，以捕捉年级水平的表现模式。基于概要文件的方法提供了跨任务结果的细粒度描述。结果揭示了三个一致的模式。首先，程序巩固在整数基数的任务中更为明显，尽管在需要表征变化时不太稳定。其次，分数基数的任务产生了很高的遗漏率和反复出现的错误模式，尤其是在图形格式中。第三，语境化的文字问题只有在语言和视觉要求相对较低时才支持表现；否则，结果往往会下降。这些发现与强调乘法关系的口头表达和发展对操作者的强大理解的代表性灵活性的理论解释相一致。该研究扩展了这一文献，大规模地记录了困难变得最明显的特定任务条件，从而为教学设计和评估实践提供了诊断价值。

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引用次数: 0

Realistic visual representations as mediators between everyday and mathematical discourses in heterogeneous classrooms 现实的视觉表征在异质教室中日常话语和数学话语之间的中介作用

IF 1.7 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2025-11-22 DOI: 10.1016/j.jmathb.2025.101300

Merav Weingarden , Ronnie Karsenty , Boris Koichu

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.

在异构数学课堂中创造有效的学习环境仍然是一个持续的挑战。这项探索性研究探讨了现实表征如何支持学习者跨越不同的成就水平。通过两部分调查，本研究首先对两种表征为导向的活动进行了先验分析，以阐明现实表征的作用，然后探讨教师如何看待这种作用，以支持低成就学生和高成就学生。初步分析确定了现实表征的四个角色：激励、启动、连接和应用——每一个角色在日常话语和数学话语之间都有不同的中介作用。随后通过对两位教师在课堂上实施这些活动的案例研究进行的检查表明，一位教师认识到激励对成绩差的学生的好处，而另一位教师则强调了他们在鼓励成绩好的学生建立联系方面的价值。我们讨论了如何阐明现实表征的多方面作用，以帮助教师和教师教育者设计更具包容性的数学教学。

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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 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2025-11-21 DOI: 10.1016/j.jmathb.2025.101296

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

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.

在学习数学方面遇到困难的孩子通常有一种根深蒂固的应对机制，即把别人视为解决方案正确性的权威。在这个试点案例研究中，我们展示了促进他们检查自己的答案可以增强他们发展的方法。具体来说，我们研究了一个认知多样化的六年级学生在解决加法任务时使用的答案检查方案。我们利用建构主义方案理论作为框架来分析一项为期一年的教学实验的数据，证明他在解决问题的方案上取得了相当迅速的进步，从数全部到分解成十。随着这种快速的概念进步，我们发现他的答案检查方案与问题解决方案是同步发展的，通常比后者落后一步，也就是说，从没有答案检查到一个数字计数方案。在解决问题的过程中，他可能会使用参与性或预期性阶段的方案，而在答疑过程中，他主要使用预期性阶段的方案。我们讨论了这些关于认知多样化学生的数字进步的新发现的理论和实践意义。

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引用次数: 0

Teachers’ knowledge and technology in the teaching of functions: A view from Portugal and Brazil 函数教学中教师的知识与技术：来自葡萄牙和巴西的视角

IF 1.7 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2025-11-21 DOI: 10.1016/j.jmathb.2025.101301

Floriano Viseu , Helena Rocha , Sandra Menezes

Technology integration although highly recommended has proved to be complex. This study aims to characterize the perceptions of secondary mathematics teachers (grades 7th to 12th), from Portugal (Braga region) and Brazil (Campinas region), regarding the teaching of Functions and the use of technology. We also intend to consider the context of the two countries, reflecting on justifications for the differences identified (if any). The study adopts a mixed methodology based on a questionnaire applied to 129 Portuguese teachers in Braga region and 136 Brazilian teachers in Campinas region (quantitative dimension) and on an interview to six Portuguese teachers and six Brazilian teachers (qualitative dimension). The results point to similarities between teachers from Portugal and Brazil, but also to some differences. Technology potential is recognized by teacher from both countries, however teachers from Brazil tend to assume a more traditional use of technology, based on showing and more teacher-centered. This difference can be related to the Portuguese curriculum, valuing a relevant use of technology for more than three decades.

技术集成虽然受到强烈推荐，但已被证明是复杂的。本研究旨在描述来自葡萄牙（布拉加地区）和巴西（坎皮纳斯地区）的中学数学教师（7年级至12年级）对函数教学和技术使用的看法。我们还打算考虑两国的背景，反思所确定的差异（如果有的话）的理由。本研究采用混合方法，基于对布拉加地区129名葡萄牙教师和坎皮纳斯地区136名巴西教师的问卷调查（定量维度）和对6名葡萄牙教师和6名巴西教师的访谈（定性维度）。结果表明，葡萄牙和巴西的教师之间存在相似之处，但也存在一些差异。两国教师都认识到技术潜力，但巴西教师倾向于采用更传统的技术使用方式，以展示为基础，更以教师为中心。这种差异可能与葡萄牙课程有关，葡萄牙课程重视30多年来对技术的相关使用。

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引用次数: 0

Leveraging simulated student interactions to learn about prospective teachers’ formative assessment practice 利用模拟的学生互动来了解未来教师的形成性评估实践

IF 1.7 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2025-11-04 DOI: 10.1016/j.jmathb.2025.101298

Meghan Shaughnessy , Timothy Boerst , D’Anna Pynes

Formative assessment practices, including eliciting student thinking, interpreting student thinking, and engaging in teaching action grounded in information from those practices, must be nurtured in teacher preparation. We present a methodological approach using simulated student interactions to learn about prospective teachers’ formative assessment practices. We used the simulated student interaction to investigate whether such a simulated student interaction could reveal important details about prospective teachers’ capabilities with formative assessment. We report on findings from a study in which 12 prospective teachers engaged in the simulated student interaction. The study findings affirm that a simulated student interaction approach can encourage prospective teachers to engage in elements of formative assessment practices and reveal aspects of prospective teachers’ mathematical knowledge.

形成性评估实践，包括引出学生的思维，解释学生的思维，以及参与基于这些实践信息的教学行动，必须在教师准备过程中得到培养。我们提出了一种方法方法，使用模拟学生互动来了解未来教师的形成性评估实践。我们使用模拟的学生互动来调查这种模拟的学生互动是否可以揭示未来教师进行形成性评估的能力的重要细节。我们报告了一项研究的结果，其中12名准教师参与了模拟学生互动。研究结果证实，模拟学生互动方法可以鼓励准教师参与形成性评估实践的要素，并揭示准教师数学知识的各个方面。

引用次数: 0