Journal of Mathematical Behavior最新文献

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Complementary dimensions of the Theory of Didactic Situations in Mathematics and the Theory of Social Interactionism: Synthesizing the Topaze effect and the funnel pattern 数学教学情境理论和社会互动理论的互补维度：综合托帕兹效应和漏斗模式

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-11-21 DOI: 10.1016/j.jmathb.2024.101194

Heidi Strømskag

This study examines the theory of didactic situations in mathematics (TDS) and the theory of social interactionism (TSI), employing strategies from the networking of theories schema to uncover potential complementarities between them. These theories provide different a priori perspectives on mathematics classroom interaction: TDS focuses on the functioning of mathematical knowledge in adidactic situations, while TSI centers on the emergence of mathematical meanings through the interactive accomplishment of intersubjectivity. The study gives rise to a hypothesis concerning complementary dimensions of the theoretical frameworks, particularly regarding social interaction and related classroom regulations. This hypothesis is empirically substantiated through theoretical triangulation of a dataset from a mathematics classroom. The TDS analysis, considering the mathematical knowledge in question, identifies a Topaze effect within the dataset, whereas the TSI analysis construes the empirical facts as exhibiting a funnel pattern of interaction. It is argued that the interpretations mutually enhance each other’s explanatory power.

本研究探讨了数学教学情境理论（TDS）和社会互动理论（TSI），采用了理论网络图式的策略来揭示它们之间潜在的互补性。这些理论为数学课堂互动提供了不同的先验视角：TDS 侧重于数学知识在说教情境中的运作，而 TSI 则侧重于通过主体间性的互动成就数学意义的产生。本研究提出了一个关于理论框架互补层面的假设，特别是关于社会互动和相关课堂规则的假设。通过对数学课堂数据集的理论三角分析，这一假设得到了实证。考虑到相关数学知识，TDS 分析确定了数据集中的 Topaze 效应，而 TSI 分析则将经验事实解释为呈现出漏斗状的互动模式。本文认为，这两种解释相互增强了对方的解释力。

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引用次数: 0

A lens for exploring which dimensions contribute to a justification’s proofiness 探索哪些因素有助于证明理由的可证明性的视角

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-11-20 DOI: 10.1016/j.jmathb.2024.101204

Dov Zazkis , Andre Rouhani

This study extends the investigation of students’ conceptions of what makes a written justification a proof by introducing a novel theoretical lens—the proofiness lens. Under a proofiness lens a justification is conceptualized as occurring in a multi-dimensional space with each dimension influencing the extent to which that justification is considered a proof. In this work, we target a single potential dimension, the proof-to-procedure continuum, although, other dimensions emerged from students’ work. Our data allows us to explore how sensitive students are to the proof-to-procedure dimension of proofiness. Additionally, all students in our study were attentive to writing style as an emergent dimension. We demonstrate that the proofiness lens and its associated methodology shed light on which dimensions of proofs students attend to and why.

本研究通过引入一个新颖的理论视角--证明度视角，扩展了对学生关于什么使书面理由成为证明的概念的调查。在证明度视角下，一个理由被概念化为发生在一个多维空间中，每个维度都会影响该理由被视为证明的程度。在这项研究中，我们的目标是一个单一的潜在维度，即从证明到程序的连续统一体，尽管学生的作品中还出现了其他维度。我们的数据允许我们探索学生对证明性的证明到程序维度有多敏感。此外，在我们的研究中，所有学生都关注写作风格这一新兴维度。我们证明，证明性视角及其相关方法揭示了学生关注证明的哪些维度以及关注的原因。

引用次数: 0

Mature intuition and mathematical understanding 成熟的直觉和数学理解力

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-11-18 DOI: 10.1016/j.jmathb.2024.101203

William D'Alessandro , Irma Stevens

Mathematicians often describe the importance of well-developed intuition to productive research and successful learning. But neither education researchers nor philosophers interested in epistemic dimensions of mathematical practice have yet given the topic the sustained attention it deserves. The trouble is partly that intuition in the relevant sense lacks a usefully clear characterization, so we begin by offering one: mature intuition, we say, is the capacity for fast, fluent, reliable and insightful inference with respect to some subject matter. We illustrate the role of mature intuition in mathematical practice with an assortment of examples, including data from a sequence of clinical interviews in which a student improves upon initially misleading covariational intuitions. Finally, we show how the study of intuition can yield insights for philosophers and education theorists. First, it contributes to a longstanding debate in epistemology by undermining epistemicism, the view that an agent’s degree of objectual understanding is determined exclusively by their knowledge, beliefs and credences. We argue on the contrary that intuition can contribute directly and independently to understanding. Second, we identify potential pedagogical avenues towards the development of mature intuition, highlighting strategies including adding imagery, developing associations, establishing confidence and generalizing concepts.

数学家们经常描述完善的直觉对于富有成效的研究和成功的学习的重要性。但是，无论是教育研究者还是对数学实践的认识论层面感兴趣的哲学家，都还没有对这一话题给予应有的持续关注。问题的部分原因在于，相关意义上的直觉缺乏一个有用的明确表征，因此，我们首先提出一个表征：我们说，成熟的直觉是对某些主题进行快速、流畅、可靠和有洞察力的推理的能力。我们用各种例子来说明成熟直觉在数学实践中的作用，其中包括一连串临床访谈的数据，在这些数据中，一个学生改进了最初误导性的协变直觉。最后，我们展示了直觉研究如何为哲学家和教育理论家提供启示。首先，它对认识论中长期存在的争论做出了贡献，破坏了认识论，即一个人对客观事物的理解程度完全由其知识、信念和可信度决定的观点。相反，我们认为直觉可以直接、独立地促进理解。其次，我们确定了发展成熟直觉的潜在教学途径，强调了包括增加想象、发展联想、建立信心和概括概念在内的策略。

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引用次数: 0

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-11-17 DOI: 10.1016/j.jmathb.2024.101209

Polly Thanailaki

This paper examines mathematics teaching and learning, specifically of Geometry, in Greek girls’ schools in the 19th century. It explores how educational laws and school practice defined its teaching. Research has proved that female students received only the basics in Geometry, substantially less than what was offered to male students in all-boys’ schools. Also, the Geometry textbooks designed for girls are discussed. The problems considered in the article are at the intersection of economic, political and ideological issues. The study draws on a wide range of primary sources such as school archives and records as well as government gazettes. In particular, the school archives of the Philekpedeutiki Etaireia provide this research with a rich source of information regarding female schooling in 19th century.

本文探讨了 19 世纪希腊女子学校的数学教学，特别是几何教学。它探讨了教育法和学校实践是如何定义数学教学的。研究证明，女学生只能学到几何方面的基础知识，远远低于男子学校为男生提供的基础知识。文章还讨论了为女生设计的《几何》教科书。文章考虑的问题是经济、政治和意识形态问题的交叉点。研究利用了大量原始资料，如学校档案和记录以及政府公报。特别是 Philekpedeutiki Etaireia 的学校档案为本研究提供了有关 19 世纪女性学校教育的丰富信息。

引用次数: 0

Different types of talk in mixed-attainment problem-solving groups: Contributions to individual students’ combinatorial thinking 混合成绩解题小组中的不同谈话类型：对学生个人组合思维的贡献

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-11-16 DOI: 10.1016/j.jmathb.2024.101206

Maria Larsson , Hanna Fredriksdotter , Nina Klang

This study contributes to previous research on collaborative approaches to instruction in mathematics. The study focuses on the relationships between the type of talk in groups and individual students’ combinatorial thinking. Four case studies of mixed-attainment groups of middle-school students working with mathematical problem solving in combinatorics were conducted. Video-recordings of dyad and group work, as well as interviews with four students (one per group), were analyzed. The results reveal how affordances and constraints in different types of talk (exploratory, cumulative, disputational talk) in mixed-attainment groups can contribute to individual students’ combinatorial thinking. The results highlight the interconnectedness of collective and individual reasoning in combinatorics, emphasizing the role of quality of group talk.

本研究对以往有关数学合作教学法的研究有所贡献。研究的重点是小组中的谈话类型与学生个人的组合思维之间的关系。本研究进行了四项个案研究，研究对象是初中学生中的不同成绩群体，他们在解决组合数学问题时的情况。研究分析了两人和小组合作的视频记录，以及对四名学生（每组一名）的访谈。结果揭示了在不同水平的小组中，不同类型的谈话（探索性谈话、积累性谈话、争论性谈话）中的承受力和制约力如何促进学生的组合思维。结果突出了组合数学中集体推理和个人推理的相互联系，强调了小组谈话质量的作用。

引用次数: 0

Mathematical understanding – Common themes in philosophy and mathematics education 数学理解 - 哲学与数学教育的共同主题

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-11-07 DOI: 10.1016/j.jmathb.2024.101202

Jessica Carter

We present different characterizations of mathematical understanding given by mathematicians, philosophers of mathematics, and mathematics educators. One purpose is to illustrate the diversity of these characterizations. Although the descriptions of understanding may seem incompatible, the paper ends by pointing to some shared themes. They include an emphasis on qualities such as relations and unification. Additionally, we note that re-presentation, including visual representations, is thought to play a role in understanding.

我们介绍数学家、数学哲学家和数学教育家对数学理解的不同描述。目的之一是说明这些描述的多样性。尽管对数学理解的描述看似互不相容，但本文最后指出了一些共同的主题。其中包括对关系和统一等特质的强调。此外，我们还注意到，包括视觉表征在内的再表征被认为在理解中发挥了作用。

引用次数: 0

On mathematics education for women in Russia prior to 1917 关于 1917 年之前俄罗斯妇女的数学教育

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-11-05 DOI: 10.1016/j.jmathb.2024.101201

Alexander Karp

This paper attempts to describe women’s mathematics education in certain types of educational institutions in Russia before 1917. The history of women’s education (inclusive of the humanities) begins effectively in the eighteenth century. This education was inevitably limited, since the role assigned to women did not imply any special study of mathematics – mathematics was needed primarily for maintaining the household. To be sure, to this was also added the problem of intellectual development, which sometimes led to girls being taught geometry, and even algebra, although this did not happen often. At the same time, women’s mathematical talents could be valued quite highly. Gradually, the situation changed, and already in the twentieth century the opinion that women’s mathematics education should not differ from men’s was very widely expressed. This paper analyzes various views expressed in surviving documents, as well as textbooks written for girls, and memoirs that make it possible to imagine to a certain degree how exactly the teaching of mathematics at women’s educational institutions was implemented and perceived.

本文试图描述 1917 年前俄罗斯某些类型教育机构中的女性数学教育。妇女教育（包括人文学科）的历史实际上始于十八世纪。这种教育不可避免地受到限制，因为赋予妇女的角色并不意味着要专门学习数学--数学主要是维持家庭生活所必需的。当然，除此以外，还有智力发展的问题，有时会让女孩学习几何甚至代数，尽管这种情况并不常见。与此同时，女性的数学才能也得到了很高的评价。渐渐地，情况发生了变化，在二十世纪，女性数学教育不应有别于男性数学教育的观点已得到广泛表达。本文分析了现存文献中表达的各种观点，以及为女孩编写的教科书和回忆录，从而可以在一定程度上想象女性教育机构的数学教学是如何实施和看待的。

引用次数: 0

Undergraduate students’ collaboration on homework problems in advanced mathematics courses 本科生在高等数学课程作业问题上的合作

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-11-04 DOI: 10.1016/j.jmathb.2024.101200

Ciara Murphy, Maria Meehan

While mathematicians and mathematics education researchers have acknowledged the importance of undergraduate mathematics students’ learning outside of class time, little is known about what students actually do. The aim of this study is to examine one aspect of students’ out-of-class learning: their collaboration with peers on homework problems. Ten interviews with recent graduates of mathematics degrees were conducted and analyzed using reflexive thematic analysis. We examine participants’ descriptions of how they collaborated on homework problems and with whom. Additionally, we explore their perceptions of the affordances of collaborating on homework, as well as the factors they perceive as constraining their engagement in the practice. Our study is an initial step towards developing a more complete understanding of undergraduate mathematics students’ engagement with homework problems and out-of-class learning practices more generally. We discuss the implications of our findings in terms of guiding future research.

虽然数学家和数学教育研究者都承认本科生在课外学习数学的重要性，但对学生的实际学习情况却知之甚少。本研究旨在考察学生课外学习的一个方面：他们与同学合作解决作业问题的情况。我们对 10 名数学专业的应届毕业生进行了访谈，并采用反思性主题分析法对访谈内容进行了分析。我们研究了参与者对他们如何就家庭作业问题进行合作以及与谁合作的描述。此外，我们还探讨了他们对合作完成家庭作业的好处的看法，以及他们认为制约他们参与合作的因素。我们的研究为更全面地了解本科数学学生参与家庭作业和课外学习实践迈出了第一步。我们将讨论研究结果对未来研究的指导意义。

引用次数: 0

Quantitative operators as an analytical tool for explaining differential equation students’ construction of new quantities during modeling 数量运算符是解释微分方程学生在建模过程中构建新数量的分析工具

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-10-30 DOI: 10.1016/j.jmathb.2024.101198

Sindura Kularajan , Jennifer Czocher , Elizabeth Roan

Theories of quantitative reasoning have taken precedence as an analytical tool to interpret and describe students’ mathematical reasonings, especially as students engage in mathematical modeling tasks. These theories are particularly useful to describe how students construct new quantities as they model. However, while using this lens to analyze Differential Equations students’ construction of mathematical models of dynamic situations, we found cases of quantity construction that were not fully characterized by extant concepts. In this theory-building paper, we present five examples of such cases. Additionally, we introduce a new construct—quantitative operators—as an extended analytical tool to characterize those cases. Our findings suggest that quantitative operators may be viewed as an extension for theories of quantity construction and complementary to symbolic forms, when localizing theories of quantity construction for mathematical modeling, especially at the undergraduate differential equation level.

定量推理理论作为一种分析工具，在解释和描述学生的数学推理，特别是学生参与数学建模任务时，已占据了主导地位。这些理论对于描述学生如何在建模过程中构建新的数量特别有用。然而，在使用这一视角分析微分方程学生构建动态情境数学模型的过程中，我们发现了一些量的构建并没有完全被现有的概念所描述。在这篇理论构建论文中，我们介绍了五个此类案例。此外，我们还引入了一个新的概念--定量算子--作为一种扩展的分析工具来描述这些案例。我们的研究结果表明，在对数学建模的数量构造理论进行本地化时，尤其是在本科生微分方程层面，数量算子可被视为数量构造理论的扩展和符号形式的补充。

引用次数: 0

Mathematizing the world: A routine to advance mathematizing in the elementary classroom 世界数学化：在小学课堂上推进数学化的常规方法

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-10-22 DOI: 10.1016/j.jmathb.2024.101196

Julia M. Aguirre , Erin E. Turner , Elzena McVicar , Amy Roth McDuffie , Mary Q. Foote , Erin Carll

The Mathematizing-the-World routine (MWR) is an efficient culturally responsive instructional routine for mathematizing that explicitly supports problem posing using an image or object. Given the under-representation of problem-posing studies in elementary school settings, our qualitative study analyzed student responses from 56 MWR enactments in grade 3–5 classrooms in two regions of the United States. Our findings include detailed examples of the MWR in action, including how three open-ended prompts engaged younger students in mathematizing and posing problems related to authentic, real-world situations. We summarize findings across the 56 MWR classroom enactments focusing on the understandings about the context and the mathematical ideas evidenced in student responses. Our findings demonstrate the potential of the MWR as a catalyst for eliciting and communicating diverse student ideas while engaged in the problem-posing process. We discuss research and practice implications for this routine to support mathematizing, and specifically problem posing in the elementary classroom.

世界数学化例行程序（MWR）是一种高效的文化响应式数学化教学例行程序，它明确支持利用图像或物体提出问题。鉴于问题摆放研究在小学环境中的代表性不足，我们的定性研究分析了美国两个地区 3-5 年级课堂中 56 个 MWR 案例中学生的反应。我们的研究结果包括 MWR 在行动中的详细实例，包括三个开放式提示如何让低年级学生参与数学化，并提出与真实世界情境相关的问题。我们总结了 56 个 MWR 课堂实践的发现，重点是学生回答中体现的对情境和数学思想的理解。我们的研究结果表明，在提出问题的过程中，MWR 有助于激发和交流学生的不同想法。我们讨论了这一常规的研究和实践意义，以支持数学化，特别是小学课堂中的问题提出。

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Journal of Mathematical Behavior

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