Journal of Mathematical Behavior最新文献

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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

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。

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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

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。

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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

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。

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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

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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Teaching practice aimed at promoting student engagement with metarules of defining 旨在促进学生参与定义元规则的教学实践

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-10-22 DOI: 10.1016/j.jmathb.2024.101197

By viewing students’ learning to define as learning about meta-discursive rules about defining and how to engage with them in their activity of defining, we investigated teaching practices that aim to promote such learning by analyzing a teaching experiment in which Calculus II students reinvented a formal definition of a limit. The teaching practices we identified addressed how students view defining tasks by providing a method to check whether their narratives satisfied the metarules that they aimed to follow and also provided guidance about how to revise their definitions to satisfy those metarules. Our results provide an example for the teaching practice that promotes student learning about and engagement with new meta-discursive rules that existing literature called for in general, especially in their reinvention of a formal definition of a mathematical object.

通过将学生的定义学习视为对有关定义的元话语规则以及如何在定义活动中运用这些规则的学习，我们通过分析微积分 II 学生重塑极限的正式定义的教学实验，研究了旨在促进这种学习的教学实践。我们所确定的教学实践通过提供一种方法来检查学生的叙述是否符合他们所要遵循的元规则，并指导学生如何修改他们的定义以符合这些元规则，从而解决了学生如何看待定义任务的问题。我们的结果为教学实践提供了一个范例，促进学生学习和参与现有文献普遍呼吁的新的元辨析规则，特别是在他们重塑数学对象的形式定义时。

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What types of insight do expert students gain during work with ill-structured problems in mathematics? 专家型学生在处理数学中结构不严谨的问题时会获得哪些类型的洞察力？

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-10-21 DOI: 10.1016/j.jmathb.2024.101199

In our study, we explored how two high-performing mathematics students gained insight while working on ill-structured problems. We followed their problem-solving process through task-based interviews and observed a similar sequence of insights in both participants’ work- (1) Spontaneous insight, (2) Passive gradual insight, (3) Sudden insight, and (4) Active gradual insight. An impasse occurred in the intersection between the second and third insight and seemed to accelerate the progression toward solution. During this insight sequence, we observed emotional transitions that appeared to impact the process in a useful manner, especially due to the participant’s interpretation of uncertainty related to the impasse as a challenge and an inspiration. Future research is needed to study the observed sequence of insights and related affects in a larger data set and in a broader spectrum of problem solvers.

在我们的研究中，我们探讨了两名数学成绩优秀的学生是如何在处理结构混乱的问题时获得洞察力的。我们通过基于任务的访谈跟踪了他们的解题过程，并观察到这两名学生在解题过程中获得洞察力的相似顺序--(1) 自发洞察力，(2) 被动渐进洞察力，(3) 突发性洞察力，(4) 主动渐进洞察力。在第二次和第三次洞察之间出现了僵局，这似乎加快了解决问题的进程。在这一洞察过程中，我们观察到情绪的转变似乎对这一过程产生了有益的影响，特别是参与者将与僵局有关的不确定性解释为一种挑战和启发。未来的研究需要在更大的数据集和更广泛的问题解决者中研究观察到的洞察力序列和相关影响。

引用次数: 0

Preservice teachers’ metacognitive process variables in modeling-related problem posing 职前教师在与建模相关的问题假设中的元认知过程变量

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-10-09 DOI: 10.1016/j.jmathb.2024.101195

For teaching mathematical modeling in schools, teachers need to create suitable problems for their students to deal with. Despite an emphasis on teaching approaches for mathematical modeling, little is known about the processes involved in posing problems based on real-world situations, referred to as modeling-related problem posing, and specifically about what has been termed “implemented anticipation” as a metacognitive process variable. To contribute to filling this research gap, this study analyzed the nature and presence of implemented anticipation among preservice teachers as they posed problems based on real-world situations. The study was conducted through qualitative research with seven preservice teachers and revealed that the decision-making process in modeling-related problem posing involves different processes of implemented anticipation, depending on the role the preservice teacher takes on. The paper discusses the implications for preparing preservice teachers to pose problems for teaching mathematical modeling.

在学校的数学建模教学中，教师需要为学生创设合适的问题。尽管数学建模的教学方法受到重视，但人们对基于真实世界情境提出问题（即与建模相关的问题提出）的过程，特别是被称为 "实施预期 "的元认知过程变量知之甚少。为了填补这一研究空白，本研究分析了职前教师在基于真实情境提出问题时实施预期的性质和存在情况。该研究通过对七位职前教师进行定性研究，揭示了在与建模相关的问题假设中，根据职前教师所扮演角色的不同，决策过程涉及不同的 "实施预期 "过程。本文讨论了准备职前教师提出数学建模教学问题的意义。

引用次数: 0

Snapshots of sameness: Characterizations of mathematical sameness across student groups 同一性的剪影：不同学生群体的数学同一性特征

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-10-03 DOI: 10.1016/j.jmathb.2024.101193

Sameness is foundational to mathematics but has only recently become an area of focus in mathematics education research. In this paper, we describe characterizations of sameness generated by four student groups: discrete mathematics students, linear algebra students, abstract algebra students, and graduate students. Based on qualitative analysis of open response surveys, we compare these groups’ characterizations of sameness; note the subcomponents discussed and variation within each dimension; and highlight experiences influential to students’ perceptions of sameness. Findings include interpretability of sameness as a big idea, nascent development of thematic connections across courses, emphases on current course material rather than connections to prior courses for students solicited from a particular course, greater reflectiveness from the graduate student group, and abstract algebra as an impactful course. Implications include a need for thoughtful examinations of how “big ideas” develop among students and what experiences might support such development.

同一性是数学的基础，但直到最近才成为数学教育研究的重点领域。在本文中，我们描述了四个学生群体对同一性的描述：离散数学学生、线性代数学生、抽象代数学生和研究生。基于对开放式回答调查的定性分析，我们比较了这些群体对同一性的描述；指出了每个维度中讨论的子部分和差异；并强调了对学生同一性认知有影响的经历。调查结果包括：同一性作为一个大概念的可解释性、跨课程主题联系的初步发展、强调当前课程材料而不是从特定课程中征集的学生与以前课程的联系、研究生群体的反思能力更强以及抽象代数是一门有影响力的课程。研究的启示包括：需要对学生如何形成 "大思想 "以及哪些经验可以支持这种发展进行深思熟虑的研究。

引用次数: 0

What are explanatory proofs in mathematics and how can they contribute to teaching and learning? 什么是数学解释性证明，它们如何促进教学？

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-09-26 DOI: 10.1016/j.jmathb.2024.101191

This paper will examine several simple examples (drawn from the mathematics literature) where there are multiple proofs of the same theorem, but only some of these proofs are widely regarded by mathematicians as explanatory. These examples will motivate an account of explanatory proofs in mathematics. Along the way, the paper will discuss why deus ex machina proofs are not explanatory, what a mathematical coincidence is, and how a theorem's proper setting reflects the naturalness of various mathematical kinds. The paper will also investigate how context influences which features of a theorem are salient and consequently which proofs are explanatory. The paper will discuss several ways in which explanatory proofs can contribute to teaching and learning, including how shifts in context (and hence in a proof’s explanatory power) can be exploited in a classroom setting, leading students to dig more deeply into why some theorem holds. More generally, the paper will examine how “Why?” questions operate in mathematical thinking, teaching, and learning.

本文将研究几个简单的例子（摘自数学文献），在这些例子中，同一定理有多个证明，但只有其中一些证明被数学家广泛视为解释性证明。这些例子将促使我们对数学中的解释性证明进行阐述。同时，本文还将讨论为什么神来之笔的证明不是解释性的，什么是数学巧合，以及定理的适当设置如何反映各种数学种类的自然性。本文还将研究背景如何影响定理的哪些特征是突出的，从而影响哪些证明是解释性的。论文将讨论解释性证明如何促进教学，包括如何在课堂上利用情境的变化（以及证明的解释力），引导学生更深入地探究某些定理成立的原因。更广泛地说，本文将探讨 "为什么？"问题如何在数学思考、教学和学习中发挥作用。

引用次数: 0

Encouraging students to explain their ideas when learning mathematics: A psychological perspective 鼓励学生在学习数学时解释自己的想法：心理学视角

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-09-24 DOI: 10.1016/j.jmathb.2024.101192

Children’s self-explanations are answers to voiced why and how questions that attempt to make sense of new information for oneself. They often are not sophisticated or generalizable. Despite this, prompting children to generate explanations often improves their learning. After providing examples of children’s explanations, this article summarizes empirical evidence for the learning benefits of prompting people to generate explanations when learning mathematics. There is strong evidence that prompting learners to explain leads to greater conceptual knowledge, procedural knowledge and procedural transfer when knowledge is assessed immediately after the learning session; there is limited evidence for greater procedural transfer after a delay. Scaffolding high-quality explanations via training or structured responses, designing prompts to carefully balance attention to important content, prompting learners to explain correct information, and prompting learners to explain why incorrect information is incorrect when appropriate, increases the benefits of prompts to generate explanations.

儿童的自我解释是对 "为什么 "和 "怎么做 "等问题的回答，目的是为自己理解新信息。它们往往不够成熟，也不具有普遍性。尽管如此，促使儿童做出解释往往能提高他们的学习效果。在举例说明了儿童的解释之后，本文总结了实证证据，说明在学习数学时，促使学生做出解释对学习有好处。有确凿证据表明，在学习课程结束后立即对知识进行评估时，促使学习者进行解释会带来更多的概念性知识、程序性知识和程序性迁移；而在延迟后进行评估会带来更多程序性迁移的证据则很有限。通过培训或有条理的回答为高质量的解释搭建脚手架，设计提示语以谨慎地平衡对重要内容的关注，提示学习者解释正确的信息，并在适当的时候提示学习者解释错误信息不正确的原因，这些都会增加提示产生解释的益处。

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