Journal of Mathematical Behavior最新文献

英文中文

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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引用次数: 0

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

Jungeun Park , Jason Martin , Michael Oehrtman , Douglas Rizzolo

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。

引用次数: 0

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

Eirin Stenberg , Per Haavold , Bharath Sriraman

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

Luisa-Marie Hartmann , Stanislaw Schukajlow , Mogens Niss , Uffe Thomas Jankvist

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

Rachel Rupnow , Rosaura Uscanga , Anna Marie Bergman , Cassandra Mohr

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。

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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

Marc Lange

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

Bethany Rittle-Johnson

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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引用次数: 0

Lecturers' use of questions in undergraduate mathematics lectures 讲师在本科数学授课中使用问题的情况

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-09-18 DOI: 10.1016/j.jmathb.2024.101190

George Kinnear , Gemma Hood , Eloise Lardet , Colette Sheard , Colin Foster

Mathematics lecturers frequently ask questions in their lectures, and these questions presumably play an important role in students’ thinking about and learning of the lecture content. We replicated and developed a coding scheme used in previous research in the US to categorise lecturers’ questions in a sample of 136 lectures given by 24 lecturers at a research-intensive UK university. We found that the coding scheme could be applied reliably, and that factual questions were predominant (as in previous research). We explore differences in the lecturers’ use of questions – both between our UK sample and the previous US work, and between individual lecturers in our sample. We note the presence of strings of related successive questions from the lecturer, which we term ‘question chains’. We explore the nature of these, examine their prevalence, and seek to account for them in terms of the lecturers’ possible intentions.

数学讲师在授课过程中经常会提出问题，而这些问题可能在学生思考和学习授课内容的过程中发挥着重要作用。我们复制并发展了之前在美国研究中使用的编码方案，对英国一所研究密集型大学的 24 位讲师在 136 场讲座中提出的问题进行了分类。我们发现，该编码方案的应用是可靠的，而且事实性问题占主导地位（与之前的研究一样）。我们探讨了讲师在使用问题时的差异--既包括英国样本与之前美国研究之间的差异，也包括我们样本中不同讲师之间的差异。我们注意到讲师提出了一连串相关的连续问题，我们称之为 "问题链"。我们探讨了这些问题的性质，研究了它们的普遍性，并试图从讲师可能的意图角度来解释它们。

引用次数: 0

Creating and sharing linear algebra metaphors as an assessment for engaging students beyond the cognitive domain 创建和分享线性代数隐喻，将其作为吸引学生参与认知领域以外活动的一种评估方法

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior

Pub Date : 2024-09-11 DOI: 10.1016/j.jmathb.2024.101189

Hortensia Soto , Jessi Lajos , Vladislav Kokushkin

In this research, we investigated how a summative assessment of creating and sharing metaphors for linear algebra concepts supported undergraduate students’ affective, behavioral, and cognitive engagement. All seven research participants were enrolled in an introductory linear algebra course designed to develop students’ geometric understanding of linear algebra concepts in R² and R³. Using embodied cognition and an engagement framework, we analyzed students’ written responses and video-taped their focus group discussion. Our findings suggest that this summative assessment (a) privileged mathematical aesthetics and the affective domain of learning, (b) engaged students in binding formal aspects of linear algebra concepts with metaphors that they enacted via embodiment, and (c) was an opportunity to demonstrate learning and higher-order cognition. Thus, illustrating that assessments can focus on aesthetic and affective domains of mathematics while simultaneously integrating serious mathematical cognition. We conclude by offering adaptations of this assessment for other mathematics courses.

在这项研究中，我们调查了创建和分享线性代数概念隐喻的终结性评估如何支持本科生的情感、行为和认知参与。所有七名研究参与者都参加了线性代数入门课程，该课程旨在培养学生对 R2 和 R3 中线性代数概念的几何理解。我们利用体现认知和参与框架分析了学生的书面回答，并对他们的焦点小组讨论进行了录像。我们的研究结果表明，这个终结性评价（a）重视数学美学和学习的情感领域，（b）让学生参与到线性代数概念的形式方面与隐喻的结合中，他们通过体现来实施这些隐喻，（c）是展示学习和高阶认知的一个机会。由此可见，评估可以关注数学的审美和情感领域，同时结合严肃的数学认知。最后，我们为其他数学课程提供了这一评估的调整方案。

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