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Journal of Mathematical Behavior最新文献

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Demands and scaffolds for explaining the connection of multiple representations: Revisiting the bottle-filling task 解释多重表象之间联系的要求和支架：重新审视装瓶任务
IF 1.7
Journal of Mathematical Behavior Pub Date : 2024-01-10 DOI: 10.1016/j.jmathb.2023.101118
Katharina Zentgraf , Susanne Prediger
{"title":"Demands and scaffolds for explaining the connection of multiple representations: Revisiting the bottle-filling task","authors":"Katharina Zentgraf ,&nbsp;Susanne Prediger","doi":"10.1016/j.jmathb.2023.101118","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101118","url":null,"abstract":"<div><p>Explaining the connections of multiple representations can enhance students’ conceptual understanding (e.g., in the bottle-filling task for functional graphs). But it poses high discursive demands that need to be further unpacked. The design research study qualitatively investigates the potentials and demands that fourteen second-language learners face when explaining the connection between functional graphs and filling glasses. The qualitative analysis of students’ pathways towards good explanations identifies (a) demands to construct a mental contextual representation of the filling process, (b) demands to unpack the holistic perspective into more refined concept elements of covariation and correspondence approaches, (c) highly intertwined demands to identify the relevant variables in view. For each of these underlying demands, we identify scaffolds to enable students – even recent second-language learners – to engage in mathematically and discursively demanding practices and to enable teachers to support them.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0732312323000883/pdfft?md5=233d8b40eaec6c747662981abc2284ba&pid=1-s2.0-S0732312323000883-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139419182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extending the covariation framework: Connecting covariational reasoning to students’ interpretation of rate of change 扩展协变框架：将协变推理与学生对变化率的解释联系起来
IF 1.7
Journal of Mathematical Behavior Pub Date : 2024-01-09 DOI: 10.1016/j.jmathb.2023.101122
Franklin Yu
{"title":"Extending the covariation framework: Connecting covariational reasoning to students’ interpretation of rate of change","authors":"Franklin Yu","doi":"10.1016/j.jmathb.2023.101122","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101122","url":null,"abstract":"<div><p>Research on covariational reasoning has continued to evolve as researchers learn more about how students coordinate two (or more) quantities’ values as covarying. In this study, I examine the connection between students’ covariational reasoning and how they interpret the value of a rate of change. The findings suggest that attending to students’ quantifications of a rate of change can provide insight into their covariational reasoning and how we might better support students in reasoning at higher levels. Additionally, this manuscript provides an update to the Carlson et al. (2002) Covariation Framework that includes two additional categories of student reasoning and an additional dimension that describes students’ interpretation of a rate value at each level of the framework.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139398869","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
Learning to teach through problem posing: A teacher’s journey in a networked teacher−researcher partnership 通过提出问题学会教学：一位教师在网络教研员伙伴关系中的心路历程
IF 1.7
Journal of Mathematical Behavior Pub Date : 2024-01-09 DOI: 10.1016/j.jmathb.2023.101120
Stephen Hwang , Ranran Xu , Yiling Yao , Jinfa Cai
{"title":"Learning to teach through problem posing: A teacher’s journey in a networked teacher−researcher partnership","authors":"Stephen Hwang ,&nbsp;Ranran Xu ,&nbsp;Yiling Yao ,&nbsp;Jinfa Cai","doi":"10.1016/j.jmathb.2023.101120","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101120","url":null,"abstract":"<div><p>This study presents a specific case of how a teacher in China learned to teach with problem posing through a collaborative, iterative design process with a researcher. Supported by a networked improvement community, at every step of the journey that they undertook, they partnered to design, deliver, and revise a mathematics lesson that fostered students’ learning through problem posing. A detailed travelogue of their journey serves to document what research on teaching through mathematical problem posing can look like and how the teacher learned to teach using this novel approach. We explore the utility of the 3H (head, heart, and hands) model as a powerful way to think about holistic, transformative teacher learning. In addition, we consider aspects of the networked improvement community in which the teacher–researcher partnership operated that enabled capacity for sustaining this kind of effort to change practice.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139398870","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 “learning to variate” to “variate for learning”: Teachers learning through collaborative, iterative context-based mathematical problem posing 从 "学习变式 "到 "变式学习"：教师通过合作、迭代的情境式数学问题学习
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-12-30 DOI: 10.1016/j.jmathb.2023.101119
Nadav Marco , Alik Palatnik
{"title":"From “learning to variate” to “variate for learning”: Teachers learning through collaborative, iterative context-based mathematical problem posing","authors":"Nadav Marco ,&nbsp;Alik Palatnik","doi":"10.1016/j.jmathb.2023.101119","DOIUrl":"10.1016/j.jmathb.2023.101119","url":null,"abstract":"<div><p>Problem posing (PP) has been found to contribute to teachers’ mathematical pedagogical knowledge. However, little is known about what and how teachers learn when engaged in continuous iterative PP. We use the variation theory of learning to conceptualize what and how teachers learn during iterative PP, illustrating these processes via a case study. The main argument is that what teachers learn from engaging in iterative PP are different task variables we refer to as “dimensions of possible variation.” Awareness of these dimensions allows teachers to skillfully generate new problems or re-formulate previously posed ones to achieve desired pedagogical goals. We show how, during a collaborative design process with the PD coordinator, a teacher-designer became aware of some new-to-her dimensions and developed corresponding techniques for diversifying tasks. These awarenesses were still evident in an interview six months after the end of the PD. Recommendations for teacher educators are suggested.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139072455","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
A symbolizing activity for constructing personal expressions and its impact on a student’s understanding of the sequence of partial sums 构建个人表达式的符号化活动及其对学生理解部分相加顺序的影响
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-12-21 DOI: 10.1016/j.jmathb.2023.101117
Derek Eckman , Kyeong Hah Roh
{"title":"A symbolizing activity for constructing personal expressions and its impact on a student’s understanding of the sequence of partial sums","authors":"Derek Eckman ,&nbsp;Kyeong Hah Roh","doi":"10.1016/j.jmathb.2023.101117","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101117","url":null,"abstract":"<div><p>This paper reports the results from a set of exploratory teaching interviews in which students constructed individualized algebraic expressions<span> (called personal expressions) to describe their meanings for partial sums. Our analysis focused on one student, Emily, who constructed two distinct personal expressions for partial sums, one novel and one based on her image of summation notation. Emily created her novel expression to denote the process of generating the summands to compute the value of a partial sum. Emily adopted summation notation to describe the value of the partial sum. After reflecting on her inscription for a series’ general term of summation, Emily constructed a single expression to describe either the process of computing an arbitrary partial sum or the value of the sum itself. Using Emily’s story, we propose three categories for students’ coordination of their meanings for partial sums with a corresponding representation.</span></p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138839390","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
Adidactical problem-posing as captured by scripting journeys: Investigating sums of consecutive integers 通过脚本旅程捕捉交互式问题：探究连续整数之和
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-12-20 DOI: 10.1016/j.jmathb.2023.101113
Andrew Kercher , Canan Güneş , Rina Zazkis
{"title":"Adidactical problem-posing as captured by scripting journeys: Investigating sums of consecutive integers","authors":"Andrew Kercher ,&nbsp;Canan Güneş ,&nbsp;Rina Zazkis","doi":"10.1016/j.jmathb.2023.101113","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101113","url":null,"abstract":"<div><p><span>Research has demonstrated that problem-posing and problem-solving mutually affect one another. However, the exact nature and full extent of this relationship requires detailed elaboration. This is especially true when problem-posing arises in order to facilitate problem-solving, such as during the investigation of an unfamiliar mathematical property or phenomenon. In this study, groups of participants used scripting to record their mathematical activity as they made conjectures and justified conclusions about sums of consecutive integers. We analyze the unprompted problem-posing found within these scripting journeys using three facets of a problem-posing framework: </span>mathematical knowledge base, problem-posing heuristics, and individual considerations of aptness. Our analysis reveals how these aspects of problem-posing emerge within a mathematical investigation, how they are related to surrounding problem-solving, and the kinds of mathematical insights and realizations that act as catalysts to promote further problem-posing activity.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138770093","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
“The theorem says…”: Engineering students making meaning of solutions to Ordinary Differential Equations "定理说......"：工科学生理解常微分方程解的意义
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-12-15 DOI: 10.1016/j.jmathb.2023.101116
Paul Hernandez-Martinez , Svitlana Rogovchenko , Yuriy Rogovchenko , Stephanie Treffert-Thomas
{"title":"“The theorem says…”: Engineering students making meaning of solutions to Ordinary Differential Equations","authors":"Paul Hernandez-Martinez ,&nbsp;Svitlana Rogovchenko ,&nbsp;Yuriy Rogovchenko ,&nbsp;Stephanie Treffert-Thomas","doi":"10.1016/j.jmathb.2023.101116","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101116","url":null,"abstract":"<div><p>There is a need for further studies on students’ learning of Differential Equations (DEs), especially in advanced undergraduate and graduate courses. Research on the mathematical education of engineers shows a conflict between students’ demands for practical, contextualized pedagogies and the need for abstract reasoning and appropriate use of mathematical results. Few papers focus on engineering students’ interpretation of theorems and their use as tools in argumentation and problem-solving. This paper takes a sociocultural stance on learning and employs <em>dialogical inquiry</em> – a methodology rooted in Bakhtinian theory, newly developed for collaborative inquiry and qualitative data analysis – to investigate the meanings that senior engineering students made while working on a task designed to evaluate their understanding of Existence and Uniqueness Theorems (EUTs) of solutions of DEs. We identified two important epistemological disconnections that explain the difficulties that some of our students faced in making meaning of solutions of DEs and the EUT.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S073231232300086X/pdfft?md5=f75789b683634c34ebd322e8c2d186ad&pid=1-s2.0-S073231232300086X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138656250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Comparing student strategies in a game-based and pen-and-paper task for linear algebra 比较学生在线性代数游戏任务和纸笔任务中的策略
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-12-14 DOI: 10.1016/j.jmathb.2023.101105
Jeremy Bernier , Michelle Zandieh
{"title":"Comparing student strategies in a game-based and pen-and-paper task for linear algebra","authors":"Jeremy Bernier ,&nbsp;Michelle Zandieh","doi":"10.1016/j.jmathb.2023.101105","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101105","url":null,"abstract":"<div><p><span>This study examines the mathematical activity<span> involved in engaging with two tasks designed for introductory linear algebra: the </span></span><em>Vector Unknown</em><span> digital game and the pen-and-paper Magic Carpet Ride task. Five undergraduate students worked on both tasks, and we qualitatively analyzed their strategies using a modified version of a framework from prior literature. In the findings, we report on the seven distinct strategies seen in our data set. We found that while our participants did use some of the same strategies on both tasks, there were also certain strategies which were more characteristic of work on one task or the other. In our discussion, we consider how the design differences in the tasks may influence the strategy differences, and how our findings can be leveraged by instructors of linear algebra in selecting tasks. Finally, we conclude by discussing broader implications for mathematics education research in comparing game-based and non-game-based tasks.</span></p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138656249","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
An exploratory mixed methods study about teacher candidates’ descriptions of children’s confusion, productive struggle, and mistakes in an elementary mathematics methods course 小学数学方法课程中教师候选人对儿童的困惑、有成效的挣扎和错误的描述的混合方法探索性研究
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-12-12 DOI: 10.1016/j.jmathb.2023.101103
Crystal Kalinec-Craig , Anthony Rios
{"title":"An exploratory mixed methods study about teacher candidates’ descriptions of children’s confusion, productive struggle, and mistakes in an elementary mathematics methods course","authors":"Crystal Kalinec-Craig ,&nbsp;Anthony Rios","doi":"10.1016/j.jmathb.2023.101103","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101103","url":null,"abstract":"<div><p><span>Recognizing and describing children's mathematical thinking in humanizing ways, especially when students engage in confusion, productive struggle, and mistakes, is a complex and challenging process. This paper describes an exploratory, mixed-methods study about how elementary teacher candidates (TCs) describe children's thinking as a right to exercise and to value their humanity when learning mathematics. The study analyzed transcripts from 64 TCs' </span>summative assessments, which consisted of mock parent-teacher conferences (MPTC). Findings suggest that TCs described children's confusion, productive struggle, and mistakes (RotL 1 and 2) as: a teacher's observation, an opportunity for students to correct or clarify their thinking, an opportunity for teachers to adjust instruction or provide support, and as a normal part of the learning process. More importantly, some TCs reassured children that learners have fundamental rights when learning mathematics, especially when feeling confused and claiming a mistake. Implications for research and teacher education are discussed. Keywords: Elementary, teacher education, mathematics, mixed methods, rehumanizing, Torres’ rights of the learner</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138577535","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
The role of topology in the construction of students’ optimization schema for two-variable functions 拓扑在学生二元函数优化图式构建中的作用
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-11-30 DOI: 10.1016/j.jmathb.2023.101106
Rafael Martínez-Planell , Maria Trigueros , Vahid Borji
{"title":"The role of topology in the construction of students’ optimization schema for two-variable functions","authors":"Rafael Martínez-Planell ,&nbsp;Maria Trigueros ,&nbsp;Vahid Borji","doi":"10.1016/j.jmathb.2023.101106","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101106","url":null,"abstract":"<div><p>This study uses Action-Process-Object-Schema theory (APOS) to examine students’ understanding of two-variable function optimization<span>. A genetic decomposition (GD) based on the notion of Schema is proposed. This is a conjecture of mental structures and relations that students may construct to understand the optimization of these functions. The GD was tested with semi-structured interviews with eleven students who had just finished an introductory multivariable calculus course. Results show that giving explicit attention during instruction to the topological structure of the domain of the function to be optimized and the use of GD-based activities was effective in promoting students’ understanding of two-variable function optimization. On the theoretical side, the study contributes to a better understanding of the APOS notions of Schema, Schema-triad, and types of relations between Schema components that have not been used extensively in the literature and that proved to be a powerful tool to model students’ learning.</span></p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138471812","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
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