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Reasoning productively across algebraic contexts: Students develop coordinated notions of inverse 在代数背景下有效地推理:学生发展反的协调概念
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-10-19 DOI: 10.1016/j.jmathb.2023.101099
John Paul Cook , Kathleen Melhuish , Rosaura Uscanga
{"title":"Reasoning productively across algebraic contexts: Students develop coordinated notions of inverse","authors":"John Paul Cook ,&nbsp;Kathleen Melhuish ,&nbsp;Rosaura Uscanga","doi":"10.1016/j.jmathb.2023.101099","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101099","url":null,"abstract":"<div><p>The concept of inverse is threaded throughout K-16 mathematics. Scholars frequently advocate for students to understand the underlying structure: combining an element and its inverse through the binary operations yields the relevant identity element. This ‘coordinated’ way of reasoning is challenging for students to employ; however, little is known about how students might reason en route to developing it. In this study, we analyze a teaching experiment with two beginning abstract algebra students through the lens of three ways of reasoning about inverse: inverse as an undoing, inverse as a manipulated element, and inverse as a coordination of the binary operation, identity, and set. In particular, we examine the implications of these ways of reasoning as students work to develop inverse as a coordination. We identify pedagogical tools and facets of instructional design that appeared to support students’ development of inverse as a coordination. We further suggest that all three ways of reasoning can support productive activity with inverses.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49875826","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
Examining the concept of inverse: Theory-building via a standalone literature review 检视逆的概念:透过独立文献回顾建立理论
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-10-14 DOI: 10.1016/j.jmathb.2023.101100
John Paul Cook , April Richardson , Steve Strand , Zackery Reed , Kathleen Melhuish
{"title":"Examining the concept of inverse: Theory-building via a standalone literature review","authors":"John Paul Cook ,&nbsp;April Richardson ,&nbsp;Steve Strand ,&nbsp;Zackery Reed ,&nbsp;Kathleen Melhuish","doi":"10.1016/j.jmathb.2023.101100","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101100","url":null,"abstract":"<div><p>Inverse is a critical topic throughout the K–16 mathematics curriculum where students encounter the notion of mathematical inverse across many contexts. The literature base on inverses is substantial, yet context-specific and compartmentalized. That is, extant research examines students’ reasoning with inverses within specific algebraic contexts. It is currently unclear what might be involved in productively reasoning with inverses across algebraic contexts, and whether the specific ways of reasoning from the literature can be abstracted to more general ways of reasoning about inverse. To address this issue, we conducted a standalone literature review to explicate and exemplify three cross-context ways of reasoning that, we hypothesize, can support students’ productive engagement with inverses in a variety of algebraic contexts: <em>inverse as an undoing</em>, <em>inverse as a manipulated element</em>, and <em>inverse as a coordination of the binary operation, identity, and set</em>. Findings also include explicating affordances and constraints for each of these ways of reasoning. Finally, we reflect on when and how standalone literature reviews can serve the purpose of unifying fragmented and obscured insights about key mathematical ideas.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49875825","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
Learners building conceptual understandings of mathematics: Conditions for promoting growth in understanding – A global perspective from ICME-14 学习者建立对数学的概念性理解:促进理解增长的条件-来自ICME-14的全球视角
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-10-04 DOI: 10.1016/j.jmathb.2023.101101
Jenni Ingram, Stéphane Clivaz, Carolyn A. Maher, Louise Wilkinson
{"title":"Learners building conceptual understandings of mathematics: Conditions for promoting growth in understanding – A global perspective from ICME-14","authors":"Jenni Ingram,&nbsp;Stéphane Clivaz,&nbsp;Carolyn A. Maher,&nbsp;Louise Wilkinson","doi":"10.1016/j.jmathb.2023.101101","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101101","url":null,"abstract":"","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49875824","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
Noticing of argumentation: A comparison between pre-service and in-service secondary-school mathematics teachers 论述注意:职前与在职中学数学教师之比较
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-09-22 DOI: 10.1016/j.jmathb.2023.101098
Samaher Nama , Maysa Hayeen-Halloun , Michal Ayalon
{"title":"Noticing of argumentation: A comparison between pre-service and in-service secondary-school mathematics teachers","authors":"Samaher Nama ,&nbsp;Maysa Hayeen-Halloun ,&nbsp;Michal Ayalon","doi":"10.1016/j.jmathb.2023.101098","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101098","url":null,"abstract":"<div><p>This study compares pre-service mathematics teachers’ (PSMTs) and in-service mathematics teachers’ (ISMTs) noticing of argumentation at the secondary-school level. Thirty-five PSMTs and 32 ISMTs engaged in analyzing argumentation classroom situations (ACSs) using an ACS-report format emphasizing two sub-skills of noticing: attending and interpretation. Analysis of the participants’ ACS reports revealed differences between the two research groups. The ISMTs paid a high level of attention to all four aspects: ‘co-constructing of arguments’, ‘critiquing arguments’, ‘mutual respect’, and ‘working toward consensus-building’, whereas the PSMTs paid a high level of attention to ‘mutual respect’ and ‘co-constructing of arguments’ aspects only. In terms of interpretation, the ISMTs outperformed the PSMTs in interpreting the argumentation through the lenses of ‘task characteristics’, ‘teaching strategies’, and ‘student cognitive characteristics’. The findings are interpreted in light of both theory and practice.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49875830","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
Applying a new framework of connections between mathematical symbols and natural language 应用数学符号与自然语言之间联系的新框架
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-09-12 DOI: 10.1016/j.jmathb.2023.101097
Ulrika Wikström Hultdin, Ewa Bergqvist, Tomas Bergqvist, Lotta Vingsle, Magnus Österholm
{"title":"Applying a new framework of connections between mathematical symbols and natural language","authors":"Ulrika Wikström Hultdin,&nbsp;Ewa Bergqvist,&nbsp;Tomas Bergqvist,&nbsp;Lotta Vingsle,&nbsp;Magnus Österholm","doi":"10.1016/j.jmathb.2023.101097","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101097","url":null,"abstract":"<div><p>A reader of mathematical text must often switch between reading mathematical symbols and reading words. In this study, five different categories of structural connections between symbols and language, which invite such switches, are presented in a framework. The framework was applied in a study of Swedish mathematics textbooks, where 180 randomly selected pages from different educational stages were analyzed. The results showed a significant change in communication patterns as students progress through school. From a predomination of connections based on proximity found in year two, there is a gradual change to a predomination of symbols interwoven in sentences in year eight. Furthermore, more qualitative investigations of the different connections complemented the quantification, both through further explanations of the quantitative results, and through more examples of differences in communication patterns. The implications for readers of mathematics texts are discussed.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49875829","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
Impact of prompts on students’ mathematical problem posing 提示对学生提出数学问题的影响
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-09-08 DOI: 10.1016/j.jmathb.2023.101087
Jinfa Cai, Hua Ran, Stephen Hwang, Yue Ma, Jaepil Han, Faith Muirhead
{"title":"Impact of prompts on students’ mathematical problem posing","authors":"Jinfa Cai,&nbsp;Hua Ran,&nbsp;Stephen Hwang,&nbsp;Yue Ma,&nbsp;Jaepil Han,&nbsp;Faith Muirhead","doi":"10.1016/j.jmathb.2023.101087","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101087","url":null,"abstract":"<div><p>This study used three pairs of problem-posing tasks to examine the impact of different prompts on students’ problem posing. Two kinds of prompts were involved. The first asked students to pose 2–3 different mathematical problems without specifying other requirements for the problems, whereas the second kind of prompt did specify additional requirements. A total of 2124 students’ responses were analyzed to examine the impact of the prompts along multiple dimensions. In response to problem-posing prompts with more specific requirements, students tended to engage in more in-depth mathematical thinking and posed much more linguistically and semantically complex problems with more relationships or steps required to solve them. The findings from this study not only contribute to our understanding of problem-posing processes but also have direct implications for teaching mathematics through problem posing.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49875831","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
Geometry, groceries, and gardens: Learning mathematics and social justice through a nested, equity-directed instructional approach 几何、食品杂货和花园：通过嵌套的、以公平为导向的教学方法学习数学和社会正义
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-09-01 DOI: 10.1016/j.jmathb.2023.101069
Frances K. Harper , Queshonda J. Kudaisi
{"title":"Geometry, groceries, and gardens: Learning mathematics and social justice through a nested, equity-directed instructional approach","authors":"Frances K. Harper ,&nbsp;Queshonda J. Kudaisi","doi":"10.1016/j.jmathb.2023.101069","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101069","url":null,"abstract":"<div><p>We addressed the call for explorations of how BIPOC students’ “experiences in secondary mathematics classrooms might advance transformative, equity-focused, pedagogical models” (Joseph et al., 2019, p. 149) by exploring how a nested, equity-directed approach created different kinds of opportunities for students to take up, shift, or resist what it means to teach, learn, and do mathematics. Specifically, we looked at efforts to engage equity-directed dominant and critical approaches through a series of three mathematics projects aimed at investigating food insecurity as a social (in)justice issue using geometry. Our analysis focused on a subset of data generated during three projects from different times of the year. Findings revealed that the teacher more readily enacted critical equity-directed practices than dominant ones; that students more readily embraced those critical practices; and that students expected their use of mathematics and exploration of social issues to align with authentic, real-world situations.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50182173","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
Students’ development of mathematical meanings while participating vicariously in conversations between other students in instructional videos 学生在教学视频中间接参与其他学生之间的对话时对数学意义的发展
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-09-01 DOI: 10.1016/j.jmathb.2023.101068
Joanne Lobato , John Gruver , Michael Foster
{"title":"Students’ development of mathematical meanings while participating vicariously in conversations between other students in instructional videos","authors":"Joanne Lobato ,&nbsp;John Gruver ,&nbsp;Michael Foster","doi":"10.1016/j.jmathb.2023.101068","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101068","url":null,"abstract":"<div><p>Although interest in using videos in educational settings has surged in recent years, researchers know little about what mathematical meanings students develop from watching these videos or how they do so. To contribute to this gap in the research, we examined how two students appropriated mathematical meanings from instructional videos. In contrast to typical instructional videos, which rely heavily on an expert’s exposition, the videos in our study featured the unscripted conversation of two high school students as they engaged with novel mathematical problems. This allowed us to examine how other students watching the videos coordinated meanings expressed by both the video participants and each other, including meanings that were initially incorrect or incomplete. To analyze these data, we adopted a Bakhtinian-inspired lens, which allowed us to conceptualize meaning as emerging from the relationships among multiple voices. Additionally, the appropriation of meanings from the videos was not straightforward. Instead, we found evidence of the repetition (mimicry) of words and actions from the video participants, revision, resistance, and the invocation of previously-appropriated voices, before the students were able to make the meanings expressed in the videos their own.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50182175","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
Secondary-school teachers’ noticing of aspects of mathematics teaching talk in the context of one-day workshops 一天工作坊背景下中学教师对数学教学话语各方面的注意
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-09-01 DOI: 10.1016/j.jmathb.2023.101084
Núria Planas , José M. Alfonso
{"title":"Secondary-school teachers’ noticing of aspects of mathematics teaching talk in the context of one-day workshops","authors":"Núria Planas ,&nbsp;José M. Alfonso","doi":"10.1016/j.jmathb.2023.101084","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101084","url":null,"abstract":"<div><p>In a research project with one-day teacher education workshops for secondary-school mathematics teachers, our study explores the potential of tool-supported discussions in helping them to notice important and critical aspects of mathematics teaching talk. Mathematical practices of naming and explaining in teaching talk, students’ content learning challenges, and noticing processes of identifying, interpreting and deciding are the components of our framework and the tools that guided the design and implementation of three workshops on linear equations, fractions and plane isometries. The data was collected during the discussions with the seven teachers and the teacher educator throughout these workshops. The coding of the discussions allowed us to see discourse moves that reveal the teachers’ noticing of: (i) challenges in the identification of mathematical naming, (ii) mathematical explaining that voices the students’ learning, (iii) classroom practice in relation to mathematical naming and explaining.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50182309","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: 1
Identifying competent problem posers and exploring their characteristics 识别有能力的问题提出者并探索他们的特点
IF 1.7
Journal of Mathematical Behavior Pub Date : 2023-09-01 DOI: 10.1016/j.jmathb.2023.101086
Ling Zhang , Andreas J. Stylianides , Gabriel J. Stylianides
{"title":"Identifying competent problem posers and exploring their characteristics","authors":"Ling Zhang ,&nbsp;Andreas J. Stylianides ,&nbsp;Gabriel J. Stylianides","doi":"10.1016/j.jmathb.2023.101086","DOIUrl":"https://doi.org/10.1016/j.jmathb.2023.101086","url":null,"abstract":"<div><p>In a study involving 66 masters and 60 sixth-grade students, we conducted Principal Component Analysis to identify more-and-less competent problem posers based on performance criteria rather than, as in prior research, relying on participants’ mathematical experience or background. Also, to cast light on characteristics of competent posers, we explored possible patterns in the problem-posing process based on the two identified groups’ self-reports and eye-movements. The results showed that: masters students had a significantly lower proportion of the more-competent group and a higher proportion of the less-competent group than sixth graders; more-competent posers perceived a better understanding of the problem-posing tasks than less-competent posers; more-competent posers exhibited significantly more fixation time on completing the entire problem-posing activities than less-competent posers, though this pattern disappeared across particular stages of the problem-posing process; and more-competent posers appeared to engage in a more purposeful search and processing to construct their problems than less-competent posers.</p></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49875828","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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