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First graders’ understanding of variables: Learning trajectories in a function machine environment 一年级学生对变量的理解：函数机环境下的学习轨迹
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-10-17 DOI: 10.1016/j.jmathb.2025.101295
Konstantinos P. Christou , Eleni Vellidou
{"title":"First graders’ understanding of variables: Learning trajectories in a function machine environment","authors":"Konstantinos P. Christou ,&nbsp;Eleni Vellidou","doi":"10.1016/j.jmathb.2025.101295","DOIUrl":"10.1016/j.jmathb.2025.101295","url":null,"abstract":"<div><div>This study examines how a function machine learning environment can instantiate a developmental trajectory of understanding variables in first-grade students. The intervention involved exploring input-output relationships and symbolic representations of indeterminate quantities. Data were collected through classroom interactions and interviews at three time points: before, immediately after, and six weeks following the intervention. The analyses revealed multiple developmental pathways. Some students progressed directly from pre-variable reasoning to advanced algebraic applications of variable notation. Others consolidated their understanding at intermediate stages or displayed misconceptions, such as treating letters as labels. Though a few students reverted, most maintained or deepened their new understandings, demonstrating the durability of learning. These results highlight the potential of function machines as instructional tools that facilitate exploration, identify misconceptions, and enable timely guidance. They also show how learning trajectories can inform instructional designs that foster early functional reasoning and challenge deficit views of young learners' algebraic capacities.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"81 ","pages":"Article 101295"},"PeriodicalIF":1.7,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145324418","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
Does understanding moderate aesthetic appraisals of proofs? 理解是否能缓和对证明的审美评价？
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-09-20 DOI: 10.1016/j.jmathb.2025.101284
George Kinnear , Matthew Inglis
{"title":"Does understanding moderate aesthetic appraisals of proofs?","authors":"George Kinnear ,&nbsp;Matthew Inglis","doi":"10.1016/j.jmathb.2025.101284","DOIUrl":"10.1016/j.jmathb.2025.101284","url":null,"abstract":"<div><div>The relationship between understanding and aesthetic appraisal in mathematics is an open question, with implications for both the philosophy of mathematics and mathematics education. In this study, we investigated how undergraduate students’ understanding of a mathematical proof relates to their perception of its aesthetic value. Participants were asked to evaluate the proof’s aesthetics and to complete three different assessments of their understanding. The results reveal that self-reported understanding was moderately associated with aesthetic appraisals, while two performance-based measures of understanding showed close-to-zero relationships. These findings challenge the view that aesthetic judgements in mathematics are merely disguised epistemic judgements, and suggest that future research should focus on exploring the non-epistemic factors that shape aesthetic judgements. We conclude by discussing the implications of these results for educational practices that seek to promote aesthetic experiences.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"81 ","pages":"Article 101284"},"PeriodicalIF":1.7,"publicationDate":"2025-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097144","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
Laying the groundwork: The grounding metaphors that build quotient groups 奠定基础：构建商群的基础隐喻
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-09-18 DOI: 10.1016/j.jmathb.2025.101294
Holly Zolt , Kathleen Melhuish
{"title":"Laying the groundwork: The grounding metaphors that build quotient groups","authors":"Holly Zolt ,&nbsp;Kathleen Melhuish","doi":"10.1016/j.jmathb.2025.101294","DOIUrl":"10.1016/j.jmathb.2025.101294","url":null,"abstract":"<div><div>Quotient groups are a foundational topic within abstract algebra courses, and they provide a context in which rich, powerful, and vivid language can be used to convey an understanding of the mathematics at play. We examined the language mathematicians used to describe their instruction on quotient groups and found that mathematicians draw on several source domains and use various metaphorical expressions when discussing their teaching practices. Of prominence in these findings was the use of a construction source domain which spanned all major aspects of quotient groups. The use of this source domain afforded the use of many metaphorical expressions that are tied to various formal content goals. We detail these findings and discuss how the metaphorical expressions used relate to the formal mathematics content that mathematicians are trying to convey.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"81 ","pages":"Article 101294"},"PeriodicalIF":1.7,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097141","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’ productive use of equivalence transformations 学生对等效变换的有效运用
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-09-13 DOI: 10.1016/j.jmathb.2025.101285
John Paul Cook , April Richardson , Zackery Reed , Elise Lockwood , O. Hudson Payne , Cory Wilson
{"title":"Students’ productive use of equivalence transformations","authors":"John Paul Cook ,&nbsp;April Richardson ,&nbsp;Zackery Reed ,&nbsp;Elise Lockwood ,&nbsp;O. Hudson Payne ,&nbsp;Cory Wilson","doi":"10.1016/j.jmathb.2025.101285","DOIUrl":"10.1016/j.jmathb.2025.101285","url":null,"abstract":"<div><div>Equivalence transformations – that is, transformations that produce an object that is equivalent to the original – are a unifying conceptual thread in K-16 mathematics. Though researchers have already established that productive reasoning about equivalence transformations hinges on an awareness that the transformed objects are equivalent to the given object, research (a) has not yet explored the various ways in which students might attend to equivalence, and (b) has primarily examined equivalence transformations on only one type of object, leaving open the question of what commonalities might be present in students’ reasoning across transformations of multiple types of objects. In this study, we present our analysis of task-based clinical interviews with university students. This paper’s primary contribution to the literature involves the description and illustration of three common, unified ways in the students productively reasoned about the equivalence of the objects they produced with transformations. Our findings extend the theoretical scope of an existing equivalence framework and suggest that these ways of reasoning can inform efforts to help students overcome the widespread reports of difficulties they experience. We conclude with a discussion of the theoretical implications for research on equivalence transformations across K-16 mathematics.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"81 ","pages":"Article 101285"},"PeriodicalIF":1.7,"publicationDate":"2025-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050152","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
Stage 3 high school students’ generalization of a cubic identity 第三阶段高中生对三次恒等式的泛化
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-09-06 DOI: 10.1016/j.jmathb.2025.101283
Erik S. Tillema , Andrew M. Gatza , Weverton Ataide Pinheiro
{"title":"Stage 3 high school students’ generalization of a cubic identity","authors":"Erik S. Tillema ,&nbsp;Andrew M. Gatza ,&nbsp;Weverton Ataide Pinheiro","doi":"10.1016/j.jmathb.2025.101283","DOIUrl":"10.1016/j.jmathb.2025.101283","url":null,"abstract":"<div><div>This paper reports on one study in a series of design research studies that have taken as a guiding design principle that combinatorial and quantitative reasoning can serve as a constructive resource for high school students to establish algebraic structure between a polynomial and its factors. Within this framing, we report on an interview study with eight 10th-12th grade students whose purpose was to investigate their progress towards generalization of the cubic identity <span><math><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mspace></mspace><msup><mrow><mn>1</mn><mo>∙</mo><mo>(</mo><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo><mo>+</mo><mn>3</mn><mo>∙</mo><mrow><mfenced><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∙</mo><mn>1</mn></mrow></mfenced></mrow><mo>+</mo><mn>3</mn><mo>∙</mo><mrow><mfenced><mrow><msup><mrow><mi>x</mi><mo>∙</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mo>+</mo><mn>1</mn><mo>∙</mo><mo>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></math></span>. The students worked on this generalization by solving cases of a 3-D combinatorics problem and representing their solutions using 3-D arrays. Findings include the identification of how differences in students’ combinatorial reasoning impacted their reasoning with 3-dimensional arrays and their progress towards a general statement of the cubic identity.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"81 ","pages":"Article 101283"},"PeriodicalIF":1.7,"publicationDate":"2025-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145005201","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
Experienced provers’ operative logical principles for evaluating proofs of conditional claims 有经验的证明者评估条件断言证明的操作逻辑原则
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-08-27 DOI: 10.1016/j.jmathb.2025.101282
Steven Ruiz , Mario Gonzalez , Paul Christian Dawkins , Kyeong Hah Roh
{"title":"Experienced provers’ operative logical principles for evaluating proofs of conditional claims","authors":"Steven Ruiz ,&nbsp;Mario Gonzalez ,&nbsp;Paul Christian Dawkins ,&nbsp;Kyeong Hah Roh","doi":"10.1016/j.jmathb.2025.101282","DOIUrl":"10.1016/j.jmathb.2025.101282","url":null,"abstract":"<div><div>Introduction to proof courses often teach mathematics majors logical principles to support their later interactions with proof. Prior research tells us little about how students draw upon these principles or their justifications in subsequent processes of comprehending proofs. This study investigates what logical principles thirteen experienced undergraduates employ while reading proofs of conditional claims. We also consider whether and how they justify any of those principles. Asking students to read various proofs related to a given conditional statement helped reveal their attention to and reasoning about logical principles. We found that student judgments were almost always logically normative, but their readiness to justify those principles varied more widely. Furthermore, we observe that students interpreted proofs in terms of inferences rather than truth-values showing some disconnect between their operative logic and the ways logic is generally taught in introduction to proof courses.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"80 ","pages":"Article 101282"},"PeriodicalIF":1.7,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144907760","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
Enhancing mathematical instruction for emergent bilinguals through collaborative situated professional development 透过合作情境下的专业发展，加强对新兴双语者的数学教学
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-08-19 DOI: 10.1016/j.jmathb.2025.101281
Jiyeong Yi , Jasmine Sourwine , Shristi Shrestha
{"title":"Enhancing mathematical instruction for emergent bilinguals through collaborative situated professional development","authors":"Jiyeong Yi ,&nbsp;Jasmine Sourwine ,&nbsp;Shristi Shrestha","doi":"10.1016/j.jmathb.2025.101281","DOIUrl":"10.1016/j.jmathb.2025.101281","url":null,"abstract":"<div><div>This study investigates the shifts in high school mathematics teachers’ instructional practices when working with emergent bilinguals (EBs) after participating in a collaborative situated professional development (CSPD) program alongside researchers. Over the course of one academic year, the teachers engaged in co-planning, co-teaching, and co-reflecting sessions with the research team while integrating research-based strategies to support EBs in learning mathematics. The CSPD also emphasized implementing cognitively demanding and contextually relevant mathematical tasks. The results reveal that the teachers’ instructional practices significantly improved after participation in the CSPD. These improvements include a stronger ability to maintain the cognitive demand of tasks throughout lessons, an increased focus on creating opportunities for students to articulate and explain their mathematical reasoning, and more effective incorporation of meaningful visual aids that enhance comprehension for EBs. These shifts highlight the potential of CSPD to positively impact mathematics instruction for linguistically diverse classrooms.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"80 ","pages":"Article 101281"},"PeriodicalIF":1.7,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865843","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
Discursive tuning: The case of digital environments 话语调优：数字环境的情况
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-08-16 DOI: 10.1016/j.jmathb.2025.101280
Giulia Lisarelli , Bernardo Nannini
{"title":"Discursive tuning: The case of digital environments","authors":"Giulia Lisarelli ,&nbsp;Bernardo Nannini","doi":"10.1016/j.jmathb.2025.101280","DOIUrl":"10.1016/j.jmathb.2025.101280","url":null,"abstract":"<div><div>This paper introduces discursive tuning as a new commognitive notion for analyzing how students engage with a digital environment and react to unexpected responses they receive from it. Discursive tuning provides a theoretical tool for describing changes in students’ discourse in response to commognitive conflicts arising during their activity with the digital environment. We discuss the notion’s analytical capacity through some episodes from a study in which upper secondary students interact with the dynagraph of the function <em>f(x)</em>= 1<em>/x</em>. Our analysis identifies specific cases of discursive tuning that occur when students attempt to include into their discourse narratives that are consistent with the behavior of the dynagraph or with canonical mathematical discourse. These cases illustrate how discursive tuning provides insight into how students can resolve commognitive conflicts in productive ways.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"80 ","pages":"Article 101280"},"PeriodicalIF":1.7,"publicationDate":"2025-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144852571","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
Choosing and not choosing: Examining students’ perceptions of not choosing a particular attribute in combinatorial tasks 选择和不选择：考察学生在组合任务中不选择特定属性的感知
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-08-06 DOI: 10.1016/j.jmathb.2025.101279
Niusha Modabbernia
{"title":"Choosing and not choosing: Examining students’ perceptions of not choosing a particular attribute in combinatorial tasks","authors":"Niusha Modabbernia","doi":"10.1016/j.jmathb.2025.101279","DOIUrl":"10.1016/j.jmathb.2025.101279","url":null,"abstract":"<div><div>As combinatorics becomes increasingly prominent in K-12 and undergraduate curricula, so does the need for research on teaching, learning, and understanding combinatorial concepts. Within general research on learning combinatorics, this study attends to students’ work on counting problems, with particular attention to how composite outcomes are constructed. As part of a larger research project, this report focuses on two undergraduate students, whose responses to several counting problems were elicited in semi-structured interviews. The analysis highlights that recognizing “not choosing” a particular attribute as a valid choice in determining and counting outcomes significantly supports students’ success in solving counting problems.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"80 ","pages":"Article 101279"},"PeriodicalIF":1.7,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780255","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
Imagination as an embodied space for engaging with abstract algebra concepts 想象是参与抽象代数概念的具体空间
IF 1.7
Journal of Mathematical Behavior Pub Date : 2025-08-02 DOI: 10.1016/j.jmathb.2025.101275
Jessi Lajos , Hortensia Soto , Francisco De Jesus Pagan
{"title":"Imagination as an embodied space for engaging with abstract algebra concepts","authors":"Jessi Lajos ,&nbsp;Hortensia Soto ,&nbsp;Francisco De Jesus Pagan","doi":"10.1016/j.jmathb.2025.101275","DOIUrl":"10.1016/j.jmathb.2025.101275","url":null,"abstract":"<div><div>The purpose of this research is to contribute to the literature at the intersection of imagination and the learning of abstract algebra through the lens of embodied cognition. We explored how imagination and perceptuomotor capacities can interact as two undergraduate students collaborated on an abstract algebra task. This task involved quotient group constructions in a research setting. Through our analysis we found that (a) imagination emerged as re-enactments of prior material and symbolic engagements, (b) different embodiments served a variety of interactive and supportive roles in imagining, and (c) joint embodiments with physical materials served to regulate and lift constraints on what can be imagined.</div></div>","PeriodicalId":47481,"journal":{"name":"Journal of Mathematical Behavior","volume":"80 ","pages":"Article 101275"},"PeriodicalIF":1.7,"publicationDate":"2025-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144757818","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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