: This paper formalizes the stepping stone problem introduced Ladoucer and Rebenstock [G] to the setting of simple graphs. This paper considers the set of functions from the vertices of our graph to N , which assign a fixed number of 1’s to some vertices and assign higher numbers to other vertices by adding up their neighbors’ assignments. The stepping stone solution is defined as an element obtained from the argmax of the maxima of these functions, and the maxima as its growth. This work is organized into work on the bounded and unbounded degree graph cases. In the bounded case, sufficient conditions are obtained for superlinear and sublinear stepping stone solution growth. Furthermore this paper demonstrates the existence of a basis of graphs which characterizes superlinear growth. In the unbounded case, properly sublinear and superlinear stepping stone solution growth are obtained.
{"title":"Stepping Stone Problem on Graphs","authors":"Rory O’Dwyer","doi":"10.54870/1551-3440.1616","DOIUrl":"https://doi.org/10.54870/1551-3440.1616","url":null,"abstract":": This paper formalizes the stepping stone problem introduced Ladoucer and Rebenstock [G] to the setting of simple graphs. This paper considers the set of functions from the vertices of our graph to N , which assign a fixed number of 1’s to some vertices and assign higher numbers to other vertices by adding up their neighbors’ assignments. The stepping stone solution is defined as an element obtained from the argmax of the maxima of these functions, and the maxima as its growth. This work is organized into work on the bounded and unbounded degree graph cases. In the bounded case, sufficient conditions are obtained for superlinear and sublinear stepping stone solution growth. Furthermore this paper demonstrates the existence of a basis of graphs which characterizes superlinear growth. In the unbounded case, properly sublinear and superlinear stepping stone solution growth are obtained.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70969739","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}
: Consider a 1 × 3 grid whose top-left vertex is connected to the bottom-right vertex of each of the unit squares. What is the sum of the three acute angles formed by the connecting segments with the unit squares’ bases? This is the so-called three-squares problem, often attributed to Gardner. In a recent academic year, the author used a video on this problem, produced by the YouTube channel Numberphile, for a group project in a first-semester undergraduate module: Mathematical Prob-lem Solving. The project involved collaborative writing on the problem and individual completion of a peer-assessment form. We report the outcomes of this project, which give rise to both theoretical and pedagogical discussions. The theoretical discussion comprises seven alternative solutions to the problem, as well as a generalisation to the case of identical parallelograms forming an arbitrary-sized grid whose top-left vertex is connected to the bottom-right vertex of each parallelogram. The pedagogical discussion highlights the peer-assessment form’s effectiveness in detecting unequal group members’ contribution, as well as the students’ inadequate communication skills. The latter, which has consistently raised concerns, subsequently led to the module being developed and renamed as Mathematical Writing and Reasoning, whose realisation commenced the following academic year.
{"title":"Using Gardner's Three-Squares Problem for a Group Project in a Mathematical Problem Solving Module","authors":"Jonathan Hoseana","doi":"10.54870/1551-3440.1638","DOIUrl":"https://doi.org/10.54870/1551-3440.1638","url":null,"abstract":": Consider a 1 × 3 grid whose top-left vertex is connected to the bottom-right vertex of each of the unit squares. What is the sum of the three acute angles formed by the connecting segments with the unit squares’ bases? This is the so-called three-squares problem, often attributed to Gardner. In a recent academic year, the author used a video on this problem, produced by the YouTube channel Numberphile, for a group project in a first-semester undergraduate module: Mathematical Prob-lem Solving. The project involved collaborative writing on the problem and individual completion of a peer-assessment form. We report the outcomes of this project, which give rise to both theoretical and pedagogical discussions. The theoretical discussion comprises seven alternative solutions to the problem, as well as a generalisation to the case of identical parallelograms forming an arbitrary-sized grid whose top-left vertex is connected to the bottom-right vertex of each parallelogram. The pedagogical discussion highlights the peer-assessment form’s effectiveness in detecting unequal group members’ contribution, as well as the students’ inadequate communication skills. The latter, which has consistently raised concerns, subsequently led to the module being developed and renamed as Mathematical Writing and Reasoning, whose realisation commenced the following academic year.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70970468","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}
: Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students’ arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants’ perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers’ conceptions of conviction with respect to counterexamples.
{"title":"On Convincing Power of Counterexamples","authors":"Orly Buchbinder, Rina Zazkis","doi":"10.54870/1551-3440.1636","DOIUrl":"https://doi.org/10.54870/1551-3440.1636","url":null,"abstract":": Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students’ arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants’ perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers’ conceptions of conviction with respect to counterexamples.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70970532","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}
{"title":"Encounters With “Love and Math” A Belated Review of Edward Frenkel’s Love and Math: The Heart of Hidden","authors":"Rina Zazkis","doi":"10.54870/1551-3440.1585","DOIUrl":"https://doi.org/10.54870/1551-3440.1585","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70969103","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}
{"title":"A Math Ed Take on a Love-Hate Relationship A Review of Reuben Hersh and Vera John-Steiner’s Loving + Hating Mathematics: Challenging the Myths of Mathematical Life","authors":"J. Holm","doi":"10.54870/1551-3440.1587","DOIUrl":"https://doi.org/10.54870/1551-3440.1587","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44784937","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}
{"title":"Algorithms and Mathematics Education A Response and Review of Hannah Fry’s Hello World: Being Human in the Age of Algorithms","authors":"Joshua T. Hertel","doi":"10.54870/1551-3440.1599","DOIUrl":"https://doi.org/10.54870/1551-3440.1599","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49111015","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}
{"title":"Overcoming Misconceptions about Probability A Review of David J. Hand’s The Improbability Principle","authors":"Keith Gallagher","doi":"10.54870/1551-3440.1607","DOIUrl":"https://doi.org/10.54870/1551-3440.1607","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43417181","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}
{"title":"Numerate Life for Whom? A Non-book-review of John Allen Paulos’s A Numerate Life: Mathematician Explores the Vagaries of Life, His Own and Probably Yours","authors":"Ofer Marmur","doi":"10.54870/1551-3440.1590","DOIUrl":"https://doi.org/10.54870/1551-3440.1590","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41627246","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}
{"title":"Queering the Stats A Review of David Spiegelhalter’s The Art of Statistics: Learning from Data","authors":"N. Radaković","doi":"10.54870/1551-3440.1592","DOIUrl":"https://doi.org/10.54870/1551-3440.1592","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42599538","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}
{"title":"Woo-Hoo! The Mathematics and Education of the D’oh-Nut. A Review of Simon Singh’s The Simpsons and Their Mathematical Secrets","authors":"Jamie S. Pyper","doi":"10.54870/1551-3440.1591","DOIUrl":"https://doi.org/10.54870/1551-3440.1591","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47098433","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}
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