This article contains a selection of problems from the American MathematicsCompetitions.
本文选取了美国数学竞赛中的一些问题。
{"title":"Engaging Students Through Math Competitions","authors":"Bela Bajnok","doi":"arxiv-2406.15460","DOIUrl":"https://doi.org/arxiv-2406.15460","url":null,"abstract":"This article contains a selection of problems from the American Mathematics\u0000Competitions.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528453","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}
We present in this article a general approach (in the form of recommendationsand guidelines) for tackling Diophantine equation problems (whether singleequations or systems of simultaneous equations). The article should be usefulin particular to young "mathematicians" dealing mostly with Diophantineequations at elementary level of number theory (noting that familiarity withelementary number theory is generally required).
{"title":"How to Solve Diophantine Equations","authors":"Taha Sochi","doi":"arxiv-2406.16919","DOIUrl":"https://doi.org/arxiv-2406.16919","url":null,"abstract":"We present in this article a general approach (in the form of recommendations\u0000and guidelines) for tackling Diophantine equation problems (whether single\u0000equations or systems of simultaneous equations). The article should be useful\u0000in particular to young \"mathematicians\" dealing mostly with Diophantine\u0000equations at elementary level of number theory (noting that familiarity with\u0000elementary number theory is generally required).","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528456","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}
We present a scoping review of published literature on ethnomathematics andIndigenous mathematics as a step towards a goal to decolonize the prevailingEurocentric view of the provenance of mathematics. Mathematical practices wereidentified globally from 169 included studies. We map three development stagesof ethnomathematical research from 1984 to 2023 and identify 20 categories ofIndigenous and traditional cultural activities that evidence mathematicaldesign and expression. We address two challenges of investigating non-Westernbased mathematics: where to look for mathematical knowledge, and how to decodeit from cultural practices. These two hurdles are overcome by cluster analysisof the keywords of included studies. Existing research falls into twocategories: I. identification of mathematical concepts used in Indigenoussocieties, and II. systematizing identified mathematical concepts. Bothapproaches are essential for research on Indigenous mathematics to flourish, inorder to empower Indigenous knowledge holders and deconstruct restrictivecolonial boundaries of mathematical knowledge and education.
{"title":"Multiple Forms of Knowing in Mathematics: A Scoping Literature Study","authors":"Hongzhang Xu, Rowena Ball","doi":"arxiv-2406.16921","DOIUrl":"https://doi.org/arxiv-2406.16921","url":null,"abstract":"We present a scoping review of published literature on ethnomathematics and\u0000Indigenous mathematics as a step towards a goal to decolonize the prevailing\u0000Eurocentric view of the provenance of mathematics. Mathematical practices were\u0000identified globally from 169 included studies. We map three development stages\u0000of ethnomathematical research from 1984 to 2023 and identify 20 categories of\u0000Indigenous and traditional cultural activities that evidence mathematical\u0000design and expression. We address two challenges of investigating non-Western\u0000based mathematics: where to look for mathematical knowledge, and how to decode\u0000it from cultural practices. These two hurdles are overcome by cluster analysis\u0000of the keywords of included studies. Existing research falls into two\u0000categories: I. identification of mathematical concepts used in Indigenous\u0000societies, and II. systematizing identified mathematical concepts. Both\u0000approaches are essential for research on Indigenous mathematics to flourish, in\u0000order to empower Indigenous knowledge holders and deconstruct restrictive\u0000colonial boundaries of mathematical knowledge and education.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"144 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528454","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}
Mathematical research is often motivated by the desire to reach a beautifulresult or to prove it in an elegant way. Mathematician's work is thus stronglyinfluenced by his aesthetic judgments. However, the criteria these judgmentsare based on remain unclear. In this article, we focus on the concept ofmathematical beauty, as one of the central aesthetic concepts in mathematics.We propose an account according to which beautiful mathematics is one in whicha great deal of the mathematical world is reflected. We argue that beauty inmathematics reveals connections between apparently non-related problems orareas and provides a wider insight into mathematical reality as a whole. Wealso analyze the close relationship between beauty and other important notionssuch as depth, elegance, simplicity and others.
{"title":"Kurt Gödel and the Logic of Concepts","authors":"Jovana Kostić, Slobodan Vujošević","doi":"arxiv-2406.05442","DOIUrl":"https://doi.org/arxiv-2406.05442","url":null,"abstract":"Mathematical research is often motivated by the desire to reach a beautiful\u0000result or to prove it in an elegant way. Mathematician's work is thus strongly\u0000influenced by his aesthetic judgments. However, the criteria these judgments\u0000are based on remain unclear. In this article, we focus on the concept of\u0000mathematical beauty, as one of the central aesthetic concepts in mathematics.\u0000We propose an account according to which beautiful mathematics is one in which\u0000a great deal of the mathematical world is reflected. We argue that beauty in\u0000mathematics reveals connections between apparently non-related problems or\u0000areas and provides a wider insight into mathematical reality as a whole. We\u0000also analyze the close relationship between beauty and other important notions\u0000such as depth, elegance, simplicity and others.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510198","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}
The question of the definition of what is an algorithm is recurrent. It isfound in teaching, at different levels and particularly in secondary educationbecause of the recent evolutions in high school, with immediate consequences inhigher education. It is found in mediation, with the different meanings thatthe word "algorithm" is charged with in the media space. It is also found inresearch, with issues in different branches of computer science, fromfoundations in computability and complexity to applications in big data. Beyondthe issue of definition, it is the raison d'{^e}tre of the notion of algorithmthat should be questioned: what do we want to do with it and what is at stake?It is by trying to specify this that we can identify didactic elements that arelikely to help teach the algorithm, in interaction with mathematics or not, andto different audiences.
{"title":"L'algorithme: pourquoi et comment le définir pour l'enseigner","authors":"Emmanuel BeffaraMeTAH, LIG, IREM","doi":"arxiv-2406.04385","DOIUrl":"https://doi.org/arxiv-2406.04385","url":null,"abstract":"The question of the definition of what is an algorithm is recurrent. It is\u0000found in teaching, at different levels and particularly in secondary education\u0000because of the recent evolutions in high school, with immediate consequences in\u0000higher education. It is found in mediation, with the different meanings that\u0000the word \"algorithm\" is charged with in the media space. It is also found in\u0000research, with issues in different branches of computer science, from\u0000foundations in computability and complexity to applications in big data. Beyond\u0000the issue of definition, it is the raison d'{^e}tre of the notion of algorithm\u0000that should be questioned: what do we want to do with it and what is at stake?\u0000It is by trying to specify this that we can identify didactic elements that are\u0000likely to help teach the algorithm, in interaction with mathematics or not, and\u0000to different audiences.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"322 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510199","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}
Researchers have demonstrated that humans are unable to generate a sequenceof random numbers that corresponds in a statistical sense to a simpledistribution such as the uniform distribution. The purpose of this article isto present the results of research on the generation of random number sequencesby humans. The article describes 10 effects found in such studies, mechanismsexplaining these effects, and 14 measures (not including modifications) used todetect deviations from randomness in the sequences. The analysis of numericalsequences is not only of academic interest; it can also be used for the purposeof data validation (auditing).
{"title":"Indicators of the human origin of numbers","authors":"Vitaliy Grigoriev","doi":"arxiv-2406.10240","DOIUrl":"https://doi.org/arxiv-2406.10240","url":null,"abstract":"Researchers have demonstrated that humans are unable to generate a sequence\u0000of random numbers that corresponds in a statistical sense to a simple\u0000distribution such as the uniform distribution. The purpose of this article is\u0000to present the results of research on the generation of random number sequences\u0000by humans. The article describes 10 effects found in such studies, mechanisms\u0000explaining these effects, and 14 measures (not including modifications) used to\u0000detect deviations from randomness in the sequences. The analysis of numerical\u0000sequences is not only of academic interest; it can also be used for the purpose\u0000of data validation (auditing).","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528455","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}
This review paper highlights research findings from the authors'participation in the SUMMIT-P project, which studied how to build and sustainmulti-institutional interdisciplinary partnerships to design and implementcurricular change in mathematics courses in the first two years of college,using the Curriculum Foundations Project (CFP) as a launchpad. The CFPinterviewed partner discipline faculty to learn about the mathematical needs oftheir students and how they use mathematics in their courses. This papersummarizes research findings from the CFP and the SUMMIT-P project, andpresents a detailed example of how these findings were implemented in thecalculus sequence at Augsburg University to improve course focus, increase therelevance of course content, and provide opportunities for student to practicetransference of the calculus to disciplinary contexts. This paper is based onthe talk "Applied and Active Calculus Built Through InterdisciplinaryPartnerships" presented at the 2022 AWM Research Symposium in the Session on"Research on the First Two Years of College Mathematics".
{"title":"Renovating Calculus through Interdisciplinary Partnerships Using the SUMMIT-P Model","authors":"Suzanne Dorée, Jody Sorensen","doi":"arxiv-2406.08508","DOIUrl":"https://doi.org/arxiv-2406.08508","url":null,"abstract":"This review paper highlights research findings from the authors'\u0000participation in the SUMMIT-P project, which studied how to build and sustain\u0000multi-institutional interdisciplinary partnerships to design and implement\u0000curricular change in mathematics courses in the first two years of college,\u0000using the Curriculum Foundations Project (CFP) as a launchpad. The CFP\u0000interviewed partner discipline faculty to learn about the mathematical needs of\u0000their students and how they use mathematics in their courses. This paper\u0000summarizes research findings from the CFP and the SUMMIT-P project, and\u0000presents a detailed example of how these findings were implemented in the\u0000calculus sequence at Augsburg University to improve course focus, increase the\u0000relevance of course content, and provide opportunities for student to practice\u0000transference of the calculus to disciplinary contexts. This paper is based on\u0000the talk \"Applied and Active Calculus Built Through Interdisciplinary\u0000Partnerships\" presented at the 2022 AWM Research Symposium in the Session on\u0000\"Research on the First Two Years of College Mathematics\".","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"214 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528409","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}
Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, Samuel Tsui
Fitch Cheney's 5-card trick was introduced in 1950. In 2013, Mulcahy inventeda 4-card trick in which the cards are allowed to be displayed face down. Wesuggest our own invention: a 3-card trick in which the cards can be face downand also allowed to be placed both vertically and horizontally. We discuss thetheory behind all the tricks and estimate the maximum deck size given thenumber of chosen cards. We also discuss the cases of hiding several cards andthe deck that has duplicates.
{"title":"Card Tricks and Information","authors":"Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, Samuel Tsui","doi":"arxiv-2405.21007","DOIUrl":"https://doi.org/arxiv-2405.21007","url":null,"abstract":"Fitch Cheney's 5-card trick was introduced in 1950. In 2013, Mulcahy invented\u0000a 4-card trick in which the cards are allowed to be displayed face down. We\u0000suggest our own invention: a 3-card trick in which the cards can be face down\u0000and also allowed to be placed both vertically and horizontally. We discuss the\u0000theory behind all the tricks and estimate the maximum deck size given the\u0000number of chosen cards. We also discuss the cases of hiding several cards and\u0000the deck that has duplicates.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259696","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}
Recent years have seen the dramatic rise of the usage of AI algorithms inpure mathematics and fundamental sciences such as theoretical physics. This isperhaps counter-intuitive since mathematical sciences require the rigorousdefinitions, derivations, and proofs, in contrast to the experimental scienceswhich rely on the modelling of data with error-bars. In this Perspective, wecategorize the approaches to mathematical discovery as "top-down", "bottom-up"and "meta-mathematics", as inspired by historical examples. We review some ofthe progress over the last few years, comparing and contrasting both theadvances and the short-comings in each approach. We argue that while thetheorist is in no way in danger of being replaced by AI in the near future, thehybrid of human expertise and AI algorithms will become an integral part oftheoretical discovery.
{"title":"A Triumvirate of AI Driven Theoretical Discovery","authors":"Yang-Hui He","doi":"arxiv-2405.19973","DOIUrl":"https://doi.org/arxiv-2405.19973","url":null,"abstract":"Recent years have seen the dramatic rise of the usage of AI algorithms in\u0000pure mathematics and fundamental sciences such as theoretical physics. This is\u0000perhaps counter-intuitive since mathematical sciences require the rigorous\u0000definitions, derivations, and proofs, in contrast to the experimental sciences\u0000which rely on the modelling of data with error-bars. In this Perspective, we\u0000categorize the approaches to mathematical discovery as \"top-down\", \"bottom-up\"\u0000and \"meta-mathematics\", as inspired by historical examples. We review some of\u0000the progress over the last few years, comparing and contrasting both the\u0000advances and the short-comings in each approach. We argue that while the\u0000theorist is in no way in danger of being replaced by AI in the near future, the\u0000hybrid of human expertise and AI algorithms will become an integral part of\u0000theoretical discovery.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141188183","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}
Many recent studies have expounded the benefits of active learning intertiary education. It can be challenging to implement these techniques atlarge scale (for example in first year mathematics courses). A common methodfor actively engaging students in large classes is through online quizzes,which may include peer instruction. In this paper, we investigate the effect ofhaving students answer quiz-style questions during class both with and withoutdiscussion in a first year mathematics course. We also investigate the short-and long-term effects of each protocol. We find that peer instruction improves student learning in mathematics in thefollowing ways: First, when the responses to questions was measured before andafter peer instruction the proportion of questions answered correctly increasedby 0.2; second, when correct responses were compared to similar questions thefollowing week the proportion correct increased by 0.34 (compared to 0.07 forthe control); finally, when measured at the end of the semester the proportionof questions answered correctly increased by 0.42 (compared to 0.2 for thecontrol).
{"title":"An assessment of peer instruction in large first year mathematics courses","authors":"Raymond Vozzo, Stuart Johnson, Jonathan Tuke","doi":"arxiv-2405.14151","DOIUrl":"https://doi.org/arxiv-2405.14151","url":null,"abstract":"Many recent studies have expounded the benefits of active learning in\u0000tertiary education. It can be challenging to implement these techniques at\u0000large scale (for example in first year mathematics courses). A common method\u0000for actively engaging students in large classes is through online quizzes,\u0000which may include peer instruction. In this paper, we investigate the effect of\u0000having students answer quiz-style questions during class both with and without\u0000discussion in a first year mathematics course. We also investigate the short-\u0000and long-term effects of each protocol. We find that peer instruction improves student learning in mathematics in the\u0000following ways: First, when the responses to questions was measured before and\u0000after peer instruction the proportion of questions answered correctly increased\u0000by 0.2; second, when correct responses were compared to similar questions the\u0000following week the proportion correct increased by 0.34 (compared to 0.07 for\u0000the control); finally, when measured at the end of the semester the proportion\u0000of questions answered correctly increased by 0.42 (compared to 0.2 for the\u0000control).","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151372","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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