Stemming from an idea put forward by Loo-Keng Hua in 1962, this article describes an original way to perform arithmetic directly on infinite decimals. This approach leads to a new and elementary construction of the real number system via decimal representation. Based on the least upper bound property, a definition of trigonometric functions is also included, which settles an issue that Godfrey H. Hardy called “a fatal defect” in his Course of Pure Mathematics.
{"title":"Real numbers as infinite decimals","authors":"Nicolas Fardin, Liangpan Li","doi":"10.54870/1551-3440.1511","DOIUrl":"https://doi.org/10.54870/1551-3440.1511","url":null,"abstract":"Stemming from an idea put forward by Loo-Keng Hua in 1962, this article describes an original way to perform arithmetic directly on infinite decimals. This approach leads to a new and elementary construction of the real number system via decimal representation. Based on the least upper bound property, a definition of trigonometric functions is also included, which settles an issue that Godfrey H. Hardy called “a fatal defect” in his Course of Pure Mathematics.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44478024","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}
What shapes do you get when you spin a cube on one of its vertices? This article explores the algebraic aspects of a spinning cube and establishes equations for the two cones and the hyperboloid using secondary school mathematics. Both the analysis and the verification take advantage of dynamic mathematics learning technologies.
{"title":"The Spinning Cube: An Algebraic Excursion","authors":"Lingguo Bu","doi":"10.54870/1551-3440.1512","DOIUrl":"https://doi.org/10.54870/1551-3440.1512","url":null,"abstract":"What shapes do you get when you spin a cube on one of its vertices? This article explores the algebraic aspects of a spinning cube and establishes equations for the two cones and the hyperboloid using secondary school mathematics. Both the analysis and the verification take advantage of dynamic mathematics learning technologies.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43902191","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}
Sampling is one of the most fundamental concepts in statistics, as the quality and accuracy of the statistical inferences made, heavily depend on the method used to obtain the sample and the sample’s ability to represent the population of inference. Despite being a simple concept, sampling presents researchers with many challenges. Generally, due to monetary and time constraints, researchers must take a smaller sample size than they would ideally use. Using statistics from these small samples, estimates for population parameters can be made, typically in the form of a confidence interval. However, the validity of these confidence intervals depends on three basic assumptions that are difficult to meet with small sample sizes. This paper traces the development of the sampling method known as bootstrapping that helps small samples to meet these assumptions. The paper touches on previous methods used before the development of bootstrapping and shows how bootstrapping has evolved over the last four decades and become widely used in the field of statistics.
{"title":"The History of Bootstrapping: Tracing the Development of Resampling with Replacement","authors":"Denise LaFontaine","doi":"10.54870/1551-3440.1515","DOIUrl":"https://doi.org/10.54870/1551-3440.1515","url":null,"abstract":"Sampling is one of the most fundamental concepts in statistics, as the quality and accuracy of the statistical inferences made, heavily depend on the method used to obtain the sample and the sample’s ability to represent the population of inference. Despite being a simple concept, sampling presents researchers with many challenges. Generally, due to monetary and time constraints, researchers must take a smaller sample size than they would ideally use. Using statistics from these small samples, estimates for population parameters can be made, typically in the form of a confidence interval. However, the validity of these confidence intervals depends on three basic assumptions that are difficult to meet with small sample sizes. This paper traces the development of the sampling method known as bootstrapping that helps small samples to meet these assumptions. The paper touches on previous methods used before the development of bootstrapping and shows how bootstrapping has evolved over the last four decades and become widely used in the field of statistics.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42465064","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}
Drawing on the Semiotic-cultural approach and the Anthropological theory of the didactic, this paper discusses how exploration of historically framed conceptualizations of mathematical objects can ...
本文借鉴符号文化方法和人类学的教学理论，探讨了如何探索数学对象的历史概念。。。
{"title":"Secondary School Mathematics Students Exploring the Connectedness of Mathematics: The Case of the Parabola and its Tangent in a Dynamic Geometry Environment","authors":"Christer Bergsten, M. Kondratieva","doi":"10.54870/1551-3440.1520","DOIUrl":"https://doi.org/10.54870/1551-3440.1520","url":null,"abstract":"Drawing on the Semiotic-cultural approach and the Anthropological theory of the didactic, this paper discusses how exploration of historically framed conceptualizations of mathematical objects can ...","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44447743","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 first of a pair of articles describes a professional development project that prepared four non-Indigenous mathematics teachers (Grades 5-12) to implement Canada’s Truth and Reconciliation Commission’s (TRC, 2016) notion of reconciliation: cross-cultural respect through mutual understanding. The researchers collaboratively mentored the teachers to enhance their mathematics teaching with Indigenous mathematizing 3 . The teachers’ focus was on developing and revising lesson plans for other teachers to teach. This process explicitly and implicitly revealed precise supports that need to be in place for a teacher to succeed at innovating with this Indigenous culture-based school mathematics (ICBSM). Part I is a template for scaling up the development of much needed Indigenous resources for mathematics teachers. Part II reports on the research results of this year-long research project.
{"title":"Indigenous Culture-Based School Mathematics in Action: Part I: Professional Development for Creating Teaching Materials","authors":"Sharon Meyer, Glen Aikenhead","doi":"10.54870/1551-3440.1516","DOIUrl":"https://doi.org/10.54870/1551-3440.1516","url":null,"abstract":": This first of a pair of articles describes a professional development project that prepared four non-Indigenous mathematics teachers (Grades 5-12) to implement Canada’s Truth and Reconciliation Commission’s (TRC, 2016) notion of reconciliation: cross-cultural respect through mutual understanding. The researchers collaboratively mentored the teachers to enhance their mathematics teaching with Indigenous mathematizing 3 . The teachers’ focus was on developing and revising lesson plans for other teachers to teach. This process explicitly and implicitly revealed precise supports that need to be in place for a teacher to succeed at innovating with this Indigenous culture-based school mathematics (ICBSM). Part I is a template for scaling up the development of much needed Indigenous resources for mathematics teachers. Part II reports on the research results of this year-long research project.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46262127","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 exact time when the mathematician Omar Khayyam lived is not well-defined, but it is generally agreed upon that he lived from the end of the 11th century to the beginning of the 12th century C.E. in Nishapur, which is in modern-day Iran and Afghanistan (Struik, 1958). Other than mathematics, Omar Khayyam also made considerable contributions to other fields, such as astronomy, philosophy, and poetry (Struik, 1958). He is probably most famous for his poem titled Rubaiyat of Omar Khayyam, which was translated by Edward Fitzgerald (Struik, 1958). Although famous for his poetry, he was professionally inclined to astronomy and mathematics. In mathematics, he is well-known for being the first individual to find positive root solutions to multiple cubic equations, and he is also known for furthering understanding of the parallel axiom (Eves, 1958, p. 285; Struik, 1958). In this report, details of Omar Khayyam’s life will be mentioned, but the focus will be on his contributions to mathematics and his role in the history of mathematics.
{"title":"The Works of Omar Khayyam in the History of Mathematics","authors":"T. Bisom","doi":"10.54870/1551-3440.1524","DOIUrl":"https://doi.org/10.54870/1551-3440.1524","url":null,"abstract":"The exact time when the mathematician Omar Khayyam lived is not well-defined, but it is generally agreed upon that he lived from the end of the 11th century to the beginning of the 12th century C.E. in Nishapur, which is in modern-day Iran and Afghanistan (Struik, 1958). Other than mathematics, Omar Khayyam also made considerable contributions to other fields, such as astronomy, philosophy, and poetry (Struik, 1958). He is probably most famous for his poem titled Rubaiyat of Omar Khayyam, which was translated by Edward Fitzgerald (Struik, 1958). Although famous for his poetry, he was professionally inclined to astronomy and mathematics. In mathematics, he is well-known for being the first individual to find positive root solutions to multiple cubic equations, and he is also known for furthering understanding of the parallel axiom (Eves, 1958, p. 285; Struik, 1958). In this report, details of Omar Khayyam’s life will be mentioned, but the focus will be on his contributions to mathematics and his role in the history of mathematics.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46756112","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 address the problem of determining what points in a field satisfy Freshman’s Dream, or equivalently, when a monomial behaves additively. It is conjectured that the only additive points over the rational numbers are trivial. In the case of finite fields, we generalize well-known results about univariate polynomials to bivariate homogeneous polynomials in order to count the number of additive points.
{"title":"When is Freshman's Dream Actually True?","authors":"M. Abramson","doi":"10.54870/1551-3440.1509","DOIUrl":"https://doi.org/10.54870/1551-3440.1509","url":null,"abstract":"We address the problem of determining what points in a field satisfy Freshman’s Dream, or equivalently, when a monomial behaves additively. It is conjectured that the only additive points over the rational numbers are trivial. In the case of finite fields, we generalize well-known results about univariate polynomials to bivariate homogeneous polynomials in order to count the number of additive points.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48969865","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}
DanceSport is a competitive form of ballroom dancing. At a DanceSport event, couples perform multiple dances in front of judges. This paper shows how a goal for a couple and the judges’ evaluations of the couple’s dance performances can be used to formulate a weighted simple game. We explain why couples and their coaches may consider a variety of goals. We also show how prominent power values can be used to measure the contributions of dance performances to achieving certain goals. As part of our analysis, we develop novel visual representations of the Banzhaf and Shapley-Shubik index profiles for different thresholds. In addition, we show that the “quota paradox” is relevant for DanceSport events.
{"title":"DanceSport and Power Values","authors":"Diana S. Cheng, P. Coughlin","doi":"10.54870/1551-3440.1528","DOIUrl":"https://doi.org/10.54870/1551-3440.1528","url":null,"abstract":"DanceSport is a competitive form of ballroom dancing. At a DanceSport event, couples perform multiple dances in front of judges. This paper shows how a goal for a couple and the judges’ evaluations of the couple’s dance performances can be used to formulate a weighted simple game. We explain why couples and their coaches may consider a variety of goals. We also show how prominent power values can be used to measure the contributions of dance performances to achieving certain goals. As part of our analysis, we develop novel visual representations of the Banzhaf and Shapley-Shubik index profiles for different thresholds. In addition, we show that the “quota paradox” is relevant for DanceSport events.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48833206","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 didactical idea of variation is exemplified with examples from convergence of series. This yields a new convergence test similar to the root test.
通过级数收敛的例子说明了变分的教学思想。这产生了一个类似于根测试的新的收敛性测试。
{"title":"Variations on Convergence Criteria","authors":"R. Oldenburg","doi":"10.54870/1551-3440.1525","DOIUrl":"https://doi.org/10.54870/1551-3440.1525","url":null,"abstract":"The didactical idea of variation is exemplified with examples from convergence of series. This yields a new convergence test similar to the root test.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42825927","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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