{"title":"Probably Approximately Correct: Nature's Algorithms for Learning and Prospering in a Complex World","authors":"Chris Arney","doi":"10.5860/choice.51-2716","DOIUrl":"https://doi.org/10.5860/choice.51-2716","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132865193","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":"The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry","authors":"Chris Arney, K. Crowley","doi":"10.5860/choice.43-4076","DOIUrl":"https://doi.org/10.5860/choice.43-4076","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132873889","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}
Steven G. Krantz Harvard University, founded in 1636, is America’s oldest institution of higher learning. It is the wellspring of many of our intellectual traditions. One of my favorite of these is the ritual of various Harvard faculty from the humanities and the sciences and the social studies getting together once per month or so to exchange ideas. It is a fascinating exercise: the humanist trying to explain to the cosmologist the current issues of deconstructionism; the homotopy theorist explaining to the philologist about toposes; the philosopher informing the geneticist about logical positivism. Barry Mazur is evidently the product of this crucible of erudition. His work, obviously a popular math book, is not the mindless gibbering of 1089 and All That [ACH], nor is it the self-important bombast of Chaos [GLE]. Barry Mazur has a mission: he wishes to explain to a humanist or a social theorist or a poet what √−15 is. This is a remarkable quest, and I am quite sure that I do not know how to carry it out myself. Bear in mind that I am a professional mathematician, an accomplished expositor, and in fact I am a complex analyst. I am supposed to know what √−15 is. But in fact I do not. The casual reader might conclude that this is what is wrong with the tenure system: Irresponsible faculty who are accountable to nobody. But that is not really the nub of the matter.
{"title":"Imagining Numbers (Particularly the Square Root of Minus Fifteen)","authors":"J. Rauff","doi":"10.5860/choice.41-0977c","DOIUrl":"https://doi.org/10.5860/choice.41-0977c","url":null,"abstract":"Steven G. Krantz Harvard University, founded in 1636, is America’s oldest institution of higher learning. It is the wellspring of many of our intellectual traditions. One of my favorite of these is the ritual of various Harvard faculty from the humanities and the sciences and the social studies getting together once per month or so to exchange ideas. It is a fascinating exercise: the humanist trying to explain to the cosmologist the current issues of deconstructionism; the homotopy theorist explaining to the philologist about toposes; the philosopher informing the geneticist about logical positivism. Barry Mazur is evidently the product of this crucible of erudition. His work, obviously a popular math book, is not the mindless gibbering of 1089 and All That [ACH], nor is it the self-important bombast of Chaos [GLE]. Barry Mazur has a mission: he wishes to explain to a humanist or a social theorist or a poet what √−15 is. This is a remarkable quest, and I am quite sure that I do not know how to carry it out myself. Bear in mind that I am a professional mathematician, an accomplished expositor, and in fact I am a complex analyst. I am supposed to know what √−15 is. But in fact I do not. The casual reader might conclude that this is what is wrong with the tenure system: Irresponsible faculty who are accountable to nobody. But that is not really the nub of the matter.","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116599965","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}
MATHEMATICS IN VICTORIAN BRITAIN Edited by Raymond Flood, Adrian Rice, and Robin Wilson Oxford University Press, 201 1 , 466 pp. ISBN: 978-0-19-960139-4Britain's Queen Victoria reigned from 1837 to 1901. During that time, Britain witnessed Hamilton's invention of the quaternions, Boole's algebraic logic, and Babbage's calculating machines. Mathematics in Victorian Britain is a collection of 18 papers that examine different but often overlapping topics and characters from this fascinating period in mathematical history.The Introduction by Adrian Rice (Randolph-Macon College) is a tantalizing name-dropping, topic-spotlighting overview of the contents to follow. The topic of the opening chapter by Tony Crilly (Middlesex University) is the famous Cambridge mathematical tripos, its examination, and the ranking of "wranglers" based on that exam. This chapter describes the status given to the top-scoring students (first or senior wranglers), and the evolution and eventual disappearance of the exam. The second chapter, by Keith Hannabus (Oxford University), looks at the mathematics and mathematicians of Cambridge's rival at Oxford. The bulk of this chapter examines the three Savilian professors of geometry credited with elevating Oxford's status in mathematics: Baden Powell (Savilian professor from 1827-1860), Henry Smith (1860-1883), and J.J. Sylvester (1883-1894).Moving from Oxbridge to the nation's capitol, the third chapter by Adrian Rice surveys the teaching of university mathematics in London. The work of several well-known names in British mathematics (Augustus DeMorgan, Karl Pearson, and J. J. Sylvester), as well as some who are not so well-known (William Clifford, Thomas Archer Hirst, and John Perry), is highlighted.The next three chapters take us out of England and into other parts of the United Kingdom and the British Empire. The chapter by Tony Mann (University of Greenwich) and Alex Craik on Victorian mathematics in Scotland introduces the triumvirate of mathematical physicists William Thomson, Peter Guthrie Tait, and James Clerk Maxwell (aka T, T-prime, and dp/dt). We are also introduced to the lesser-known Scottish mathematicians Alexander Bain, Philip Kelland, and Mary Somerville. Chapter Five, by Raymond Flood (University of Oxford), takes us to Ireland. This chapter centers on William Rowan Hamilton, but also attempts to identify the characteristics of Irish mathematics during the Victorian period. June Barrow-Green (Open University) finishes the excursion through the British Empire with a fascinating exposition of high wranglers who found themselves teaching in Australia, Canada, South Africa, India, and New Zealand. The information in this wide-ranging chapter is not easily accessible elsewhere. Thus ends the geographical portion of Mathematics in Victorian Britain.An interesting chapter on Victorian mathematical journals and societies by Sloan Despeaux (Western Carolina University) follows. Up next are ten mathematical field-focused
{"title":"Mathematics in Victorian Britain","authors":"J. Rauff","doi":"10.5860/choice.50-0932","DOIUrl":"https://doi.org/10.5860/choice.50-0932","url":null,"abstract":"MATHEMATICS IN VICTORIAN BRITAIN Edited by Raymond Flood, Adrian Rice, and Robin Wilson Oxford University Press, 201 1 , 466 pp. ISBN: 978-0-19-960139-4Britain's Queen Victoria reigned from 1837 to 1901. During that time, Britain witnessed Hamilton's invention of the quaternions, Boole's algebraic logic, and Babbage's calculating machines. Mathematics in Victorian Britain is a collection of 18 papers that examine different but often overlapping topics and characters from this fascinating period in mathematical history.The Introduction by Adrian Rice (Randolph-Macon College) is a tantalizing name-dropping, topic-spotlighting overview of the contents to follow. The topic of the opening chapter by Tony Crilly (Middlesex University) is the famous Cambridge mathematical tripos, its examination, and the ranking of \"wranglers\" based on that exam. This chapter describes the status given to the top-scoring students (first or senior wranglers), and the evolution and eventual disappearance of the exam. The second chapter, by Keith Hannabus (Oxford University), looks at the mathematics and mathematicians of Cambridge's rival at Oxford. The bulk of this chapter examines the three Savilian professors of geometry credited with elevating Oxford's status in mathematics: Baden Powell (Savilian professor from 1827-1860), Henry Smith (1860-1883), and J.J. Sylvester (1883-1894).Moving from Oxbridge to the nation's capitol, the third chapter by Adrian Rice surveys the teaching of university mathematics in London. The work of several well-known names in British mathematics (Augustus DeMorgan, Karl Pearson, and J. J. Sylvester), as well as some who are not so well-known (William Clifford, Thomas Archer Hirst, and John Perry), is highlighted.The next three chapters take us out of England and into other parts of the United Kingdom and the British Empire. The chapter by Tony Mann (University of Greenwich) and Alex Craik on Victorian mathematics in Scotland introduces the triumvirate of mathematical physicists William Thomson, Peter Guthrie Tait, and James Clerk Maxwell (aka T, T-prime, and dp/dt). We are also introduced to the lesser-known Scottish mathematicians Alexander Bain, Philip Kelland, and Mary Somerville. Chapter Five, by Raymond Flood (University of Oxford), takes us to Ireland. This chapter centers on William Rowan Hamilton, but also attempts to identify the characteristics of Irish mathematics during the Victorian period. June Barrow-Green (Open University) finishes the excursion through the British Empire with a fascinating exposition of high wranglers who found themselves teaching in Australia, Canada, South Africa, India, and New Zealand. The information in this wide-ranging chapter is not easily accessible elsewhere. Thus ends the geographical portion of Mathematics in Victorian Britain.An interesting chapter on Victorian mathematical journals and societies by Sloan Despeaux (Western Carolina University) follows. Up next are ten mathematical field-focused ","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115523596","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":"The Honors Class: Hilbert's Problems and Their Solvers","authors":"Chris Arney","doi":"10.5860/choice.39-5863","DOIUrl":"https://doi.org/10.5860/choice.39-5863","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"23 11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125782840","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":"Gödel's Theorem: An Incomplete Guide to Its Use and Abuse","authors":"J. Rauff","doi":"10.5860/choice.43-3434a","DOIUrl":"https://doi.org/10.5860/choice.43-3434a","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116621363","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}
COINCIDENCES, CHAOS, AND ALL THAT MATH JAZZ: MAKING LIGHT OF WEIGHTY IDEAS by Edward B. Burger and Michael Starbird W. W. Norton and Company, 2005, 276 pp. ISBN: 0-393-05945-6 Come on, babe, We 're gonna brush the sky; I betcha lucky Lindy Never flew so high 'cause in the stratosphere, How could he lend an ear To all that jazz? ("Velma," All Thai Jazz, from Bob Fosse's musical Chicago; Marshall, 2002). Mystery, curiosity, chaos, beauty, jazz music, and - math? Yes! Like Velma, Burger and Starbird invite their audience on a whirlwind tour of a world not often seen by the average individual. In this case, it is the world of truly jazzy mathematical ideas that reveal often astounding patterns and truths. But readers, unlike Lindbergh in the song above, will soar through the stratosphere of fun and fascinating facts and concepts while attuning their ears to the earthly mathematical riffs that underpin our abilities to make our planes and our imaginations fly, swoop, and barrel roll through the universe ... and maybe even beyond. First and foremost, while maintaining a lighthearted, humorous, and extremely accessible sense about the beauty and wonder of mathematics, Burger and Starbird do an excellent job of instructing the reader about how fundamental concepts produce startling observations. Readers learn how small variations can result in chaos; about Fibonacci numbers and nature; what a big number really is; fractals and art; cryptography; the fundamentals of computing; the transcendence of the fourth dimension; and many other fascinating mathematical concepts. In their Opening Thoughts (preface), the authors state: Many people think mathematics is the mechanical pursuit of solving equations. In truth, mathematics is an artistic pursuit .... But no-one should be fooled into believing that the lighthearted tone implies that we are not pursuing lofty goals. Within these pages is authentic mathematics, often of a rather advanced kind, but presented in a way that enlists the help of our (and your) everyday experiences, (p. viii) It is the lofty goals of engaging and educating the reader that the authors do achieve, early and often. By the end of Chapter One, I was convinced that math is fun. I wanted to learn more - and soon became convinced that I had been taught mathematics in the wrong way my entire academic life! As a developmental/educational psychologist, I couldn't help but wish that teachers would use the examples set forth in this book to introduce children in the third grade to the fascination of Fibonacci pineapples, coneflowers, and golden ratio rectangles. These ideas are very engaging and could be easily taught to young children, providing ideal opportunities for hands-on discovery learning activities that could be completed in cooperative groups. The noted developmental psychologist Jean Piaget [3] (and many others - e.g., Gelman and Gallistel [1] - who have since expanded, tested, and refined Piaget's initial theories) has shown quite c
{"title":"Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas","authors":"K. Crowley","doi":"10.5860/choice.43-3430","DOIUrl":"https://doi.org/10.5860/choice.43-3430","url":null,"abstract":"COINCIDENCES, CHAOS, AND ALL THAT MATH JAZZ: MAKING LIGHT OF WEIGHTY IDEAS by Edward B. Burger and Michael Starbird W. W. Norton and Company, 2005, 276 pp. ISBN: 0-393-05945-6 Come on, babe, We 're gonna brush the sky; I betcha lucky Lindy Never flew so high 'cause in the stratosphere, How could he lend an ear To all that jazz? (\"Velma,\" All Thai Jazz, from Bob Fosse's musical Chicago; Marshall, 2002). Mystery, curiosity, chaos, beauty, jazz music, and - math? Yes! Like Velma, Burger and Starbird invite their audience on a whirlwind tour of a world not often seen by the average individual. In this case, it is the world of truly jazzy mathematical ideas that reveal often astounding patterns and truths. But readers, unlike Lindbergh in the song above, will soar through the stratosphere of fun and fascinating facts and concepts while attuning their ears to the earthly mathematical riffs that underpin our abilities to make our planes and our imaginations fly, swoop, and barrel roll through the universe ... and maybe even beyond. First and foremost, while maintaining a lighthearted, humorous, and extremely accessible sense about the beauty and wonder of mathematics, Burger and Starbird do an excellent job of instructing the reader about how fundamental concepts produce startling observations. Readers learn how small variations can result in chaos; about Fibonacci numbers and nature; what a big number really is; fractals and art; cryptography; the fundamentals of computing; the transcendence of the fourth dimension; and many other fascinating mathematical concepts. In their Opening Thoughts (preface), the authors state: Many people think mathematics is the mechanical pursuit of solving equations. In truth, mathematics is an artistic pursuit .... But no-one should be fooled into believing that the lighthearted tone implies that we are not pursuing lofty goals. Within these pages is authentic mathematics, often of a rather advanced kind, but presented in a way that enlists the help of our (and your) everyday experiences, (p. viii) It is the lofty goals of engaging and educating the reader that the authors do achieve, early and often. By the end of Chapter One, I was convinced that math is fun. I wanted to learn more - and soon became convinced that I had been taught mathematics in the wrong way my entire academic life! As a developmental/educational psychologist, I couldn't help but wish that teachers would use the examples set forth in this book to introduce children in the third grade to the fascination of Fibonacci pineapples, coneflowers, and golden ratio rectangles. These ideas are very engaging and could be easily taught to young children, providing ideal opportunities for hands-on discovery learning activities that could be completed in cooperative groups. The noted developmental psychologist Jean Piaget [3] (and many others - e.g., Gelman and Gallistel [1] - who have since expanded, tested, and refined Piaget's initial theories) has shown quite c","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"126 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134417250","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":"Legacy of the Luoshu: The 4,000 Year Search for the Meaning of the Magic Square of Order Three","authors":"J. Rauff","doi":"10.5860/choice.40-0346","DOIUrl":"https://doi.org/10.5860/choice.40-0346","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127697959","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":"The Pea and the Sun: A Mathematical Paradox","authors":"J. Rauff","doi":"10.5860/choice.43-2262","DOIUrl":"https://doi.org/10.5860/choice.43-2262","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130880252","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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