Pub Date : 2023-11-13DOI: 10.1080/07468342.2023.2273195
Ellen Lehet, Kuo-Liang Chang
SummaryIn this paper, the authors present a new approach for teaching completing the square that does not rely on geometrical or algebraic models. Instead the authors’ novel approach relies on purely numerical reasoning. Additional informationNotes on contributorsEllen LehetEllen Lehet (elehet@alumni.nd.edu) works in mathematics curriculum and is interested in the intersection of mathematical practice, philosophy, and education. She earned her Ph.D. in philosophy from the University of Notre Dame in 2020 with a focus in philosophy of mathematics. Her research interests are focused in mathematical explanation and understanding from both a philosophical and a pedagogical perspective.Kuo-Liang Chang Kuo-Liang Chang (kchang@uvu.edu) is currently a professor in the department of Mathematical and Quantitative Reasoning at Utah Valley University. He earned his Ph.D. degree in mathematics education (2010) and master’s degree in applied mathematics (2003) from Michigan State University. His research interests center on the originality and flexibility of mathematical reasoning, and mathematical problem solving. He believes offering students alternative perspectives in problem solving and helping students complete their own reasoning attempts (for obtaining their perspectives) could be a way of enhancing students’ learning experience.
本文提出了一种不依赖几何或代数模型的正方形补全教学新方法。相反，作者的新方法依赖于纯粹的数值推理。关于贡献者sellen LehetEllen Lehet (elehet@alumni.nd.edu)从事数学课程的工作，对数学实践，哲学和教育的交叉感兴趣。她于2020年获得圣母大学哲学博士学位，专注于数学哲学。她的研究兴趣集中在从哲学和教育学的角度对数学的解释和理解。张国良(kchang@uvu.edu)现任美国犹他谷大学数学与定量推理系教授。2010年获美国密歇根州立大学数学教育学博士学位，2003年获应用数学硕士学位。他的研究兴趣集中在数学推理的独创性和灵活性，以及数学问题的解决。他认为，为学生提供解决问题的不同视角，并帮助学生完成自己的推理尝试(以获得他们的观点)，可能是增强学生学习体验的一种方式。
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Pub Date : 2023-11-13DOI: 10.1080/07468342.2023.2273181
Roger Nelsen
SummaryWe derive the inequalities in the title and illustrate their use in establishing a variety of inequalities encountered in undergraduate mathematics.
我们推导了题目中的不等式，并说明了它们在建立本科数学中遇到的各种不等式中的应用。
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Pub Date : 2023-11-13DOI: 10.1080/07468342.2023.2271821
Greg Oman, Charles N. Curtis
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Pub Date : 2023-11-10DOI: 10.1080/07468342.2023.2276651
Robert L. Lamphere
SummaryWe give two formulas for finding the volumes of solids of revolution in hyperbolic geometry. We also prove each formula. These formulas and their proofs are analogous to the ones in Euclidean geometry. We also provide several examples of their use. These formulas may be useful in college geometry courses that include a section on hyperbolic geometry. Additional informationNotes on contributorsRobert L. Lamphere Robert L. Lamphere (robert.Lamphere@kctcs.edu) received his Masters in mathematics from University of Illinois and his Masters in computer science from Northern Illinois University. He is an emeritus professor at the Elizabethtown Community and Technical College. His research interests are Non-Euclidean geometry and Newton’s Principia.
摘要给出了双曲几何中求旋转立体体积的两个公式。我们还证明了每个公式。这些公式及其证明类似于欧几里得几何中的公式。我们还提供了几个使用它们的例子。这些公式在包含双曲几何部分的大学几何课程中可能很有用。Robert L. Lamphere (robert.Lamphere@kctcs.edu)获得伊利诺伊大学数学硕士学位和北伊利诺伊大学计算机科学硕士学位。他是伊丽莎白镇社区和技术学院的名誉教授。他的研究兴趣是非欧几里得几何和牛顿原理。
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Pub Date : 2023-11-10DOI: 10.1080/07468342.2023.2274250
Thomas E. Cooper
SummaryWe extend a result from triangles to quadrilaterals and prove without words that circles externally tangent to a side and tangent to the extended adjacent sides can be used to bisect the perimeter of the quadrilateral.
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Pub Date : 2023-11-07DOI: 10.1080/07468342.2023.2270891
Rus May
SummaryThe weight of a ten-pound bag of potatoes is almost certainly not exactly ten pounds. Rather, it is a random variable with a minimum of ten pounds. We investigate the distribution of these weights in the context of a renewal theorem from the theory of stochastic processes and provide a straightforward demonstration of this theorem using basic tools from calculus. Additional informationNotes on contributorsRus MayRus May (r.may@moreheadstate.edu) teaches math at Morehead State University. He is interested in probability and combinatorics, especially the burgeoning field of analytic combinatorics in several variables. Outside of the classroom he enjoys cooking and eats a lot of potatoes.
一袋10磅重的土豆的重量几乎肯定不是10磅。相反，它是一个最小值为10磅的随机变量。我们在随机过程理论中的一个更新定理的背景下研究了这些权重的分布，并使用微积分的基本工具提供了一个简单的证明。srus MayRus May (r.may@moreheadstate.edu)在莫尔黑德州立大学教授数学。他对概率和组合学很感兴趣，尤其是新兴的多变量分析组合学。在课堂之外，他喜欢做饭，吃很多土豆。
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Pub Date : 2023-11-03DOI: 10.1080/07468342.2023.2268494
Timo Tossavainen, Pentti Haukkanen, Jorma K. Merikoski, Mika Mattila
SummaryWe survey the history of the arithmetic derivative and more recent advances in research on this topic. Among other things, we discuss a few generalizations of the original arithmetic derivative and some arithmetic differential equations that are related to Goldbach’s conjecture and the twin prime conjecture. Our primary purpose is to give an overview of this field, but we also aim at providing supplementary material for an introductory course on discrete mathematics or number theory. Therefore, our survey contains ten exercises. Additional informationNotes on contributorsTimo TossavainenTimo Tossavainen (timo.tossavainen@ltu.se) is professor of mathematics education at Lulea University of Technology in Sweden. He received his Ph.D. in mathematics from Jyväskylä University, Finland, under the supervision of Pekka Koskela. He is interested in recreational mathematics, nonfiction literature, progressive rock, and cross-country skiing.Pentti HaukkanenPentti Haukkanen (pentti.haukkanen@tuni.fi) received his Ph.D. in mathematics from Tampere University, Finland, under the supervision of Seppo Hyyrö. Currently, he is university lecturer of mathematics at his alma mater. In his free time, he enjoys various sports and culture.Jorma K. MerikoskiJorma K. Merikoski (jorma.merikoski@tuni.fi) is emeritus professor of mathematics at Tampere University. He received his Ph.D. in mathematics from this university under the supervision of Seppo Hyyrö. Besides mathematics, he enjoys running, cross-country skiing, and literature.Mika MattilaMika Mattila (mika.mattila@tuni.fi) is university teacher of mathematics at Tampere University, where he also received his Ph.D. in mathematics under the supervision of Pentti Haukkanen. In addition to mathematics and some physical activities, he is interested in literature, movies, and retro console games.
{"title":"We Can Differentiate Numbers, Too","authors":"Timo Tossavainen, Pentti Haukkanen, Jorma K. Merikoski, Mika Mattila","doi":"10.1080/07468342.2023.2268494","DOIUrl":"https://doi.org/10.1080/07468342.2023.2268494","url":null,"abstract":"SummaryWe survey the history of the arithmetic derivative and more recent advances in research on this topic. Among other things, we discuss a few generalizations of the original arithmetic derivative and some arithmetic differential equations that are related to Goldbach’s conjecture and the twin prime conjecture. Our primary purpose is to give an overview of this field, but we also aim at providing supplementary material for an introductory course on discrete mathematics or number theory. Therefore, our survey contains ten exercises. Additional informationNotes on contributorsTimo TossavainenTimo Tossavainen (timo.tossavainen@ltu.se) is professor of mathematics education at Lulea University of Technology in Sweden. He received his Ph.D. in mathematics from Jyväskylä University, Finland, under the supervision of Pekka Koskela. He is interested in recreational mathematics, nonfiction literature, progressive rock, and cross-country skiing.Pentti HaukkanenPentti Haukkanen (pentti.haukkanen@tuni.fi) received his Ph.D. in mathematics from Tampere University, Finland, under the supervision of Seppo Hyyrö. Currently, he is university lecturer of mathematics at his alma mater. In his free time, he enjoys various sports and culture.Jorma K. MerikoskiJorma K. Merikoski (jorma.merikoski@tuni.fi) is emeritus professor of mathematics at Tampere University. He received his Ph.D. in mathematics from this university under the supervision of Seppo Hyyrö. Besides mathematics, he enjoys running, cross-country skiing, and literature.Mika MattilaMika Mattila (mika.mattila@tuni.fi) is university teacher of mathematics at Tampere University, where he also received his Ph.D. in mathematics under the supervision of Pentti Haukkanen. In addition to mathematics and some physical activities, he is interested in literature, movies, and retro console games.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818872","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}
Pub Date : 2023-11-03DOI: 10.1080/07468342.2023.2273197
James Schultz
"Proof Without Words: The American Flag Inspires a Proof that the Sum of the First n Positive Odd Integers is n2." The College Mathematics Journal, ahead-of-print(ahead-of-print), p. 1
“不用语言的证明:美国国旗启发了一个证明:前n个正奇数的和是n2。”《大学数学学报》，第1页
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Pub Date : 2023-11-03DOI: 10.1080/07468342.2023.2273194
Otis F. Graf
SummaryIt is shown that there is a logarithmic spiral that segments a straight line through its pole in the golden ratio ϕ. This is the same spiral that is often referred to in the literature as the “Golden Spiral” Using the tools of analytic geometry and parametric vector equations, it is shown that the golden spiral is actually a member of a family of spirals defined by a generalized equation. The spirals exist as a continuum defined by a real number q > 1. When q=ϕ, the spiral segments a line through its pole in the golden ratio. That is the characteristic feature that sets it apart from all the other spirals in the continuum. Additional informationNotes on contributorsOtis F. GrafOtis F. Graf Jr. (otis.graf@hccs.edu) received a BS in physics and math and a Ph.D. in aerospace engineering from the University of Texas at Austin. He began his career doing trajectory planning, mission analysis and software development for NASA at the Johnson Space Center in Houston, TX. Later he joined IBM and designed large data storage systems for US and international research organizations. After retirement from IBM he joined the adjunct faculty at Houston Community College where he tutors math and physics students who are on an academic path to university engineering degrees.
它表明，有一个对数螺旋分段直线通过其极点在黄金比例φ。这与文献中经常提到的“黄金螺旋”相同，使用解析几何和参数向量方程的工具，证明了黄金螺旋实际上是由广义方程定义的螺旋族的成员。螺旋以实数q > 1定义的连续体存在。当q= φ时，螺旋通过其极点在黄金比例中分割一条线。这是使它有别于连续体中所有其他螺旋的特征。otis F. Graf Jr. (otis.graf@hccs.edu)获得德克萨斯大学奥斯汀分校物理和数学学士学位以及航空航天工程博士学位。他的职业生涯开始于在德克萨斯州休斯顿的约翰逊航天中心为美国宇航局做轨迹规划、任务分析和软件开发。后来他加入IBM，为美国和国际研究组织设计大型数据存储系统。从IBM退休后，他加入了休斯顿社区学院的兼职教师，在那里他指导数学和物理学生，这些学生正在攻读大学工程学位。
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