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Association of Christians in the Mathematical Sciences Proceedings Association of Christians in the Mathematical Sciences PROCEEDINGS ACMS May 2018, Volume 21 Editor’s Introduction The Resolved and Unresolved Conjectures of R. D. Carmichael “Big Idea” Reflection Assignments for Learning and Valuing Mathematics Start a Math Teacher Circle: Connect K-12 Teachers with Engaging, Approachable, and Meaningful Mathematical Problems A Pre-Calculus Controversy: Infinitesimals and Why They Matter Blended Courses Across the Curriculum: What Works and What Does Not Mentoring as a Statistical Educator in a Christian College Developing the Underutilized Mathematical Strengths of Students The Daily Question: Building Student Trust and Interest in Undergraduate Introductory Probability and Statistics Courses The Set of Zero Divisors of a Factor Ring The Topology of Harry Potter: Exploring Higher Dimensions in Young Adult Fantasy Literature Using Real-World Team Projects: A Pedagogical Framework Ten Mathematicians Who Recognized God’s Hand in Their Work (Part 2) Variations on the Calculus Sequence Reading Journals: Preview Assignments that Promote Student Engagement Finding Meaning in Calculus (and Life) Axioms: Mathematical and Spiritual: What Says the Parable? Cultivating Mathematical Affections through Engagement in Service-Learning Appendix 1: Conference Schedule and Abstracts Appendix 2: Conference Attendees Charleston Southern University Conference Attendees, May 31 - June 2, 2017 Table of Contents Editor’s Introduction Russell W. Howell....................................... iv The Resolved and Unresolved Conjectures of R. D. Carmichael Brian D. Beasley........................................1 “Big Idea” Reflection Assignments for Learning and Valuing Mathematics Jeremy Case, Mark Colgan..................................9 Start a Math Teacher Circle: Connect K-12 Teachers with Engaging, Approachable, and Meaningful Mathematical Problems Thomas Clark, Mike Janssen, Amanda Harsy, Dave Klanderman, Mandi Maxwell, Sharon Robbert............................... 22 A Pre-Calculus Controversy: Infinitesimals and Why They Matter Karl-Dieter Crisman...................................... 30 Blended Courses Across the Curriculum: What Works and What Does Not Ryan Botts, Lori Carter, Catherine Crockett........................ 42 Mentoring as a Statistical Educator in a Christian College L. Marlin Eby......................................... 58 Developing the Underutilized Mathematical Strengths of Students Patrick Eggleton........................................ 71 The Daily Question: Building Student Trust and Interest in Undergraduate Introductory Probability and Statistics Courses Matthew A. Hawks....................................... 77 The Set of Zero Divisors of a Factor Ring Jesus´ Jimenez´ ......................................... 90 The Topology of Harry Potter: Exploring Higher Dimensions in Young Adult Fantasy Literature Sarah Klanderman, Alexa Schut, Dave Klanderman, William Boerman-Cornell.... 101 Using Real-World Team Projects: A Pedagogical Framework Mike Leih............................................ 108 – ii – Ten Mathematicians Who Recognized God’s Hand in Their Work (Part 2) Dale McIntyre......................................... 117 Variations on the Calculus Sequence Christopher Micklewright................................... 134 Reading Journals: Preview Assignments that Promote Student Engagement, Productive Struggle, and Ultimate Success in Undergraduate Mathematics Courses Sarah Nelson.......................................... 141 Finding Meaning in Calculus (and Life) Doug Phillippy......................................... 157 Axioms: Mathematical and Spiritual: What Says the Parable? Melvin Royer.......................................... 167 Cultivating Mathematical Affections through Engagement in Service-Learning Joshua B. Wilkerson...................................... 181 Appendix 1: Conference Schedule and Abstracts ......................... 194 Appendix 2: Conference Attendees ................................ 226 – iii – Editor’s Introduction Russell W. Howell (Westmont College) The twenty-first biennial conference of the Association of Christians in the Mathematical Sciences was held at Charleston Southern University from May 31 until June 3, 2017. Thanks go to Jamie Probin and his colleagues at CSU for all the organizational efforts that went into hosting it. Many thanks also to the three invited speakers, each of whom prepared two engaging presentations: Sloan Despeaux (Western Carolina University) • – De Morgan’s Budget of Paradoxes – Fit to Print? Referees’ Reports of Mathematics in Nineteenth-Century London Dominic Klyve (Central Washington University) • – The Mathematics of Faith: Euler’s Anonymous Work on the Limits of Mathematics, Science, and Faith – Translation and Betrayal: Euler’s “Letters to a German Princess” Derek Schuurman (Calvin College) • – Shaping a Digital World: Connecting Bytes and Beliefs – Responsible Automation: Faith and Work in an Age of Intelligent Machines There were a total of 66 papers presented by the 126 conference attendees. The relevant abstracts, as well as the entire conference schedule, are given in Appendix 1 of these Proceedings. Appendix 2 lists the participants, home institutions, and e-mail addresses. Prior to the gathering at Charleston Southern University the ACMS board decided to initiate a peer- review procedure for the papers submitted to all conference Proceedings, and asked Russell Howell to serve as the editor for this first effort. Of course, not every attendee submitted an exposition of his or her talk. The following pages contain the papers that made their way through the single-blind review process, each having been scrutinized by a minimum of two reviewers. The editor expresses thanks to the authors for their cooperative spirit. Too numerous to mention are all the referees that were involved, but heartfelt thanks go to them for their diligent work. The twenty-second biennial conference for ACMS, set to be hosted by Indiana Wesleyan University, is scheduled for May 29 – June 1, 2019. Details can be found at acmsonline.org, which is the official website for ACMS. – iv – The Resolved and Unresolved Conjectures of R. D. Carmichael Brian D. Beasley (Presbyterian College) Brian Beasley (B.S., Emory University; M.S., University of North Car- olina; Ph.D., University of South Carolina) has taught at Presbyterian Col- lege since 1988. He became a member of the Mathematical Association of America in 1989 and joined ACMS in 2007. Outside the classroom, Brian enjoys family time with his wife and two sons. A big fan of the var- ious writings of C. S. Lewis, he is also an enthusiastic Scrabble player, a somewhat less than enthusiastic jogger, and a very shaky unicyclist. Abstract Even before heading to Princeton University to work on his doctoral degree, Robert Carmichael started influencing the path of number theory in the 20th century. From his study of Euler’s totient function to his discovery of the first absolute pseudoprime, he set the stage for years of productive research. We present a brief overview of Carmichael’s life, including his breadth of mathematical interests and his service on behalf of the Mathematical Association of America. The main focus is upon his two most famous conjectures - which one has been settled, and which one remains open to this day? 1 Early Years Robert Daniel Carmichael was born in Goodwater, Alabama in 1879. Carmichael spent the first thirty years of his life in Alabama, never too far away from Goodwater. He graduated from nearby Lineville College in 1898, and three years later he married Eula Smith Narramore from Randolph (south of Birmingham). Robert and Eula settled in Hartselle with their four children, Eunice, Erdys, Gershom, and Robert Leslie. Carmichael began training to become a Presbyterian pastor, presumably with the intention of keeping his home and family and ministry in Alabama [30]. Yet starting in 1905, the situation began to change for Carmichael. Between 1905 and 1915, he sub- mitted dozens of problems to The American Mathematical Monthly. For example, in 1908 Carmichael contributed the following problem [7] to the “Number Theory and Diophantine Analysis” section: If p and q are primes and m and n are any integers, find the cases in which the equation pm qn = 1 may be satisfied. − Meanwhile, in October of 1906, Carmichael became professor of mathematics at Presbyterian College in Anniston, Alabama. The school was in just its second year when he arrived, and it continued as a college until 1918. As noted in [2], during its brief history, it offered both a Classical Course (B.A.) and a Scientific Course (B.S.), awarding a total of 37 degrees. Presbyterian College also somehow managed to field a football team, playing local high schools and colleges and even taking on the University of Georgia squad in 1909 and in 1911. The team’s nickname? The Predestinarians. ACMS 21st Biennial Conference Proceedings, Charleston Southern University, 2017 Page 1 In 1909, Carmichael moved his family to Princeton, where he started graduate work in mathematics. Under the direction of George Birkhoff, he wrote his thesis (Linear Difference Equations and their Analytic Solutions) and received his doctorate in 1911. Carmichael then accepted a professorship at Indiana University, teaching there from 1911 to 1915 [30]. Of his eventual 35 doctoral students, only the first earned the degree at Indiana, but this represented a significant milestone: In 1912, Cora B. Hennel became the first person, male or female, to receive
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