Moving Things Around

by Bowen Kerins, Darryl Yong, Al Cuoco

Designed for pre-college teachers by a collaborative of teachers, educators, and mathematicians, Moving Things Around is based on a course offered in the Summer School Teacher Program at the Park City Mathematics Institute. But this book isn't a "course" in the traditional sense. It consists of a carefully sequenced collection of problem sets designed to develop several interconnected mathematical themes, and one of the goals of the problem sets is for readers to uncover these themes for themselves. The goal of Moving Things Around is to help participants make what might seem to be surprising connections among seemingly different areas: permutation groups, number theory, and expansions for rational numbers in various bases, all starting from the analysis of card shuffles. Another goal is to use these connections to bring some coherence to several ideas that run throughout school mathematics-rational number arithmetic, different representations for rational numbers, geometric transformations, and combinatorics. The theme of seeking structural similarities is developed slowly, leading, near the end of the course, to an informal treatment of isomorphism. Moving Things Around is a volume of the book series IAS/PCMI-The Teacher Program Series published by the American Mathematical Society. Each volume in this series covers the content of one Summer School Teacher Program year and is independent of the rest.

134 pages, Paperback

Published October 6, 2016

Reviews

[Tom Schulte · Rating 4 out of 5]

...Analysis of in- and out-shuffles of various sized decks exemplifies modular arithmetic and permutations. Returning often to this card shuffle theme, the material brings together number and group theoretic concepts in a truly hands on approach. Problem Set 2 can be used to exemplify the pace and breadth of the problems: After opening with “Can perfect shuffles restore a deck with 9 cards”, #2 asks for the least significant digit of three progressively more complex arithmetic expressions; #13 questions if “… .99999… was equal to 1”; and in #21 the student is challenged to “find all solutions to x2 – 6x + 8 = 0 in mod 105 without the use of technology.” In my experience, roughly two thirds of this material would be new to first-year college students, especially abstract algebra topics such as generators, dihedral groups, isomorphism, etc. While the target is precollege, this advanced material is applicable to many first-year undergraduate programs. The problems have helpful sidebar prompts supplying hints and guidance. The problem sets are laid out as worksheets with permission given to “copy select pages for use in teaching”. The middle section of Facilitator Notes offers presentation and pedagogical advice. This is often drawn from observations with students: “At PCMI 2012, the opener proved difficult to understand for participants. We suggest working through at least one or two examples from both tables…”. The concluding section is Solutions, often multistep and detailed.

[Look for my entire review up at MAA Reviews.]