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Moving Things Around

Bowen Kerins, Darryl Yong, Al Cucoco, Glenn Stevens, and Mary Pilgrim
American Mathematical Society/Institute for Advanced Study/Park City Mathematics Institute
Publication Date: 
Number of Pages: 
IAS/PCMI--The Teacher Program Series 5
Problem Book
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Tom Schulte
, on

Moving Things Around is based on courses offered in the Summer School Teacher Program at the Park City Mathematics Institute. The courses for precollege teachers are facilitated by instructors and mathematicians. This book is based on the 2012 program, which had a decade of previous programs to draw upon. The experience from past sessions becomes visible in the Facilitator Notes. It becomes obvious in reading — and this can be of value for independent reading and study outside of the classroom — that the years of application and breadth of contributions produced a well-organized, cogent body of material. Here, a sequenced and carefully curated collection of problem sets — not a grab-bag problem bank — guides students toward the awareness of connections between group theory and number theory. I feel that to realize the full potential for students to experience discovering these connections requires committing to a semester-length course; fourteen problem sets at roughly one per week. This would be more directed groupwork than lecture presentations, although strategic selections among this material could make for enlightening diversions in a traditional, lecture-based course.

Analysis of in- and out-shuffles of various sized decks exemplifies modular arithmetic and permutations. Returning often to this card shuffling 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, isomorphisms, etc. While the target is precollege teachers, 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 has Solutions, often multistep and detailed.


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Tom Schulte teaches first-year undergraduates in mathematics at Oakland Community College in Michigan.

See the table of contents in the publisher's webpage.