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Some Applications of Geometric Thinking

Bowen Kerins, Darryl Yong, Al Cuoco, 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 4
Problem Book
[Reviewed by
Tom Schulte
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This resource is designed for secondary education teachers. It is based on a course offered in the Summer Teacher Program at the Park City Mathematics Institute. Much of the material is applicable to many first-year college finite mathematics and algebra courses.

This is a sequenced collection of problem sets to help teachers apply geometric ideas to teaching concepts in algebra, number theory, arithmetic, and more. Leafing through the pages one sees examples, problems, and solutions in the back. That may suggest that this has the makings of a slim textbook, but this teachers’ resource is lacking the detail necessary detail. It is not something one could put into a student’s hands to introduce topics.

For instance, \(i\) is introduced with complex number arithmetic problem set Exercise 5.6 as, “When you see \(i^2\), make it \(-1\). The number \(i\) is the imaginary square root of \(-1\).” In only one of the six problems would \(i^2\) actually make an appearance, but all of them require some understanding of complex numbers. This brevity is consistent through the first collection of problem sets. Naturally, this still works as a source for an interactive course and assignments.

Chapter 2 is much more complete in pedagogical suggestions. This chapter is the Facilitator’s Guide, clearly stating goals and directing approaches to the problem sets of Chapter 1. It is not clear why the chapters are ordered in this fashion. Exercise 5.6 is here explained as designed to have complex plane analogies to triangles placed in the coordinate plane in Problem Set 2.

Combing through the Facilitator Guide first is the key to ascertaining the value and even intent of this resource. Many geometric concepts configured for pre-college presentation target gaps in understanding I routinely see in first-year college students, such as knowledge of the complex plane, modeling area or perimeter problems with conic sections, and a motivation for least-squares approaches. Augmenting the detailed Facilitator Guide are rich, colored illustrations in the third and final solutions chapter. There is much here I myself plan on using in an upcoming semester.


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Tom Schulte teaches finite mathematics as well as beginning and intermediate algebra at Oakland Community College in Michigan.

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