*Applications of Algebra and Geometry to the Work of Teaching* is based on the 2008 Summer School Teacher Program (SSTP) at the Park City Mathematics Institute (PCMI) and the PROMYS for Teachers Program (PfT) at Boston University. As pointed out in the preface, PCMI is sponsored by the Institute for Advanced Study and is a three-week summer program for a wide range of people involved in mathematics: research mathematicians, graduate students, undergraduate faculty, undergraduate students, and precollege teachers. The SSTP has been a part of PCMI since 2001. The SSTP meets for 15 two-hour sessions in which teachers investigate a selected aspect of mathematics related to the overall goal of the summer’s PCMI.

The major goal of this book is to present a “course” that introduces readers to the use of complex numbers, with a focus on arithmetic of Gaussian and Eisenstein integers, and to study some questions involving the intersection of algebra and geometry, such as the classification of Pythagorean triples and the number of representations of an integer as the sum of two squares. With all the PCMIs, the overall goal is to put the mathematics that teachers present in their home schools into a broader view of mathematics as a scientific structure.

It should be emphasized that the “course” in the title is not a traditional course. The book provides materials and problems that will engage the student in deep and rich discovery of mathematics. The problem sets presented in the book fall into three categories, Important Stuff, Neat Stuff, and Tough Stuff. Problems located in the Important Stuff are problems that present the fundamental knowledge students need. Problems in the Neat and Tough Stuff are meant to have the students think outside the box and to creatively solve complex problems. All of the problems are related to the *Common Core* Standards for Mathematical Practice. As pointed out in the Preface, the overall goal of the PCMI teacher program is to provide teachers with the opportunity to:

- Deepen their understanding of mathematics.
- Reflect on the practice of teaching.
- Serve as mathematical resources for their colleagues.

These main ideas have been very beneficial to mathematics teachers across the country. As evidenced by one evaluation report, many teachers have been motivated by PCMI to change the amounts of time they lecture vs. how much group work is given to the students. The ideas of constructivist teaching are growing as a result of efforts by the PCMI.

When reading through the problems, it is clear that they are carefully crafted and are not designed to be an entire traditional course. They are designed for students to make connections, draw on various concepts, and to develop the logic and critical thinking skills needed to solve problems. For students who may have never seen certain topics before, the problems lead students to the big idea. For example, in Problem Set 3 in Chapter 1, the first question asks students what numbers \(n\) can be written in the form \(n=x^2+y^2-xy\)? In the Neat stuff section, the question comes back in the form, for each prime \(p=\) 101, 127, 419, 421, and 10009, find integers \(x\) and \(y\) that satisfy \(p=x^2+y^2-xy\) or show that it’s impossible. This is a great number theory question that not only is a fun activity for the curious student, but can motivate the student to study more mathematics during college years.

The last two chapters are a Mathematical Overview, which serves as a guide for teachers to gain some ideas of how to teach applications and Solutions. The former outlines some very interesting applications as in the geoboard problems presented in Chapter 1. Counting all the segments of different lengths from one corner of a square geoboard, it appears at first that the number of distinct lengths on an \(n \times n\) geoboard is related to triangular numbers, but the pattern breaks down at \(n = 6\) since \(25\) can be written as a sum of squares in two different ways: \(25 = 5^2+0^2=3^2+4^2\). So there are two different segments of length \(5\) that would get counted twice in the counting scheme. This will lead the student to the main question, which integers can be written as a sum of two squares? Teachers can guide students to make a list of primes that are sums of two squares, which points out the fact they will all seem to be of the form \(4k + 1\). In addition, students can show that a number of the form \(4k + 3\) *cannot* be written as the sum of two squares. Lastly, teachers can make some initial suggestions to try 13, 21, 45, 101, 202, 105, and 260. Teachers can then look at the SOTS Conjecture.

This is a book for both parties, teachers and students. Both groups can learn from reading the problems and solving them to develop a greater appreciation of mathematics and the art of solving problems. It is refreshing to see a book with a wealth of problems and also a discussion how they can be used for multiple learning levels. I see the book being used in many areas of instruction, such as projects, competitions, and for practice. I highly recommend this book if you want more interaction with your students and to develop their logical thinking skills.

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Peter Olszewski is a Mathematics Lecturer at Penn State Behrend, an editor for Larson Texts, Inc. in Erie, PA, and is the 362^{nd} Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He has written several book reviews for the MAA and his reach fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks along with various online homework software’s. He can be reached at [email protected]. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.