Despite its awkward title, An Introduction to Abstract Algebra with Notes to the Future Teacher is exactly the kind of text that math teacher training programs need. Nicodemi, Sutherland, and Towsley provide a solid, rigorous introduction to abstract algebra, but at the same time help future teachers connect the subject to the high school curriculum.
Most sections of this book end with a note “To the Teacher” and each chapter has a section called “In the Classroom,” in which the text addresses aspiring teachers in two main ways. First, if topics come up that are appropriate for the high school classroom, the authors “point these out and point to how they might be implemented.” For example, after a section on primes and unique factorization, Nicodemi, Sutherland, and Towsley encourage future teachers to challenge a class with a lesson about the Twin Prime Conjecture and the Goldbach Conjecture, as they are ideas that intrigue mathematicians and curious students alike.
Second, and perhaps even more important, by studying more advanced topics in abstract algebra, the future teacher gains a “deeper knowledge that will allow you to see the high school curriculum in larger context.” One example of this is when the authors examine the typical high school algebra topic of factoring completely, examining the meaning of this in different rings. Many textbooks are vague in their directions to the students, and usually expect the student to know that they are factoring over the ring of integers. If factoring x4 – 25, Nicodemi, Sutherland, and Towsley show that the answers are distinct if factoring over integers, real numbers, or complex numbers. The text helps prospective teachers to clear up these ideas in their own minds, so that they can explain it more clearly to their students.
While these connections to the familiar territory of high school algebra will obviously serve the future teacher, they will also enhance all learners’ understanding of abstract algebra. In other words, the text is, simply put, a well-written introductory abstract algebra text, regardless of whether the reader wants to be a high school math teacher. As an introductory text, its use of numbers and concrete examples before moving to abstract theorems will definitely benefit the beginning math student. For instance, before proving Euler’s Theorem, the authors work through the steps of the proof with numbers, thus “anticipating the steps in the proof that follows.”
In accordance with its introductory nature, the book spends some developing the basics in number theory (like the division algorithm and modular arithmetic) before moving onto rings, groups, and fields. Still, it is rigorous enough to reach the unsolvability of the general fifth degree polynomial in the final section.
Another aspect of the text which is simple yet so helpful is the chapter summaries and chapter exercises. They highlight the main points of the chapter and allow the student to check how well they understood the chapter. The authors provide thoughtful homework problems: simple calculations, open-ended questions, and classic ones, like an ancient Chinese problem about a band of 17 pirates divvying up coins.
Nicodemi, Sutherland, and Towsley frequently make connections with theorems learned earlier in the text, so that the reader can visit and revisit ideas, gaining the ability to see one idea from multiple vantage points. For example, first the reader learns about the expressing the greatest common multiple of two integers as an equation. Later they connect this to solving Diophantine equations, then to solving modular equations, and finally to units in Z mod m.
The only critique I have of this wonderful book relates to the section on the Fundamental Theorem of Algebra. Early sections hint at the profound implications of this theorem, building a bit of excitement for this section. When readers arrive at the section, they find a proof based on multivariable calculus, followed by a note “To the Teacher” saying “It´s okay to skip the proof of the Fundamental Theorem of Algebra.” I found myself thinking, “Then why did you include it?” Since there are hundreds of proofs of this theorem, could they not find one that they thought future teachers would be able to read?
A professor using this book will have lots of options for assigning projects and group work. In addition to the straightforward math sections, there are two additional types of sections, a history section called “From the Past” and a “Worksheet.” Many texts include a history section, where readers learn that, for example, Galois died in a duel before his profound mathematical achievements were even published — these sections are interesting side notes to entertain the reader. Nicodemi, Sutherland, and Towsley´s history sections are different, and much better. Their “From the Past” sections are an integral part of the math text because they include math, as well as math exercises at the end. The reader learns not just about the mathematicians’ lives, but also how the mathematics developed, from the mathematicians’ point of view. For example, readers learn that quadratic equations first appeared in purely geometric terms. One example they give is from ancient Mesopotamia around 1800 B.C.: “I have the area, 60. I have subtracted the width from the length, 7. Find the length and the width.” The “Worksheet” sections sketch the outline of a theme and expect the reader to fill in the details. These sections often take readers down interesting sideroads, allowing them to explore deeply concepts like RSA public key cryptography and the frieze groups (symmetry groups of the repeating patterns often found on ornate buildings).
Going through An Introduction to Abstract Algebra with Notes to the Future Teacher is the most fun I have ever had reading a math text. I could not help but solve some of the interesting problems, such as one about a unique social security number or proving why the divisibility trick for seven works. Their history sections are brilliant, giving the reader the chance to see math the way the mathematicians saw it. The text gives future teachers a firm grounding in abstract algebra, while building strong connections to the high school curriculum. Often, professors have to search for resources to make a class interesting. Nicodemi, Sutherland, and Towsley have created an innovative text that contains so many interesting sections that a professor using it almost can’t help but have an exciting class.
Kara Shane Colley studied physics at Dartmouth College and math education at Teachers College. She is currently taking a break from teaching math to manage a meditation center in Mexico.