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Algebra in Context

Amy Shell-Gellasch and J. B. Thoo
Johns Hopkins University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Tom Schulte
, on

This book approaches the teaching of algebra to first year undergraduate students with a unique use of the art’s history and development. Students that have already encountered many of these topics in a traditional format in high school or college may find this engaging framework a boon to understanding. The mathematical development of early civilizations in Babylonia, Greece, China, Rome, Egypt, and Central America figures in highly, along with the roles of famous persons making Western advances, such as Descartes, Fibonacci, Napier, Cardano, and more.

The book is wide in scope and covers the most common fundamental algebra concepts, including number bases, notation, real numbers, complex numbers, factoring of numbers and polynomials, solving equations from first to third order, nth roots, set theory, logarithms, exponential functions, and more. What appear to some students to be merely abstract procedures may become connected to their original historical motivations in an engaging and accessible manner. Thus the “context” of the title is the history of mathematics selectively presented to support and augment the learning of elementary algebra.

There are easily two semesters of material here. I have yet to hear of a curriculum broad and progressive enough to populate two semesters of elementary algebra presented in a historical context. For that practical reason, I feel the most common use of this text will be to augment more traditional courses. It contains a rich lode of classroom capsules and attention-getting historical examples.

Approaching this book for a single semester’s worth of material, there is much to pick from. Starting with Part III, Foundations, (set theory, real numbers and subsets, factorization, etc.) and including Part IV, Solving Equations, (linear through cubic polynomials, proportions, variation, logarithms, etc.) will cover most of the topics of a typical first year course.

The text offers many thought-provoking exercises for the student, without solutions. These range from specific cases, such as applying logarithm properties to rewrite a logarithm expression to essay questions like “Write about Leonard Euler: his life and impact on mathematics.”

I would not set up the typical student to dive into this text from Chapter 1. I applaud the authors for challenging the reader right at the beginning to decide what is meant by “number.” But launching from there immediately into various bases and their conversions in several antique cultures struck me as a supporting example of the point made by Richard Feynman after he served on the State of California’s Curriculum Commission for choosing math textbooks for use in California’s public schools. In his memoir of that experience, entitled “Judging Books by Their Covers,” Feynman observed

I understood what they were trying to do... The purpose was to enhance mathematics for the children who found it dull… They would talk about different bases of numbers — five, six, and so on — to show the possibilities. That would be interesting for a kid who could understand base ten — something to entertain his mind. But what they turned it into, in these books, was that every child had to learn another base! And then the usual horror would come: ‘Translate these numbers, which are written in base seven, to base five.’ Translating from one base to another is an utterly useless thing. If you can do it, maybe it’s entertaining; if you can’t do it, forget it. There’s no point to it.

For the student at the appropriate level, later chapters will prove much more likely to spark interest and insight, while an initial foray into the tedium of base changing is an area of rocky shoals that will wreck the rudders of many students already laden with weighty doubts.

Being that the history of mathematics is core to the structure of this text, the authors take permissible forays into the history of various cultures. The cultural setting for mathematical discovery allows the need for the innovation to be better understood. These sections are well illustrated, typically with maps and reproductions of commemorative postage stamps. Such side treks can be a relaxing sorbet prior to more rigorous dishes for the student to consume.

There are the beginnings of rigor here for the student, without confining the reader to strict constructions of proofs or theorems. This occurs in “Think About It” questions which liberally season the text. For instance,

Do the definitions of prime and composite in Euclid’s Elements agree with definition 17.6? In the Chinese method, the deficit is not recorded as negative, as is done in the method given in Liber Abaci… Compare the Chinese and European methods to determine why this is so.

A reader confronting these questions, or a student challenged with them in assigned coursework is taking steps in preparation for a transition to advanced math beyond what I see typically in texts at this level. To use Bloom’s Taxonomy of Learning, most textbooks at this level ask the reader to understand and evaluate, where this text asks the reader as often to go one level higher and analyze.

Of course, for confidence in such abstract reasoning, I would want to see the student succeed in these questions — at this level great care must be taken that all the pieces are at hand. I would not expect students at this level to move on their own to independent reading in other texts covering the same subject, to know how to verify online resources, etc. I find some topics very well explained, such as counting board multiplication from Kushyar ibn Labban and the Sun Zi method. However, the grating or lattice method for multiplication is one topic that lacks the same level of exposition.

Some chapters, such as the first two chapters on number bases and number theory, are more about the history than the application of those topics. In the classroom, they may need to be added to. Other chapters, such as Chapter 15 on “Rational, Irrational, and Real Numbers,” have no more on the history than may be found in any comparable text. The section on rules for exponents is another that can be used in place as is and meet typical curriculum goals. The section on continued fractions is applicable to the average classroom and is better done with more detail than I have seen done, when the challenge is even taken up, by other algebra textbooks.

While it may seem in the cases lacking significant historical content that the authors are short of their declared goal, I applaud them all the same, as they fill out such chapters with more abstract topics than I find in such texts and they invariably choose the abstract topics I find students receptive to and intrigued by at this level. In Chapter 15, that means an introduction to Cantor and the taxonomy of infinities.

Some chapters hit upon a perfect blend of the historical and the abstract. Such is Chapter 19 on “Linear Equations” which details the development of false and double false position in a way that brings the reader to the foundations of numerical analysis and thus fills a gap about approaches to numerical approximation which I find sadly lacking in texts at this level. This is unfortunate in our computation age. Also very well balanced is the section on the Quadratic Formula; it is a successful blend of history, application, and theory appropriate to the audience and appealing in its presentation.

Chapter 17, “The Higher Arithmetic,” covering the advances of Pythagoras, Euclid, Diophantus, and others on figurate numbers, number parity, primality, and more, is another triumph of using history to introduce a higher level topic to the audience. In this case, topics in number theory are brought up that usually not explored to such a degree at this level.

In Section 20.2, I find another coup by the authors. Here the important point that any extension of a number system should be performed in accordance with Hankel’s principle. Certainly this can lead to a very productive exchange with properly motivated students; it is something I do not believe I have ever seen tackled to this degree for this intended audience. Bringing in the complex plane is another very good thing. I very rarely see this in other texts at this level, but there is much to be gained by impressing upon the student that the Cartesian plane is not a unique concept as well as giving a portal through which topics like absolute value can be explored in another setting, as is done successfully here.

As a final example of a winning blend of history and theory, the chapter on logarithms develops the properties of logarithms less abruptly and awkwardly than I typically encounter. Also, using historical development to introduce the geometric mean is enlightening for the student in a largely painless depiction of a topic often confounding to first year undergraduates.

The text has an ample bibliography of sources that can also serve as a departure point for further study. There are plenty of examples in the text with chapters introducing topics punctuated with “Now You Try It” exercises. Chapters are collected into four Parts, each of which is terminated by a chapter wholly contained of review exercises over the Part’s content.

Tom Schulte teaches algebra at Oakland Community College and loves to entertain his students with relevant historical asides.