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Algebra: Form and Function

William G. McCallum, Eric Connally, and Deborah Hughes Hallett
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Richard J. Wilders
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This text is the latest offering from the Calculus Consortium. The statement on Student Background in the preface suggests that students should have completed “high school algebra” and that upon completing the text they should be well-prepared for precalculus. This makes placing the text into the standard curriculum a little difficult since precalculus is a part of the standard high school curriculum. The text appears to be pitched at about the same level as standard Intermediate Algebra texts (Marvin Bittenger’s Intermediate Algebra 11th Edition being the best known of these).

The goal of this text (an admirable one) is to provide a higher-level approach to the topics covered. As such it seeks to avoid the step-by-step approach in texts such as Bittenger’s. Additionally, the authors seek to provide a better set of examples and problems which place the algebraic techniques in a broader context. Based on the respective tables of contents, the two books are roughly equivalent in coverage (Form and Function covers sequences and series as well as matrices and vectors, which are lacking in Bittenger. Bittenger covers the conic sections)

Based on a close reading of sections from each text, I would claim that the main points of difference center around the general approach — “theoretical” in Form and Function versus mechanical in Bittenger) and the choice of problems. I think the problems in Form and Function are very creative and likely to prepare stronger students for further mathematical study. On the other hand, the careful, precise description of each algorithm in Bittenger is likely to be a better fit for students with weaker backgrounds.

One other important difference should be pointed out — there seems to be no reference to technology (calculators or CAS software). That seems a significant omission in view of the fact that most of the main line textbooks in precalculus and calculus make substantial use of technology.

Finally, there are very few historical vignettes provided. I think (and the literature seems to support this claim) that even a few short references to the history of mathematics can enhance student interest and help to undo the feeling that mathematics has always been around just as it is now.

Here are some comments about individual chapters.

Chapter One, The Key Concepts of Algebra, discusses expressions and equations. There are lots of examples asking students to create an expression to represent a particular real-world situation as well as lots of routine problems involving simplifying or evaluating expressions. I’m not sure if we would all agree that these ideas represent the key concepts of algebra.

Example: Bags of chips cost $c each and bottles of soda cost $s each. Find an expression for the cost of 5 bags of chips and 10 bottles of soda.

There is a brief section on reading algebraic expressions, but the notion of order of operations is not explicitly discussed. I think students should be told that the way we evaluate an expression such as 2k + w is based on an arbitrary choice as to which operation we perform first.

In Chapter Four, Functions, Expressions, and Equations, a function is defined as a rule which assigns exactly one output number to each input number. After two examples, the evaluation of functions is considered, including expressions such as f(122) and h(a–2). There is a brief section on graphs of functions, with no mention of graphing calculators or CAS software. The terms domain and range do not appear, nor does the vertical line test for the graph of a supposed function. This material is presented in chapter 8 along with scaling and the concept of inverse functions. The chapter concludes with a nice discussion of the average rate of change of a function.

Chapter 11, Introduction to Logarithms, begins by creating a table of values for in an attempt to solve the equation . It includes entries such as  with no explanation of how this number was obtained. The natural logarithm is defined without any explanation of why it might be considered “natural.” This is consistent with the introduction of e in the previous chapter: “The mathematician Leonhard Euler first used the letter e to stand for the important constant 2.71828… Many functions in science, economics, medicine and other disciplines involve this curious number.” (Form and Function, page 330). No date for Euler is provided, nor is there any hint as to why this “curious number” is used. It seems appropriate to explain e as arising from the limiting process for compound interest so as to provide at least a modicum of justification for the use of such a complicated base for the “natural” logarithm.

In summary I would describe Form and Function as a very novel approach to intermediate algebra. The writing is bright and interesting and the problems and examples really are different from the standard mix. On the other hand, I think the lack of technology (most importantly the graphing calculator) is a crucial shortcoming. Instructors are unlikely to want to take the time to construct technology supplements for this text when most current textbooks already provide them.

Richard Wilders is Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences and Professor of Mathematics at North Central College in Naperville, IL. He teaches calculus as well as courses in the history of mathematics and of science and has been a member of the MAA for over 30 years.

Chapter 1: The Key Concepts of Algebra

Chapter 2: Rules for Expressions and the Reasons for Them

Chapter 3: Rules for Equations and the Reasons for Them

Chapter 4: Functions, Expressions, and Equations

Chapter 5: Linear Functions, Expressions, and Equations

Chapter 6: Rules for Exponents and the Reasons for Them

Chapter 7: Power Functions, Expressions, and Equations

Chapter 8: More on Functions

Chapter 9: Quadratic Functions, Expressions, and Equations

Chapter 10: Exponential Functions, Expressions, and Equations

Chapter 11: Logarithms

Chapter 12: Polynomials

Chapter 13: Rational Functions

Chapter 14: Summation Notation

Chapter 15: Sequences and Series

Chapter 16: Matrices and Vectors

Chapter 17: Probability and Statistics