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Functions and Graphs: A Clever Study Guide

James Tanton
Publication Date: 
Number of Pages: 
AMS/MAA Problem Books 29
[Reviewed by
Tom Schulte
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Volume 29 in the AMS/MAA Problem Books series contains more than the expected content related to problem-solving and enlightening approaches for teaching or study. Teachers of secondary mathematics or first-year college algebra will find ready classroom capsules on functions and their graphs. For a student, these lessons offer revealing insights into the subject in ways different from and more broadly illuminating than most textbooks. Sandwiched between this content and concluding each chapter is a thorough analysis of solving a problem from a past year in the MAA American Mathematics Competitions program. Following that are related AMC problems. This completes the chapter format of engaging subject review, hallmark problem solution, and additional problems with the previously published solution at the back of the book.

In bringing together all the questions that link to one topic, a coherent mathematical landscape, ripe for a guided journey of study, emerges.

That “guided journey” can easily be an Algebra I or II course. I do not feel there is enough coherence to suggest this for a textbook. That is not the stated intent, so it can be forgiven that inverse functions are unnecessarily handled differently in two places thirty pages apart. (The content would be best presented in the same chapter.) Generally, the chapters meet abruptly like railroad cars without the smooth transition of material development generally seen in textbooks designed to cover one or two semesters. Lacking this natural linkage gives the book a grab bag feel while retaining the careful arrangement of a reference work.

What does come across enthusiastically is the intent to deepen comprehension of subjects expected to appear in an AMC over specific problem-solving techniques. This places the book rather midway between a textbook and competition preparation material. Two audiences may be well-served in augmenting its own core text. Personally, I feel there is more value here for the class attendees than the math team.

Chapter 1 is “What Is a Function? A Swift Conceptual Overview”. This chapter is sixteen pages of first-rate exploration of the topic that could aid in student comprehension while seeming a bit off the mark and unnecessarily abstract to mathletes drilling to prepare for problems as the sequences one featured in this chapter. It is not only in the initial paragraphs that this “natural approach” of non-numeric introductions of topics occurs. Mapping persons to mothers is the gentler, more illustrative first function exemplar here. The topic of functions while also starting more basic than may be expected, goes further than typical, for example into “well-defined”.

Basically self-contained, this introduction to sequences, conics, polynomials and more does at a couple of points expect some knowledge of basic trigonometry. There are several references to video and other content on the author’s site Aimed at moving from “grade school arithmetic” to “advanced high school algebra”, the site provides a qualitative sampling of the approaches used in this entertaining gateway from advanced high school algebra into college algebra. The success here is two-fold, one of tone and explanation while also tackling topics not typically seen at this level, yet key to later success, such as the long-term behavior of polynomials and the pre-calculus notion of limits


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Tom Schulte resides outside of New Orleans and generally starts his students out by confronting them with “What is mathematics?” before confronting them with symbols and ciphering.

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