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Review Essays of Academic, Professional & Technical Books in the Humanities & Sciences


Functioning in the Real World: A Precalculus Experience, Second Edition by Sheldon P. Gordon, Florence S. Gordon, Alan C. Tucker, and Martha J. Siegel (Pearson Addison Wesley) is a text designed to prepare students for calculus with a focus on ideas and reasoning, manipulation and decision-making.

Suppose a doctor wants to study a patient's heartbeat using an EKG or a patient's brainwaves using an EEG. There are no known formulas to ex­press these quantities algebraically, but the doctor certainly can get critical infor­mation about a patient by interpreting the graphs produced by these devices. Suppose an engineer develops a new tread design for automobile tires and wants to test its braking effectiveness for a car going 20, 30, 40, 50, and 60 miles per hour. There is no exact formula for either the braking distance or the time until the car comes to rest – too many unpredictable factors are involved. All the engineer has is a set of measurements from the experimental runs, and he or she must make decisions based on an understanding of what information the data provides.

Both cases illustrate themes that run through Functioning in the Real World. The authors focus on the appli­cations of mathematics to situations all around us and on the function concept that allows us to study these phenomena. In Functioning in the Real World students learn to use a combination of algebraic, graphical, and numerical methods, depending on which is the most helpful tool in any given context.

The mathematics curriculum is in the process of change to establish a better bal­ance among geometric, numerical, symbolic, and verbal approaches. There is a much greater emphasis on understanding fundamental mathematical concepts, on realistic applications, on the use of technology, on student projects, and on more active learning environments. These approaches tend to make greater intellectual demands on the students compared to traditional courses that place heavy empha­sis on rote memorization and manipulation of formulas.

Led by authors, Sheldon P. Gordon, Farmingdale State University of New York; Florence S. Gordon, New York Institute of Technology; Alan C. Tucker, SUNY at Stony Brook; and Martha J. Siegel, Towson State University, and with support from the National Science Foundation, the Math Modeling/ PreCalculus Reform Project developed a new precalculus or college algebra/ trigonometry experience with the following goals:

  • Extend the common themes in most of the calculus reform projects.
  • Reflect the common themes for new curricula and pedagogy as called for in the MAA/CUPM recommendations, the AMATYC Crossroads standards.recommendations, and the NCTM Standards recommendations.
  • Focus more on mathematical concepts and mathematical thinking by achieving a balance among geometric, numerical, symbolic, and verbal ap­proaches rather than focusing almost exclusively on developing algebraic skills.
  • Provide students with an appreciation of the importance of mathematics in a scientifically oriented society by emphasizing mathematical applications and models.
  • Introduce some modern mathematical ideas and applications that usually are not encountered in traditional courses at this level.
  • Provide students with the skills and knowledge they will need for subse­quent mathematics courses.
  • Reflect the major changes in the mathematical needs of other disciplines to  provide students with the skills and knowledge that they will need for courses in the disciplines.
  • Make appropriate use of technology without becoming dominated by technology to the exclusion of the mathematics.

To accomplish these goals, the authors have adopted several basic principles advocated by most leading mathematics educators.

  • Students should see the power of mathematics.
  • Students should focus on mathematical ideas, not mathematical calculations.
  • Students should DO mathematics, not just passively watch mathematics.
  • Students should be exposed to a broad view of mathematics.

What's New in the Second Edition

The second edition contains a wealth of new applications, examples and problems, and all real-world data sets have been updated. All concepts and methods are approached using the Rule of Three: graphically, symbolically, and numerically.

The new edition has been reorganized and completely rewritten to provide a slower pace through topics that some students find challenging. It also contains a more prominent role for algebraic topics, where the algebraic steps involved in der­ivations are now highlighted to assist students who may have forgotten some of the algebra they learned in prior courses. Many of the problem sets now include col­lections of problems, called Exercising Your Algebra Skills, to give those students who need it some practice with routine algebra. The book also includes a consider­ably expanded treatment of trigonometry and the use of the trig functions as mod­els of periodic behavior; there are now three chapters devoted to these ideas and methods.

Major changes in the second edition include:

Chapter 2: Families of Functions

The long section on linear function has been split into two shorter sections to slow the pace. Similarly, the treatment of expo­nential functions has been slowed by presenting the material in two sections, one on exponential growth functions and the other on exponential decay functions.

Chapter 3: Fitting Functions to Data

Many new examples, particularly relating to power functions, have been added and considerably more emphasis has been placed on the judgmental issues and mathematical reasoning. A new optional section on multivari­able linear regression has been added.

Chapter 4: Extended Families of Functions

The material has been reorganized to bring together all the discussions related to polynomial functions. New sections have been added that relate the ideas on shifting and stretching functions to oper­ations on tables of data. A new section has been added on the logistic and surge functions as applications of the material on building new functions from old.

Chapter 5: Modeling with Difference Equations

The introduction to difference equation models has been totally rewritten and reorganized. The more challenging material has been moved to Chapter 12.

Chapter 6: Introduction to Trigonometry

A new chapter on right angle tri­gonometry has been written for those students who need an exposure to this mate­rial. The development starts with the tangent ratio. The chapter in­cludes the law of sines and the law of cosines.

Chapter 7: Modeling Periodic Behavior

This chapter presents the use of the trigonometric functions as models for periodic behavior.

Chapter 8: More About the Trigonometric Functions

This chapter presents more advanced ideas on the trigonometric functions, especially trigonometric identities, complex numbers and DeMoivre's theorem.

Chapter 9: Geometric Models

The material on the conic sections has been split into several sections. Additional applications of the hyperbola have been added.

Chapter 10: Matrix Algebra and its Applications

A new section introducing geometric and physical vectors has been added. Additional examples and problems have been added that link matrix methods more directly to previous topics in the book.

Chapter 11: Probability Models

All of the material on probability models, which had been scattered throughout the text, has been collected into this chapter. A new section introducing the normal distribution and its uses has been added and made available for downloading off the web at www.aw.com/ggts as supplemental material.

Chapter 12: More About Difference Equations

The more so­phisticated ideas and methods on difference equations have been combined into this chapter. Moreover, additional sections have been added introducing sever­al models based on systems of difference equations; these include the predator-prey model and a model for competitive species. This supplementary chapter is available on the website.

Functioning in the Real World is approprate for use as an alternative to:

  1. The usual one- or two-semester precalculus course de­signed to prepare students for calculus in the spirit of the MAA's CUPM recommendations and AMATYC's Crossroads Standards.
  2. A one­ or two-semester course in college algebra and trigonometry.
  3. Traditional high-school precalculus courses in the spirit of the NCTM Standards and the recommendations of the Pace­setter curriculum project.
  4. Courses that are used as a terminal or cap­stone mathematics course.
  5. A precalculus-level course for education majors.

Functioning in the Real World finally we have a math book to teach students to use their heads as something more than a hat rack. And since it is the second edition, feedback from users of the first edition has helped clear up the problem of too much material proceeding at too fast a pace. – Anna Washington , M.A.T., M.Ed.

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