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Understanding Our Quantitative World

Janet Andersen and Todd Swanson
Publisher: 
Mathematical Association of America
Publication Date: 
2005
Number of Pages: 
320
Format: 
Hardcover
Price: 
51.50
ISBN: 
0-88383-738-3
Category: 
Textbook
[Reviewed by
William J. Satzer
, on
08/23/2005
]

Understanding Our Quantitative World is a textbook intended for use in college-level general education mathematics courses to develop "quantitative literacy." The authors intend to address the question: "What mathematical skills and concepts are useful for informed citizens?" They define quantitative literacy indirectly via the contents of their book, and more directly, though vaguely, through their stated goals for students. In the authors' words these are to:

  • Realize that mathematics is a useful tool for interpreting information.
  • See mathematics as a way of viewing the world that goes far beyond memorizing formulas.
  • Become comfortable using and interpreting mathematics so they will voluntarily use it as a tool outside academics.

I think it's useful to ask more specifically what quantitative literacy means if we are going to assess the ability of this textbook to provide it. After all, quantitative literacy per se only has a meaning by analogy with verbal literacy. There is widespread belief, backed by a large literature, that quantitative literacy is of critical importance. Unfortunately, few writers get specific about what knowledge or skills it entails.

One of the clearest definitions I could find comes from an early report from an MAA committee [1] that offers the following:

A quantitatively literate college graduate should be able to:

  1. Interpret mathematical models such as formulas, graphs, tables, and schematics, and draw inferences from them.
  2. Represent mathematical information symbolically, visually, numerically, and verbally. Use arithmetical, algebraic, geometric and statistical methods to solve problems.
  3. Estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results.
  4. Recognize that mathematical and statistical methods have limits.

I would argue that practical quantitative literacy for the "informed citizen" should put highest priority on interpretation and evaluation of mathematical and statistical information provided by others, and so concentrate on areas (1) and (4), followed by (3) and (2) in that order. There is obviously a lot of room for difference of opinion here. Consensus anytime soon on the specifics of quantitative literacy seems pretty unlikely.

Let's provisionally adopt the MAA committee's interpretation as our definition of qualitative literacy and take a look at the book in that light. How does it measure up? In summary, it does OK on (1) and (2), but considerably less well on (3) and (4). First, let's discuss what the authors do well.

The authors choose to emphasize the concept of function throughout, and I'll discuss some of the ramifications of this later. One thing they do well is to show how functions can naturally be represented in many forms, including symbolic, graphical, tabular and verbal. A tabular representation of a function makes the only-one-element-of-the-range for each element of the domain a more natural idea; after all, a table of student names versus heights can hardly have more than one height per student. There are several good examples, exercises and class activities. Later on, the authors discuss applications and interpretations of graphs and include examples with a motion detector. These motion detector graphs (especially if they are demonstrated with a simple motion detector in the classroom) can directly connect a student's kinesthetic sense with the more abstract concept of a graph.

Throughout the book there are many excellent examples and exercises that are far better than I have seen in comparable texts. They involve a huge variety of applications likely to be interesting to students, and they are presented in meaningful contexts, not as isolated problems. It's also good to see exercises showing bad graphs from newspapers and magazines. I have found examples like this work very well to awaken awareness and a quantitative critical sense in students. The book could use even more examples like these. (USA Today is a wonderful source.)

Another strength is the presentation of contour graphs in the chapter on multivariable functions. Contour graphs give another opportunity to connect with a student's kinesthetic sense. It would probably help to emphasize this by using the idea of gradually filling the space containing a graph with water and identifying contours with water levels.

Unfortunately, there are several things in the book that don't work very well. Some of them relate specifically to the handling of topics I would agree are important for quantitative literacy. Others arise from the authors' apparently different view of quantitative literacy, one that I think overemphasizes some topics and slights others.

I was struck by the first words of the first chapter: "A function is a mathematical object..." Many students who would take a course using a book like this are people I think of as "math-damaged". However that damage occurred, it makes its subjects very uncomfortable with mathematical language. Those first words from Chapter One would certainly crank up their anxiety. I don't want to overemphasize one phrase, but it definitely got my attention. If the authors felt that they really needed to emphasize that a function is a mathematical object, why not first discuss how mathematical use of the word differs from its use in ordinary discourse? Throughout most of the book the authors use an informal and conversational style that should appeal to students. That first sentence, though, is an odd false step.

Another concern is that prerequisites are unclear. In places, it appears that some facility with algebra is expected; in others, algebraic steps are very carefully articulated. Expectations are uneven. An exercise in the first chapter asks the student to write linear equations for cell phone and phone card usage versus cost. But linear equations are not discussed until Chapter 7. More generally, the order of topics treated is rather odd. After the beginning chapters on graphical and tabular representation of data and one chapter on descriptive statistics, the next chapter describes multivariable functions and contour diagrams! Succeeding chapters then take up — in order — linear, exponential, logarithmic, periodic and power functions, with a chapter on regression and correlation tucked in after linear functions.

My greatest problem with this textbook, however, is its selection and emphasis of topics. While it is important that students understand the differences between linear, polynomial and exponential growth, I simply don't understand the reason for an overwhelming emphasis on functions in a quantitative literacy course. My main concern is what gets left out or slighted, and that includes much of topics (3) and (4) above. Probability and statistics are taken up in the last two chapters of the text, and the treatment is much too brief. A basic understanding of probability and an ability to interpret statistical results are especially important for "informed citizens." Consider the number of newspaper articles over the past year discussing, for example, the probability of developing a disease, of a false positive result in a diagnostic test, or of a catastrophic failure of the space shuttle. Think of our daily bombardment with questionable statistical information and what it takes to evaluate it critically.

Calculator use is another concern. Anyone who has taught a course at this level knows that the use of calculators is problematic. There are many benefits, but the cost is generally a lot of classroom time devoted to the mechanics of calculator use. This text makes fairly extensive use of a graphing calculator. The guidance — especially regarding selection of scale and viewing window — is simply inadequate. The authors seem to suggest that the students need to understand the behavior of the functions they're graphing before they graph them. At the level of this course, that's a bit much. An instructor using this text would have to spend much more time demonstrating practical use of the calculator with a variety of functions.

The chapters on exponential and power functions have several nice examples of such functions with data from realistic examples. They also provide calculator procedures for exponential regression to fit the data. Although it is not a big issue, I am uneasy with this. Regression, when it is presented at this level, is treated as a black box with data going in and a fitting function coming out. Students need to know that there are pitfalls, and perhaps how they might be recognized.

The text has a real gap in the area of quantitative literacy described in (4) above. There is a lot of doing in the book, but not enough does this make sense. There is a worrisome problem in Chapter 3 that deals with the price of DVD players. A graph shows the prices of DVD player as a function of time from 1998 to 2002 showing piecewise linear decreases in price. The problem asks, among other things, what price might be expected in 2005? What is the student to make of this? I could find no discussion in the book about the hazards of extrapolation. The student could reasonably, but unhelpfully, answer that the price of DVD players is likely to be even lower in 2005. A simple linear extrapolation from the graph would suggest that the DVD player would have a negative price in 2005. What did the authors have in mind? Perhaps they mean to use this exercise as a vehicle to discuss extrapolation of data, but surely this is important enough to deserve some serious attention in the text. The exercise by itself is a minor issue, but it does inadvertently identify a big gap.

I would have a lot of misgivings about using this textbook for a course in developing quantitative literacy. The heavy emphasis of the text on functions and the omission or limited coverage of the areas described in areas (3) and (4) above — aspects that I think contribute a level of mathematical "street smarts" — are serious concerns.


Reference:

[1] Quantitative Reasoning for College Graduates: A Complement to the Standards, Mathematical Association of America, 1998, available at http://www.maa.org/past/ql/ql_toc.html#summary.


Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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