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Volume 5, Issue 4; December 2004

To Build a Better Mathematics Course

Bernard L. Madison*

     Over the past quarter century mounting evidence has pointed to the need for reform of U.S. undergraduate mathematics, especially the first two-year courses. For a century or so these courses have been largely the same at all U.S. colleges and universities, now numbering over 3000. With about three million enrollments each semester, the enterprise is large, complex, and firmly institutionalized in local, state, and national policies. Thus the stakes of any reform are high, but there are promising ideas for constructive change.

     About three years ago I began implementing some of these ideas in our mathematics courses for future elementary teachers and that now is bearing fruit. At about the same time I began work on education for quantitative literacy (QL). Also called numeracy or quantitative reasoning, QL is the ability to identify, understand, and use quantitative arguments in everyday life and therefore a cultural field where language and number merge and are no longer one or the other. My focus here is the search for mathematics courses that are effective in education for QL. To better understand such a search and why it is necessary, it is helpful to look back a few centuries

Measuring Reality and Risk

     About 750 years ago the idea of comprehending the world in quantitative ways—using numbers—was born (Crosby 1997). Mechanical clocks, marine charts, and double-entry bookkeeping—along with many other developments—for the first time provided ways to measure various aspects of civilization.

     About 400 years after this beginning of quantification of Western society, in 1654, a French nobleman, Chevalier de Méré, who was fond of both gambling and mathematics, challenged the famed French mathematician Blaise Pascal to solve the two-hundred year old puzzle of how to divide the stakes of an unfinished game of chance between two players when one of them was ahead. Pascal turned to Pierre de Fermat, a lawyer and brilliant mathematician, for help, and the theory of probability ensued. For the first time people were able to make decisions and forecast the future with the help of numbers (Bernstein 1996).

     In the 300 years following the introduction of probability, the management of risk and a multitude of other quantification systems became hallmarks of the new U.S. society. Historian Patricia Cohen wrote in the concluding pages of her 1982 book A Calculating People , “ Numbers have immeasurably altered the character of American society. Our modern reliance on numbers and quantification was born in the scientific and commercial worlds of the seventeenth century, under the twin impacts of republican ideology and economic development” (Cohen 1982). In the two decades since Cohen’s book, fueled by the development of computers, reliance on numbers and quantification has increased well beyond what could have been imagined, and no end is in sight.

     In 1953, the Nobel Prize-winning economist and mathematician Kenneth J. Arrow described the complete market, a situation where every possible outcome of some scenario would be a commodity, for sale at a price (Surowiecki 2001). By 2004, much of Arrow’s idea had been realized in the financial marketplace in what Mary Poovey calls the culture of finance where numbers and mathematics are used to reorganize the relationship between value and temporality (Poovey 2003). For example, risk, once time-dependent, is objectified, divided, and reassembled so that it can be traded. Stock options, derivatives, day trading, mark to marketing accounting, and adjustments to bad debt reserves are now among almost unlimited investment and accounting instruments available to individuals and companies. Understanding these very real almost everyday concepts is miles away for the student who struggles with meaning of odds or rates of change.

Mathematics and Measurement

     Development of mathematics preceded the introduction of quantification – measurement – by more than 2000 years, but it was not until the fourteenth and fifteenth centuries that there was intermingling of mathematics and measurement. However, this intermingling was superficial, and over the subsequent centuries, until the present day, the real mathematics that mathematicians study, appreciate, and extend has remained essentially apart from commercial and other useful mathematics of the real world (Madison 2004).

     Formal U.S. school mathematics from grades 8 or 9 through the first two years of college is dominated by a sequence of geometry, algebra, trigonometry, and calculus (GATC, for short). This sequence has been the essential offering in high school and college for over a century, with only minor changes in content coverage and grade-level offerings. During the middle part of the twentieth century courses for liberal arts students were constructed, mostly from selections from the GATC courses, and in the last twenty years data analysis and probability has been added to school mathematics. One of the courses originally conceived for liberal arts students, finite mathematics, now a standard offering for business students, is a mixture of probability and statistics, matrix algebra, sets, and logic.

     Over the past century, while introductory college mathematics courses have changed little, major changes have occurred around them. First, U.S. society of the 21 st Century is vastly different from that of a century ago. Second, the college population now consists of the majority of typically eligible Americans while a century ago only a select few even finished secondary school. Third, remarkable technological developments have added potential cognitive power along with educational challenges about how to use the extra power. The quantitative demands on Americans for work, personal welfare, and citizenship have increased enormously. No longer is it acceptable to be mathematically or quantitatively illiterate, but there is convincing evidence that many, if not most, college graduates are unequipped for the quantitative demands they will face daily.

Quantitative Literacy

     The QL that we should want for our graduates is what Lawrence Cremin referred to as liberating literacy—the power and habit of mind to search out quantitative information, critique it, reflect upon it, and apply it in their public, personal and professional lives (Cremin 1988). QL and mathematics are related, but they are not the same. Mathematics is an “abstract, deductive discipline, created by the Greeks, refined through the centuries, and employed in every corner of science, technology, and engineering” (Steen 2004). Although QL is not the same as mathematics or statistics, nonetheless, school and college mathematics and statistics, along with all other academic disciplines, bear responsibility for providing better education for QL. Part of that is developing better mathematics courses— ones that assist the learning of mathematics in real-world contexts and ones that help students develop the necessary habits of mind to handle the myriad of quantitative situations they will face. Better mathematics courses are necessary but not sufficient. Education for QL must be distributed across all disciplines in school and college in a coordinated way. Schools and colleges are responsible for both our students’ quantitative education and the creation of the complexities of the society they face. Therefore, it behooves us to deal with our own handiwork.

The Search Begins

     The mathematics courses required for The University of Arkansas Bachelor of Arts (B.A.) degree are College Algebra (MATH 1203) plus one of Finite Mathematics (MATH 2053), Survey of Calculus (MATH 2043), and Calculus I (MATH 2554). None of these courses was designed with

     B.A. students in mind, and all four courses are dominated by mathematical methods and procedures. Very often students do not see the relevance of the content of these courses to their chosen major. Further, students are unlikely to develop the mental conceptual structures that appear necessary for long-term retention and use of the ideas and techniques of the course material. Since the courses focus on components of mathematics— e.g ., algebraic manipulation, matrix operations, derivatives and integrals—the courses are not strong in developing processes such as logical reasoning and problem solving, processes that are often more important than content knowledge in confronting unpredictable real-world situations. In spite of these apparent shortcomings, the situation is not very different in most U.S. colleges and universities. I believe we can do better, and last summer I was finally in position to try something that I believe will be better.

     Over the past four years, I have worked with a national initiative to create some consensus on what constitutes QL in current U.S. society and how it can be achieved. In the process I collected many examples from U.S. newspapers and magazines of articles that require mathematics or statistics to understand and critique. I selected twenty or so of these articles and arranged them into eleven lessons entitled percent, petty thrift and buying stocks, condensed measures and indexes, lower math by Dave Barry, linear and exponential growth, measurement, visual representation of quantitative information, rates of change, weather maps and indexes, the “odds of that,” and risk. Beginning in August, I conducted an experimental course (taught as a section of Finite Mathematics) based on these lessons. We are just finishing the first semester with twenty-six B.A. students, about half Journalism majors. I will use what I have learned this first semester and offer a second iteration of the course to 35-40 B.A. students in the spring of 2005, this time almost all Journalism students.

     The course has several characteristics that distinguish it from many mathematics or statistics courses. These include:

• Mathematics (including statistics, without saying it every time) is confronted, developed, and used as it occurs in the articles. The course is not organized by mathematical topics.
• Mathematical concepts recur repeatedly, often cloaked in context dependent terminology.
• Almost all the problems are ill defined in the sense that assumptions are made that are not specified in the articles.
• Estimation is often the most important lesson of problems.
• Almost all exercises consist of gleaning information from the articles, formulating a mathematics problem, doing the mathematics, and reflecting the results back into the article. (The first, second, and fourth of these are difficult for students; doing the mathematics is easier for the students probably because they have practice at it. How to do the mathematics is also the easiest to teach because it is more structured and we have more practice at teaching that.)
• Class sessions are casual and interactive. Students often work on group exercises. Every class begins with a discussion of quantitatively oriented newspaper or magazine articles that students have brought. My experience in this course indicates to me that education for QL requires that we change the way we teach mathematics as well as other subjects. The changes I see are rather substantial and include the following:

• Mathematics should be encountered in many contexts such as political, economic, entertainment, health, historical, and scientific. Teachers will require broader knowledge of many of the contextual areas.
• Pedagogy is changed from presenting abstract (finished) mathematics and then applying the mathematics to developing or calling up the mathematics after looking at contextual problems first.
• Material is encountered as it is in the real world, unpredictably. Unless students have practice at dealing with quantitative material in this way they are unlikely to develop habits that allow them to understand and use the material. Productive disposition is critical for the students.
• Considerably less mathematics content is covered thoroughly. Knowing arithmetic and being able to use it is better than knowing the techniques of calculus but having no ability to utilize them. Indeed, less can be more.
• The mathematics used and learned is often elementary but the contexts are sophisticated.
• Technology—at least graphing calculators with Computer Algebra System (CAS)—is used to explore, compute, and visualize.
• QL topics must be encountered across the curriculum in a coordinated fashion. If I can coach writing then literature faculty can coach QL.
• An interactive classroom is essential. Students must engage the material and practice retrieval in multiple contexts. This experimental course is only one of many steps we need to take to reform school and college education, especially general education. The complex society that we have created with the accomplishments of higher education has outstripped our facility at educating our students to live in this society. Our discipline-dominated higher education has served us well for a century but seems to need major restructuring to meet the challenges of educating all our students appropriately. The stakes are very high. As Carnevale and Desrochers (2003) point out, “The wall of ignorance between those who are mathematically and scientifically literate and those who are not can threaten democratic cultures.

Peter L. Bernstein, Against the Gods: The Remarkable Story of Risk , Wiley, New York, 1996. Anthony P. Carenvale and Donna M. Desrochers. “The democratization of mathematics”, in Bernard L. Madison and Lynn Arthur Steen, eds.

Quantitative Literacy: Why Numeracy Matters for Schools and Colleges , National Council on Education and the Disciplines, Princeton, NJ, 2003, pp. 21-31 and 203.

Patricia Cline Cohen, A Calculating People , University of Chicago Press, Chicago, 1982.

Lawrence A. Cremin, American Education: The Metropolitan Experience 1876-1980 , Harper & Row, New York, 1988.

Alfred W. Crosby, The Measure of Reality: Quantification in Western Society 1250-1600 , Cambridge University Press, Cambridge, UK, 1997.

Bernard L. Madison, “Two Mathematics: Ever the Twain Shall Meet?,” Peer Review ( Association of American Colleges and Universities, Washington, DC), 6 (4), 9-12 (2004).

Mary Poovey, “Can Numbers Insure Honesty? Unrealistic Expectations and the U.S. Accounting Scandal,” Notices of the American Mathematical Society 50 (1), 27-35 (2003).

Lynn Arthur Steen, “Everything I Needed to Know about Averages . . . I Learned in College,” Peer Review 6 (4), 4-8 (2004).

James Surowiecki, “The Financial Page: What Weather Costs,” The New Yorker , 29, July 23, 2001.

* Bernard L. Madison is Professor of Mathematics at The University of Arkansas