Talk About Teaching ------------------- 2 2 2 2 -x - y z = (x + 2y ) e ========================== (A graph of the equation above accompanied this piece.) By Dennis DeTurck What is mathematical literacy? How is it achieved? Why bother with advanced mathematical education? The immediacy of these questions is emphasized by the nature of the debate over the book The Bell Curve, by Murray and Herrnstein. The book brims with data, statistical analyses and interpretation. Its conclusions concerning race and IQ have been criticized widely. Unfortunately, much of the commentary is emotional rather than rational, even though there is plenty of room for analytical criticism of the work. There is a story about the great eighteenth-century Swiss mathematician Leonhard Euler. He was summoned to the court to debate an esteemed but nameless philosopher about the existence of God. The philosopher offered a long, eloquently-worded argument to refute the existence of a deity. Then, Euler stepped up to a blackboard and wrote some complicated mathematical equation (perhaps the title of this essay), stepped back, and intoned, "Therefore, God exists." The philosopher was speechless in the face of the mathematics because he was not mathematically literate enough to recognize its irrelevance. Similarly, when we hear that "Housing starts were up 3% in October," how many of us know what this statistic means? How is it measured? Up 3% from what? From September housing starts? From October 1993 housing starts? Is this good news? How does all this relate to freshman Calculus? It all comes back to the issue of why one learns mathematics. Probably, if a student intends to be a physician, an engineer, a lawyer, a businesswoman or a humanist, she will never need to calculate derivatives and integrals. But she must understand that mathematics provides useful tools and language for describing, measuring and predicting all manner of natural and human phenomena. Moreover, she should know how problems particular to her discipline get translated into the language of mathematics, to what extent realism is lost in the translation, and how accurate, reliable and relevant the results of mathematical analyses are to the original problems. College students arrive having taken mathematics for many years. Most Penn students who need Calculus for their majors have studied some in high school. They say they "know Calculus." But anyone who has taught Calculus has heard the student lament: "I understand the math, it's those word problems!" These students have missed the whole point of studying mathematics in general, and Calculus in particular. Their mathematical education consisted primarily of drill on problems that were stated in mathematical terms ("Solve the equation...") and ended with "the answer in the back of the book." It's as though one learned the conjugations of all the regular and irregular French verbs, but didn't bother to understand the meaning of any of them. Developing facility with symbols and getting correct answers to drill problems is important, but is only a small part of learning Calculus (or any mathematics) for two reasons. One is described above: for all but a few of us, mathematics is pointless unless it is done in a non-mathematical context. We must be using it to solve real problems. Thus, we must be able to translate something from the real world into mathematical language, solve the resulting mathematics problem, and interpret the answer in the language of the real world. We might do this to compare mortgages or investments, to decide to elect a risky surgical procedure to avoid not-quite-certain dire consequences, or to form opinions about public policy (should medical professionals be tested randomly for AIDS?). The other reason that "symbol-pushing" is such a small aspect of mathematics education is that we can now relegate most of it to a machine. This causes controversy at every level. In elementary school, one wonders whether students should use calculators (although, in the words of Peter Drucker, "Life is too short for long division"). For Calculus students, the issue is the use of computers and programs like Maple. Computing has evoked many essays that begin like this one: What is mathematical literacy? etc. This question is only half-answered in much that is said and written around the University. Many students come to Penn having used word-processing and publishing programs, e-mail, perhaps a spreadsheet, and games. They say they "know computers". Similarly, many faculty (see the Provost's essay "The electronic environment" in October's Penn Printout) view the computer primarily as a tool for communication: we use it to "talk" to each other more, and to gain access to more information. This is a limited view, which ignores the fact that computers are also used for computing. What do we do with information once we have it? What kinds of thought take place before we post the message to the newsgroup? What are the tools for organizing, analyzing and interpreting textual, graphical and numerical data? Against this background, the use of Maple in Calculus courses can be discussed. Maple is a programming language that can solve most of the "rote" problems in a typical Calculus book: Without much thought, the user types in the problems with only minor notational adjustments and a few decorative commas and semi-colons, and the computer types back the answers. Maple can draw two and three-dimensional graphs (the one on this page [in the print version of Almanac] is the graph of the title), and can solve problems in many parts of advanced mathematics. Like any programming language, Maple has a fairly rigid syntax, and demands of the user a high degree of precision of thought. So using Maple comes with a cost. What is the return for paying this cost? Many students will attest that the return is not that Maple makes Calculus easier to learn. Calculus is a profound achievement of human thought, and takes a great effort to master. Maple's graphics can help students visualize some difficult concepts, but making Calculus "easier" or helping students finish their homework faster is not the point. Rather, we return again to the theme of this essay: why we learn mathematics. Tedious details of mathematical calculations should be relegated to the computer, so that humans can focus on the intellectual and difficult part of the enterprise: the translation of the problem into mathematical language and the interpretation of the results of the mathematical analysis. In fact, this has been one of the biggest sources of student frustration. Since the machine can do the routine calculations, courses now place more emphasis precisely where the students are the least comfortable: translating word problems into math problems. The return for using the computer comes as students develop a deeper appreciation for the power and usefulness of the mathematics they are studying, and a greater ability to apply mathematics and quantitative thought when it is appropriate. This is evidenced as students and faculty in Biology, Chemistry, Physics, Engineering, and Economics classes begin to use Maple (or one of its cousins) to do mathematical work. Another effect of the use of Maple in Calculus is the experience students gain in dealing with the software. Although most of our students will not be using this software in the future (indeed, it will undoubtedly be obsolete by the time this year's freshmen graduate), many will be faced with computer languages of one form or another (whether for the operating system of a machine, or statistical analysis, or specialized no-tation for music, etc.). Basic programming skills learned in one environment transfer to others, and are becoming more and more valuable. Penn's Mathematics Department continues to look to the future. Mathematicians, together with faculty from the natural sciences, engineering, medicine, and the social sciences are working to develop a collection of classroom modules and interdisciplinary courses that combine the teaching of mathematical techniques with applications drawn from the research of the Penn faculty and from the work of users of mathematics in industry. The Society for Industrial and Applied Mathematics (whose national headquarters is in Philadelphia) is assisting with industry links, and is part of a Penn-led consortium of institutions pursuing this "holistic" approach to mathematics and its applications throughout the curriculum. Our new administration has committed itself to a significant new initiative to position Penn as a national leader in undergraduate education. The goal of this interdisciplinary effort is to make Penn a national leader in innovative undergraduate mathematics education. Dr. DeTurck is professor and undergraduate chair of mathematics. His column is the second in a new Almanac series developed by the Lindback Society and the College of Arts and Sciences.

Almanac

Volume 41, Number 11

November 8, 1994

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