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TEACHING RESOURCES - TEACHING MATHEMATICS ARTICLE General Education Mathematics: New Approaches for a New Millennium Jeffrey O. Bennett, University of Colorado at Boulder
Introduction When it comes to mathematics, we know what we must teach students majoring in science, engineering, or mathematics ("SEM" for short). These students need particular calculus-based skills that they will use regularly in their careers. But what about the rest of the collegiate population, the "non-SEM" students that easily comprise half of all students we teach in mathematics courses in this country? These students outnumber their SEM peers among all college students, and they represent a large majority among students at two-year colleges (National Science Board, 1998). Nevertheless, many institutions have given little serious thought to the development of appropriate mathematics curricula for non-SEM students. One result is that approaches to non-SEM mathematics courses are all over the board. Another, of which we are constantly reminded in conversations with mathematics teachers across the United States, is widespread dissatisfaction with these courses among both faculty and students. We have spent much of the past twelve years working on the problem of mathematics for non-SEM students. In this paper, we will present some observations regarding the nature of the problem, along with our opinions regarding its solution. Initial Goals: Ours and Theirs The first step in developing any kind of curriculum must be to identify the goals to be achieved. As mathematicians, we resonate with the above quotation from Galileo and would like our students to feel the same sense of awe when they think of mathematics. Thus we might state an initial goal for mathematics courses for non-SEM students as follows: Show our students some of the beauty and utility of mathematics. Unfortunately, we almost immediately encounter a problem in trying to achieve this goal: the vast majority of non-SEM students enter mathematics courses with very negative attitudes toward mathematics. In our courses at the University of Colorado, we have found that almost all of our students will openly declare themselves - with no hint of embarrassment - to be either "math-phobics" or "math-loathers." The math-phobics are afraid of mathematics and have done poorly in past (high school) mathematics classes. The math-loathers have done well in past mathematics courses and therefore are not afraid of mathematics; they just believe that it is worthless and irrelevant to their lives. Thus there is a conflict between "our goal" of demonstrating the beauty and utility of mathematics and "their goal" of getting away from mathematics as far and as fast as they can. An Overarching Responsibility This conflict between our initial goal and their initial goal requires that we find a way to bring the two goals into line. The key involves thinking about our obligations as teachers. As mathematicians, we possess skills and knowledge that are of critical importance to our students. It is our responsibility to transmit this information to our students. If we succeed, we will find that, in the process, we have also shown students some of the beauty and utility of mathematics. Moreover, the students will find that what they fear or loathe is not real mathematics, but rather a caricature of mathematics (such as the common student belief that mathematics is nothing more than a set of skills) that has been set up for them by bad experiences in past courses. Once they see the relevance of mathematics to their own lives, their goals and ours will be one and the same. Refined Goals: Mathematics for College, Careers, and Life We are now ready to transform our initial goal of showing students some of the beauty and utility of mathematics into a set of goals that also meets our overarching responsibility of transmitting to students the mathematics that they need. The only question is, what type of mathematics do students need to know? We believe the answer is three-fold:
Note that these goals - mathematics for college, careers, and life - are goals that we and our students can share. Moreover, we have found that enunciating these goals to students almost immediately begins to break down the negative attitudes that they bring to mathematics. For the first time in most of their lives, students see teachers as being on "their side," working with them to help them learn crucial and relevant skills. It is important to recognize that the latter goal, mathematics for life, is more pervasive and more subtle than is commonly realized. In our courses, we frequently use current events to illustrate how thoroughly mathematics permeates everyday life. Virtually every major issue of our day is at least partly quantitative in nature; consider, for example, the issues surrounding the 2000 census, the federal budget, "saving" social security, finding an appropriate government settlement with tobacco companies, the statistics of crime rates, the science and economics of global warming, and even the recent impeachment of the President. The mathematics in these subjects may be fairly explicit or quite subtle, but in all cases it is necessary for a full understanding of the issues. What Course Meets These Goals? Now that we have a clear set of goals, it is possible to begin designing a course for non-SEM students. A brief description of how we arrived at our own course is instructive. In 1987, one of us (Bennett) was a member of an interdisciplinary faculty committee charged with developing a new mathematics requirement for non-SEM students at the University of Colorado at Boulder. The committee consisted of a dozen members; about half of whom were drawn from the Mathematics Department, while the rest represented disciplines such as astronomy, biology, psychology, geography, and economics. Early in the discussions, several committee members suggested that all students should be expected to complete a course in calculus. While this may be a noble objective, it was quickly discarded as impractical: most non-SEM students are unprepared for calculus upon entry to college and therefore could not meet this objective within the constraints of a one- or two-semester general education mathematics requirement. With calculus ruled out, the committee next considered the only other existing mathematics courses taught at the University of Colorado at the time: pre-calculus courses such as College Algebra or Trigonometry. But after only brief discussion, the faculty committee also rejected these courses for the new requirement, largely for the following three reasons:
Thus the committee decided that a new course would need to be created, and set about the task of identifying specific content. Content Areas Designing a new course means deciding upon content, so the University of Colorado committee focused most of its discussion on identifying appropriate content. In the end, the committee identified four major content areas. The original statement from the committee was a bit more terse, but we now identify the four areas as follows:
These areas and their associated skills form the core of what is often called quantitative reasoning. Interestingly, we have found that almost every group that identifies appropriate content areas for non-SEM students reaches similar conclusions. For example, the AMATYC standards (AMATYC, 1995) list the following standards (identifiers in parentheses are from the AMATYC report): (I-1) problem solving; (I-2) modeling; (I-3) reasoning; (I-4) connecting with other disciplines; (I-5) communicating; (I-7) developing mathematical power; and (C-1) number sense. Although this list is organized differently from our list above, we maintain that both say essentially the same thing. A similar set of goals was enunciated in the MAA report on quantitative reasoning (MAA, 1995; p. 10). It's Not Remedial Before we continue, it's important to point out that our four quantitative reasoning content areas are certainly not remedial, even though they do not necessarily include much in the way of formal algebra. As a first example, consider the following quotation taken from a front-page article in the New York Times (4/20/97): Teen-age smoking rates are still lower than in the 1970's. But the percentage of 12th graders who smoked daily last year jumped 20 percent since 1991, to 22 percent. The rate among 10th graders jumped 45 percent, to 18.3 percent, and the rate for 8th graders is up 44 percent, to 10.4 percent. Most non-SEM students (and many SEM students as well) have a very difficult time interpreting the use of percentages in this statement. But simply teaching percentages in a remedial sense - that is, divide two numbers and multiply by 100% - will not solve the problem. Instead, students need well-developed critical thinking skills (Content Area 1) and number sense (Content Area 2) to interpret this statement, and these aptitudes are often not emphasized in standard high school mathematics curricula. Thus learning to interpret this statement is not remedial, even though it deals only with percentages. As a second example, consider the federal budget (as it stands in March 1999). The politicians are very proud that, after decades of deficits, the government ran a $69 billion surplus in 1998. But looking deeper, you'll find that some people claim there was no surplus, but rather a $30 billion deficit. Moreover, while you might expect a surplus to reduce the national debt, the debt actually rose in 1998 by $113 billion! From the standpoint of mathematical manipulation, reconciling these budget numbers requires nothing more than addition and subtraction. But that does not make it easy or remedial. In fact, understanding federal budget numbers requires skills from all four quantitative reasoning content areas: logic to follow the convoluted path by which the numbers are derived; number sense to understand the meaning of the numbers; statistical interpretation to understand how economic data are measured; and modeling to understand how the government forecasts future surpluses or deficits. The Key to Success: A Context-Driven Approach There are many possible ways to integrate the four content areas into a course syllabus, but we believe most of them can be categorized either as "content-driven" or "context-driven." A couple of examples should clarify the difference and illustrate why we believe the latter approach is superior. Consider the topic of logic and critical thinking. A "content-driven" approach looks at logic as a mathematical content area that students should study for its own sake. This approach therefore begins by establishing the important mathematical ideas behind logic, such as sets, truth tables, and Venn diagrams, and then shows students how these ideas are useful. This approach is common in textbooks for non-SEM students. However, this content-driven approach immediately sets up the conflict discussed earlier between instructors' initial goals and students' initial goals: the instructor is trying to teach some mathematics, which the students would rather avoid. In contrast, a "context-driven" approach begins by establishing a context that helps students understand why they should care about logic. In our course, for example, we begin by discussing common logical fallacies and critical thinking problems that appear in everyday situations (such as how to choose between the fare alternatives when buying an airplane ticket), thereby showing students the immediate relevance of logic to their lives. Only then do we introduce the essential mathematical ideas. In the end, both approaches to logic cover the same mathematical content. But the context-driven approach is far more successful because it sets up a shared goal between instructor and students - discussing a topic that all can agree is important to college, careers, and life. As a second example, consider payments on loans, such as student loans, credit cards, or mortgages. In a content-driven approach, loan payments are taught as an application of exponential growth; after all, the formula for loan payments can be derived from compound interest considerations. Thus a content-driven approach starts with the mathematics of exponential growth, and eventually shows students that this has relevance to loan payments. Again, because students are shown the mathematics before being shown its relevance, this approach fails to bridge the gap between the initial goals of students and instructors. A context-driven approach recognizes that most students have loans of some type (usually credit cards or student loans, and sometimes mortgages) and the topic of loan payments therefore can engage students in the more general topic of exponential growth. Thus, in our course, we teach students about loan payments before we cover exponential growth more generally. Moreover, whereas the content-driven approach is usually finished once it reaches the loan payment formula (which students can rightly argue is something they will not use, since banks or real estate agents usually do the calculations for them), the formula is only the beginning in the context-driven approach. We continue discussions of loan payments to show students how they can avoid getting into credit card trouble, how they can make decisions between adjustable-rate and fixed-rate mortgages, and how closing costs and fees can affect the cost of a loan. These are mathematical topics that involve logic, problem solving, number sense, and modeling. (Loan payments are only one of many contexts in which exponential growth can be introduced.) In fact, when it comes to actual course construction, the four content areas identified earlier can be covered through an almost endless variety of specific applications. For example, if you want to cover the mathematics of voting, do it in the context of actual elections rather than starting from mathematical theory. If you want to cover geometry, do it in the context of art or architecture that students will appreciate, rather than as a set of abstract ideas. If you want to cover exponential modeling, begin by observing a population with a fixed doubling time. If you want to solve algebraic equations, do it in the context of linear models or inverse percentage problems. Whatever the topic, a context-driven approach will allow you to convey the beauty and utility of mathematics because your students will perceive that they are working with you toward common goals. Practical Teaching Considerations We now address several of the most frequently asked questions about teaching courses of the type described in this paper.
Summary In conclusion, we believe that all non-SEM students - including liberal arts students, business students, and pre-service elementary teachers - should take a mathematics course (one or two semesters) with the content and approach described in this paper. All of these students need to be competent in the four content areas described, and a course using a context-driven approach is the best way to build that competence. Such a course is rich and rigorous, and can serve as the cornerstone in the collegiate mathematical training of our students. By touching on topics of interdisciplinary interest, the course provides lasting benefits for students in their future courses, careers, and lifetimes. References AMATYC (American Mathematical Association for Two-Year Colleges), 1995, Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus Bennett, Jeffrey O. and Briggs, William L., 1999, Using and Understanding Mathematics - A Quantitative Reasoning Approach, Addison Wesley Forman, Susan L., 1997, "Afterword: Through Mathematicians' Eyes," in Why Numbers Count: Quantitative Literacy for Tomorrow's America (ed., Lynn Arthur Steen), The College Board MAA (Mathematical Association of America), 1995, Quantitative Reasoning for College Graduates: A Complement to the Standards National Science Board, Science and Engineering Indicators - 1998. Arlington, VA: National Science Foundation, 1998 (NSB 98-1). Paulos, John A., 1995, A Mathematician Reads the Newspaper, Basic Books |
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