Entry Level College Mathematics: Algebra or Modeling

In the past several years there has been increasing discussion of elementary mathematical modeling as an entry level college course. In several institutions, modeling is now offered as an alternative to the more traditional college algebra course, and students can choose to complete a modeling course in fulfillment of a general education requirement. Of course, not everyone agrees with this approach. How can math teachers make an informed choice between college algebra and modeling? This paper argues that no such choice is necessary, for many of the instructional goals of the college algebra course can be addressed in a modeling course. College Algebra or Mathematical Models? The entry-level college mathematics course in many institutions has for many years been college algebra. Typically, this is the lowest level course that is taught for college credit, and with which students can fulfill a general education requirement in mathematics. In the past decade, however, an increasing number of schools have introduced courses in mathematical modeling at this same level. In another article in this issue of The AMATYC Review, Lumley (2001), describes the evolution of these modeling courses. As discussed in that article, there have been ongoing discussions regarding the merits of college algebra versus modeling courses. For non-science and engineering students, modeling courses are supposed to be more interesting to students and cover material that is more applicable in their other courses. On the other hand, some mathematics teachers have been concerned that replacing college algebra with a modeling course might water down the curriculum. The thesis of this paper is that there is no need to choose between the benefits of a modeling course and the mathematical skills of the traditional college algebra course -it is possible to have both. A modeling course can easily be designed with many of the same instructional goals as a traditional college algebra course. But the modeling approach provides a context in which the algebraic methods become more meaningful and applicable. To illustrate these ideas, I will present below a progression of three topics from a modeling course I developed at American University in the mid 1990's. The course is called Elementary Mathematical Models (EMM). For a detailed look at the content and organization, refer to the companion text (Kalman, 1997). The main emphasis of the presentation here will be on mathematical topics that are common to both the modeling course and the traditional college algebra course. It should be emphasized that there are many variations on entry level mathematics curricula. In some institutions college algebra is the most common choice to fulfill a general education requirement. In others, college algebra is not permitted to count for general education credit. Moreover, there is considerable variety in the structure and content of college algebra courses, and mathematical modeling in one form or another appears in an increasing number of college algebra texts. Variety exists, as well, among entry level modeling courses. It is not my purpose here to survey all of these possibilities, nor to review their relative merits. Rather, the goal is to demonstrate in one specific example how core elements of a traditional college algebra curriculum arise in a course with a narrow focus on modeling. In this way, I hope to encourage college algebra teachers to consider a modeling alternative, and to seek out the core algebra topics that are generally to be found in modeling curricula. Before proceeding to the main discussion, a short account of the philosophy and rationale for EMM will be presented. Rationale and Philosophy of EMM The EMM course seeks to serve the same students as traditional college algebra courses. These are generally students who have completed three years of college preparatory mathematics in high school, but who are not headed for calculus. They may need to take a mathematics course for a general education requirement. They may also need familiarity with the main ideas of college algebra for quantitative aspects of general education courses in areas outside of mathematics, such as science, economics, and business. For most of these students this will be the only mathematics course completed in college. The development of EMM was guided by two main goals. The first was to create a course that students would perceive as intrinsically interesting and worthwhile. The EMM course tries to present a coherent story throughout the semester, and to convey something of the utility, power, and method of mathematics. Among other things, this depends on genuine and understandable applications. The second main goal was to emphasize the topics that students are most likely to meet in mathematical applications in other disciplines. I am thinking here of the most elementary applications, formulated in terms of arithmetic and simple algebraic operations: linear, quadratic, polynomial, and rational functions; square roots; exponential and logarithmic functions. These functions are the building blocks for the simple models that appear in first courses in the physical, life, and social sciences. They are also the central focus for the traditional college algebra course. That is why EMM shares so many of the instructional goals of college algebra. Combining the two goals is complicated by the diversity of the student audience. Entry level college math courses serve students with a broad range of backgrounds. At one extreme are students for whom this course is a first exposure to college level mathematics. At the other extreme are students who have studied more advanced topics (even calculus) and are looking to refresh rusty skills, or for an easy way to fulfill a requirement. The challenge is to make the material fresh and interesting for the more advanced students, and at the same time, accessible as a first exposure. EMM attempts to do this by introducing each mathematical topic in the context of an application. The applications are all analyzed using a common methodology, involving difference equations. The difference equation methods are applicable to a number of problems that have obvious significance and relevance. The desired mathematical topics come up in a natural way as difference equations are used to study the applications. The algebraic emphasis in EMM is restricted to what is really required for working with these simple models. Here there is a true departure from the traditional college algebra course. For example, logarithms are used to solve simple exponential equations, but only natural and base 10 logarithms are discussed. Equations involving several logarithmic terms with arbitrary or unknown bases, a standard exercise in college algebra, are not covered in EMM. Similarly, square roots and fractional exponents are included in EMM, but not radical expressions for indices other than 2. As these examples suggest, most of the traditional topics of college algebra appear in EMM, but they are not carried as far. This is a deliberate decision based on the two goals mentioned earlier. By focusing only on the mathematical manipulations that appear in realistic and understandable applications, EMM can convincingly demonstrate the utility of the algebra that does appear. At the same time, there is an increased emphasis on conceptual understanding of how mathematics really does get applied, and fundamental aspects of the modeling framework. Sample Progression of Three Topics As mentioned at the outset, the main point of this paper is to show that a modeling course can include a great deal of standard college algebra. Toward that end, a progression of three topics from EMM will be presented, with a discussion of the algebra that goes along with these topics. The discussion is not intended to be exhaustive -there are many other topics in EMM, and many other related areas of college algebra, beyond what is considered here. This presentation should be considered merely as a sample of college algebra material that arises naturally in a modeling course. The three topics discussed below are Geometric Growth, Mixed Geometric and Arithmetic Growth, and Logistic Growth. To provide an overall context for these topics, there follows a brief discussion of difference equations, and the methods of analysis that are emphasized in EMM, as well as some general remarks relating this discussion to college algebra. Then the narrative will proceed to the three sample topics. Difference Equations Difference equations concern recursive patterns in number sequences. For example, in the famous Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, . , each term is the sum of the two preceding terms. That is a recursive pattern because it can be applied repeatedly, always using terms just computed to find the succeeding terms. Recursive patterns can be represented symbolically using difference equations. Let fn be the n th term of the Fibonacci sequence. Since each term is the sum of the two preceding terms, we have