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	Preparing Students for Postsecondary Education
	Harvey B. Keynes
	University of Minnesota

The goals of High School Mathematics at Work are broad and ambitious as well as somewhat novel. This collection of essays discusses issues and potential themes for mathematical curricula that might be appropriate for both those students heading to the world of work and those headed into postsecondary education in the mathematical sciences. These issues and themes are illustrated by tasks that are intended to "exemplify central mathematical ideas" and "convey the rich explanatory power of mathematics."

     One can hardly argue with any of these goals. But we also expect that the use of mathematics in the world of work by students who have completed their mathematics education in high school will probably be different (though not necessarily easier) than for students continuing in postsecondary education. In the first instance, technical workers might be expected to do concrete multi-step computations using numerical methods, probably with technological support, and to understand and use some algebraic and geometric methods, and symbolic arguments that are job-specific. They generally will not be expected to abstract and symbolically model mathematics embedded in work situations, to reason and communicate symbolically, or to use abstract mathematical reasoning or advanced mathematical tools in applications to other disciplines.

     On the other hand, students moving on to postsecondary education, especially in careers that use mathematics in a professional capacity, will be expected to have these more conceptual skills as well as the some of the same concrete skills of students who enter the work force. Certainly in college courses, many of these more conceptual and abstract skills will be prerequisites. So a major issue for postsecondary preparation is whether tasks such as those in this volume can be used to effectively prepare students to engage in symbolic and abstract mathematical reasoning in algebra, geometry, and analysis as well as to explore concrete and numerical methods.

     When selecting or designing tasks for inclusion in a curriculum, one must ask not only whether the tasks are based on rich and deep mathematics but also whether they can be used effectively in the typical classroom to exemplify central mathematical ideas and to contribute to an integrated whole. Can the rich and deep mathematical ideas embedded in tasks be exposed and effectively explored conceptually, visually, and analytically, as well as numerically and technologically, so that they contribute in meaningful ways to students' preparation for college calculus, combinatorics, and linear algebra? These questions depend on (a) the classroom teacher's interest and capability, (b) the mathematics curriculum, (c) classroom dynamics, (d) school and family expectations, and (e) the inherent mathematical ideas embedded in the tasks themselves. Any task must be viewed in this larger perspective to see if it can really be useful in helping students learn mathematics at both the concrete/computational and symbolic/conceptual levels.

     It does not take very long to realize the difficulty of finding tasks that can effectively illustrate the major objectives of this document. Such tasks must, at a minimum,

• be presented in a practical context in language that is easily understood but precise;

• be amenable to analysis on several different levels: numerically, geometrically, symbolically, and conceptually;

• be based at least partially on mathematics that is of central importance in the high school curriculum; and

• allow for more extended mathematical interpretations.

     The first point, which concerns linguistic style and clarity of mathematical goals, needs some amplification. Poorly worded and mathematically vague tasks actually discourage students from seeking to develop and analyze the mathematical models behind these questions and encourage them simply to resort to ad hoc or strictly computational solutions. If high school students were really able to interpret mathematically these verbal descriptions, many of the widespread student difficulties with "story" problems would suddenly vanish. One needs to remember that the abstraction of the mathematical phenomena described by seemingly straightforward language is one of the most difficult tasks of applied mathematics, even for professional mathematicians.

     In examining a task, our primary concern is to determine what mathematics students can learn from it. Many of the tasks in this document have the capacity to be mathematically analyzed both concretely and conceptually at levels that support both work force and post-secondary goals. Lottery Winnings (p. 111) is an excellent illustration: one can feel reasonably confident that many teachers will encourage students both to explore numerical solutions and also to conceptualize the important mathematical ideas embedded in this task.

     Given a collection of tasks, one important measure is the breadth of mathematics present in the tasks. Clearly, any small collection of tasks will necessarily need to make choices and de-emphasize certain aspects. As a whole, the tasks in this volume use classical geometric patterns and some level of pictorial representations. Algebraic reasoning at a classical level is also addressed. On the other hand, newer uses of geometric and visual reasoning--information embedded in pictures or graphs--are downplayed. Moreover, the breadth required for vocational training or direct entry into the work world is certainly different for students who will become professional users of mathematics. These tasks can provide a piece of the picture but not the entire spectrum of mathematical expectation for all postsecondary students.

     Many of the tasks in this document meet conditions listed above. Here are three more tasks:

• You are installing track lighting in an old warehouse that is being remodeled into a restaurant. The lights can adequately illuminate up to 15 feet from the bulbs and, at that distance, illuminate a circle with a 6 foot diameter. Figure out where to place the tracks and the bulbs for maximum illumination of the customer area. This task uses geometry, trigonometry, solid geometry (looking at the cone of illumination from a constrained light bulb), and proportional reasoning. It could be modeled with computer software or solved analytically. All of the mathematics involved is within the scope of the high school curriculum.

• Your employer at your first job has given you a choice of where to invest your retirement funds: in a mutual fund that is expected to grow at 10% per year or at the local bank, which charges a 1% yearly service rate for a similar fund, also rated at 10%. You would like to deal with your local bank but don't want to lose too much money. Suppose you expect to put $1,000 each year into the fund. How much will you lose over 10 or 20 or 30 years if you invest at your local bank? This task, which can be modeled in many different ways, illustrates the famous rule that a 1% difference in interest grows very rapidly in compounding over time. It can also lead to some interesting graphs and comparisons of growth rates of functions. Finally, it can be explained in language appropriate for a high school classroom.

• Analyze a contour map with peaks and flat areas. This will provide an opportunity to study curves and shapes in two-dimensions, explore rates of change (closely packed contours), preview functions and graphs on the plane, and examine the geometry of three-space. Practical aspects from cartography and local area maps can provide an everyday context.

     The overall goal of High School Mathematics at Work--to call attention to rich and compelling manifestations of high school mathematics all around us--is enticing and potentially very important. And, in addressing a broad and diverse set of students, it is reasonable to downplay the role of abstraction. Yet many mathematicians and mathematics educators would argue that mathematics without abstraction is not mathematics. While this dictum could be argued as applying to all students, it is probably less controversial to apply it to postsecondary students who will be professional users of mathematics.

     The process of developing tasks with "real life" contexts that are relevant and mathematically significant both for students directly entering the technical work force and for students going on to mathematics-based careers is both difficult and daunting. We must continue to discuss issues and directions, such as scope, breadth, language, and complexity of the mathematics. The tasks in this volume do, however, provide an excellent core framework and standard of quality in which to continue the discussion.

Harvey B. Keynes received his B.A. in 1962 from the University of Pennsylvania and the Ph.D. in 1966 from Wesleyan University. His research interests are in dynamical systems. He has directed the following projects: University of Minnesota Talented Youth Mathematics Program (UMTYMP, state and private funding); the National Science Foundation's (NSF) Teacher Renewal Project; the NSF-supported Minnesota Mathematics Mobilization; the Ford Foundation Urban Mathematics Collaborative; the NSF-supported Mathematics in Education Reform Network; the NSF-supported Young Scholars Project; the Bush Foundation to increase female participation in UMTYMP; and the NSF-funded Early Alert Initiative. He is the recipient of the 1992 Award for Distinguished Public Service of the American Mathematical Society. He is published extensively in mathematics education journals.