Revitalizing Precalculus with Problem-Based Learning

Introduction

With reform sweeping through the mathematics curriculum, problem-based learning (PBL) is uniquely poised to facilitate multiple changes across the curriculum. In particular, the shift from the traditional, technique-intensive approach to calculus to a more conceptual, applications-oriented approach requires a different level of mathematical maturity. The problem-based approach to precalculus discussed in this paper develops this maturity and prepares students for calculus by emphasizing problem-solving and by layering calculus ideas throughout the course instead of following the common approach of simply providing students a repeat of high school algebra and trigonometry.

The motivation behind the calculus reform movement (Douglas, 1986; Schoenfeld, 1995) is the desire for students to develop their conceptual understanding of calculus and to move beyond mere symbol manipulation and memorization of a prescribed set of steps in solving problems. The underlying principle of the reform movement is that each concept is presented numerically (often through tables of data), graphically, verbally, and algebraically. Although computational techniques are addressed, conceptual understanding, rather than manipulation of symbols, is stressed.

Precalculus typically has involved a combination of college algebra and trigonometry, and has provided little in the way of actual calculus preparation beyond the introduction of difference quotients. This has led to a difficult transition to calculus for many students who feel comfortable with exercises involving a prescribed series of steps but who are unable to think conceptually. For example, a typical student response to the question "What is a derivative?" is often "It's when you multiply by the exponent and take one away from the exponent." This response, however, merely describes a procedure rather than answering the question. With the increased emphasis on concepts over calculations that characterizes reform calculus, students have an even greater need for calculus preparation.

Course Goals and Objectives

Precalculus is one of four courses that Arts & Sciences students may take in order to satisfy the General Education mathematics requirement at Samford University. This course is a prerequisite to calculus, which is required for some majors (e.g., accounting, chemistry, management, physics, and sports medicine). Although the majority of the students who take precalculus are not math majors, about half will advance to take calculus. In a typical semester all sections of precalculus, both PBL and non-PBL, have 30-35 students enrolled, and most precalculus students are freshmen.

Students are informed that the course is neither an introduction to nor a review of college algebra and trigonometry, but instead is a course to prepare them for calculus. The course includes linear, exponential, trigonometric, and polynomial and rational functions, and emphasizes the behaviors that are particular to each family of functions. While students are expected to demonstrate proficiency in working with these families of functions, students are also expected to mature mathematically during the course. In particular, students are expected to develop the ability to make informed, deliberate choices about which mathematical concepts could be applied to a particular problem. Students are also expected to learn to determine how and when to use calculators and software, and to understand the power and limitations of these technologies. In my course syllabus I also list several behaviors that I would like the students to work towards including: to actively search for links between different mathematical ideas; to pose new problems and ask "What if...?" questions; to listen to and learn from others; to assess their efforts; and to become more persistent problem solvers.

In order to meet these goals and objectives, students need exposure to problems and activities that will enable them to mature mathematically. Students' writing abilities are expected to improve throughout their coursework, but mathematical competency is often measured by the size of a student's repertoire of memorized algorithms. Students need to be able to make connections between different mathematical ideas. The real-world problems of PBL provide many opportunities for integrating different mathematical concepts to arrive at a solution. Moreover, students need to...