Integration of Precalculus Review into the First Course in Calculus

PRINCIPAL MENTOR: Doris Schattschneider, Mathematics Department

I. Nature of the Innovation

    Students required to complete a course or sequence in calculus and who come to college unprepared to enroll immediately in a calculus course characteristically take a precalculus course first. They study a number of topics they will need in calculus, but often with no sense of the relationship of those topics to their later study in mathematics. By the time the students need to apply their prior learning in the calculus sequence, they have often forgotten what they learned earlier, or do not recognize the applicability of algebra skills to problems couched in the language and notation of calculus. Repeating precalculus, taught in much the same manner as in high school, seems to have little effect: typically less than 40% of those who enroll in a college precalculus course actually complete Calculus I.

    To address this problem, Moravian College developed a year-long two-course sequence, "Calculus I with Review," integrating precalculus and the first semester of calculus. Precalculus topics are introduced at the time they are needed to understand the calculus concepts and solve problems using those skills. This "just-in-time" approach is supported by a published text, A Companion to Calculus, developed by members of the Moravian faculty and Northampton Community College. Adapters used this text in their newly-established integrated sequence along with whatever calculus text they used in Calculus I. The calculus content of the integrated course is the same as that covered in the "usual" Calculus I course.

II. Outcomes

    At four of the six adapting institutions the new integrated course is fully institutionalized. Its status is somewhat uncertain at the two University of Puerto Rico campuses, where the mathematics department has somewhat less control over course placement and course approval processes are more cumbersome.

    This innovation apparently responds to widespread frustrations of mathematics faculty members about the precalculus sequence. Though each campus experienced some skepticism about the new course from some faculty members, the superior results for students in the integrated sequence have been hard to argue with. Faculty who teach the course prefer it to the traditional model. And where students have a choice of pursuing an integrated sequence rather than the traditional precalculus-calculus sequence they overwhelmingly choose the former.

    Of all the innovations disseminated through DPR (Disseminating Proven Reforms), this one seems the most readily and broadly transferable. It deals with courses virtually all institutions offer on a large scale and addresses a widely-felt need. It is accompanied by strong, straightforward assessment questions and methods that seem always to show better student outcomes for the integrated course than the separate courses. It requires faculty to change the logic and organization of their course, but it does not require them to change their teaching styles, or the manner in which technology is used, if at all.

III. Assessment

    The adapters used the same assessment strategy that the mentors had used in assessing the outcomes of the new program when they first instituted it. They looked at:

• student persistence rates from the first semester of the sequence to the second and rate of completion for the second course.

• performance of students in the integrated course compared to students in the traditional sequence on a set of problems included in the final examination of both courses.

• instructor attitudes, including comparative assessment of the integrated and traditional sequences.

• student attitudes.

    In the traditional model, a significant percentage of the students who complete the Precalculus course choose not to continue to Calculus I. That, coupled with a high withdrawal and failure rate in Calculus I, translates into a low completion rate of the two-course sequence of Precalculus followed by Calculus I. Here is completion data from the project institutions, prior to the adoption of an integrated course (and in two instances, from a control group during the year the integrated course was introduced on a pilot basis).

Integrated Calculus with Precalculus (CWP I & II)

    Both during the project period and afterward, the Moravian College mentors (a team of four) assisted all institutions who requested information and advice; that process will continue. Several other institutions not part of the dissemination project have developed their own integrated calculus with precalculus courses, including the University of Wisconsin-Milwaukee and Ursinus College.

    During the time of the project the mentors and the adapters made eight conference presentations among them. As a result of the work at Hudson Valley Community College, the Moravian mentors were invited to work with faculty at the State University of New York-Albany, which is the main transfer institution for HVCC students. Each subsequent year additional presentations have been made at professional conferences and on other campuses. Thus a pattern of the adapters becoming disseminators is developing.

    A brochure that explains the integrated course--its purpose and success, was developed and widely distributed by mail and at conference presentations. A web site was developed by Ann Marie Murray at HVCC, and is an important first source of information for others who inquire about the course.