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Mathematics & Statistics

## Taming the Tiger: Calculus at USF

Calculus has long had the reputation as one of the hardest courses in the college curriculum — not just at USF, but around the world. Big data, tracking the number of students who drop, withdraw, or fail (DWF), has confirmed this reputation. How To Teach Calculus has become a major topic in mathematics education.

USF is making progress using a discovery- and inquiry-based teaching method in which students work in small organized groups that solve conceptual problems under the supervision and with the guidance of “peer leaders” (usually upper division undergraduates). This method was adapted from an approach called Process Oriented Guided Inquiry Learning (POGIL), which originated in chemistry programs, where introductory chemistry courses faced the same problems.

For a course that meets for four fifty-minute sessions a week, POGIL activities will take up one of those sessions. It will not involve the nitty-gritty material covered in the other three sessions, but instead focus on concepts. For example, the session introducing derivatives will start with intuitive questions involving distance and velocity, average velocity (as represented by the slope of a secant line) versus instantaneous velocity (as represented by the slope of a tangent line), and culminate with a “critical thinking question” like: “Explain how to find the instantaneous velocity of a car at a given time (say, at $$t=1$$), if you are given a graph or a formula of distance the car has travelled versus time.”

Peer group sessions are led by junior and senior undergraduates — “peer leaders” — who help the groups of calculus students learn how to solve problems. Here, peer leaders prepare for one of their sessions. In the forefront, left to right, Rebecca Gonzalez, Joseph McElroy, and Serena Rampersad. Coordinator (and mathematics graduate student) Scott Grizzard is behind Ms. Rampersad.

POGIL is one of several pedagogical approaches that were developed as researchers studying knowledge and skills retention found that lecturing is inadequate. For example, one study found that students retained 5% of the material presented to them in lectures and 30% of the material presented to them in classroom presentations. On the other hand, students engaged in “cooperative learning” and applying new knowledge and skills immediately retained 75–90% of the material. Such studies encouraged innovations like Discovery Learning and Small Group Learning, and as such innovations were tried and analyzed, techniques like POGIL were developed.

At USF, Jennifer Lewis of Chemistry brought POGIL to USF, and Catherine Bénéteau of Mathematics & Statistics adapted it for Calculus I. Students were assigned roles in their group: the Leader / Manager keeps the group on task, the Presenter / Spokesperson presents the finished work to the teacher or to the class, the Recorder / Secretary keeps notes and writes things reports and digests when necessary, and the Strategist / Reflector analyzes how the group works on problems and advises more effective problem-solving techniques.

In their article in the Journal of STEM Education, Bénéteau et al reported that passing rates for Life Science Calculus I for sections with peer-leading was 68%, as opposed to 60% for other sections. The passing rate for Engineering Calculus I sections with peer leading was 63%, as opposed to 59% for other sections. Peer leading was subsequently expanded, and this year all mathematics major Calculus I sections have peer leading.

During the past few decades, public universities have gotten a new charge: teach calculus to an unprecedented number of students so that they can master it. The problem is not unique to Calculus and Chemistry: the National Science Foundation reported a first year retention rate of 64% in STEM fields during the 1990s, and at USF, comparing the number of declared majors versus the number of degrees awarded, approximately half of the Biology, Chemistry, Engineering, Physics and Mathematics majors eventually get baccalaureate degrees in their chosen subject. At the same time, financial constraints limit the number of instructors, so classes are getting larger.

The bottom line is that learning calculus is not really about learning a body of knowledge — it is about acquiring a skill set. Being proficient in calculus means being able to solve real problems with one of the most powerful (if difficult) toolkits ever developed. And USF is at the forefront in applying new pedagogical techniques to help our students succeed.