Retention of Concepts and Skills in Traditional and Reformed Applied Calculus

Abstract

A fundamental question is currently being asked throughout the collegiate mathematics education community: "How can we help students understand and remember calculus better?" There seems to be general dissatisfaction with the knowledge and abilities of students who have completed calculus courses. Reformers in the calculus arena are striving to change instruction to help students understand better and remember longer what they have learned. The Calculus Consortium based at Harvard (CCH) recently published new textbooks for applied calculus which embody a major switch in the philosophy of calculus teaching. The CCH texts, in which applications are the central motivation and not coincidental afterthoughts, emphasize concepts more than symbol manipulation and encourage student-driven discovery of fundamental ideas. Is this reformed way of teaching applied calculus more effective than the traditional method? Which method leads to better long-term understanding and ability? The purpose of this study was to shed light on these questions by characterizing and comparing the skills and conceptual understandings of students of traditional and reformed methods several months after they completed their applied calculus course. A sample of 108 students of applied calculus (57 reformed, 51 traditional) who completed their course in April of 1997 were given a written test in November of 1997 to assess their conceptual understandings and computational skills. Sixteen of these students (8 traditional, 8 reformed) were interviewed to ascertain more about their conceptual understandings as well as their motivation, commitment and attitudes with respect to their applied calculus courses. Test results indicate that although there was no significant difference in overall performance between the two groups, students of the reformed method performed better on conceptual problems, while students of the traditional method performed better on computational problems. Interview results indicate that of the two groups, reformed course students were more confident in their ability to explain derivatives. Reformed course students mentioned graphs and applications more, and they also were more inclined to use estimation techniques than traditional course students. The traditional course students had a clearer idea of the connection between the derivative and the integral.