Collaborative Research: Project DIRACC: Developing and Investigating a Rigorous Approach to Conceptual Calculus

Arizona State University, Scottsdale AZ

Investigators

Abstract

Calculus is an essential tool, and provides a conceptual foundation, in each of the STEM disciplines. Acquiring deep conceptual understandings when first learning calculus poses many difficulties for students and creates many challenges for teachers. As a consequence, many concerted efforts in calculus reform appear to have failed to make educationally significant differences in student understanding that their proponents predicted. Project DIRACC: Developing and Investigating a Rigorous Approach to Conceptual Calculus will build on the documented success at Arizona State University's already-redesigned and already-deployed Calculus I to redesign and implement Calculus II, and will research students' learning in both the redesigned and traditional calculus sequences. In addition, Project DIRACC will develop short-answer assessment tools (also known as concept inventories) for Calculus I and Calculus II that other institutions can use to assess, compare and contrast their students' progress in understanding central ideas of the calculus. Students taking this redesigned calculus sequence will be better prepared for, and will be more likely to apply, their comprehension of rate-of-change functions and accumulation functions in other STEM courses and to learn calculus-dependent material in subsequent mathematics courses. The textbook produced by this project will be made available as an open resource for others to use or build upon. Finally, the results on students' calculus learning, and the calculus concept inventories built to investigate students' learning, will inform future research in these areas. Results from Project DIRACC will be disseminated widely at regional, national and international conferences and published in peer reviewed mathematics education journals and conference proceedings. One possible reason for the lack of effectiveness of calculus reform is that the fundamental structure of the underlying curriculum has remained unchanged in the vast majority of contexts where the subject is taught. Specifically, reform projects have not focused on students' development of richly connected meanings for rate-of-change functions and accumulation functions, which are essential to any introductory calculus. The Project DIRACC courses are highly conceptual because they are based upon students' development of coherent meanings for ideas applicable throughout calculus and that also facilitate the learning of mathematical ideas beyond calculus. The first course addresses this challenge by making the fundamental theorem of calculus central to every aspect of students' experience of calculus in the course. At the same time, the Project DIRACC calculus sequence will be explicitly computational because, as in the existing first course, students will use computers to represent processes that define functions as models of dynamic situations, which then become objects of study themselves. The courses will also acknowledge and address known weaknesses in students' preparation for calculus. The calculus concept inventories will be developed using standard instrument-development techniques, given at the beginning and end of their respective courses, and their psychometric properties will be established with approximately 600 students per course.