Flexibility of differentiation procedures in calculus

Abstract

Calculus is perhaps the most widely taught and researched upper-secondary/post-secondary mathematics subject the world over. The research literature is amassing greater clarity around students’ understandings of calculus, yet calculus instruction tends to be at odds with this literature, maintaining a focus on procedural aspects of the subject. This current work explores one central procedural topic, differentiation, through the broader lens of procedural flexibility. Both students and experts calculated the derivatives to a collection of functions in multiple ways, selected their preferred method and commented on their preference. Many students demonstrated a knowledge of multiple procedures and an ability to select appropriate solutions. Experts produced a greater number of distinct solution methods, including some not found among the student responses. The primary impact of this paper is a more nuanced understanding of ‘flexibility’ as a construct, buttressed by contrasting student and expert data, especially as it concerns procedures in higher-level mathematics. These results are expected to contribute to a bridge between the research on and practice of calculus instruction.