The Learning and Teaching of Multivariable Calculus: A DNR Perspective
Keywords: Multivariable Calculus; DNR-Based Instruction; Inhibitive Shortcut; Catalytic Shortcut
Abstract: The paper presents analyses of multivariable calculus (MVC) learning-teaching phenomena through the lenses of DNR-based instruction, focusing on several foundational calculus concepts, including cross product, linearization, total derivative, Chain Rule, and implicit differentiation. The analyses delineate the nature of current MVC instruction and theorize the potential consequences of the two types of instruction to student learning. One of the crucial parameters differentiating between the two types of instruction is the approach to "shortcuts": inhibitive shortcuts —those which inhibit knowledge growth—and catalytic shortcuts— those which evolve out of learning trajectories that catalyze knowledge growth.
Among the consequences addressed are: understanding functions in terms of covariational processes; understanding derivative as the best linear approximation; understanding the relation between composition of functions and their Jacobian matrices ; understanding the idea underlying the concept of parametrization ; understanding the rationale underpinning the process of implicit differentiation; thinking in terms of structure, appending external inputs into a coherent mental representation, and making logical inferences. (Click here for article)