DEVELOPING A QUANTITATIVE UNDERSTANDING OF U-SUBSTITUTION IN FIRST-SEMESTER CALCULUS

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DEVELOPING A QUANTITATIVE UNDERSTANDING OF U-SUBSTITUTION IN FIRST-SEMESTER CALCULUS

Abstract

In much of calculus teaching there is an overemphasis on procedures and manipulation of symbols and insufficient emphasis on conceptual understanding of calculus topics. As such students to struggle to understand and use calculus ideas in applied settings. Research shows that learning calculus topics from a quantitative reasoning-perspective results in more powerful and flexible conceptions of calculus topics like integration. However, topics beyond introducing integrals and the Fundamental Theorem of Calculus, like u-substitution, have yet to be explored from a quantity-based perspective. In this study, I conducted a set of two clinical interviews where we discussed quantitative meanings of integrals, derivatives, and differentials and used those meanings to quantitatively develop u-substitution. This study suggests that given the scaffolding of the quantity-based tasks students can develop the u-substitution structure (substitution of the bounds, the function, and the differential) by applying quantitative reasoning. It also suggests that two-quantity quantitative relationships are critical to students’ productive thinking about substitution. Finally, this study offers a theoretical and quantitatively grounded framework for understanding u-substitution.

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