Measuring the derivative using surfaces

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Abstract Students need a robust understanding of the derivative for upper-division mathematics and science courses, including thinking about derivatives as ratios of small changes in multivariable and vector contexts. In Raising Calculus to the Surface activities, multivariable calculus students collaboratively discover properties of derivatives by using tangible tools to solve context-rich problems. In this paper, we present examples of student reasoning about derivatives during the first of a sequence of three Raising Calculus activities. In this sequence, students work collaboratively on dry-erasable surfaces (tangible graphs of functions of two variables) using an inclinometer, a tool that can measure derivatives in any direction on the surfaces, to invent procedures to determine derivatives in any direction. Since students are not given algebraic expressions for the underlying functions, they must coordinate conceptual and geometric notions of derivatives, building on their understandings from introductory differential calculus. We discuss examples of student reasoning that demonstrate how the activity supports student realization of the need to define a path, attend to direction and the orientation of the coordinate axes and recognize covariation between related quantities. This first activity enables students to initially recognize the ratio of small changes approach to derivatives. We briefly describe how students utilize the ratio of small changes approach in subsequent activities as they measure partial derivatives on surfaces (using an inclinometer) and on contour maps.