• Daniel Lee McGeeEmail author
  • Deborah Moore-Russo


In two dimensions (2D), representations associated with slopes are seen in numerous forms before representations associated with derivatives are presented. These include the slope between two points and the constant slope of a linear function of a single variable. In almost all multivariable calculus textbooks, however, the first discussion of slopes in three dimensions (3D) is seen with the introduction of partial derivatives. The nature of the discussions indicates that authors seem to assume that students are able to naturally extend the concept of a 2D slope to 3D and correspondingly it is not necessary to explicitly present slopes in 3D. This article presents results comparing students that do not explicitly discuss slopes in 3D with students that explicitly discuss slopes in 3D as a precursor to discussing derivatives in 3D. The results indicate that students may, in fact, have significant difficulty extending the concept of a 2D slope to a 3D slope. And that the explicit presentation of slopes in 3D as a precursor to the presentation of derivatives in 3D may significantly improve student comprehension of topics of differentiation in multivariable calculus.

Key words

differentiation multivariable calculus representations semiotic registers slopes 


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Supplementary material

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Copyright information

© Ministry of Science and Technology, Taiwan 2014

Authors and Affiliations

  1. 1.Kentucky Center for MathematicsHighland HeightsUSA
  2. 2.Department of Learning and InstructionsSUNY - BuffaloUSA

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