PROCEEDINGS OF THE 22ND ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2019
Oklahoma City, Oklahoma
Discovering the linearity in directional derivatives and linear approximation
Page: 204
Linear functions of more than one variable exhibit the property that changes in the dependent variable are linear combinations of changes in the independent variables. Although multivariable calculus makes frequent use of this linearity condition, it is not known how students reason about linearity within this context. This report addresses this question by analyzing how three students incorporate linearity into their schemas for linear approximation and directional derivative. The students in this report showed a progression in their understanding from not using linearity within their reasoning to incorporating linearity into first their scheme for linear approximation and finally into their scheme for directional derivative. The results indicate that the context of linear approximation was useful for developing concepts of linearity and aiding their development of the concept of directional derivative.