Calculus students' early concept images of tangent lines

Abstract

This study explored first-semester calculus students' understanding of tangent lines as well as how students used tangent lines within the context of Newton's method. Task-based interviews were conducted with twelve first-semester calculus students who were asked to verbally describe a tangent line, sketch tangent lines for multiple curves, and use tangent lines within the context of Newton's method. We examined students' graphical illustrations and the language they used to describe tangent lines and identified six prominent categories that described students' concept images of tangent lines. Our data show that individual students often possessed multiple concept images of tangent lines, and often these multiple concept images were conflicting. Furthermore, students were usually willing to modify their concept images of tangent lines depending on the task presented to the students.

Keywords:

• calculus
• tangent line
• Newton's method
• derivative

Notes

1. We do not mean to suggest that instructors should spend time discussing the relationship between the tangent function and a tangent line. Instead, we simply mean that instructors should state clearly that a tangent line is not the same as the tangent function. For further explanation of the relationship between tangent lines and tangent functions, we direct readers to the video at https://www.youtube.com/watch?v=MFytqBe7VDE.

Additional information

Funding

Partial support for this work was provided by the National Science Foundation's ADVANCE IT Program under Award HRD-1007978. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.