In-Service Mathematics Teachers' Conceptions of Tangent Lines

Previous investigations on calculus students' misconceptions of tangency have noted that many students tend to include properties that are consistent with the treatment of the subject in algebra or geometry in their definitions of a tangent line (Biza et al., 2008; Vincent et al., 2015). One might suspect that these students had developed these misconceptions in a prior mathematics class, however, in our previous research on the subject (Hogue & Scarcelli, 2021), we found that students could hold these algebraic or geometric conceptions of tangency despite the fact that they could not remember learning about tangent lines in any other course. Whether students develop these misconceptions in algebra, geometry, or entirely within their calculus class, the cause of these misconceptions is still not entirely clear. Some potential causes include an overuse of certain examples of tangency (e.g., a tangent line to a circle or a parabola), improper language use when discussing tangent lines (e.g., the use of the phrases “barely touching, just touching, or hugging), and student exposure to tangent lines in other mathematics or non-mathematics courses. Our current research examines in-service mathematics teachers' conceptions of tangency in an attempt to establish a link between teachers' understandings of the subject and common student misconceptions about tangent lines. We interviewed 16 mathematics teachers that are currently teaching at the secondary level to come to a better understanding of what teachers know about the subject. Our findings show that these teachers held many of the same misconceptions that the students in our previous study held, which sheds some light onto the question of where these misconceptions originate.