Connecting Representations and Ways of Thinking about Slope from Algebra to Calculus

While slope is a topic in the algebra curriculum, having a robust understanding of slope is needed for students to truly understand several single and multivariable calculus topics with any depth. We begin with a review of the topic of slope and present what is known from its existing corpus of literature. We then outline the tenets of APOS theory. Building from there, we suggest what a robust, flexible understanding of slope involves, as well as how slope is used, with the APOS-slope framework acting as a theoretical lens. This is followed by the cases of two hypothetical students built from amalgamations of research and experience to emphasize why moving easily between different ways of thinking about and the various uses of slope is vital to successfully transition into calculus. We offer suggestions as to how university instructors might consider slope understanding when teaching calculus, then conclude with suggestions for future research on slope.