Instructional Conventions for Conceptualizing, Graphing and Symbolizing Quantitative Relationships

Abstract

Quantitative and covariational reasoning characterize essential ways of thinking for conceptualizing and representing growth patterns in pairs of quantities in dynamic situations. Despite the broad body of research documenting the central role of quantitative reasoning in students’ ability to construct meaningful function formulas and graphs relating pairs of quantities’ values, little progress has been made to operationalize this theory in precalculus curricula and instruction. This chapter describes conventions for supporting instructors transitioning their students to spontaneously engage in quantitative reasoning toward the goal of conceptualizing the quantitative structure described in an applied problem context. The Pathways conventions for supporting quantitative reasoning reinforce: (i) consistent patterns of referencing and speaking about quantities and how they relate to other quantities in the problem context (speaking with meaning); (ii) physical motion for conceptualizing quantities and considering how the values of pairs of quantities vary together (quantity tracking tool); (iii) use of consistent representational conventions when making a drawing to represent the quantitative structure of a problem context (quantitative drawing); and (iv) consistent expectations and patterns for defining variables, and constructing algebraic expressions and formulas (emergent symbolization). These conventions emerged in the context of a 15-year research and development project focused on supporting instructors in making their precalculus teaching more engaging, meaningful, and coherent. We provide a rationale for introducing each convention and explain how each convention supports students in constructing and representing quantitative relationships symbolically and graphically. We describe our approach to motivating and supporting instructors to adopt these conventions and conclude by illustrating how these instructional conventions might impact an instructor’s mathematical connections and instructional practices.