This paper addresses how secondary students might reason about amounts of change in covarying quantities. Two empirically based forms of covariational reasoning are distinguished. The first form— reasoning about quantities as varying simultaneously and independently—supports tandem comparison of amounts of change. The second form—coordination of change in one quantity with change in a related quantity—supports coordinated comparison of amounts of change. By expanding the mental actions of Carlson et al.’s (2002) covariation framework, these forms of reasoning provide finer grained distinctions in the “Quantitative Coordination” level of covariational reasoning. Distinctions made between these forms of reasoning might help to explain how students could begin from informal reasoning to transition to more formal reasoning about average and instantaneous rate of change.
Johnson, Heather L., "Two Forms of Reasoning About Amounts of Change in Covarying Quantities" (2012). STEM Faculty Presentations. 43.