Value-thinking and location-thinking: Two ways students visualize points and think about graphs

The purpose of this study is to examine the characteristics of students’ thinking about aspects of graphs in the context of evaluating statements about real-valued functions from Calculus. We conducted clinical interviews in which undergraduate students evaluated mathematical statements using graphs to explain their reasoning. From our data analysis, we found two ways students think about graphs, value-thinking and location-thinking. These two ways of thinking were rooted in students’ attention to different attributes of points on graphs we provided: either the values represented by the points or the locations of the points in space. In this paper, we report our classification of students’ thinking about aspects of graphs in terms of value-thinking and location-thinking. Our findings indicate that students’ thinking about aspects of graphs accounts for key differences in their understandings of mathematical statements. We discuss some implications of our findings for instruction and curriculum development in Calculus and beyond.