Srri #111 Misconceptions in Graphing

This paper attempts to organize some recent findings in the literature concerning the errors students make on tasks involving graphs and proposes some initial cognitive explanations for patterns in these errurs. Some basic characteristics of a competence model are proposed for knowledge structures used in comprehending and generating graphs. Two types of common misconceptions, treating the graph as a picture and slope-height confusions, are discussed in terms of this model. Competence Model The ability to interpret graphs is important for mathematical literacy and for understanding the concepts of function and variable, as well as for developing basic concepts in calculus. I will first propose a partial model of four types of knowledge structures needed to comprehend graphs, shown in Figs. 1a and lb. This should not be viewed as a Platonic "given", but as a plausible competence model which provides an initial theoretical framework for interpreting students' errors. I will consider some of the conceptions needed to solve the problem of drawing the qualitative shape of the graph of speed vs. time for a bicycle rider coming down from the top of a hill. In particular, I want to incorporate (a) the connection to a real world context and (b) the idea of variation. Static model. Conception (1) in Fig. 1a is a naive practical representation incorporating everyday knowledge about the problem situation based on one's concrete experience with watching and riding bicycles, including the sensation of speeding up as one rides down a hill. Here I will simply refer to each knowledge structure as a conception, but they can also be thought of as occurring at different levels of representation. Conception (2) represents the idea that at a particular time, the bicycle is at a particular speed. Thus, the subject must have adequately developed concepts for speed and time and must be able to isolate these variables in the problem situation. In conception (3), I show the subject forming a spatial distance