Graphical representations of speed: Obstacles preservice K-8 teachers experience

School Science and Mathematics, Dec 2000 by Billings, Esther MH, Klanderman, David

This study examines the difficulties college students experience when creating and interpreting graphs in which speed is one of the variables. Nineteen students, all preservice elementary or middle school teachers, completed an upper-level course exploring algebraic concepts. Although all of these preservice teachers had previously completed several mathematics courses, including calculus, they demonstrated widespread misconceptions about the variable speed This study identifies four cognitive obstacles held by the students, provides excerpts of their graphical constructions and verbal interpretations, and discusses potential causes for the confusion. In particular, misconceptions arose when students interpreted the behavior and nature of speed within a graphical context, as well as in situations where they were required to construct a graph involving speed as a variable. The study concludes by offering implications for the teaching and learning of speed and its interpretation within a graphical setting.

Introduction and Review of Related Literature Mathematicians and educators often assume that

college students have a firm understanding of how to interpret and utilize graphs after they have taken courses in calculus. When the first author of this article was teaching a 300-level Oum'or-level) mathematics course to math majors and minors, all of whom had taken at least one semester of undergraduate calculus and were seeking K-8 certification, the researchers observed that these prospective teachers encountered a variety of cognitive obstacles as they interpreted various graphs. Speed graphs were especially problematic. Because an understanding of functions and their applications impacts numerous aspects ofthe mathematics curriculum, particularly differential and integral calculus, the researchers decided to explore these obstacles *in more depth.

The National Council of Teachers of Mathematics (NCTM, 1989, 2000) emphasized that the understanding of functions is critical in K- 12 mathematics curriculum. Students encounter and are expected to analyze finctions in a variety of representational modes, including symbolic, tabular, and graphical. Philipp, Martin, and Richgels (1993) argued that, until recently, functions, particularly graphical representations, have received little attention in the mathematics curriculum. Thompson (1994) stressed that understanding functions is foundational at the collegiate level. He stated that a firm understanding of functions is essential for a variety of undergraduate courses, especially the first year of calculus. Other studies (e.g., Moschkovich, Schoenfeld, & Arcavi, 1993) have discussed the ways in which students' understanding of the interconnectedness of the multiple representations of functions enriches their understanding of the function concept.

This study examined the hurdles or "cognitive obstacles" students faced when interpreting speed graphs. Herscovics (1989) first used the term "cognitive obstacle" to describe individual learning experiences that in some way hinder the understanding of some other mathematical concept. Thompson (1994) broadened this use of the term to "instructional obstacle" in which instructional devices are designed to create cognitive obstacles in the mind of the learner. Thus, not all obstacles are inherently bad, but they do tend to block the understanding of mathematical concepts, at least temporarily.

Since the present study focused primarily on fimctions within the graphical representation mode, a few research studies documenting a variety of cognitive obstacles students experience as they interpret graphs will be examined. Graphs may be either quantitative or qualitative in nature, so issues relating to both were included in the study. Qualitative graphs, unlike traditional graphs, do not include quantitative scales. Rather than focus on specific numerical cases of the function, students must more abstractly relate the independent and dependent variables. For example, two articles published in The Mathematics Teacher (Geuther- Graham & Ferrini-Mundy, 1990; Van Dyke, 1994) have provided teachers several classroom activities involving qualitative graphs. In most cases, the independent variable was time, and the dependent variable was either distance or speed. Some of these problems asked the student to interpret a given graph, while others provided a qualitative graph and asked the student to interpret it or even write a story consistent with the given graph.

Kerslake (1981) referredto these graphs as "travel graphs" and highlighted several difficulties exhibited by students when working with situations in which the independent variable was time and the dependent variable was distance. She found that few students could accurately interpret these situations and were often misled by the physical appearance of the graph. In addition, Janvier (1981 a) found that students confused the shape of the graph with the physical phenomenon it was representing.