Toward a Model of Learning Data Representations

Toward a Model of Learning Data Representations Ryan Shaun Baker (rsbaker@cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University Pittsburgh, PA 15213 USA Albert T. Corbett (corbett+@cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University Pittsburgh, PA 15213 USA Kenneth R. Koedinger (koedinger@cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University Pittsburgh, PA 15213 USA Abstract The use of graphs to represent and reason about data is of growing importance in pre-high school mathematics curricula. This study examines middle school students’ skills in reasoning about three graphical representations: histograms, scatterplots and stem-and-leaf plots. Students were asked to interpret graphs, select an appropriate graph type to represent a relationship and to generate graphs. Accuracy levels varied substantially across the three tasks and three graph types. The overall pattern of results is largely explained by the varying ease of transfer of student knowledge from a simpler graph type, based on surface similarity. Introduction External graphical representations are of considerable importance in problem solving. Considerable research has taken place over the last two decades on the different mechanisms through which graphical representations assist their users in drawing inferences (Larkin & Simon, 1987; Stenning, Cox, and Oberlander 1989). In this paper we take up the use of representations at a very early point – at the point when a student is just learning to generate and interpret a representation – and ask what some of the major challenges are in learning these skills. There has been growing interest in attempting to teach these skills to students as young as those in the third through eighth grades 1 (NCTM 2000), but there is considerable evidence as well that these skills have not yet been developed by many undergraduates (Tabachneck, Leonardo, and Simon 1994). We take up this subject in the context of developing a cognitive model of how novices generate and interpret some of the simpler representations used in data analysis. This model is designed with production-rule logic, in ACT-R (Anderson 1993). In this process, we hope to follow in the footsteps of some of the successful cognitive models of novices developed in other domains such as algebra problem solving (Koedinger & MacLaren 1997). One area which might considerably influence students’ performance on these tasks is transfer of the knowledge students already have of generating and interpreting other Between the ages of 7 and 13. representations. Since students are taught different sets of representations at different grade levels (NCTM 2000), it is quite plausible that an important model for learning new representations will be the representations encountered earlier. Previous research into when transfer occurs shows that transfer can happen between exercises taking place in different representations, through mechanisms such as analogy, and that transfer can occur between similar processes (Novick 1988, Novick and Holyoak 1991, Singley and Anderson 1989). Hence, we seek to find out if and how these processes extend to the very first stages of learning how to use and generate a representation. We are interested both in positive transfer, and in overgeneralization, where knowledge is transferred inappropriately. Scanlon’s (1993) research in the use of representations for physics problem-solving provides some excellent examples of overgeneralization in the interpretation of different graphical representations. Additionally, other research has shown that misconceptions in physics, arising from overgeneralization of previously learned knowledge, causes long-term difficulties in correctly learning new material. How best to deal with such misconceptions is an active question in the research literature, with some arguing for a curricular strategy which acknowledges the appropriate contexts for certain conceptions and helps students see when they are inappropriate (NRC 1999). In this paper, we present results and analysis of a empirical study we conducted in this domain, investigating novice performance (with an eye towards transfer effects) on interpreting, generating, and selecting representations important to early data analysis. Domain Representations This study focuses on three graphical representations of data: histograms, scatterplots, and stem-and-leaf plots. A histogram depicts a frequency distribution, as displayed in Figure 1. A set of interval categories (as in Figure 1) are represented in the X axis, and the frequency of each category is represented by the height of the corresponding vertical bar. A stem-and-leaf plot, shown in Figure 2, also