The Real Story Behind Story Problems: Effects of Representations on Quantitative Reasoning
The Bystander Effect: A Lens for Understanding Patterns of Participation
We explored how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, we found that students were more successful solving simple algebra story problems than solving mathematically equivalent equations. Contrary to some views of situated cognition, this result is not simply a consequence of situated world knowledge facilitating problem solving performance, but rather a consequence of student difficulties with comprehending the formal symbolic representation of quantitative relations. We draw on analyses of students' strategies and errors as the basis for a cognitive process explanation of when, why, and how differences in problem representation affect problem solving. In general, we conclude that differences in external representations can affect performance and learning when one representation is easier to comprehend than another or when one representation elicits more reliable and meaningful solution strategies than another.
A number of studies have shown that students are often more willing to participate in educational conversations online than in the classroom (Beauvois, 1992a, 1992b, 1994/1995, 1997; Beauvois & Eledge, 1995/1996; Bruce, Peyton, & Batson, 1993; Hudson & Bruckman, 2002; Kelm, 1992; Kern, 1995). However, other studies have shown that online environments have poor student participation (Davis & Huttenlocher, 1995; Guzdial, 1997a; Guzdial et al., 2001; Palonen & Hakkarainen, 2000). Why is this the case? What causes participation to vary from one environment to another? To help explore these phenomena, we borrow a concept from social psychology: the bystander effect (Latan� & Darley, 1970). Briefly, the bystander effect explains why individuals are less likely to help in an emergency if others are present. While the bystander effect specifically applies to helping behavior in emergency situations, we use this construct as a lens through which to view non-emergency situations such as educational environments. The bystander effect has four key components: self-awareness, social cues, blocking mechanisms, and diffuse responsibility. Focusing on these mechanisms can help us more fully characterize participation patterns observed in different educational environments and leverage this knowledge in the design of such systems. We present a case study of two students in both classroom and online French learning environments. We show how the psychological mechanisms of the bystander effect help us understand observed behavioral changes.