This descriptive qualitative study examines six eighth-grade students' thinking while solving probability problems. This study aimed to gather direct information on students' problem solving processes informed by the heuristics and biases framework. This study used purposive sampling (Patton, 1990) to identify eighth-grade students who were knowledgeable about probability and had reached the formal operational stage of cognitive development. These criterion were necessary to reduce the likelihood of students' merely guessing answers and important so that the researcher could distinguish between reasoning and intuition.
The theoretical framework for this study was informed by Kahneman and Fredrick's (2002) recent revision to the heuristics and biases framework grounded in the research of Amos Tversky and Daniel Kahneman. Kahneman and Fredrick (2002) drew on dual process theory to explain systematic and predictable heuristic ways of thinking. Dual process theory hypothesizes that human thinking is divided into two different modes of processing. One mode, called System 1, is fast and linked to intuition, and the other, called System 2, is slow and linked to reasoning (Evans, 2008; Stanovich and West, 2000). Within dual process theory, System 1 thinking provides a credible system for explaining why people use heuristic thinking (Kahneman and Frederick, 2002). The recent revision to the heuristics and biases framework is focused on three heuristics, representativeness, conjunction fallacy, and availability. These three heuristics are believed to share the same mental process identified by Kahneman and Fredrick (2002), as the attribute substitution process.
The clinical task based interview method was used in this study. This technique allowed the researcher to better observe and interact with the participants while exploring the students' probability thinking. The researcher also used think-aloud protocols to better reveal the organic thinking patterns of the students in real time (Ericsson and Simon, 1980; Fox, Ericsson, and Bets, 2010; Van Someren, Barnard, and Sandberg, 1994). The data from the interviews were analyzed using the constant comparison method (Glaser, 1965). This analysis revealed three categories that were combined with other analyses to create profiles for various thinking patterns observed by the researcher.
The researcher identified patterns of thinking by students that were consistent with System 1 thinking and associated with the attribute substitution process (Kahneman and Fredrick, 2002). There were also situations in which students demonstrated ways of thinking consistent with System 2 thinking. However, unexpected ways of thinking were also identified by the researcher. For example, there were occasions when students substituted their fraction knowledge when solving probability problems and even seemed to equate probability with fractions. This type of thinking was referred to as the content substitution process in this study. This process occurred when students were using System 1 thinking as well as other types of thinking. In addition, the researcher observed students with thinking patterns that contained characteristics of both System 1 and System 2, which is referred to as slow intuition in this study. Slow intuition seemed to affect students' problem solving strategies as they wavered between multiple problem solving strategies that included either of the two substitution processes: attribute substitution and content substitution.
This study contributes to the body of knowledge related to probabilistic thinking. In particular, this study informs our understanding of heuristic thinking used by eighth-grade students when solving probability problems. Further, teaching practices that draw on Fischbein's (1975, 1987) general notion of intuition might be developed and used to improve probability reasoning skills. These teaching practices target students that depend on the attribute substitution process and/or the content substitution process. Each of these heuristic ways of thinking may require different instructional techniques to help students develop more sound ways of thinking about probability. Regardless, teachers need to be informed of the extent that some students rely on their fraction knowledge when solving probability problems.||en_US