People of all ages have been found to have misconceptions and lack sound intuition in situations of uncertainty (Kahneman, Slovic, & Tversky, 1982; Bar-Hillel & Falk, 1982; Shaughnessy, 1981; Shaughnessy, 1992). Further, many people hold to misconceptions even after being presented with evidence contradicting their intuitions (Kahneman & Tversky, 1982a; Tversky & Kahneman, 1982) or beliefs (Anderson, Lepper, & Ross, 1980; Slusher & Anderson, 1989). In order to overcome these misconceptions and build sound probabilistic understandings in school-age children, the National Council of Teachers of Mathematics (NCTM) has recommended that students be involved in hands-on activities and experiments, such as simulations, to model situations of uncertainty while determining probabilities and solving problems (NCTM, 1989; 2000; also see Konold, 1991,1994; Shaughnessy, 1981; 1992; Watson, 1995). Further, it has been suggested that using a computer to carry out simulations may help students overcome misconceptions, because students can generate large amounts of data and analyze sample distributions that are closer to actual population distributions (NCTM, 2000; also see Biehler, 1991). However, there is little research that systematically investigates the effects of different pedagogical techniques on students' misconceptions (Shaugnessy, 1992). Zhonghong and Potter (1994) found that computers can be a helpful tool for overcoming misconceptions and some studies have investigated the role of computers in students' understanding of distribution (Cohen & Chechile, 1997; Wilensky, 1997), however, beyond this, little research exists on the effectiveness of computer simulations in overcoming probabilistic misconceptions.
The purpose of this study is to investigate the persuasiveness of computer-based Monte Carlo simulations on students' decisions in situations of uncertainty. This study 1) compares the use of computer simulations as an investigative-pedagogical tool (teacher-facilitated) to more traditional instructional methods (teacher-directed) for teaching probability, 2) considers the impact of simulation on students' psychological attachment to their misconceptions, and 3) compares the impact of computer simulations on students of different achievement levels. While the main purpose of this study is to consider the instructional impact of computer simulations on students' misconceptions and decision making, the psychological barriers associated with these misconceptions are also discussed.
The work of Kahneman and Tversky (e.g., Kahneman, Slovic, & Tversky, 1982) was devoted to identifying misconceptions and notions of probability that people possess and the associated heuristics that they use to make probabilistic decisions. Misconceptions in conditional probability are some of the most interesting and have been found to be prevalent in school-aged children as well as adults (Bar-Hillel & Falk, 1982; Shaughnessy, 1992). In situations involving conditional probability, one must determine the probability of an event when certain information has been given or something has happened. Many people are confused by the information that they are given. They have difficulties determining what is the conditioning event and are further confused by the additional information that is presented in conditional situations (Shaughnessy, 1992).
Monty's Dilemma is a problem frequently associated with the teaching of conditional probability (Shaughnessy, 1992; Shaughnessy & Dick, 1991). Monty's Dilemma is an age-old problem whose interest was revitalized by the Parade Magazine's column "Ask Marilyn" (vos Savant, 1990a, 1990b, 1991a, 1991b, 1991c).
In a certain game show, a contestant is presented with three doors. Behind one of the doors is an expensive prize, behind the others are junk. The contestant is asked to choose a door. The game show host, Monty, then opens one of the other doors to reveal a junky gift behind it. The contestant is then asked if she/he would like to stick with the original door or switch to the remaining door.
Monty's Dilemma has been studied for years and subsequently explained by numerous mathematicians and statisticians (Gardner, 1959; Gillman, 1992; Morgan, Chaganty, Dahiya, & Doviak, 1991a, 1991b, 1991c; Mosteller, 1965; Selvin, 1975a, 1975b). This problem has often been used to trigger misconceptions associated with conditional probability (Shaughnessy & Dick, 1991) and provides a good situation for assessing students' willingness to change their mind when allowed to investigate problems using a computer simulation.
Shaughnessy (1992; also see Shaughnessy and Dick, 1991) found that many students (upper secondary through graduate) believe that as soon as Monty opens a door that the chances of winning increase from 1/3 to 1/2, since now there are only two doors left. Therefore, many students assess that they might as well just stick. However, unless students physically "flip" a coin, thus making it a 50-50 choice, their chances of winning remain 1/3. Although, flipping a coin does increase a contestant's probability of winning to 50-50, the counterintuitive Switch strategy actually offers the contestant the best chance of winning. This is because the only way that they can lose by switching is if they originally chose the correct door and the probability of that happening is 1/3. Therefore, by switching, the probability of winning the prize is 2/3.
Listing all possible scenarios involving switching provides another way of determining the probability of winning when using the Switch strategy (Selvin, 1975a). From Table 1, of the nine equally likely switching scenarios listed, six of them result in a win. Therefore, the probability of winning when switching is 6/9 or 2/3.
In an anecdotal account of past experience, Seymann (1991) noted that many undergraduate students maintain that they would not switch even knowing that switching doubles their chances of winning. Granberg and Brown (1995) found that only 13% of 228 undergraduate students presented with Monty's Dilemma as a word problem chose the Switch strategy. With the dilemma presented as a computer game to a different group of 114 students, Granberg and Brown (1995) found that initially only 10% of these undergraduate students chose the Switch strategy. Granberg and Brown (1995) further allowed these 114 undergraduates to play 50 trials of Monty's Dilemma using the computer game. Most of these students were subsequently not persuaded by the computer-based Monty's Dilemma. By the end of the 50 trials, although 55% of the decisions on the final 10 trials were Switches, only 7% of the students consistently chose the Switch strategy. Granberg and Brown (1995) concluded that students probably could not inductively come to understand Monty's Dilemma even if they were allowed to play 100 or 1000 times. However, students in their study played 50 individual games on the computer. Although some students did attempt to find patterns, there was no explicit opportunity to analyze past success as a whole or to see what happens on average in the long run. In other words, this particular study did not allow students to use or experience the "law of large numbers," the idea that as the number of experiments increases the expected values estimated from the game approaches the true or theoretical values. In essence there was no explicit evidence that switching actually doubles the chances and therefore it is not clear if students actually came to this conclusion.
The law of large numbers (1) loosely states that the relative frequency of occurrence of an event produced from a random experiment closely estimates or converges to the probability of occurrence as the number of trials in the experiment increases. This notion provides the basis for learning about probability through experimentation or simulation (e.g., Steinberg, 1991) that stems from a frequentist view of probability (for discussion of different views see e.g., Borovcnik, Bentz, & Kapadia, 1991). In other words, in this view, probabilities are defined based on relative frequencies.
Notions of the law of large numbers (e.g., the importance of repeated trials or larger sample size) have been shown to exist in children as early as 7 years old (although at a rudimentary level) and most children by the age of 12 are able to recognize the importance of sample size in statistical tasks (Piaget & Inhelder, 1951/1975; Nisbett, Krantz, Jepson, & Kunda, 1983). In general, people tend to use intuitive statistical heuristics in everyday inductive reasoning (Nisbett, et al. 1983). In particular, they tend to appreciate the importance of sample size when considering statistical tasks, especially those involving frequency distributions (2) (e.g., Piaget & Inhelder, 1951/1975; Nisbett et al., 1983; Kunda & Nisbett, 1986; Sedlmeier & Gigerenzer, 2000; for a review see Sedlmeier & Gigerenzer, 1997). These intuitions are believed to develop informally through everyday exposures to the law of large numbers (Fong, Krantz, & Nisbett, 1986). Through statistical training, it has also been shown that peoples' statistical sophistication related to this law can be increased, including an ability to later apply the law across different domains (Fong et al., 1986; Fong & Nisbett, 1991; Kosonen & Winne, 1995). In summary, it would seem that people even without statistical training have an intuitive sense of the law of large numbers that can be applied to statistical tasks. Thus, it would seem that providing opportunities for students to use this intuition when investigating Monty's Dilemma could enhance their understanding of the problem or at least provide information that may help students derive the underlying probabilities associated with the problem.
When considering Monty's Dilemma, students' tendency to maintain their original strategy choice for no apparent reason, in particular the Stick strategy, is very interesting from a psychological perspective because psychological barriers can cause students to maintain their misconceptions in spite of instruction. Students insisting on holding on to their original door may suffer from belief perseverance (Slusher and Anderson, 1989; cf. Granberg & Brown, 1995). In studies of belief perseverance it has been shown that once people have formed a belief, even if the belief has been discredited, they will hold fast to it. In the case of Monty's Dilemma, students may hold to …