The persuasiveness of computer-based simulations on students' probabilistic misconceptions.

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.

Background

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 beliefs associated with luck, or instinct, asserting that "it is always better to go with your first choice."

It has been suggested (Granberg & Brown, 1995; Seymann, 1991) that students' propensity to maintain their strategy might also be attributed to counterfactual judgements based on the use of the simulation (heuristic) (Kahneman & Tversky, 1982). That is, some students may mentally simulate alternative outcomes to a given situation and be concerned about the negative affect resulting from switching and being wrong (Landman, 1987; Kahneman & Tversky, 1982; Kahneman & Miller, 1986; Gleicher, F., Kost, K.A., Baker, S.M., Strathman, A.J., Richman, S.A., & Sherman, S.J., 1990). In the case of Monty's Dilemma, they may simultaneously simulate the occurrences of sticking and losing and switching and losing. Past research (Landman, 1987; Kahneman & Tversky, 1982; Kahneman & Miller, 1986; Gleicher et al., 1990) suggests that judgements of negative affect are greater when a decision involves an action, such as switching in the case of Monty's Dilemma, as opposed to an inaction, such as sticking. This suggests that in situations of uncertainty like Monty's Dilemma, some people may choose to stick (an inaction) to avoid anticipated negative feelings associated with changing one's mind and being wrong. Granberg & Brown (1995) found that students assigned to a "switch-and-lose" situation were significantly more likely to think that they would feel frustrated and angry than a group of students assigned to a "stick-and-lose" situation.

Little research to date has considered the effects of simulation on students' misconceptions. Even fewer studies have considered the role of computer simulations. Several mathematics educators have suggested that misconceptions in probability might best be overcome by modeling problems through simulation (Konold, 1994; Shaughnessy, 1981, 1992). With the ever increasing availability of computers, graphing calculators, and simulation software (e.g., Resampling Stats, [Resampling Stats, Inc, 1996]; ProbSim, [Konold, 1995]), it becomes important to consider the role of computers and computer simulations in overcoming students' misconceptions. Because the computer can create large amounts of data, students are able to work with sample distributions that tend to more closely represent the actual population distributions (NCTM, 2000). More specifically, with large amounts of data, frequency distributions associated with specified outcomes more closely represent the actual probabilities of occurrence. Computers also create opportunities for "... richer, deeper, and broader analyses of data" through visual and interactive representations of problems (Schwartz, 1999, p. 109).

The uncertainty associated with Monty's Dilemma and the propensity for students to Stick in spite of evidence that switching doors offers a better chance of winning makes it an ideal situation to analyze students reactions to a computer-based Monte Carlo simulation. In addition, prior studies that have investigated Monty's Dilemma have reported responses of older students (upper secondary through graduate). Little attention has been given to students who may have had very little formal training in probabilistic concepts, or even a course in Algebra. Further, it is also important to consider students who may have only recently developed (or are still developing) formal probabilistic reasoning, which according to Piaget and Inhelder (1951/1975) occurs in most children by the age of 12. In other words, reasoning capabilities of students in previous studies may have been better developed.

Considering a younger sample of students, this study investigates the persuasiveness of computer simulations both from an instructional and a psychological perspective. In other words, this study tests whether students allowed to investigate Monty's Dilemma using a computer-based simulation are subsequently more persuaded to choose the Switch strategy than those students taught through teacher-directed instruction. In addition, this study investigates the likelihood of students initially choosing the Stick strategy to change their strategy (either to the Flip or Switch strategy). Finally, this study compares the impact of the computer simulation on students in different curricular tracks.

Methods

Subjects

A total of 107 eighth graders participated in the study (47 girls, 60 boys). The participants were from four classrooms, two high (algebra) track and two low track, within one middle school serving a lower middle to middle class neighborhood in the Midwestern region of the United States.

Treatment Groups

The four classrooms were divided into two treatment groups, the Simulation Treatment (henceforth, ST) and the Classroom Treatment (henceforth, CT), with one high track classroom and one low track classroom in each group. The ST group was made up of 55 students, and the CT group 52 students.

Monty's Dilemma

The problem. The previous description of Monty's Dilemma will be the focus problem for the study. In addition, because previous research has found that some students express that there is no difference in whether you stick or switch (i.e., with two doors the chances are 50-50) the following extension to the problem, allowing for the possibility of actually flipping a coin, was added to Monty's Dilemma (Shaughnessy & Dick, 1991) and presented to the students.

If you were the contestant, which strategy would you choose?

1. Stick with the original door (STICK strategy).

2. Switch to the other door (SWITCH strategy).

3. Flip a coin to decide which of the remaining doors to choose (FLIP strategy).

Game situation. Two identical props were used to act out the Monty's Dilemma game show. Each consisted of three "cigar" boxes, fastened vertically to a board, with the numerals 1,2, and 3 on the lids (doors). In two boxes (i.e., behind two doors) was placed a picture of a goat. Behind the third door was placed the prize (a Snickers[TM] candy bar). The prize was randomly placed in both props before each session. These props were used for both treatment groups.

Computer simulation. An interactive computer simulation of Monty's Dilemma was developed by the second author. In this simulation, students are presented with three doors and asked to click on the door they think contains the prize. Monty (the computer) then opens one of the remaining doors to reveal a GOAT and offers the player the chance to "STICK" with their original door, "SWITCH" to the remaining door, or "FLIP" a coin to decide (see Figure 1). Upon the student's choice of strategy, the computer opens the doors to reveal the prize and thus whether the player won or lost. Tallies of success are kept in the RESULTS section, which can be cleared at any time using the "Start Over (Clear All)" button. Different from Granberg and Brown's (1995) game, players can run the simulation 10, 100, or 1000 times at once for any of the strategies using the "Repeated Trials" component of the simulator.

[FIGURE 1 OMITTED]

Student Probability and Statistics Achievement

Background information on students' probability and statistics achievement was collected using a 25 item multiple choice test (see Appendix). Most of these items were drawn from achievement tests used in the Longitudinal study of American youth (see Miller, Kimmel, Hoffer, & Nelson, 2000). Test items dealt with concepts such as elementary probability and statistics, conditional probability, reading charts and graphs, and percentages. A reliability coefficient, based on the Kuder Richardson-20, was calculated for the statistics test, [[rho].sub.-20], = .88, and along with the face validity of the test items provides good evidence that the test is a reasonably valid and reliable measure of statistics achievement.

Procedures

Students were involved in one of two treatments, the Simulation Treatment (ST) or the Classroom Treatment (CT). The treatments proceeded in three phases which are described below.

Phase 1

Each student (either individually in the ST group, or as a class in the CT group) was presented with a description of Monty's Dilemma and asked to participate in a game situation with a chance to win a candy bar for a prize. They were asked to pick the door, from the three available, that they believed contained the prize. At this point, one of the other doors was opened to reveal a picture of a goat. They were then asked to choose from among the three possible strategies ("Stick", "Flip", or "Switch") the one that they felt would give them the best chance of winning. Once students had made their choice, the strategy was recorded, either by the experimenter, in the case of the ST group, or individually by the students and taken up by the experimenter, in the case of the CT group. The game was then postponed to be continued later.

Phase 2

Simulation treatment (ST). Each ST student individually investigated Monty's Dilemma using an interactive computer based Monte Carlo simulation. The simulation could either be played using individual trials (one at a time) or repeated trials (in which the computer would run the simulation 10, 100, or 1000 at a time). The computer tallied and displayed the number of wins and losses for each of the three strategies.

During a period of familiarization, students were guided by the researcher through the different parts of the simulation until they felt comfortable with the computer simulation. The role of the computer in the repeated trials was discussed, explaining that the computer played the game just as they would, only many times very quickly. At this point students were allowed to play the game independently, that is, students were allowed to investigate the different strategies using the computer. They could play individual games using each of the strategies or they could analyze long run results for each of the strategies using the repeated trials component which has the computer to simulate 10, 100, or 1000 trials at one time. Students were able to alternately analyze the results of their experiments (which were tallied for them) and continue playing (or testing) the different strategies. The researcher served as a facilitator during this investigation answering questions when asked but otherwise allowing the students to explore. However, students who tended to use the individual trials only were encouraged to run some repeated trials. After the student had time to investigate the problem (usually 10-15 minutes) the student was asked to run 2,000 trials for each strategy, using the repeated trials component of the simulator (which, based on the law of large numbers, should provide adequate evidence). Students were asked to read the results of this simulation and verbally state (based on the simulation) which strategy won the most and which strategy won the least. This was done to make sure that students had understood the results of the simulation.

Classroom treatment (CT). The first author taught a lesson on Monty's Dilemma to each of the two classes in the CT group explaining and discussing each of the different strategies and the probability of winning using each of the strategies. Such an explanation might proceed as follows. After asking the students as a class to assume that they chose, say, door number 1, an explanation of the Stick strategy would proceed by asking the class the probability of winning given that they Stick. Upon hearing "one third," without discussion, 1/3 was written in a table of strategy probabilities displayed on an overhead transparency. Next, the class was asked, "What is the probability of losing if you stick, or what is the probability that the prize is behind one of the other doors (doors 2 or 3)?" Upon hearing "two thirds," the students were directed to assume that door number 3 was opened to reveal a goat. As a reminder students were asked, "What is the probability that the prize is behind doors 2 or 3." And upon hearing "two thirds," the class was asked, "so, what is the probability that the prize is behind door number 2?" Upon hearing "two-thirds," 2/3 was written in the table of probabilities for the Switch strategy. In similar fashion, an explanation for the probability of winning using the Flip strategy was given. The lesson did not involve the use of computers. There was no opportunity for interaction with classmates or hands-on simulation or investigation of Monty's Dilemma. To assess students' attention, students were asked to write down which strategy would provide them with the best and worst chance of winning.

Phase 3

After the treatments were completed students were presented with a new Monty's Dilemma game situation in a different set of cigar boxes completely independent from the first, and asked to play the game again, as in Phase 1. During Phase 3, interviews were conducted with students using open ended questions. For the ST group, these interviews were conducted individually and student responses were audio-taped (and later transcribed). For the CT group responses to questions were written by the student. In both groups, students were asked to explain why they chose their particular strategy. In the ST group, if students changed their strategy they were asked to explain their change. If they did not change their strategy they were asked why they used the same strategy. They were also asked if the computer affected their choice and, if so, how. In both groups, after questioning was complete, then, this time, the prize was revealed.

Results and Discussion Quantitative Analyses

Descriptive Analyses

A comparison of treatment groups found no significant difference in probability and statistics achievement (see Table 2). These results provide evidence that the two treatment groups are similar with regard to their statistical background. When comparing achievement across tracks, the high tracked students were found to have significantly higher scores on the achievement test than the low track students (t = 9.38, p<.001).

Pre Treatment

Considering the total sample, in the initial game situation, 81 (76%), 18 (17%), and 8 (7%) students chose the Stick, Switch, and Flip strategies respectively (see Tables 3 and 4). For the Simulation Treatment group (ST), 45 (82%), 4 (7%), and 6 (11%) students chose the Stick, Switch, and Flip strategies, respectively (see Table 3). For the Classroom Treatment group (CT), 36 (69%), 14 (27%), and 2 (4%) students chose the Stick, Switch, and Flip strategies, respectively (see Table 4). In all three situations, the data overwhelmingly suggest that despite the extra information, in general, students stick with their original choice (for the total sample, [chi square] (2, N= 107) = 87.84, p<.001, see Tables 3 and 4). The overall percentage of students choosing the switch strategy (17%) is similar to the 13% found by Granberg and Brown (1995). However, given the opportunity to flip a coin, some students seem to have decided to (re) play the odds, and flip, which may explain some of the discrepancy with the percentage of stickers in this study (81%) and the two studies in Granberg and Brown (1995; 87% and 90%).

Comparing different tracks, in the initial game situation, 50 (75%), 14 (21%), and 3 (4%) of the high track students chose the Stick, Switch, and Flip strategies, respectively. For the low-track students, 31 (78%), 4 (10%), and 5 (12%) chose the Stick, Switch, and Flip strategies, respectively. A comparison of tracks found no significant difference in initial choice of strategy ([chi square] (2, N = 107) = 3.95).

Post Treatment

In order to assess the effect of different instructional treatments on students' desire to choose the Switch strategy, students were combined into two groups: those who initially chose the switch strategy and those who did not (i.e., those that chose the Stick or Flip strategy). Using a Chi-squared test of independence with a correction for continuity, a comparison of the pre treatment marginals (see Table 5) found the proportion of switchers in the CT group greater than the ST group, [chi square] (1, N= 107) = 6.04, p<.05. However, despite this difference, a comparison of the post-treatment marginals found significantly more switchers in the ST group [chi square] (1, N= 107) = 4.05, p<.05. Further, a McNemar test of change found that students in the CT group were not significantly persuaded to choose the Switch strategy, [chi square] (1, N = 52) = 3.05. However, students in the ST group were overwhelmingly persuaded to choose the Switch strategy, [chi square] (1, N = 55) = 30.03, p<.05. These results suggest that students who were allowed to investigate Monty's Dilemma using a Monte-Carlo computer simulation were, subsequently, more persuaded to choose the Switch strategy.

In order to assess the effect of treatments on students' propensity to maintain their original Stick strategy (i.e., remain attached to their misconceptions), students were combined into two different groups: those initially choosing the Stick strategy and those that did not (i.e., those that chose the Flip or Switch strategy). Although thirty six percent of the total sample of students, reacting as predicted by Seymann (1991), continued to stick with their original choice of door, a majority of students (64%, 52 out of 81, see Tables 3 and 4) did change from their initial Stick strategy (11 to flip and 41 to switch). This finding suggests that many students, given sufficient evidence, will abandon their original choice of door either to (re) play the odds in the case of flipping or to go with the odds in the case of switching.

A higher proportion of students (71%, 32 out of 45, see Table 6) in the ST group did change from the Stick strategy than in the CT group (56%, 20 out of 36). However, the difference in the percentage of students changing for the two groups was not statistically significant [chi square] (1, N = 81) = 2.10. Yet, what is most important about these data is that a large majority of the students (71%) using the computer simulation was subsequently persuaded to change from the Stick strategy. It is not surprising that many students in the CT group were persuaded through direct instruction to change their strategy as this is the model for learning often experienced by students in the US. In other words, student learning is often equated with teacher's imposed understanding. But the fact that a larger proportion of the students in the ST group, on their own, were able to analyze data and subsequently reason (possibly through their notions of the law of large numbers) to abandon their initial Stick strategy provides promising evidence that the computer simulation can provide a rich and persuasive context for helping students overcome their probabilistic misconceptions. Furthermore, the increased percentage of students that changed from the Stick strategy in the ST group may be reflective of the fact that some students need or benefit from an interactive, hands on experience beyond a teacher's explanation when dealing with statistical tasks.

Post treatment comparisons by track found no significant overall or within treatment effects. That is, students in different tracks were not impacted differently by the different treatments. This finding suggests that instructional practices involving computer simulations are equally valuable for all levels of students.

Qualitative Analyses

In addition to making a decision about which strategy to use, students were also asked to explain their choice of strategy. Analysis of these explanations provide some valuable insights into students' thinking and reasoning as they made their decisions. This information better helps us understand why some students changed their strategy and why others maintained their initial strategy and the impact of the computer on these decisions. Further, this information provides evidence of psychological barriers that do exist that may impede or mask student understanding.

The Impact of the Computer Simulation

A majority of students were persuaded by the computer simulation to Switch. Furthermore, based on their general comments and eagerness to investigate, many students in the simulation group seemed to gain an appreciation for their experimental results: "I've done ... research on it and it's shown that if you switch it the odds are better to get the prize." Another student noted, "If these two games are the same [computer and props] switching should ... be the better choice because it has a lot more probability of winning." Students who worked with the computer were basing their decisions on information that they had gathered, instead of depending on teacher provided information. Observations of the students' investigations led us to believe that students were making conjectures, testing them, and drawing conclusions: "It seems like as long as you're not running the ten [option] ... this one's [Stick] around ... thirty something, and this one's [Switch] sixty, around sixty, and this one's [Flip] around fifty."

Many students would proceed by first testing their initial choice of strategy. Next, they would play a few individual trials, then they would try each strategy 100 or 1000 times. Then they might return to the individual trials to test their findings. Later, they may return to the repeated trials to try a few more experiments or continue playing the individual trials for further confirmation of their findings. When asked why they chose the Switch strategy most of them based their decision on the results of their experiment: "... as you can see, the Switch won more, so, it probably has a better chance." This comment suggests that students were able to apply their intuitive notions of the law of large numbers to make conclusions. This type of reflection did not occur in the CT group. Students in the CT group who chose the Switch strategy most often referred to the higher probability of winning as their reason, however, they had to depend on their belief in the teacher's explanation without any self reflection or investigation. As was found in the quantitative analyses presented earlier, for some students the teacher's explanation is enough to convince them to change from their Stick strategy. However, the computer simulation provided students with a richer and more reflective experience with the opportunity to analyze data and make their own conclusions about the problem.

The following excerpt from one student's interview provides evidence that through investigation and reflection that students not only can be convinced that the Switch strategy provides the best chance of winning but that they can actually come to understand the underlying mathematics associated with the problem. Initially, this student adamantly declared that he would stick with his original door, even interrupting the interviewer to make his intentions clear. At one point he stated, "This is all random chance.... It's luck, if you get it or not get, it's really just random chance. Or is that what you're trying to prove here?" After completing his investigation and allowing the computer to run 2000 trials, the following transpired.

  Interviewer: Now, based on the results, which strategy won most often?
  Student: The Switch strategy, and I think I know why. Because it's
  most likely you'll choose a goat because there's two of them out of
  three and so when they open one, and that gets rid of a goat, you're
  most likely on a goat. So, if you switch you're probably on the prize.
  Interviewer: Hmmm. Based on the results, which strategy won least
  often?
  Student: Stick, for the same reason <Student was interrupted to read
  results from computer. After doing so he continued to describe his
  thinking>.
  Student: And uh, the reason for the flip is because there's a 50-50
  chance for you staying or going.
  Interviewer: Okay.

After being presented with Phase 3 of the simulation treatment, that is, another Monty's Dilemma, he was again asked about his choice of strategy.

  Interviewer: Would you like to stick with your original door, switch
  to the remaining door, or flip a coin to decide?
  Student: I'll go ahead and switch.
  Interviewer: You'll switch. Why will you switch?
  Student: Because of what I said earlier, because it's most likely I
  picked a goat and since you opened one I'm most likely on a goat. So,
  it's most likely the one I didn't pick is the prize.

Through investigation of this problem using the computer this student was able to provide a sound explanation of Monty's Dilemma.

Students who initially chose to flip or switch (i.e., Flip-Switch, Switch-Switch, see Table 3) were overwhelmingly convinced by the computer simulation to switch. This might suggest that students who tend to be more "risky," may be more willing to base their decisions on statistically sound information (for a discussion of risk perception see Slovic, Fischoff, & Lichtenstein, 1982). In other words, they might be willing to take a "good risk." As one student said, "... the first time I just switched, just to switch, and this time [second time], 'cause the best chance I have at winning would be to switch." For some of these students (i.e., flippers), results from the simulation may resolve a cognitive conflict between their instincts and indecision. Similar students in the Classroom Treatment were not as easily convinced of the chances associated with the Switch strategy as several subsequently chose to stick or flip (see Table 4).

For some students, in particular the Stick-Flip students, the simulation seemed to create cognitive conflict between their instincts and what the computer results suggest. Although the simulation did not convince them to switch, it did seem to create doubt. As one student who flipped explained, "Well I kinda want to stick with the original one [door], but if we're playing the game ... switching wins more often, but I'm afraid I'll switch and then [door] two will be the right one." Doubt can be a great motivator for inquiry and eventual understanding (e.g., Dewey, 1933), and therefore such simulation activities may prime students for a "teachable moment," which might be further facilitated through additional opportunities to investigate similar problems.

Recognizing and Overcoming Psychological Barriers

Overall, 36 percent of the students initially using the Stick strategy subsequently continued to use this strategy. While a majority of students were convinced by the computer simulation to change to the Switch strategy, understanding why other students continue to hold to their misconceptions can be helpful in designing curriculum and instruction to further help these students. Based on comments and explanations from students that maintain their original Stick strategy some may in fact suffer from belief perseverance. Several students seemed to hold to beliefs associated with luck, or instinct, asserting that "my 1st [sic] choice is usually right," or "I just trust my instincts." Choice perseverance was probably best represented by the statement of this student: "I don't know, probably the fact that my grandma always said that I have a strong will and once I make something ... that's what I go for." It seems that these students believe that it is important to go with their instinct.

As suggested by prior research, it may also be the case that students' propensity to maintain their strategy might result from counterfactual judgements based on the use of the simulation heuristic (Kahneman & Tversky, 1982; Granberg & Brown, 1995). Some students seemed to be concerned about the negative affect associated with switching and being wrong. One student said, "I probably should have switched since it has a greater probability of winning, but my gut feeling said to stay with it." Or like the student mentioned earlier, who ultimately chose to flip, who remarked: "well I kind of want to stick with the original one [door], but if we're playing the game ... switching wins more often, but I'm afraid I'll switch and then [door] two will be the right one." Such a feeling was mirrored by yet another student who remarked: "whenever ... I think I know it and switch it ... it ends up wrong, so when I stick with it, sometimes its better." Such feelings are hard to overcome for some students even when they seem to understand the problem. They are more concerned about how they would feel if they switch and lose than whether they are making a statistically sound decision.

Alternatively, it seems that many of the students that chose to stick even after using the computer may be relying on what Konold (1989) referred to as an "outcome approach" to situations of uncertainty. The goal of such an approach, according to Konold, is to successfully predict the outcome of single trials. For example, given a thumbtack and asked to predict whether the point or the head of the tack is more likely to land up, people using an outcome approach tend to try and predict the next outcome. After one toss of the tack, such individuals tend to evaluate their prediction as right or wrong with little attention to possible distributional characteristics or frequency information that may be obtained by repeatedly tossing the tack.

Despite the uncertainty inherent in Monty's Dilemma, several students seemed to feel that they could actually figure out which door contained the prize. Many of these students felt that their instinct was as good a reason as any to choose a particular door: "I just trust my instinct," or "[I] just think it's there." Other students felt that their success depended on their ability to pick numbers. As one student remarked, "I just got a good feeling about the numbers I pick." Such explanations suggest that some students believe they can actually predict each outcome. All of these situations occur in spite of the frequency information presented by the computer simulation for students in the simulation treatment. Even the student who concurred with the results of the computer that "If I stick a lot, I'll lose a lot," and "Switchin' you're making a lot of profits," still had a "gutsy feeling" to stick. Based on the outcome approach, this student might have interpreted the computer as being incorrect "a lot" when it chose to stick, and correct "a lot" when it chose to switch, instead of considering the long run results as a whole from which to base future predictions.

General Discussion and Conclusions

Based on the results of this study, the use of computer based Monte Carlo simulations can be very persuasive in helping students make statistically sound decisions in situations of uncertainty. In the case of Monty's Dilemma, overall, students who were allowed to investigate the problem using the computer simulation were, subsequently, more persuaded to choose the statistically sound Switch strategy than students involved in teacher directed instruction. Further, a large majority of students who initially chose the Stick strategy were subsequently persuaded through the use of the computer simulation to abandon their initial Stick strategy. In other words, the computer helped many students overcome their psychological attachment to their misconceptions. Interestingly, many students were convinced by the teacher directed instruction to abandon the Stick strategy as well, but qualitative evidence suggests that students' experiences with the computer simulation were much richer, more reflective, and autonomous. That is, students' using the computer simulation made decisions based on their own conclusions drawn from analyzing and thinking about the data instead of based on an external explanation from someone else. However, the study did find that some students do hold to their choices and beliefs in spite of information that discredits these beliefs or choices. In addition, misconceptions were found to exist and persist irrespective of students' achievement level. However, investigating Monty's Dilemma using a computer simulation seems to provide an equally positive experience for both high- and low-achieving students.

In this study, similar to Granberg and Brown (1995), a majority of students did initially choose the Stick strategy. However, different from the Granberg and Brown experiments, students using the computer in this study were encouraged to run many more trials. They were explicitly given the opportunity to summarize their data from their investigations and to think about the implications of the results. Students were able to investigate large amounts of data on more than one occasion and were possibly able to witness the stability of estimates based on frequency distributions and thus may have become more aware of the power of data in determining probabilities. In other words, notions of the law of large numbers may have had an influence on students' conclusions. Results of this study would suggest, inconsistent with Granberg and Brown, that students can inductively come to conclusions based on the law of large numbers, and in some cases actually come to an understanding of the problem.

However, of the students in the ST group, twenty nine percent of those initially choosing the Stick strategy maintained a Stick strategy even after investigating with the computer. Qualitative evidence suggests that some students possibly do suffer from belief perseverance and hold to their beliefs in luck or instinct, while others may mentally simulate the results of switching and being wrong and decide not to take this chance (cf., Granberg & Brown, 1995). Still others may have based their decisions on the ideas associated with the outcome approach (Konald, 1989). They believe that they can predict each outcome. Consistent with the outcome approach, students who are trying to predict each outcome would not be overly convinced by frequency or distributional information which was the type of evidence provided in this study. To overcome such a barrier, Konold (1989) suggested that giving students the opportunity to make distributional predictions instead of individual outcome predictions might help build a better appreciation of frequency information. In the case of Monty's Dilemma, one might present students with a situation in which the game is played 10 times and the contestant must pick a strategy to use for all 10 games, the object being to win a majority of times. Here students would have to base their judgments on long run results or frequency of wins. Konold (1989) also identified several problems that can be used to identify students relying on the outcome approach and the level at which they depend on the approach. Further study of the misconceptions associated with Monty's Dilemma (and similar situations) might combine an investigation of Monty's Dilemma and several of these problems to associate "Monty categories" with levels of outcome orientation.

Another interesting finding from this study was the lack of differences across tracks. In studies on tracking, Oakes (1985, 1990) found that students in high tracked classrooms are given many more opportunities to learn than their lower tracked peers. Such opportunities included better teachers, more challenging content, and more interesting instruction. Increased opportunities to learn has been shown to be related to increased achievement (e.g., Wiley ad Harnischfeger, 1974; Barr & Dreeben, 1983; Gamoran, 1987; Wilkins, 1997; Wang, 1998). Policy decisions based on this research might suggest that all students should be given similar opportunities in the classroom. Tracking research (e.g., Oakes, 1985, 1990) has considered differences across tracks and, subsequently, recommended that changes be made in instruction in low track classrooms. It is important, however, to make comparisons of intact tracks being given the same instruction and subsequent effects assessed. This study provides such a comparison, and provides results that support the belief that similar opportunities to learn can produce similar learning for all levels of student. Moreover, students at all levels can learn from the same level of inquiry based instruction. Results suggest that misconceptions are prevalent in all students irrespective of prior statistical knowledge. Students in the high track classrooms maintained the use of the Stick strategy for the same reasons as students in the low track classrooms. After the treatments, students in the low track classrooms chose the Switch strategy for the same reasons as high track students. It is important, however, to conduct further studies with a larger sample of classrooms to verify the findings presented here. However, these findings offer promising evidence that given similar opportunities to investigate problems through simulation, particularly in the case of situations of uncertainty, high- and low-achieving students can construct similar understandings and potentially overcome existing misconceptions.

For some students, although they believe what the computer presents, the computer may represent a "black box" from which they can gain little understanding of the process. Some people may have more success understanding the problem if they are allowed to physically carry out the simulation using spinners and pennies (e.g., Shaughnessy & Dick, 1991). Shaughnessy and Dick suggested that hands on simulations should precede computer simulations, however, it might be the case that because of its black box nature, computers may supply necessary doubt and cognitive conflict and thus better prime students for the use of spinners and pennies leading to better understanding. In any case, the goal of this study was to assess the persuasiveness of computer simulation on changing students' misconceptions. In general, it was shown that the computer simulation used in this study was persuasive. Moreover, it was found to be a useful instructional tool for helping many students think about their misunderstandings.

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Author Note

Jesse L. M. Wilkins, Department of Teaching and Learning, Virginia Polytechnic Institute and State University; George C. Reese, Department of Curriculum and Instruction, University of Illinois at Urbana Champaign.

The authors would like to thank Michelle Wilkins and Barry Sloane for reading and making suggestions for improvement to earlier drafts of this article.

Correspondence concerning this article should be addressed to Jesse L. M. Wilkins, Department of Teaching and Learning, Virginia Polytechnic Institute and State University, 300 C War Memorial Hall, Blacksburg, Virginia 24061. E mail: [e-mail]

Notes

(1) Freudenthal (1972; also see Sedlmeier & Gigerenzer, 1997) made a distinction between Bernoulli's 'mathematical' law of large numbers and his more intuitive notion that Freudenthal referred to as the 'empirical law of large numbers.' In this paper the notion of the law of large numbers aligns more closely with the later.

(2) A recent distinction has been made between frequency distribution tasks and sampling distribution tasks that have been used to evaluate peoples' appreciation of sample size (Sedlmeier & Gigerenzer, 1997). In a review of the literature Sedlmeier and Gigerenzer found that in the former tasks people tend to take sample size into account, but, in the later they tend to be less sensitive to sample size.

Appendix

1. Allen's batting average is 0.425. What is his batting average as a percent?

(A) 0.0425%

(B) 4.25%

(C) 42.5%

(D) 425%

2. In a certain school, 10% of the students bring their lunch, 60% buy a lunch in the cafeteria, and the remaining students go home for lunch. What percent of the students go home for lunch?

(A) 30%

(B) 50%

(C) 70%

(D) More facts are needed to answer.

MUSICAL FAVORITES

Singers          Number of Votes

Michael Jackson  20
Diana Ross       10
Julio Igesias    10
Willie Nelson     5
Culture Club      5

3. According to the chart above, what percent of all the votes went to Michael Jackson?

(A) 20%

(B) 40%

(C) 50%

(D) 66 2/3%

[ILLUSTRATION OMITTED]

4. The jar shown above contains 2 black and 3 white marbles. Al picks one marble without looking. What is the probability that he picks a black marble?

(A) 1/5

(B) 2/5

(C) 2/3

(D) 5

5. If a fair coin is tossed, the probability that it will land heads up is 1/2. In four successive tosses the coin lands head each time. What happens when it is tossed a fifth time?

(A) It will most likely land tails up.

(B) It is more likely to land tails up than heads up.

(C) It is more likely to land heads up than tails up.

(D) It is equally likely to land heads up or tails up.

6. A jar contains 5 red, 6 blue, and 7 green marbles. One marble is drawn from the jar. What is the probability that the marble drawn at random is red or green?

(A) 1/12

(B) 1/5

(C) 1/2

(D) 2/3

7. When Carlos shoots free-throws left-handed he makes 33% of his shots, when he shoots right-handed he makes 66% of his shots, and when he shoots underhanded he makes 50% of his shots. If Carlos is about to attempt a game winning free-throw which method of shooting should he use?

(A) left-handed

(B) right-handed

(C) underhanded

(D) It depends on how he feels

8. A coin is tossed and a die is rolled. What is the probability that the coin comes up heads and the die comes up 3?

(A) 1/12

(B) 1/8

(C) 1/5

(D) 6/12

9. Dawn has 3 skirts and 5 blouses. How many different skirt-blouse outfits can she make with these?

(A) 3

(B) 5

(C) 8

(D) 15

10. Bill made the lowest score on the test. He only got 27 points. The teacher said the class mean was 63 and the range was 61. Jane made the highest score on the test. What score did Jane make?

(A) 61

(B) 63

(C) 88

(D) 90

11.

[TABLE OMITTED]

Which graph below best fits the data in the tally chart above?

(A)

[GRAPHIC OMITTED]

(B)

[GRAPHIC OMITTED]

(C)

[GRAPHIC OMITTED]

(D)

[ILLUSTRATION OMITTED]

12. Jan has 3 dimes in her pocket and nothing else. If she takes 1 coin from her pocket, what is the probability that it will be a dime?

(A) 1/10

(B) 3/10

(C) 1/3

(D) 1

13. Carlos' basketball team won 75% of its games last season. If they played 80 games, how many games did they win?

(A) 20

(B) 60

(C) 68

(D) 75

14. In a game you are given these 5 cards.

[ILLUSTRATION OMITTED]

[ILLUSTRATION OMITTED]

[ILLUSTRATION OMITTED]

[ILLUSTRATION OMITTED]

[ILLUSTRATION OMITTED]

A rule says you must select 2 cards and form a 2-digit number such as:

[ILLUSTRATION OMITTED]

[ILLUSTRATION OMITTED]

How many different 2-digit numbers can you form including the one above?

(A) 10

(B) 15

(C) 20

(D) 25

(E) 30

*Questions 15-16 refer to the following picture.

[ILLUSTRATION OMITTED]

15. Scott is rolling a number cube with 1, 2, 3, 4, 5 and 6 dots on its faces. What is the probability of Scott getting a 4 on his next roll?

(A) 0

(B) 1/6

(C) 2/6

(D) 3/6

(E) 4/6

(F) 5/6

16. Scott rolls five 5's in a row. What is the probability of getting a 5 on his next roll?

(A) 0

(B) 1/6

(C) 2/6

(D) 3/6

(E) 4/6

(F) 5/6

17. According to the table below, what is the total amount of protein contained in two boiled eggs and one-half cup of whole milk?

NUTRITIVE VALUE OF CERTAIN FOODS

               Measure  Calories  Protein (grams)  Carbohydrates (grams)

Banana, raw    1        100        1               26
Beef hambuger  3 oz.    245       21                0
Whole milk     1 cup    160        9               12
Doughnut       1        125        1               16
Eggs, boiled   2 eggs   160       13                1

(A) 30.5 grams

(B) 22 grams

(C) 17.5 grams

(D) 7 grams

According to the graph above, which car reached 60 miles per hour in the shortest time?

[GRAPHIC OMITTED]

(A) Car A

(B) Car B

(C) Car C

19. It took 3 games for a basketball player to score a total of 51 points. If the player keeps this scoring average, how many total points will the player have scored by the end of the seventh game?

(A) 17

(B) 51

(C) 119

(D) 153

(E) 170

(F) 357

20. In a pet shop there are 12 animals. Seven are dogs and the rest are cats. What is the ratio of dogs to cats?

(A) 12:7

(B) 5:7

(C) 7:12

(D) 7:5

21. In a basketball game, George made 70% of his free-throws, Jay made 50% of his free-throws, and Jim made 30% of his free-throws. Based on these results who is the best free-throw shooter?

(A) Jim

(B) Jay

(C) George

(D) There is no difference

22. What is 8% of 25?

(A) 2

(B) 20

(C) 31.25

(D) 200

23. A penny was tossed 20 times. Which of the following is most likely to be the number of times heads came up?

(A) 0

(B) 2

(C) 5

(D) 9

(E) 19

POPULATIONS OF DETROIT AND LOS ANGELES 1920-1970

             CITY
YEAR  DETROIT    LOS ANGELES

1920    950,000    500,000
1930  1,500,000  1,050,000
1940  1,800,000  1,500,000
1950  1,900,000  2,000,000
1960  1,700,000  2,500,000
1970  1,500,000  2,800,000

24. Considering the chart above, what was the first year listed in which the population of Los Angeles was greater than the population of Detroit?

(A) 1920

(B) 1930

(C) 1940

(D) 1950

(E) 1960

(F) 1970

25. Susie has a batting average of .750, Jane has a .500 batting average, and Karen has a .250 batting average. Which softball player is most likely to get a hit next time at bat?

(A) Karen

(B) Susie

(C) Jane

(D) They are all equally likely to get a hit.

Jesse L.M. Wilkins

Virginia Polytechnic Institute and State University

George C. Reese

University of Illinois at Urbana-Champaign 

TABLE 1 All Possible Monty's Dilemma Scenarios that Involve Switching
and Associated Outcome

Prize behind  Initial choice  Door
door          of door         opened  Switch to:             Outcome

1             1               2 or 3  3 or 2 (respectively)  Lose
2             3               1                              Win
1             3               2       1                      Win
2             1               3       2                      Win
2             2               1 or 3  3 or 1 (respectively)  Lose
2             3               1       2                      Win
3             1               2       3                      Win
3             2               1       3                      Win
3             3               1 or 2  2 or 1 (respectively)  Lose

Note: From "A Problem in Probability," by S. Selvin, 1975, American
Statistician, 29, p. 67. Copyright 1975 by Steve Selvin. Adapted with
permission of the author.

TABLE 2 A Comparison of Statistics Achievement

                       Statistics Achievement
               N       M      SD

ST Group       55(47)  19.8   4.0
CT Group       52(50)  18.2   5.7
t-value                -1.62  ns
High-Track     67(61)  21.6   2.6
Low-Track      40(36)  14.5   4.9
t-value                 9.38  ***

Note. The parenthetical N represents the number of students on which the
background statistics were calculated. This group of students is a
subset of the total N which represents the number of students that
received treatment. ***p<.001.

Table 3 Pre- and Post-Treatment Choice of Strategy for the Simulation
Treatment (ST) Group

                  Post-treatment
Pre-Treatment  Stick  Switch  Flip  Total

Stick          13     27      5     45
Switch          0      4      0      4
Flip            1      5      0      6
Total          14     36      5     55

Table 4. Pre- and Post-Treatment Choice of Strategy for the Classroom
Treatment (CT) Group

                 Post-treatment
Pre-Treatment  Stick  Switch  Flip  Total

Stick          16     14      6     36
Switch          4      8      2     14
Flip            0      1      1      2
Total          20     23      9     52

Table 5. A Comparison of the Simulation and Classroom Treatment on
Persuasiveness to Choose the Switch Strategy

                                  After
Treatment   Before      non-SWITCH  SWITCH  Totals

SIMULATION  non-SWITCH  19          32      51
            SWITCH       0           4       4
            Totals      19          36      55

CLASSROOM   non-SWITCH  23          15      38
            SWITCH       6           8      14
            Totals      29          23      52

Table 6. A Comparison of the Simulation and Classroom Treatment on
Persuasiveness to Change from the Stick Strategy

                           After
Treatment   Before     STICK  non-STICK  Totals

SIMULATION  STICK      13     32         45
            non-STICK   1      9         10
            Totals     14     41         55

CLASSROOM   STICK      16     20         36
            non-STICK   4     12         16
            Totals     20     32         52