• Students of all ages have a difficult time understanding and using randomness, with “no marked differences” in understanding within this wide age range. Teachers need to give students multiple and diverse experiences with situations involving randomness and help them understand that randomness “implies that a particular instance of a phenomenon is unpredictable but there is a pattern in many repetitions of the same phenomenon” (Green, 1987; Dessert, 1995).
• Students tend to interpret probability questions as “requests for single outcome predictions.” The cause of this probability misconception is their tendency to rely on causal preconceptions or personal beliefs (e.g., believing that their favorite digit will occur on a rolled die more frequently despite the confirmation of equal
probabilities either experimentally or theoretically) (Konold, 1983).
• Students estimating the probability of an event often ignore the implications of the sample size. This error is related to an operational misunderstanding of the law of large numbers (Kahneman and Tversky, 1972).
• Students have poor understandings of fundamental notions in probability: the use of tree diagrams, spinners using the area model, random vs. nonrandom distributions of objects, and the general idea of randomness itself. To overcome these understandings, students need more exposure to the ratio concept, the common language of probability (e.g., “at least,” “certain,” and “impossible”), and broad, systematic experiences with probability throughout their education (Green, 1983, 1988).
• Appropriate instruction can help students overcome their probability misconceptions. Given an experiment, students need to first guess the outcome, perform the experiment many times to gather data, then use this data to confront their original guesses. A final step is the building of a theoretical model consistent with the experimental data (Shaughnessy, 1977; DelMas and Bart, 1987).
• Students’ growth in understanding probability situations depends on three abilities that can be developed. First, they need to overcome the “sample space misconception” (e.g., the ability to list events in a sample space yet not recognize that each of these events can occur). Second, they need to apply both part-part and
part-whole reasoning (e.g., given four red chips and two green chips, “part-part” involves comparing the two green chips to the four red chips while “part-whole” involves comparing the two green chips to the six total chips). And third, they need to participate in a shared adoption of student-invented language to describe probabilities (e.g., “one-out-of-three” vs. “one-third”). (Jones et al., 1999a).
• Students construct an understanding of probability concepts best in learning situations that (1) involve repeatable processes and a finite set of symmetric outcomes (e.g., rolling a die), (2) involve outcomes produced by a process involving pure chance (e.g., drawing a colored chip from a bag of well-mixed chips), and (3) are well recognized as being “unpredictable and capricious” (e.g., predicting the weather). When probability situations deviate from these three prototypes, students experience great difficulty and revert to inappropriate reasoning (Nisbett et al., 1983). These claims appear valid if students are asked to determine the most likely outcomes but are not valid if students are asked to determine the least likely outcomes, a discrepancy due to students’ misunderstanding the concept of independence (Konold et al., 1993).
• Six concepts are fundamental to a young child trying to reason in a probability context. These six probability concepts are sample space, experimental probability of an event, theoretical probability of an event, probability comparisons, conditional probability, and independence (Jones et al., 1999b).
• As students progress through the elementary grades into the middle grades, their reasoning in probability situations develops through four levels:
1. Subjective or non quantitative reasoning: They are unable to list all of the outcomes in a sample space and focus subjectively on what is likely to happen rather than what could happen.
2. Transitional stage between subjective reasoning and naïve quantitative reasoning: They can list all of the outcomes in a sample space but make questionable connections between the sample space and the respective probability of an event.
3. Naïve quantitative reasoning: They can systematically generate outcomes and sample spaces for one- and two-stage experiments and appear to use quantitative reasoning in determining probabilities and conditional
probabilities, but they do not always express these probabilities using conventional numerical notation.
4. Numerical reasoning: They can systematically generate outcomes and numerical probabilities in experimental and theoretical experiments, plus can work with the concepts of conditional probability and independence (Jones et al., 1999b).
• Students often compute the probabilities of events correctly but then use incorrect reasoning when making an inference about an uncertain event. The problem is the students’ reliance on false intuitions about probability situations that overpower their mathematical computations (Garfield and Ahlgren, 1988; Shaughnessy, 1992).
• Students will focus incorrectly on the single events making up the series when given probability information about a series of events. For example, told that that there is a predicted 70 percent chance of rain for ten days, students will claim that it should rain on every one of the ten days because of the high 70 percent value. The underlying problem is known as “outcome orientation” and is prompted by an intuitive yet misleading model of probability (Konold, 1989).
• Students need to make probability predictions about possible events in diverse situations, then test their predictions experimentally in order to become aware of and confront both personal misconceptions and incorrect reasoning. Too often, students will discredit experimental evidence that contradicts their predictions
rather than restructure their thinking to accommodate the contradictory evidence (DelMas, Garfield, and Chance, 1997).
• Students tend to categorize events as equally likely because of their mere listing in the sample space. An example is the student who claims that the probability of rolling a prime number is the same as the probability of rolling a composite number on a single role of a single die (Lecoutre, 1992).
• Student errors when estimating probabilities often can be traced to the use of two simplifying techniques which are misleading: representativeness and availability. Using the technique of representativeness, students estimate an event’s probability based on the similarity of the event to the population from which it is drawn (e.g., students see the coin flip sequence [HHHHHTHHHH] then claim that H is more probable on the next toss in order to even out the overall probabilities). Using the technique of availability, students estimate an event’s probability based on the “ease” with which examples of that event can be produced or remembered (e.g., students’ estimations of the probability of rain on a fall day in Seattle will differ if they have recently experienced several rainy days). A residual of the two techniques are these specific errors in making probability prediction:
1. Disregard for the population proportions when making a prediction.
2. Insensitivity to the effects of sample size on the ability to make accurate predictions.
3. Unwarranted confidence in a prediction based on incorrect information.
4. Fundamental misconceptions of chance, such as the gambler’s fallacy.
5. Misconceptions about the “speed” with which chance data regress to a mean (Shaughnessy, 1981).
• Fischbein and Schnarch (1997) confirmed Shaughnessy’s conclusions and added these additional probability misconceptions:
1. The representativeness misconception decreases with age.
2. The misleading effects of negative recency (e.g., after seeing HHH, feeling the next flip will be a T) decreases with age.
3. The confusion of simple and compound events (e.g., probability of rolling two 6’s equals probability of rolling a 5 and a 6) was frequent and remained.
4. The conjunction fallacy (e.g., confusing the probability of an event with the probability of the intersection of that event with another) was strong through the middle grades then decreased.
5. The misleading effects of sample size (e.g., comparing probability of two heads out of three tosses vs. probability of 200 heads out of 300 tosses) increased with age.
6. The availability misconception increased with age.
• Students often will assign a higher probability to the conjunction of two events than to either of the two events individually. This conjunction fallacy occurs even if students have had course experiences with probability. For example, students rate the probability of “being 55 and having a heart attack” as more likely than the probability of either “being 55” or “having a heart attack.” An explanation for the error is that students may confuse the conjunction form (e.g., “being 55 and having a heart attack”) with the conditional form (e.g., “had a heart attack given that they are over 55”) (Kahneman and Tversky, 1983).
• Students have difficulties with conditional probabilities Prob(A|B), attributed to three types of errors:
1. The Falk Phenomenon (Falk, 1983, 1988) arises when the “conditioning event” occurs after the event that it conditions (e.g., If two balls are drawn without replacement from an urn [WWBB], what is the probability that the first ball was white given that the second ball was white?).
2. Confusion can arise when trying to identify the correct “conditioning event.”
3. Confusion, especially when diagnosing diseases (Eddy, 1982), between a conditional statement and its inverse (e.g., “the probability that it is raining given that it is cloudy” versus “the probability that it is cloudy given that it is raining”). Student experiences with real world examples of probability situations will help overcome these misconceptions (Shaughnessy, 1992).
• Student misconceptions of independent events in probability situations can be impacted by exposure to real-world experiences that help the students:
1. Realize that dependence does not imply causality (e.g., oxygen does not cause life yet life depends on oxygen to keep breathing).
2. Realize that it is possible for mutually exclusive events to not be complementary events.
3. Realize the distinction between contrary events and contradictory events (Kelly and Zwiers, 1988).
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