A second area of research conducted primarily by statistics educators, is focused less on general patterns of thinking, and more on how statistics is learned. Some of these studies have contradicted implications of the psychological studies described earlier (e.g., Borovcnik, 1991; Konold et al., 1991; Garfield &delMas, 1991). For example, some of these studies indicate that students' use of heuristics (such as representativeness and availability) seems to vary with problem context.

Garfield and delMas(1991) examined performance of students in an introductory course on a variety of parallel problems, designed to elicit use of the representative heuristic. Their results suggest that students do not rely exclusively on the representativeness heuristic to answer all problems of a similar type (Garfield &delMas, 1991). Konold et al. (1991) hypothesized that inconsistencies in student responses are caused by a variety of perspectives with which students reason. Students appear to understand and reconstruct a problem in different ways, leading them to apply different strategies to solve them. Borovcnik and Bentz (1991) discuss other reasons for inconsistencies in student responses, such as the constraints imposed by artificial experiments and ambiguity of questions used.

Additional research on learning probability and statistics suggests ways to help students learn, as well as problems that need to be considered.

**What helps students learn:**

- Activity-based courses and use of small groups appear to help students
overcome some misconceptions of probability (Shaughnessy, 1977) and enhance
student learning of statistics concepts (Jones, 1991).
- When students are tested and provided feedback on their
misconceptions, followed by corrective activities (where students
are encouraged to explain solutions, guess answers before computing
them, and look back at their answers to determine if they make sense),
this ``corrective-feedback" strategy appears to help students overcome
their misconceptions (e.g., believing that means have the same properties
as simple numbers) (Mevarech, 1983).
- Students' ideas about the likelihood of samples (related to the
representativeness heuristic) are improved by having them make
predictions before gathering data to solve probability problems, then
comparing the experimental results to their original predictions.
(Shaughnessy, 1977; del Mas &Bart, 1987; and Garfield &delMas, 1989).
- Use of computer simulations appears to lead students to give more correct answers to a variety of probability problems (Garfield &delMas, 1991; Simon, Aktinson, &Shevokas; 1976; Weissglass &Cummings, 1991).
- Using software that allows students to visualize and interact with data appears to improve students' understanding of random phenomena (Weissglass &Cummings, 1991) and their learning of data analysis (Rubin, Rosebery, &Bruce, 1988).

**Problems to be considered:**

- Training involving application of the Law of Large Numbers may
improve students' reasoning about samples of data. (Nisbett et al., 1987).
Other studies contradicted these results and showed that students'
responses to a narrow type of probability problem improved, but thinking
did not (Shaughnessy, 1992).
- Students may answer items correctly on a test because they know
what the expected answer is, but still have incorrect ideas. In a
study involving students in various courses, students were able to say
that different sequences of coin tosses were all equally likely when
asked which was most likely to occur. However, when asked which was
**least likely**to occur, they unperturbedly selected one or another particular sequence! (Konold, 1989b). - Students' misconceptions are resilient and difficult to change. Instructors can't expect students to ignore their strong intuitions merely because they are given contradictory information in class (Konold, 1989b; Well et al.,1990; delMas and Garfield, 1991).

Wed Jun 29 13:57:26 EDT 1994