Proceedings

The present investigation is concerned especially with the influence that the teaching of probabilities may have indirectly on intuitive probabilistic judgments. There is very little information available about this problem. In an earlier work, Ojemann et al. have reported a positive, indirect effect of probability lessons on predictions made by their subjects (8-10 years old) in probability learning tasks. In this study the authors found a clear increase with age in proportions of correct answers to probability problems. They also found that by emphasizing (via systematic instruction) specific probability viewpoints and procedures, one may disturb the subjects proportional reasoning, still fragile in many adolescents.

The aims of this project, sponsored by the Social Science Research Council from November 1978 to October 1981, were: a) to survey the intuitions of chance and the concepts of probability possessed by English school pupils aged 11 to 16 years. b) to establish patterns of development of probability concepts and to relate these to other mathematics concepts. c) to investigate pupil responses to GCE 'O' Level and CSE probability items, with particular references to sex differences. A summary of findings is presented, based on responses of 2930 children.

We view understanding of mathematical material as a function of (1) connections of text concepts and formulas to real-world referents; (2) integration of concepts and formulas within the text; and (3) explanation of formulas. This view has provided a basis for constructing three written treatments of elementary probability, presumably varying in the degree to which they convey understanding. Our view of the processes involved in solving problems has led us to use both formula and story problems, and to emphasize analyses of error protocols. A research study is described involving 48 undergraduate students, randomly assigned to three text conditions. Results indicated that very different patterns of knowledge appear to be present in subjects in the non-explanatory (standard and low-explanatory texts) and explanatory conditions. Subjects in the two non-explanatory groups performed considerably less well on story than on formula problems, and often used the correct formula for a problem (that is, met the lenient criterion) but failed to solve it. A closer look at answers to story problems revealed that subjects in the non-explanatory conditions often required the explicit presence of key words which unambiguously pointed to certain operations, tended to misclassify problems in the presence of irrelevant or redundant information, and made many errors when the values in the story required modification before insertion into the formula. In contrast, subjects in the highly explanatory condition performed equally well on story and formula problems, tended to solve whenever they showed evidence of knowing the appropriate formula, and were considerably less hindered by absence of key words, the presence of irrelevant information, and the need to translate values in the story.

As part of a study on the natural interpretations of probability, experiments about elementary "purely random" situations (with dice of poker chips) were carried out using students of various backgrounds in the theory of probability. A prior study on cognitive models which analyzed the individual data of more than 600 subjects had shown that the most frequent model used is based on the following incorrect argument: the results to compare are equiprobable because it's a matter of chance; thus, random events are thought to be equiprobable "by nature". In the present paper, the following two hypotheses are tested: 1) Despite their incorrect model, subjects are able to find the correct response. 2) They are more likely to do so when the "chance" aspect of the situation has been masked. An experiment testing 87 students showed, as expected, that there is a way to induce the utilization of an appropriate cognitive model. However, the transfer of this model to a classical random situation is not as frequent as one might expect.