Chapter

Probability and statistics (stochastics) are viewed as necessary for all students no matter their ambitions. However, there are barriers to the effective teaching of both stochastics and problem solving: 1) getting stochastics into the mainstream of the mathematical science school curriculum; 2) enhancing teachers' background and conceptions of probability and statistics; 3) confronting students' and teachers' beliefs about probability and statistics. Psychologists and mathematics educators should work collaboratively to diminish misconceptions. Doing so combines the roles of observer, describer, and intervener. Research in stochastics suggests that heuristics that are used intuitively by learners impede the conceptual understanding of concepts such as sampling. This paper reviews the research on judgemental heuristics and biases, conditional probability and independence (i.e., causal schemes), decision schema (i.e., outcome approach), and the mean. Learners have difficulties in these areas, however, evidence is contradictory as to whether training in stochastics improves performance and decreases misconceptions. The conclusion emerging from this research is that probability concepts can and should be introduced into the school at an early age. Instruction that is designed to confront misconceptions should encourage students to test whether their beliefs coincide with those of others, whether they are consistent with their own beliefs about other related things, and whether their beliefs are born out with empirical evidence. Computers can be used to provide both an exploratory and representational aspect of the discipline. The role of teachers in this type of environment and the issue of whether stu- dents should use artificial or real data sets should be considered.

The main aim of our research has been to contribute to an elucidation of the following problem: from what age, and in what sense, is it possible to speak of an intuition of chance and probability in the child? Taking Piaget's view as our starting-point, we have assumed that the central feature is that of the relationship between the possible and the necessary. At the same time, we have looked at other aspects which seemed likely to throw light on the problem as a whole: (a) Probability can be expressed either as a prediction of a single isolated event (without specifying the hypothetical multiplicity of its origin), of as a prediction of several events - which may be repetitions of a single event, or a succession of different events. We therefore need to know to what extent children of different ages understand the concept of relative frequency, and the extent to which this enters into their understanding of probabilistic situations. (b) How far is the child able to abstract a common probabilistic structure from different specific contexts and situations? It seemed important to us to include series of events described by unequal probabilities. Reprinted from Enfance 2 (1967), 193 - 206.

This article described three heuristics that are employed in making judgments under uncertainty: (i) representativeness, which is usually employed when people are asked to judge the probability that an object or event A belongs to class or process B; (ii) availability of instances or scenarios, which is often employed when people are asked to assess the frequency of a class or the plausibility of a particular development; and (iii) adjustment from an anchor, which is usually employed in numerical prediction when a relevant value is available. These heuristics are highly economical and usually effective, but they lead to systematic and predictable errors. A better understanding of these heuristics and of the biases to which they lead could improve judgments and decisions in situations of uncertainty.

Daniel Kahneman and Amos Tversky have proposed that when judging the probability of some uncertain event people often resort to heuristics, or rulers of thumb, which are less than perfectly correlated (if, indeed, at all) with the variables that actually determine the event's probability. One such heuristic is representativeness, defined as a subjective judgment of the extent to which the event in question "is similar in essential properties to its parent population" or "reflects the salient features of the process by which is is generated" (Kahneman & Tversky, 1972b, p. 431, 3). Although in some cases more probable events also appear more representative, and vice versa, reliance on the representativeness of an event as an indicator of its probability may introduce two kinds of systematic error into the judgment. First, it may give undue influence to variables that effect the representativeness of an event but not its probability. Second, it may reduce the importance of variables that are crucial to determining the event's probability but are unrelated to the event's representativeness.