# Research

• ### Empirical research on probability concepts

A frame of an interplay between intuitions of individuals and abstract theories is developed. Within this frame difficulties in learning/understanding a theory may be described by a breakdown in communication between the intuitive level of individuals and the "official" language of theory (used in class). It is, however, the scope of pertaining empirical research on probability concepts to investigate such intuitive imaginations. We will show that the interpretation of results is a precarious task and cannot successfully be done without regarding this interplay between intuitions and mathematics frame. Among the major problems are: Understanding of concepts and problem solving strategies are on different levels. It is hard to get information about the concept level in probands. The spectrum of intuitions covered be the investigations so far is very narrow, a huge percentage of diagnosing items referring to the symmetry view, few being related to the frequency aspect, hardly any to subjectivist ideas. The narrowness of the view on probability leads you inextricably into puzzles and troubles and may cause a breakdown in communication between interviewer and interviewed. The results of the discussion are directly transferred to the teaching situation. Some clues for improving teaching in class may be drawn from it.

• ### The use of statistical heuristics in everyday inductive reasoning.

"In reasoning about everyday problems, people use statistical heuristics, that is, judgmental tools that are rough intuitive equivalents of statistical principles. Statistical heuristics improved historically and they improve ontogenetically. Use of statistical heuristics is more likely when (a) the sample space and the sampling process are clear, (b) the orle of chance in producing events is clear, (c) the culture specifies statistical reasoning as normative for the events. Perhaps because statistical procedures are part of people's intuitive equipment to begin with, training in statistics has a marked impact on reasoning. Training increases both the likeli- hood that people will take a statistical approach to a given problem and the quality of the statistical solution. These empirical findings have important normative implications."

• ### From thumbtacks to inference

During the three years 1978-1981 a research project based at Loughborough in the East Midlands region of England investigated probability concepts of 11-16 year olds. A test of twenty-six questions was administered to a stratified sample of 2930 pupils from comprehensive mixed schools. These pupils were also given a test of general reasoning ability. the project's findings have been summarized in a 40 page booklet which includes all the test questions and analyses of responses. Nearly all test items, displayed an improvement in performance with increasing chronological age and with intellectual ability. Items requiring only the comparison of two direct quantities were well done by all ages tested (11 to 16 years), but those requiring comparison of two ratios were very poorly done, especially below the age of 15 years. This contrast is exemplified by the results for the two items. In this article we shall look at the development and test results of just one of the questions used.

• ### A bibliography on the teaching of probability and statistics

Recently, one of the compilers of this bibliography published a review of selected publications on the teaching of probability and statistics. A computerized listing of the complete bibliography was made available as a supplement on request. This report is an expanded and updated version of that work. It presents a listing of available literature on the teaching of probability and statistics.

• ### Teaching reasoning

Twentieth-century psychologists have been pessimistic about teaching reasoning, prevailing opinion suggesting that people may possess only domain-specific rules, rather than abstract rules; this would mean that training a rule in one domain would not produce generalization to other domains. Alternatively, it was thought that people might possess abstract rules (such as logical ones) but that these are induced developmentally through self-discovery methods and cannot be trained. Research suggests a much more optimistic view: even brief formal training in inferential rules may enhance their use for reasoning about everyday life events. Previous theorists may have been mistaken about trainability, in part because they misidentified the kind of rules that people use naturally.

• ### Use of the Representativeness Heuristic in a Situation of Simulation

The research of psychologists, in particular from Kahneman and Tversky, has shown that in many situations of everyday life, people estimate the probability of random events using certain heuristics, specially representativeness. Much of the subsequent research in this area supports their thesis. Nevertheless, most of this work has used verbal problems as the means of studying people's conceptions and thinking, whether in a questionnaire or in an interview. In this work we present the results of a study of the pupils' use of representativeness in a situation of simulation of one of the classical problems related to the subject. The experience consisted of an individual interview with the students while simulating this situation and graphically representing the results, in order to answer some predetermined questions posed by the researcher. The analysis of student's pattern of responses before and after the realization of the simulation shows a wide variety of conceptions and the influence of the result of this simulation on the initial arguments of the pupils. As a result of this we conclude the didactical possibilities of simulation both as a means of exploring pupil's probabilistic intuitions and as an educational tool to overcome some of the misconceptions concerning these intuitions.

• ### The child's intuition of probability

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.

• ### Students' Understanding of the Significance Level in Statistical Test

In this paper the initial results of a theoretical- experimental study of university students' errors on the level of significance of statistical test are presented. The "a pripri" analysis of the concept serves as the base to elaborate a questionnaire that has permitted the detection of faults in the understanding of the same in university students, and to categorize these errors, as a first step in determining the acts of understanding relative to this concept.

• ### The ups and downs of the hope function in a fruitless search

In order to investigate the nature of probabilistic reasoning, we devised two experimental problems, each of which involved two hope questions (long- and short-term). We will present the two standard problems along with the Bayesian solution. Then we will discuss a number of features of the solution by applying it to a variety of situations. After describing how our subjects reasoned about the standard problems, we will present a didactic device we developed to make the search problem more conducive to resolution. Finally, we will explore subjects' ability to transfer the lesson learned from the didactic device to the analogous wait problem.

• ### Elementary teachers' representations of relationships among statistics concepts

This study examined ways the elementary school teachers represented their understanding of the broad area of statistics. Special attention was paid to their understanding of the relationships among four critical statistics concepts in the North Carolina Standard Course of Study for Grades K-6. Relationships were examined through concept maps that the inservice teachers drew at the beginning of a three-week summer workshop on statistics.