# Unpublished Manuscript

• ### Probability and statistics in the junior secondary school: The proposal of the Nucleo Di Ricerca Didattica of Pavia

The project for the teaching of elements of Probability and Statistics at the ages 11 - 14 which I intend to describe is the result of the work of the Didactic Research Group of Pavia (1), formed of 5 university researchers and about 20 in-service teachers. In this paper I will refer in particular to the Mathematical contents of our proposal. But in order to reach a good understanding of the proposed work I should like to stress that the classroom activity we propose should include these very important steps: the presentation of a problematic situation, the intuition and explanation of one or more solutions, the collective discussion, the generalization and sometimes the formalization of the results and finally, if it is possible, their use in interesting real-life applications.

• ### Principles of Learning Probability and Statistics

Probability judgments may have to be revised if new information is available. From the mathematical perspective probability revisions are intimately connected to the notion of conditional probability and Bayes' formula, a subsidiary concept and a trivial theorem respectively. Nevertheless empirical investigations in subjects' understanding of probability do indicate that people do not cope adequately with situations involving probability revisions, if they have been taught the mathematical concepts or not does not matter. In what follows I will try to sketch some phenomena of misunderstanding, give some comments on the interplay between mathematics and intuitions which I think represents the origin of lack of comprehension. A brief overview on the favor concept should enable the impression that by way of teaching this concept probabilistic reasoning could be improved.

• ### Children's Understanding of Symmetry

Freudenthal (1982) has observed that Symmetry as a source of stochastic understanding is a virtually unknown and badly neglected intuitive and didactic tool. Saying this should be a platitude, but didacticians appear to be no faster learners than their students. In the teaching of probability three common aids which assume symmetry are frequently used: coins, dice and urns. This paper will discuss some aspects of children's understanding of the first two of these aids.

• ### Report on a study tour

During the period of August-September 1983 I carried out a study-tour of parts of Canada and the U.S.A. This tour was supported by the polytechnic, the Institute of Statisticians (to visit the ASA), the Open Tech Project (to attend a Quality Assurance Continuing Education seminar) and, primarily, a Royal Society study grant. I would like to express my appreciation for all this support. I talked to many people and visited many organizations. I have tried to report on my visit in a list below the papers and I enclose papers as appropriate. If you wish to have copies of any other papers please let me know. Clearly the views expressed are based on one person's contacts with specific people and organizations. I hope however that, as I met many of the most respected people in the areas covered, the view presented is not too distorted.

• ### Evaluation of the Quantitative Literacy Project: Teacher and student surveys

An initial evaluation plan was outlined by a team of QL project directors and evaluation consultants. This plan had seven key components to be investigated by different individuals and teams. One component was a survey to gather descriptive information on participating teachers. the DESCRIPTIVE INFORMATION survey was designed and mailed out by QL staff. Some of the information gathered through the survey is summarized in this report. Another component of the plan was a sample survey, designed to obtain information on both attitude and pedagogy from a large group of teachers. In addition, teachers were to be surveyed about their reactions of the QL training sessions. Students from selected classes of the teachers were also to be surveyed with specially designed instruments. Questions were posed to structure this part of the evaluation and to guide development of appropriate instruments. This report describes how answers to these questions were sought with the teacher and student survey components of the evaluation.

• ### Another Look at the Probabilities of the Notorious Three Prisoners

The "Problem of the Three Prisoners," a counterintuitive puzzle in probability, is reanalyzed, following Shimojo and Ichikawa (1989). Several intuitions that are examined represent attempts to find a simple and commonsensical criterion to predict whether and how the probability of the target event will change as a result of obtaining evidence. However, despite the psychological appeal of these attempts, none proves to be valid in general. A necessary and sufficient condition for change in the probability of the target event, following observation of new data, is proposed. That criterion is an extension to any number of alternatives, of the likelihood principle, that holds in the case of only two complementary alternatives. It is based on comparison of the likelihood of the data, given the truth of the target possibility, with the weighted average of the other likelihoods. This criterion is shown to be psychologically sound, and it may be assimilated to the point of becoming a secondary intuition.

• ### Children's subjective notions of probability

This paper presents the results of an instrument designed to probe children's intuitive notions of probability. The test consists of 16 questions, a few of which are analyzed in depth either qualitatively or quantitatively. A good selection of pupils' responses are included to illuminate children's intuitions. It is suggested that a mixture of teaching approaches should be used to help children develop probabilistic concepts coherently.

• ### Representing Probabilities with Pipe Diagrams

In this article I introduce a way of representing probabilities which has shown promise both in developing a quantitative interpretation of probability and in helping students understand basic arithmetic operations on probabilities. I refer to this representational device as a "pipe diagram." Many will recognize pipe diagrams as modified tree diagrams. By way of introduction, I first show a standard tree-diagram solution to a typical probability problem.

• ### Initial considerations concerning the understanding of probabilistic and statistical concepts in Australian students

Following the questions raised by Watson (1992) concerning research in probability and statistics education in Australia in the 1990's, this paper reports on the initial trailing of items with 64 Grade 9 and 10 Grade 6 students. The analysis supports the belief that misconceptions observed in other countries also are present in Australia. Further, the application of a developmental cognitive model offers promise for classifying responses to items and structuring remediation procedures. Suggestions are made for the next stage of research in the area.

• ### The theoretical nature of probability and how to cope with it in the classroom

In this paper the role of the theoretical character of stochastics is considered for the teaching process and its organisation. The consequence of the theoretical character of stochastic knowledge - or the fact that the basic concepts cannot completely define probability theory, but are conversely only formed while developing probability theory - is that the teaching process must be organized differently. It is inappropriate to start from ready-made concepts, gradually adding further knowledge. Conversely, it is necessary to begin with meaningful situations which permit the forming and developing of concepts. Freudenthal characterizes this inversion of the teaching method by introducing mental objects.