Literature Index

The main objective of our research has been to study the intuitive biases corresponding to certain fundamental concepts and methods of the theory of probability. This is not simply a question of the general concepts of chance, or randomness, on which there are data already available. Research which has already been done by psychologists indicates the existence of a natural intuition of chance, or even of probability (cf. Fischbein et al., 1967, 1970a, b). However, A. Engel, a mathematician, has written: "...we have a natural geometric intuition but no probabilistic intuition". In order to elucidate this problem, we decided to go beyond the notion of chance, and try to follow the course of what, in fact, happens during the systematic teaching of certain concepts in the theory of probability. We therefore decided to study the intuitive responses of subjects to certain concepts and calculational precedures which were introduced during some experimental lessons on probability, viz. chance, and probability as a metric of chance; the multiplication of probabilities in the case of an intersection of independent events, and the addition of probabilities in the case of mutually exclusive events. Reprinted from Educational Studies in Mathematics 4 (1971), 264 - 280.

This chapter proposes design principles for developing statistical reasoning in elementary school. In doing so, we will draw on a classroom design experiment that we conducted several years ago in the United States with 12-year-old students that focused on the analysis of univariate data. Experiments of this type involve tightly integrated cycles of instructional design and the analysis of students' learning that feeds back to inform the revision of the design. To ground the proposed design principles, we first give a short overview of the classroom design experiment and then frame it as a paradigm case in which to tease out design principles that address five aspects of the classroom environment that proved critical in supporting the students' statistical learning:<br>o The focus on central statistical ideas<br>o The instructional activities<br>o The classroom activity structure<br>o The computer-based tools the students used<br>o The classroom discourse

The present study used both quantitative and qualitative methods to investigate the learning and motivational strategies used by students in a beginning-level statistics course. The research questions that guided the investigation are: (1) Do motivational variables account for unique variance in the academic performance of statistics students?, (2) Do deeper-level processing strategies account for unique variance in the academic performance of statistics students?, and (3) Do successful students report using different motivation and learning strategies than unsuccessful students in a beginning-level statistics course? Ninety-four students enrolled in six sections of the same course over a two-year period completed measures designed to assess attitudes about statistics, motivation and learning strategies use as well as previous math and statistics knowledge. In addition, randomly selected participants were interviewed about how they prepared for their midterm exam. The results of the study show that both motivation and learning strategies variables influenced performance in the introduction to statistics class. These results help to expand our understanding of what is involved in the process of learning statistics. Also, suggestions for teaching statistics are explored.

I have selected the stated problem which may be found in Takacs (1960,p.21) with a brief reference to statistical mechanics, for several reasons: its source is an important test used by many engineering students; the problem lends itself well to illustrate the ideas on imagination presented in this paper and, with imagination unfettered, it can serve to generate quite exciting purely mathematical questions.