Literature Index

The teaching of probability is currently being reinforced in many countries, as it is visible from new curricular documents such as the NCTM Standards (2000), where the acquisition of a precise language in connection to chance and probability is considered to be a main learning goal. On the other hand, textbooks are an important resource for teachers who can find in these books ideas and activities to facilitate students' learning. In recent Spanish curricular documents (M.E.C., 1992) the teaching of probability is introduced at earlier ages with a teaching methodology based on simulation and experimentation. A main concern is children's progressive acquisition of a precise language in connection to chance and probability. This curriculum is not an exception, since we find similar concerns in curricular documents from other countries, such as the United Kingdom or the United States. On the other hand, when children are first taught probability, they have frequently used terms and expressions to refer to randomness, sometimes with a meaning different to what is usual in the mathematics classroom. All these reasons suggest the interest to carry out an empirical study to determine the specific language that about chance and probability is presented in the textbooks.

This paper examines how two twelve-year-old students built up their recognition and understanding of relationships in a set of data. Using a small multivariate dataset created by Watson, Collis, Callingham and Moritz (1995), the students conducted an investigation of their choice in a pencil-and-paper environment. The students' thinking across the three representations of cards, tables and graphs is analysed from the perspectives of transnumeration, consideration of variation, reasoning with statistical models, and integrating the statistical with the contextual, which were identified as fundamental statistical thinking elements in empirical enquiry in the framework of Wild and Pfannkuch (1999). The ways of thinking within each element across the representations are identified. In the analysis, references are also made to the types of statistical thinking present in the other ten students in the study. From the analysis we identified five issues that should be considered for determining how students construct meanings from data. They are: prior contextual and statistical knowledge; thinking at a higher level than constructed representations; actively representing and construing; the intertwinement of local and global thinking; and the changing statistical thinking dialogue across the representations.

The computer algebra system, MathematicaTM, is used to determine the exact distributions for sums and means of small random samples taken from a specific probability density function. The method used is the Inverse Laplace Transform for real-valued functions. These distributions are used to compare exact probabilities for probability interval statements for sums and means with normal approximations for these probabilities using the Central Limit Theorem. The maximum normal approximation errors are determined for probability intervals for various sample sizes.

In Italy the knowledge both of the environments in which the teachers work and of their attitudes towards the teaching of mathematics in general and of probability and statistics in particular, is in extremely short supply. The survey of which the broad outlines are presented here aims to fill this gap. The intention is to provide material for policies of reform for the school levels considered. The outstanding result is in the way it brings out the great differences, not only in basic knowledge and training of teachers, but also in their attitude towards the teaching of mathematics and in particular probability and statistics. This makes it particularly difficult to propose a standard syllabus for the subjects previously mentioned at the Upper Secondary School level., and yet this tendency of policy-makers, at least for the first to be reformed. It is quite clear that serious problems that arise at the level of teacher retraining derive from this.