# Theory

• ### Subjective probability and everyday life

Recent research on probability judgment indicates that people's ability to estimate probabilities is very limited. It is argued that people may lack the cognitive apparatus necessary for processing probabilistic information, in so far as probability judgments play an unimportant role in everyday life. When probability judgments occasionally are made in everyday life it is argued that they are not based on frequency data but on some more or less well grounded theory.

• ### Common difficulties with probabilistic reasoning

In the past two decades several influential organizations, including the national Council of Supervisors of Mathematics, NACOME, UNESCO, CEEB, and the Cambridge Conference on School Mathematics, have acknowledged the role that probability and statistics play in our society. Consequently, each has recommended that probability and statistics be included as part of the modern mathematics curriculum. Probabilistic reasoning may not be an easily acquired skill for most students, however. Several recent studies have reported that even after instruction, many students have difficulties developing an intuition about the fundamental ideas of probability. Without this intuition they fail miserably when forced to reason about probable events.

• ### A curriculum to improve thinking under uncertainty

Recent evidence indicates that people's intuitive judgments are sometimes affected by systematic biases that can lead to bad decisions. Much of the value of this research depends on its applicability, i.e., showing people when and how their judgments are wrong and how they can be improved. This article describes one step toward that goal, i.e., the development of a curriculum for junior high school students aimed at improving thought processes, specifically, those necessary in uncertain situations (probabilistic thinking). The relevant psychological literature is summarized and the main guidelines in the curriculum development are specified: (a) encouraging students to introspect and examine their own (and other's) thought processes consciously, (b) indicating the circumstances in which common modes of thinking may cause fallacies, and (c) providing better tools for coping with the problems that emerge. Two detailed examples are given. In addition, the problem of training teachers is briefly discussed and a small-scale evaluation effort is described.

• ### Broadening the scope of statistics and statistical education.

The profession of statistics has adopted too narrow a definition of itself. As a consequence, both statistics and statisticians play too narrow a role in policy formation and execution. Broadening that role will require statisticians to change the curriculum they use to train and develop their own professionals and what they teach nonstatisticians about statistics. Playing a proper role will require new research from statisticians that combines our skills in methods with other techniques of social scientists.

• ### Statistical inference

The core concept, around which all statistics teaching should be based, is probability. The basic ideas of probability and utility should be taught to everyone, because every citizen is forced to make decisions in an uncertain world.

• ### Future statistics education

Statistics has rapidly developed with the help of modern probability theory and with the effective use of computers. Such an accomplishment marks a new era for statistics education. It is now time to think of how to teach statistics, taking into account both the subjects to be chosen and the actual method of education. At present we can see many places where mathematical statistics is efficiently used and where people are requested to learn statistics, not only in academic institutions, but also in daily life. The remarkable fact is that, compared to the past, the need for statistics has changed, and its appearances in actual subject fields have become highly modernised. We are therefore led to discuss now to teach statistics and to think of what topics should be taught. At this juncture, we are going to look over the present stage of practical use of statistics and to propose some ideas of statistics education by focussing our vision on high school level mathematics.

• ### Graphicacy in the primacy curriculum

A hierarchical model is developing graphicacy in the primary curriculum. The model stresses the importance of a progression in graphical work. As indicated, it was not meant to be a rigid, finalised version; rather it is designed to provide an initial framework for discussion. Readers interested in teaching statistics to young children will find this article an excellent basis for developing their own curriculum plan.

• ### Counting noses and scary things: Children construct their ideas about data

At every point in their development, students are engaged in serious intellectual work as they attempt to construct their own understanding of the world and their relation to it. As part of this work, they are immersed in mathematical ideas which are just at the edge of their understanding. In this paper, I will first discuss the nature of the mathematics in which the child in the primary grades can engage in the context of data analysis, and then give some examples of children's work in this area to illustrate how young children must construct for themselves key processes which are the building blocks of collecting, describing, and interpreting data.

• ### The central role of the PC in teaching statistics at school

Usually the PC is used in statistics to do quickly and conveniently what we have always been doing. This is a misuse of the PC since it has the potential to change statistical practice fundamentally. Historically, statistics was developed when computation was hard and expensive. To avoid massive computation a lot of sophisticated theory based on asymptotics was developed. Now computation is cheap and easy. The PC should replace sophisticated theory by simple computations, and make statistics more comprehensible. We should strive for understanding. Sophisticated statistical software is pedagogically harmful. We do not want to solve a dozen problems a day by using recipes. We want to solve a few paradigmatic examples in a few weeks. School statistics should solve a small number of fundamental problems, not quickly, but leisurely. We should derive programs for the solutions, which are general enough, so that they solve a whole class of problems. I do not advocate much deep programming, which is quite difficult. On the other hand, statistical software is for professionals. To learn its use requires an effort comparable to the effort of learning a new programming language. Design of a program is an important part of the learning process. A problem is solved if you have an efficient algorithm for its solution, which you understand.

• ### Teaching statistics to 16-18 year olds: Dreams and reality

Many statisticians are convinced that statistics, as an independent and compulsory subject, should be included in secondary-level general education. There are different (sometimes quite ambitious) proposals for the subject matter. Our real possibilities seem, however, not to match them. In this paper we discuss some aspects of this confrontation. Our conclusion is, that to a certain extent, statistics can be dealt with within mathematics. Examples are given to show how one can embed statistical notions into traditionally taught mathematics. Still much effort has to be taken to improve conditions for a widespread introduction of an independent statistics course.