This paper addresses the topic of inadequate preparation of the students in the mathematical and logical concepts which form the basis of statistical reasoning. First, a rationale for the assessment of students' entering skills is developed. This is followed by a discussion of the skills that should be included in the assessment. The final section discusses ways to use the results of these assessments.
Probability and Statistics are increasingly being given an important place in the K-12 mathematics curriculum. According to the NCTM Curriculum and Evaluation Standards (1989), students should learn to apply probability and statistics concepts to solve problems and evaluate information in the world around them. The statistics standards suggest using hands-on activities involving collecting and organizing data, representing and modeling data including the use of technology, and communicating ideas verbally and in written reports. Teachers are encouraged to help students develop important ideas (for example, about distributions, randomness, and bias) and gain experience in choosing appropriate techniques to use in analyzing data. Many teachers are currently using materials from recent projects or projects in development that have developed curricula and software to implement the NCTM Standards (e.g., the Quantitative Literacy Project, the Reasoning under Uncertainty Project, and the ChancePlus Project). These new materials encourage teachers to have students work on statistical projects: formulate research questions, collect and analyze data, and write up the results. Working on statistical projects individually or in groups engages students in learning about statistics and helps them to integrate the knowledge they have learned.
This chapter discusses the attempts to included probability and statistics in the curriculum at the secondary level.
We shall present four important areas in teaching data analysis: the selection of problems and data sets; the critical concepts underlying the process of data analysis; the development of pupils' facility with a range of representations, including the role of technology; and the management of data analysis activities in the classroom.
In this short paper important strands of research in probabilistic notions will be critically presented, followed by an indication of the author's own research.
In the following article some examples used in empirical investigations are discussed to show how difficult empirical research really is. A problem catalogue should have an impact on critical analysis of major research work in this field which is still to be done.
In the tradition of Kahneman and Tversky some interesting heuristics have been outlined and discussed, that might explain and predict failures in probabilistic situations. In a series of examples we will give a short description of some of these intuitive strategies. The so called representativeness heuristic is discussed at more depth. It is linked to the idea of quota sampling. Random sampling is nothing but a trick to have a good chance of getting representative samples. All in all the representativeness idea is shown to be a fundamental statistical idea.
Descriptive statistics plays a subsidiary role in the stochastics curriculum, it is used to introduce notions in probability theory by analogy to corresponding notions in descriptive statistics and to motivate problems in inferential statistics. New ideas are to be developed and discussed in this paper which might enrich teaching descriptive statistics and change its status, namely exploratory data analysis, "open mathematics", and the idea of visualization. Descriptive statistics can and should be taught as interesting subject of its own right. Furthermore the new ideas have consequences on the view on statistics as a whole.
While solving stochastical problems one often notices a certain discrepancy between the intuitive reasoning of the person involved, and the "objective" causes given by the mathematical theory. So, the paths to follow in either direction will usually turn out to be different ones and will not always lead to the same final answer.We have the greatest difficulties to grasp the origins and effects of chance and randomness. Also, the history of probability reports some problems and paradoxical examples which support the suspicion that stochastics is a rather exceptional science even within the mathematical fields. I shall introduce and discuss a small collection of problems of that kind i.e. problems which carry certain counterintuitive aspects. My objections here are manifold. First of all, the discussion of such problems, especially in the classroom, helps (i) to clarify ambiguous stochastical situations, (ii) to understand basic concepts on this field, (iii) to interpret formulations and results. Then, we, the teachers and professionals, can use them to test our own intuitive level of understanding. Finally, as those "paradoxes" and teasers have an entertaining aspect too, we should make use of this to increase the motivation of the students occasionally. Six of these problems were chosen to be discussed in the sequel. Here, I have tried to present them in a unique form. First, the problem will be formulated, an illustration included. Then, a "hint" is given, which adds (or stresses) some information about hidden processes or about strategies, which I recommend to follow. Thirdly, one solution is outlined, although very often several different approaches are known. Where possible I have chosen the one which follows a general idea. Finally, some variations, comments and references are added.
Some standard probability distributions (like the binomial or Poisson) are especially attractive for teaching. They can be decomposed into smaller "moduls". These moduls are much easier to handle, a check of them could be a statistical test, but a judgement by mere reflection of the situation is also possible. It is argued that the process of decomposing into moduls and synthesizing them to the original situation enables insight into the stochastic structure of the standard situations.