# Theory

• ### Children's conceptions of probability - A psychological and pedagogical review

This paper identifies a number of key questions concerning children's understanding of probability.

• ### What gets assessed is what gets learned

Performance based assessments of statistical thinking provide teachers with a window to students' conceptual development. With this aim in mind, we examined the performance of a pair of fourth/fifth-grade students working collaboratively on a probabilistic assessment task developed by the California Department of Education. To account for student performance, we developed a cognitive model of the knowledge guiding student performance. Using this cognitive model, we revised the assessment to provide opportunities for assisted performance. We believe that performance assessments of statistical thinking should be designed with ways to assist performance, a view consistent with visions of teaching and learning as assisted performance.

• ### Some remarks on empirical investigations on probability and statistics concepts

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.

• ### On 'Representativeness' - A fundamental statistical strategy

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.

• ### The role of descriptive statistics

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.

• ### Why certain probability distributions do offer stochastical insight and others do not

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.

• ### How to make standard deviation meaningful

Variability is a somewhat difficult concept. There is no internal relation in it which gives you insight into the concept of standard deviation at an intuitive level. Some relations that are very important to standard deviation are discussed. They relate standard deviation to mean and other notions at an abstract level. In a broader context of "statistics (mathematics ) as a tool to communicate" they loose their fascination. Maybe we will overcome this problem by techniques of exploratory data analysis and thus cope with the idea of variability more successful in the future.

• ### A note on the birthday problem

Formal calculations often do not yield insight into "how your model solves your problem". A strategy in connection to the birthday problem is discussed that does give intuitive orientation. Furthermore some situations do not seem "stochastical" at first sight but can be structured by stochastical models. Even in these models do not fit the situation you can get sensible results from that models. The birthday problem is shown to be a problem of that type.

• ### Research in probability and statistics: Reflections and directions

Probability and statistics (stochastics) are viewed as necessary for all students no matter their ambitions. However, there are barriers to the effective teaching of both stochastics and problem solving: 1) getting stochastics into the mainstream of the mathematical science school curriculum; 2) enhancing teachers' background and conceptions of probability and statistics; 3) confronting students' and teachers' beliefs about probability and statistics. Psychologists and mathematics educators should work collaboratively to diminish misconceptions. Doing so combines the roles of observer, describer, and intervener. Research in stochastics suggests that heuristics that are used intuitively by learners impede the conceptual understanding of concepts such as sampling. This paper reviews the research on judgemental heuristics and biases, conditional probability and independence (i.e., causal schemes), decision schema (i.e., outcome approach), and the mean. Learners have difficulties in these areas, however, evidence is contradictory as to whether training in stochastics improves performance and decreases misconceptions. The conclusion emerging from this research is that probability concepts can and should be introduced into the school at an early age. Instruction that is designed to confront misconceptions should encourage students to test whether their beliefs coincide with those of others, whether they are consistent with their own beliefs about other related things, and whether their beliefs are born out with empirical evidence. Computers can be used to provide both an exploratory and representational aspect of the discipline. The role of teachers in this type of environment and the issue of whether stu- dents should use artificial or real data sets should be considered.