Theory

• Teaching statistics to the rest of humanity

Statistics is taught widely in the UK as an element of mathematics up to age 16. Over a quarter of a million children taking examinations involving statistics or mathematics with statistical elements. At the age of 18 the examinations involving the use of statistics show a different pattern. Some 50 thousand pupils take statistics or mathematics with statistical elements, but over 100 thousand take examinations in geography, biology, psychology, etc. involving significant elements of statistics. Thus the prime interest in statistics comes not from those with a specialist interest but from the "rest of humanity" who need statistics to support their other interests. The aim of this paper is to explore some of the implications of this fact.

• Statistical Reasoning: What is the Role of Interential Rule Training? Comment on Fong and Nisbett

Fong and Nisbett (1991) argued that, within the domain of statistics, people possess abstract rules; that the use of these rules can be improved by training; and that these training effects are largely independent of the training domain. Although their results indicate that there is a statisically significant improvement in performance due to training, they also indicate that, even after training, most college students do not apply that training to example problems.

• How Students Learn Statistics

Research in the areas of psychology, statistical education, and mathematics education is reviewed and the results applied to the teaching of college-level statistics courses. The argument is made that statistics educators need to determine what it is they really want students to learn, to modify their teaching according to suggestions from the research literature, and to use assessment to determine if their teaching is effective and if students are developing statistical understanding and competence.

• Inevitable Illusions: How Mistakes of Reason Rule our Minds

This book proposes to set out the recent scientific discovery of an unconscious. Not the unconscious or subconscious explored by psychoanalysis, but one that always and unbeknownst to us involves the cognitive: that is, the world of reason, of judgment, of the choices to be made among different opportunities, of the difference between what we consider probably and what we consider unlikely. The material we deal with in this book derives from wherever we make decisions "under uncertainty." In short, our examples are based on phenomena found almost anywhere, in almost anyone, and just about at any moment. Chapters include: Probability Illusions, Calculating the Unknown, or Bayes' Law; The Fallacy of Near Certainty, and The Principle of Identity and the Psychology of Typicality.

• The hot hand in basketball: On the misperception of random sequences

People's intuitive conceptions of randomness have been found to depart systematically from the laws of chance. This is illustrated in the game of basketball. Players and fans have been found to believe in "streak shooting," a phenomenon involving the belief that players have a better chance to get a basket after a few successful attempts despite the statistical odds against such an occurrence. This misconception seems to affect how the game is played as well since many coaches and players believe that it is important to pass the basketball to a player that has successfully attempted most shots. This finding is consistent with "gambler's fallacy." It is suggested that the belief in the law of small numbers could be due to performance heuristics since strings of successful shots are more memorable than mixed ones.

The authors state the view that many intuitive judgements, right and wrong, are produced by the application of heuristics such as the representativeness and availability. Moreover, observed errors of judgement have two kinds of implications: they may illustrate a judgement heuristic or they may indicate a failure to correct the error once the intuition was articulated.

• Understanding graphs and tables

This article states that there are two considerations for learning to graph data: 1) the structure of the phenomenon, and 2) the limitations of the format of graphical representation used. This paper provides historical examples of how graphic data has been used, highlights aspects of a display theory, and identifies concrete steps to improve the tabular quality of graphs. According to the author, a graph has the power to answer most commonly asked questions about data, and invite deeper questions as long as it is properly drawn. Bertin's (1973) levels of questions that can be asked from a graph are described. The first level deals with elementary questions which involve simple extractions of data from the graphs. The second level deals with intermediate questions. These refer to trends among multiple points in the data, and the identification of outliers. The third level involves overall questions which requires an understanding of the deep structure of the data in its totality, often comparing trends or groups in the data, and the overall message of what is being said in the picture. Questions that involve retrieving data from tables are almost always elementary questions. These only require that students understand discrete units of data. Wainer states that most tables do a disservice by confusing the kinds of data presented. Columns are often placed without much thought to the relevancy of their order. The order within the columns is similarly vulnerable to irrelevance. For example, criterion variables (like countries) are often presented in alphabetical order when they should be arranged on a concept that is more useful (like size of GNP or population). The author therefore recommends that rows and columns are ordered in a way that makes sense and that numbers be rounded off as much as possible.

• Summary and conclusions

The development of scientific thinking is centered around the development of skills in the coordination of theories and evidence. Three skills are required to achieve ideal coordination: 1) the ability to think about a theory rather than to think with it (i.e., awareness and control of a theory, to use and contemplate it- the ability to evaluate the bearing of evidence on a theory is due to such awareness., and the ability to see that a theory may be false and that alternative theories exist); 2) theory and evidence must be differentiated; and 3) ability to temporarily set aside one's own acceptance (or rejection) of a theory in order to assess what the evidence itself would mean for the theory. Two factors are required for the development of skills in coordinating theory and evidence: 1) exercise in relating evidence to multiple theories and 2) development in the skills involved in the interpretation of evidence given that it is sufficiently differentiated from theory. In conclusion, the main finding in all these studies was that older children, adolescents, and adults are limited in their understanding of covariation and its connection to causality.

• A methodology to study the students' interaction with the computer

The students' interaction with the computer poses new problems of research in Mathematics Education and also offers new methodological resources. One of these is the possibility to employ the recording of the students' interaction with the computer as a technique to gather data on the processes that the students follow to solve the proposed tasks. In this work, different examples of the use of these records in research on Mathematics Education are analyzed, showing the diversity of the obtainable data and the dependence of the same with respect to the "roles" carried out by the computer and the student. We end by presenting the method of analysis of these records that we have used in our own research, that combines qualitative and quantitative elements and can be easily adapted to other research.

• The Theoretical Status of Judgmental Heuristics

Past empirical research on judgmental heuristics and biases has focused on questions of classification, with relatively little attention given to the development of general theoretical principles. It is the latter, however, that ultimately will lead to conclusions of greater generality and usefulness. A selected review of the literature on representativeness indicates some of the effects for which any complete theory must account and some of the limitations in many of the current experimental designs. The manner in which people use probabilistic information depends, in part, in (i) information specificity, (ii) information salience, (iii) individual differences, and (iv) problem wording. Theories are needed to elucidate these phenomena, and experimental designs are required that pay more attention to them.