# Chapter

• ### Why teach statistics and probability - a rationale

This is an introduction to the book on how to teach statistics to high school students. The authors suggest using your own data collection in weather charts and business stories. Another suggestion is to have students find erroneous uses of statistics and statistical reasoning in newspaper articles and to bring them into class for analysis.

• ### 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.

• ### 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.

• ### The Base Rate Fallacy Controversy

The article attempts to sketch a conceptual and experimental history of the base rate issue. The review distinguishes between a social judgment paradigm and a textbook paradigm. Theoretical explanations of the base rate phenomenon, i.e. the representativeness heuristics, confusing or inverting conditional probabilities, the specific factor, the causality factor, the vividness factor etc. are discussed with respect to these paradigms. On the basis of a report on various studies, the author hypothesizes that the different paradigms elicit different response tendencies, matching base rates on the one hand and judging representativeness on the other hand. She argues that on the causal structures accepted illustrating this by several examples. The consequences of this phenomenon for the role of normativeness and its (in-) determinacy are discussed.

• ### On the Testability of the Availability Heuristic

TVERSKY and KAHNEMAN's availability heuristic, although originally intended to account for only frequency and probability judgments, has been used to explain almost all kinds of social judgments. Accordingly, the process of judgment formation is mediated by the availability of memorized information, that is, by the ease with which relevant material can be recalled at the time when the judgment is made. Recall operations - either pure retrieval or reconstructive recall - are regarded as the determining subprocess within the process of judgment formation. In this article, empirical evidence is presented which is hardly compatible with such an account. A reaction time experiment, on the egocentric attribution phenomenon is described suggesting that Ss' claim to contribute more to various social activities than their partner is not caused by Ss' tendency to predominantly recall examples of their own activities. Within-judge correlations of recall and judgment latencies are rather low., and an analysis of the facilitating effect of prior judgments on subsequent recall latencies (for the same issues) does not reveal the kind of priming effect that would be expected if recall operations were already involved in the preceding judgments. Negative evidence from some other experiments on illusory correlations is also mentioned. These results are discussed in the context of methodological problems inherent in testing so-called judgmental heuristics.

• ### Conceptual and Theoretical Issues in Developmental Research on the Acquisition of the Probability Concept

This article provides a critical review of psychological theories and research approaches on the ontogenesis of the probability concept. An analysis of the conceptualization and interpretation or probability within developmental research reveals that with only one exception on objectivistic interpretation of probability has been made. The reviewed (theoretical) research approaches, the cognitive developmental theory of PIAGET &amp; INHELDER, FISCHBEIN's learning-developmental approach, and verious information processing model's differ in two main aspects. Firstly on the question whether the development should be considered to be a continuous or discontinuous process, and secondly the role of conceptual versus strategic knowledge for coping with probability problems is disputed. The discussion tries to point out what progress could be gained by skipping a one-sided objectivistic interpretation of the probability concept and turning to a conceptualization of probability encompassing both sides, the objectivistic and subjectivistic view. This integration might also lead to a deeper understanding of the individual's conceptualization of uncertainty.

• ### The Perception of Probability Situations in Pupils Aged 12 to 14

The probability concept is not simple: It can be disolved into many components and we shall describe some of them. From the general point of view, the student does not acquire all these components simultaneously, but rather crosses one treshold of the other. Hence, he/she always possesses but a partial understanding. He/she will be able to answer some questions which are compatible with the tresholds already attained; but he/she will systematically fail in other tests. Besides, the mere crossing of treshold is not sufficient. fixation is indispensible to secure the knowledge acquired. Otherwise, the pupil can be asked diverting questions which reveal the instability of the knowledge acquired. This paper is a personal view of a study of Jesus ALARCON (Mexico), who presented his doctoral thesis in Strasbourg in Juni, 1982. This research is not in a genetical perspective, as this would require comparing the results of children from different age groups. The study used approx. 300 12- to 14-year -old pupils who were distributed to 3 samples of approx. equal size, i.e., 3 questionnaires were presented with slight alterations of the items to be tested. The results of only one of these samples (106 pupils) will be discussed here.

• ### Randomness and Stochastic Independence - on the Relationship between Intuitive Motion Mathematical Definition

By means of historical investigations, epistemological reflections, and didactical analysis with respect to the notion of independence, we shall try to provide insights into the problem of a theoretical term and its applications. This will be the starting-point for stating some didactical theses about treating the notion of independence in the curriculum of Sekundarstufe I (lower secondary level) and will yield examples of their realization. The difference between intuitive notion and mathematical definition reflects the insoluble tension between mathematics and reality. This should not be seen as a shortcoming, rather this tension has been one of the productive sources for the development of mathematics, and it ought to be the same for mathematics instruction.

• ### Doing the impossible: A Note on Induction and the Experience of Randomness

In this article I discuss the fundamental relation between people's ability to do induction and their beliefs about randomness or noise, and I illustrate the special difficulties that psychologists face when they try to evaluate the rationality of these beliefs. The presentation is divided into four sections. The first describes the traditional experimental approach to evaluating people's conceptions of randomness and summarizes the data that have been taken to support the conclusion that people have a very poor conception of randomness. The second contrasts the relatively narrow conception of randomness that one finds in philosophical and mathematical treatments of the topic. The third outlines some benefits for psychologists to be gained from thinking about induction as a problem in signal detection., and the fourth presents the argument that any adequate evaluation of ordinary people's conceptions of randomness must consider the role that these conceptions play an inductive inference, that is, in distinguishing between random and nonrandom events. Originally appeared in the Journal of Experimental Psychology; Learning, Memory, and Cognition, 1982, 8 (6), 626 - 636