# Theory

• ### Issues in assessing conceptual understanding in probability and statistics

Research has shown that adults have intuitions about probability and statistics that, in many cases, are at odds with accepted theory. The existence of these strongly-held ideas may explain, in part, why learning probability and statistics is especially problematic. One objective of introductory instruction ought to be to help students replace these informal conceptions with more normative ones. Based on this research, items are currently being developed to assess conceptual understanding before and after instruction.

• ### Intuitive strategies for teaching statistics

Individual thinking is driven by intuitions which have nearly no counterpart in the concepts which one learns from theory. This is especially true for stochastics teaching and is the main cause that is not very effective. The author reports about powerful strategies that might bridge the gap between individual intuitions and formal concepts. This gives a clear insight into difficult concepts and changes the behaviour of the learners.

• ### Random means hard to digest

Different sequences are reproduced or memorized with varying degrees of difficulty, depending on their structure. We obtained preliminary support for the hypothesis that difficulty of encoding is correlated with the perceived randomness of the sequence. Since the randomness of a sequence can be defined by its complexity, namely, the length of the shortest computer program for reproducing the sequence, we suggest that introducing randomness in terms of complexity may foster students' understanding. Subjective complexity, however, is maximal for sequences with exaggerated alternations, as is apparent-randomness. Thus, misperceptions of randomness cannot be corrected by the complexity approach. They can only be better understood.

• ### Computers in the statistics curriculum

This article is, in essence, a condensed extract from the comprehensive report Computers in the mathematics Curriculum (1992) recently published by the Mathematical Association and produced by a subcommittee of the Teaching Committee.

• ### A feminist approach to the introductory statistics course

That women on the average tend to suffer from math anxiety and to perform less well in advanced mathematics classes, when they are found there at all, are repeatedly documented facts that operate as highly effective barriers to women's achievement in a variety of domains. As a math anxious individual, I avoided all math in high school and agonized through the necessary courses as a traditionally aged student in college, and again as a returning student in graduate school. It seems ironic that one of the first courses I instituted when I became a college professor at a small liberal arts college for women was an introductory statistics course. Social psychology is my discipline, however, and one of the changes I noted between the time I earned my bachelors degree in 1964 and the time I entered graduate school in 1977 was that women had become a great deal more visible in psychology, even powerful in some instances. It seemed to me that many of these women also tended to be first-rate statisticians; in fact, rather than being intimidated by numbers, these women were actually using sophisticated statistics to help write women back into psychology. I decided to do what I could to work through my own math anxiety, and, in turn, to try to teach statistics in such a way that others, regardless of their discipline, would find the subject approachable, useful, even fun from their first exposure at the college level. In the beginning, I conceived of the course as simply taking a math-anxious approach. As time has gone by, I have learned more about Feminism as a philosophy/ideology and have begun to recognize that what I had called a math-anxious approach to statistics was actually a Feminist approach. With that recognition, I have begun to apply those principles even more consciously.

• ### Belief, knowledge, and uncertainty: A cognitive perspective on subjective probability

This paper presents a cognitive analysis of subjective probability judgments and proposes that these are assessments of belief-processing activities. The analysis is motivated by an investigation of the concepts of belief, knowledge, and uncertainty.

• ### Cognitive Strategies in Stochastic Thinking

Stochastic thinking denotes a person's cognitive activity when coping with stochastic problems, and/or the process of conceptualization, of understanding, and of information processing in situations of problem coping, when the chance or probability concept is referred to, or stochastic models are applied. In accordance with our view on stochastic thinking in decision making under uncertainty, three different aspects may be emphasized in psychological research: (1) the behavioral analysis, which may focus on analyzing or improving the product of the decision process. (2) The procedural analysis, which may, for instance, attempt to identify cognitive strategies of heuristics when tracing the process of thinking; and (3) the semantic or conceptual side, for instance, when analyzing the individuals's knowledge about probability or his/her use of probability to conceptualize uncertainty. Chapters: 1. Toward an understanding of individual decision making under uncertainty 2. The "Base-Rate Fallacy" - heuristics and/or the modeling of judgmental biases by information weights 3. A conceptualization of the multitude of strategies on base-rate problems 4. Modes of thought and problem framing in the stochastic thinking of students and experts (sophisticated decision makers) 5. Stochastic thinking, models of thought, and a framework for the process and structure of human information processing

• ### Visualization in Teaching and Learning Mathematics

This volume explores the role of visualization in mathematics education, especially undergraduate education.

• ### Mathematics in teaching processes- The disparity between teacher and student knowledge

When analyzing episodes of mathematics instruction from an epistemological perspective, it is seen that the disparity between teacher and student knowledge is not simply due to their knowing more or their knowing less. The independent and frequently incompatible levels of understanding knowledge which are peculiar to teachers and to students show how essential it is to make allowance for conceptual as opposed to material aspects, and how the condition of classroom processes nevertheless always tend to regress to a form of mathematical knowledge strongly determined by subject matter and method.

• ### Obstacles to Effective Teaching of Probability and Statistics

This paper describes four main categories of issues related to the effective teaching of probability and statistics at the precollege level. These issues relate to: 1. The training or retraining of mathematics teachers to teach statistics. 2. The role of probability and statistics in the mathematics curriculum. 3. The need for connecting research in difficulties students have learning probability and statistics concepts to classroom instruction, and 4. Assessment of student learning. Recommendations for dealing with these issues are offered.