Research

• The success of graphic models to visualize conditional probabilities

Why is training of conditional probabilities necessary? - Conditional probabilities play an important role within the statistics curriculum and in decision making under uncertainty (Bayesian inference). - People often have problems to use conditional probabilities in the right way (many examples in the literature of cognitive psychology). So it is important that - although there exist a lot of didactical suggestions - there has been no empirical research dealing especially with the improvement of the understanding of conditional probabilities. So we made a first try to close this gap.

• Experience of students' participation in household surveys - A case study

This paper tries to investigate the feasibility of such a supplementary programme including 1) the feelings and opinions of the students who have participated in such projects, 2) whether there exists a sufficient supply of such research projects, and 3) the possible benefits the participating students can give to the research investigators.

• Solving problems is not enough: Assessing and diagnosing the ways in which students organize statistical concepts

In many statistics courses homework exercises and examinations focus primarily on solving problems. Marks are assigned to students' responses according to the degree to which a problem solution is correct and/or to which a student's procedure employed in the solution is correct. When a statistics course has an applications or data analysis orientation, instructors often find that even though students can solve textbook and examination problems, they are frequently unable to apply probability and statistics to solve "real world" research problems in which judgments have to be made about the technique(s) to be used and in which substantive interpretations of the results of statistical analyses need to be made. The paper reviews several formats of examination questions and assessment procedures which have been used over the years in noncalculus courses in applied statistical methods which focus on data analyses, parameter estimation, and hypothesis testing. Among the types of assessment techniques reviewed are short-answer questions, essay questions, yes-no questions with student-provided justifications, concept-oriented multiple-choice items, masterlist items, analogical reasoning items, graphic inference items, free association tasks, and concept mapping tasks. The paper also reports on a computer-assisted test of knowledge structure called MicroCAM. This test allows students to create on a computer screen a spatial representation of the way in which they perceive key statistical concepts to be linked one to another. The test also permits students to specify the type of relationship which links two or more concepts together. In this way a student's unique knowledge structure is revealed. Implications of these different assessment methods for diagnosing students' learning difficulties and for teaching statistics to mathematically naive students are discussed.

• Investigating the relationship between conceptual and procedural errors in the domain of probability problem-solving

The purpose of the study was to investigate the relationships among types of errors observed during probability problem solving.

• Feeling Certain About Uncertain Things: The Role of Beliefs and Attitudes in Learning Statistics

This paper is organized in three sections. First, we examine the justification for attending to non-cognitive issues within the larger context of the goals of statistics education Next, we briefly review and critique existing approaches to research on students' beliefs and attitudes towards statistics. Finally, we explore implications for assessment practices in statistics. Finally, we explore implications for assessment practices in statistics education and for further research. (By research we refer both to "academic" research done for increasing general knowledge, as well as to local research that individual teachers or statistics departments can, and in our view should, undertake in order to be informed about where their students stand and to be able to provide a better service to learners.)

• A longitudinal study of the effects of undergraduate education on reasoning

Does an undergraduate education improve reasoning about everyday-life problems? Do some forms of undergraduate training enhance certain types of reasoning more than others? These issues have not been addressed in a methodologically rigorous manner (Nickerson, Perkins, &amp; Smith, 1985). We consequently have little knowledge of whether reasoning can be improved by instruction, yet this question has long been regarded as central the theories of cognitive development.

• A Survey on the Place of Statistics in General Education in Turkey

The Turkish system of education is undergoing great structural changes. The centralized system is becoming more flexible, giving schools and students the opportunity to develop and select new courses based on the needs and interests of the students and the environment at the secondary level. Statistics appears for the first time as a separate four-hour per week elective course in the secondary programs. The changes require the development of new curricula on different subject areas and the revision of the present curricula. The mathematics curricula at all grade levels (K-11) have been revised by the National Mathematics Curriculum Development Committee established by the Ministry of Education in February 1990 and the revised curricula are being implemented in the 1991-92 school year. The place of statistics has not changed much in the revised mathematics curricula. Being a member of the above mentioned Committee, the researcher has observed that the mathematicians and the mathematics teachers do not seem to consider statistics as an area to be taught in mathematics. They mainly concentrate on algebra and geometry. It can easily be concluded that since those people do not have the necessary background and training in statistics, they view statistics topics, which interfere with their conception of mathematics, critically. Many modes of teaching and interpretation which are particular to statistics, as compared to traditional mathematics, are viewed skeptically by the mathematicians and the mathematics teachers. Considering the above situation, the researcher undertook a survey of the place of statistics in general education in Turkey. The results of this study show that the mathematics teachers who are presently responsible for teaching statistical topics at the primary and secondary levels do not have necessary background in statistics and do not possess the necessary skills for teaching statistics.

• Effects of Age and Instruction on Combinatory Ability in Children

The aim of the study was to investigate the effect of direct instruction on the ability to handle permutations and arrangements, as an example of a problem at the level of formal operations. 60 Bucharest school children, 20 aged 10 years, 20 aged 12 and 20 aged 14, tested individually, were first asked to estimate the number of possible permutations with 3, 4 and 5 objects. Results showed that these subjective estimates improved with age, with a threshold (or marked improvement) at age 12, though there was serious underestimating at all ages. A step-by-step teaching strategy using generative "tree diagrams" was then used. Even the 10-years-olds learned the use of the tree diagrams and the appropriate procedures for permutations and arrangements. Reprinted from The British Journal of Educational Psychology 40 (1970), Part 3.

• ELASTIC and reasoning under uncertainty: Final Report to NSF

Our analysis identified problems both with the subject matter of statistics (e.g. multiple levels of abstraction, difficulty mapping statistical representations to real-world situations) and with its pedagogy (which typically does little to help concertize abstract concepts or illuminate the mapping process). Drawing on research in education, cognitive psychology and statistical computing, we designed, implemented, and pilot-tested software (ELASTIC) and a curriculum (Reasoning Under Uncertainty) to address these problems. Our approach was successful in many of the problem areas identified above; in addition, our experiences in classrooms helped us better understand the difficulties students have in understanding and applying statistical reasoning.

• Belief in the law of small numbers

Apparently, most psychologists have an exaggerated belief in the likelihood of successfully replicating an obtained finding. The sources of such beliefs, and their consequences for the conduct of scientific inquiry, are what this paper is about. Our thesis is that people have strong intuitions about random sampling; that these intuitions are wrong in fundamental respects; that these intuitions are shared by naive subjects and by trained scientists; and that they are applied with unfortunate consequences in the course of scientific inquiry.