# Research

• ### Assessment of performance in introductory linear models courses

We have been teaching a General Linear Models approach to problem formulation and analysis to university students and applied researchers for over 25 years. The students and researchers have come from a wide variety of application areas such as education and all of the social sciences, biology, management, and operations research. The objectives of the course range from the very clearly defined to the very unclear. The range of objectives and the heterogeneity of backgrounds create difficulties in the assessment of performance. Among these difficulties are defining the objectives of the instruction, developing instruments to evaluate student performance, and methods of awarding grades.

• ### Children's concepts of average and representativeness

This paper addresses two major questions about children's understanding of average. The first question deals with children's own understanding of representativeness within the context of data sets. When asked to describe a data set, how do children construct and interpret representativeness? The second question focuses on how children think about the mean as a particular mathematical definition and relationship. It deals with the underlying issue of how children develop mathematical definitions and how they connect these definitions with their informal mathematical understanding. This question, which has been considered by other researchers primarily in the context of experimental research designs, is addressed here in an open-ended, descriptive manner.

• ### Student views on effective, and ineffective learning in graduate level applied statistics courses

What is it that makes the difference for students? What is their perspective on what benefited or hindered their learning in a first statistics class? Additionally, can students' reasons be categorized based on different student characteristics or learning styles? To answer these questions, the present investigation is a qualitative follow-up to a quantitative analysis that was aimed at determining relationships among learning styles, academic programs, background variables and attitude toward statistics.

• ### Children's choice behavior in probabilistic situations

Most decisions in uncertain situations involve comparison of the probabilities of success under the alternative choices, and selection of that alternative where the chances of success are higher. Hence the ability to discriminate between probabilities of varying discrepancies is crucial in such tasks. The development of that capacity was examined in two previous experiments. The purpose of the present study is to identify the principles of choice in individual response patterns. One of the educational implications of this study is that the teaching of probability to children should be carefully planned. The lesson to be learned from the results of the present research, in conjunction with earlier studies, is that the concept of proportion (and probability) is very elusive.

• ### Does the teaching of probability improve probabilistic intuitions?

The present investigation is concerned especially with the influence that the teaching of probabilities may have indirectly on intuitive probabilistic judgments. There is very little information available about this problem. In an earlier work, Ojemann et al. have reported a positive, indirect effect of probability lessons on predictions made by their subjects (8-10 years old) in probability learning tasks. In this study the authors found a clear increase with age in proportions of correct answers to probability problems. They also found that by emphasizing (via systematic instruction) specific probability viewpoints and procedures, one may disturb the subjects proportional reasoning, still fragile in many adolescents.

• ### A survey of probability concepts in 3000 pupils aged 11-16 years

The aims of this project, sponsored by the Social Science Research Council from November 1978 to October 1981, were: a) to survey the intuitions of chance and the concepts of probability possessed by English school pupils aged 11 to 16 years. b) to establish patterns of development of probability concepts and to relate these to other mathematics concepts. c) to investigate pupil responses to GCE 'O' Level and CSE probability items, with particular references to sex differences. A summary of findings is presented, based on responses of 2930 children.

• ### Identifying and Alleviating Problems Novices Have While Learning Formulae

In this study I investigate what elements of statistical formulae cause people to perceive said formulae as difficult. The perception of difficulty is important because it affects how and what people study as well as the amount of time they devote to study. By understanding how people perceive formulae we can design our instruction to accommodate these perceptions. I approach this investigation from three major theoretical bases. The first theoretical base will concern cognitive load. Are students intimidated by certain formulae because the formulae contain to much information to be held in working memory? The second theoretical base will concern motivational perceptions. Are there aspects of certain formulae which cause students to perceive certain formulae as more difficult and thereby stimulate negative emotions that interfere with the efficient use of working memory. The third theoretical base involves impasse drive learning. When learners reach a cognitive impasse, are they able work around it. If learners can work around certain impasses and not others, how do the impasses they can work around and those they cannot work around differ?

• ### Factors affecting probabilistic judgments in children and adolescents

Six hundred and eighteen pupils, enrolled in elementary and junior-high-school classes (Pisa, Italy) were asked to solve a number of probability problems. The main aim of the investigation has been to obtain a better understanding of the origins and nature of some probabilistic intuitive obstacles. A linguistic factor has been identified: It appears that for many children, the concept of "certain events" is more difficult to comprehend than that of "possible events". It has been found that even adolescents have difficulties in detaching the mathematical structure from the practical embodiment of the stochastic situation. In problems where numbers intervene, the magnitude of the numbers considered has an effect on their probability; bigger numbers are more likely to be obtained than smaller ones. Many children seem to be unable to solve probability questions, because of their inability to consider the rational structure of a hazard situation: "chance" is, by itself, an equalizing factor of probabilities. Positive intuitive capacities have also been identified; some problems referring to compound events are better solved when addressed in a general form than when addressed in a particular way.

• ### The Affective Component of Attitude in Statistics Instruction

Suggestions from the literature for dealing with students' statistics anxiety are listed in this paper. The role of motion in learning is reviewed and results of a small correlational study are used to illustrate the association between the affective component of attidude as a measure of anxiety and statistics performance for classes of education students. Strategies the author has found helpful are presented.

• ### Authentic Assessment of Statistical Reasoning in Cooperative Learning Environments

This paper describes two case studies that examined statistical reasoning skills in an authentic environment in which learning occurred in small groups. Multiple forms of assessments were developed to obtain a detailed profile of reasoning demonstrated by individual students and groups of students on projects. Two conditions were developed by individual students and groups of students on projects. Two conditions were developed to exemplify assessment criteria on such projects: a library of exemplars condition and a text condition. Qualitative analyses of verbal protocols during group discussions and presentations indicated that the library of exemplars was effective in promoting (a) reasoning about data analysis and data presentation issues (b) planning and (c) aligning students ratings of projects with experimenter ratings. The importance of student participation and request for assistance in group situations was highlighted in this study. Moreover, information about the reasoning demonstrated by individual students is required to ensure that group assessments do not overestimate student performance.