Jean-Claude Falmagne - US grants

Researchers in many of the social and behavioral sciences often need to analyze the data of a complex experiment into a few essential features or trends, in order to ascertain the essential dimensions of variations in the phenomenon under investigation. Methods involving the mathematics of geometry can be very helpful in this regard, because they can provide the experimenter with potentially enlightening graphic representations of the data. There has been relatively little development of truly geometrical methods tailored to those fields. Useful techniques have been developed (e.g., factor analysis, multidimensional scaling) which are sometimes labeled `geometrical'. However, these techniques are based on extremely strong models (i.e., based on very restrictive assumptions), in which the geometry of the representing structure plays only a minor role. Moreover, the fundamental assumptions of these models are not easily testable, so that it is never quite clear how much of the results of the analysis have been forced upon the experimenter by the method itself. This research will remedy this situation and provide a collection of purely geometrical models suitable for a very general experimental setting in use by social and behavioral scientists. The experimental method is a variant of the so- called `method of triads', but the interpretation of the data differs drastically from traditional ones. The mathematical developments are largely based on the methods and concepts of modern geometry. One of the advantages of this approach is that the models tend to be weaker than traditional ones (i.e., to involve less restrictive assumptions) while being easily testable.

This work will continue ongoing research concerning the construction of a theory and the related computer algorithms for the efficient assessment of knowledge. The key concept of the theory is that a knowledge structure, that is, a basic set of Q of questions or problems in a given domain of information, equipped with a distinguished family of subsets called the knowledge states. The interpretation of these concepts is that the knowledge state of a student is the set of all those problems in the basic set Q that the student is capable of solving. A probabilistic version of these concepts has been developed, permitting an analysis of the data with standard statistical techniques. The knowledge assessment routines were developed in the framework of Markov chains theory. The focus of this work will be the application of the theory in one particular case, namely, the Regents Competency Test in Mathematics of June, 1987. Other aspects of the research include investigation of a number of technical questions regarding the combination of distinct knowledge structures and practical adaptation and implementation of knowledge assessment procedures.

This project is to lay the groundwork for a practical implementation of the knowledge assessment techniques, developed by the PI and his collaborators, in some selected schools in Orange County California. This involves gathering a team of collaborators, including teachers, securing the appropriate physical conditions, and beginning the construction of the pedagogical materials. The proposed work is broken down into five phases: 1. contact with local elementary, junior, and senior high schools; 2. assemble the team of researchers, including the PI's and 4 expert teachers; 3. selection of a large collection of items to form the domain of the knowledge structures, and covering grades 1-12; 4. analyze the selected items for conceptual content and skills for mastery; 5. use the QUERY routine to make a first pass at building the knowledge structures, using the teachers as subjects;

ABSTRACT 9350484 Falmagne This project develops an interactive student assessment computer system to be used in grades 4 to 8. This expert system, where subsequent questions are based on previous responses, provides a report for each student delivered in everyday language and which describes the student's current strengths and weaknesses relative to the appropriate mathematics curriculum. The report makes specific recommendations and provides a printout of critical question missed with annotated solutions. The results for each student are advisory to the classroom teacher. Each student session is completed in a short amount of time. The system considers cultural and social diversity and is constructed to accept "open" responses to the questions. The system requires the identification and construction of a "knowledge space" which includes a suitable family of "knowledge states," revised and refined through peer review. Both the development and initial implementation take place in exemplary schools in Orange County California. ***

This project will continue the development of a new theory of learning in which the progress of students learning over time is modeled probabilistically as a Markov stochastic process. This theory is designed to provide responses to key questions about learning processes such as : What is the distribution of the knowledge states at the beginning of a particular grade in school, such as grade 4? What are the most frequently traveled learning paths? What are the easiest items for a student to learn given a particular knowledge state? How long does it take, on the average, for a student with a particular learning rate to go from one knowledge state to another? Do the students in school district A tend to follow different paths from those in school district B? This theory is a potentially powerful tool for the precise monitoring of students' learning. However, crucial work remains to be done for this theory to be easily applied by practitioners. So far it has been applied only to the special case of a single test given at a particular time. The cases of two or more tests (on the same sample of students) raise a number of technical issues (some of which are related to parameter estimations) which will be addressed in the research. The PI will also conduct an application of the theory.

Abstract


Falmagne, Jean-Claude E.
University of California, Irvine
SES-9986269

Whereas considerable research has been devoted to the study of individual preferences, both from empirical and theoretical standpoints, the evolution of preferences over time has been given only scant attention. The proposed research focuses on this question. Various classes of preference relations (total orders, partial orders, semiorders, biorders, etc.) are regarded as special cases of the abstract concept of a "medium", i.e., a particular kind of semigroup of transformations on a set of states. The research will develop the algebraic and stochastic aspects of media theory, and will apply the results to substantial data, such as the evolution of opinions regarding candidates in an important election.

The study of opinions and their evolution over time is a natural concern in a democratic country. It is important to develop scientific tools capable of faithfully interpreting the judgements expressed by the members of a community regarding critical issues. The aim of the research is to make such tools not only available, but also easy to use. The results will provide, among other benefits, more accurate techniques for the analysis of opinion polls.