INTUITIVE STRATEGIES AND PRECONCEPTIONS ABOUT ASSOC IATION IN CONTINGENCY TABLES

The aim of this research was to identify students' preconceptions concerning statistical association in contingency tables. An experimental study was carried out with 213 pre-university students, and it was based on students' responses t o a written questionnaire including 2x2, 2x3 and 3x3 contingency tables. In this article, the st udents' judgments of association and solution strategies are compared with finding of previous ps ychological research on 2x2 contingency tables. We also present an original classification of students' strategies, from a mathematical point of view. Correspondence analysis is used to s h w the effect of item task variables on students' strategies. Finally, we include a qualita tive analysis of the strategies of 51 students, which has served to characterize three misconceptio ns c ncerning statistical association. The concepts and procedures involved in the study o f correlation and regression are intended to determine statistical dependence relati onships between numerical variables. The extension of the idea of correlation to qualita tive variables has originated the general concept of association, the teaching of whi ch is a fundamental topic in the statistics curricula of many different university d egrees. In secondary education and preuniversity courses, three different topics are incl uded in the teaching of association: the analysis of contingency tables, the determination o f correlation between quantitative variables, and the comparison of a numerical variab le in two or more samples. The concept of association or statistical dependen ce has great relevance to mathematics education, because it extends functiona l dependence and it is fundamental for many statistical methods, for it allows us to m del numerous phenomena in different sciences (e.g., biology, economics, medicine, educa tion). This topic has significant connections with research on functional thinking an d other areas of mathematics education, such as probability and proportional rea soning. The main goal in many of these applications is to find causal explanations, that permit us to understand our environment. However, the association does not nece ssarily imply a causal relationship. Sometimes, due to the influence of concurrent facto rs, it is possible to find a high coefficient of association among variables when the re is no causal link (spurious correlation). Besides this epistemological difficulty, psycholog ical research has shown that judging association is not an intuitive ability. Ad ults sometimes base their judgment on their previous beliefs about the type of associatio n that ought to exist between the variables that are to be studied rather than on the empirical contingencies presented in the data. The existence of these preconceptions abo ut the nature of the empirical relationship in problematic situations presents ano ther difficulty for the teaching of association. Despite these epistemological and psyc hological issues, no previous research on this topic has been carried out in math ematics education, and most psychological research has only concentrated on 2x2 contingency tables. The experimental study reported here, which was c arried out with a sample of 213 pre-university students, sought to identify stu dents' preconceptions concerning association in contingency tables. Their written re sponses to a questionnaire concerning 2x2, 2x3 and 3x3 contingency tables are analyzed fr om different points of view. First, we discuss the type of association perceived by the s udents for the different items, relating their association judgments to the task va riables. Second, we analyze the students' solution strategies, comparing our result s wi h previous research into 2x2 contingency tables. We present an original classifi cat on of these strategies, from a mathematical point of view, in which we identify co ncepts and theorems in action as described by Vergnaud (1982). Finally, we discuss t he qualitative analysis of an additional sample of 51students' responses and char a terize different misconceptions concerning statistical association. As Confrey (1990) has pointed out, the relevance o f research on students' conceptions lies in the fact that sometimes these c onceptions differ in fundamental ways from the scientific concepts that we try to teach. Furthermore, students' conceptions are resistant to change in spite of instruction. Althou gh many students in our study demonstrated correct or partially correct judgments and solutions strategies, the misconceptions and incorrect strategies presented i this article indicate a gap between the meaning of association that we try to teach, an d the subjective meaning that students may attribute to this concept (Godino & Batanero, 1 994). In the following sections all of these aspects will be described, starting with a summary of the psychological research on which our study was based. PSYCHOLOGICAL RESEARCH ON CONTINGENCY TABLES Piaget and Inhelder's Study The study of reasoning about statistical associati n started with Inhelder and Piaget (1955), who considered the understanding of association as the final step in developing the idea of probability. According to th em, the evolutionary developments of the concepts of association and probability are related, and understanding association has as prerequisites the comprehension of proportio nality, probability, and combinatorics. Consequently, they only studied reas oning about association with children in their formal operation stage (IIIa and IIIb). They proposed to the subjects the problem of the association between eyes and hair co lo , and used a set of colored cards with drawings of faces as an experimental device (f air and brown hair; blue and dark eyes). In order to explain their results, Inhelder and Piaget classified the four possibilities of combining eyes and hair color acco rding to the layout presented in Table 1, in which a, b, c and d represent the absolute fr equencies in four cells (fair hair, blue eyes; fair hair, dark eyes; brown hair, blue eyes; brown hair, dark eyes). This is the simplest form of a contingency table or cross-tabul tion involving two variables that is used to summarize the frequencies in a population o r sample. Table1. Typical Format for a 2x2 Contingency Table