The Development of Abstract Reasoning in Mathematics

Abstract

The historical antecedents to current practice in mathematics education have been outlined and contemporary theories of learning mathematics have been described. The aim of the research was to find evidence for a developmental spiral hypothesis which extends Piaqet's theory by making provision for levels and stages in the development of abstract reasoning. A thirty five minute test covering a total of 39 items was developed and used with a total of over 10,000 Secondary School students including 15 whole school populations from schools representing low, average and high socioeconomic areas of Adelaide, South Australia and Wellington, New Zealand. A smaller sample from Geelong, Victoria, was also included in the study. Validity and item fit of the questions was checked with item characteristic curves and Rasch analysis respectively. Kendall analysis was used to find question groupings in the response patterns of a sub-sample of mid- scoring pupils from one school. Success in each group of questions implies achievement at a particular stage and level of cognitive development. This grouping of questions was found to be applicable to the whole sample of students from both countries. Any gaps observed in the attainment of question groupings could be predicted on the basis of incorrect responses expected because of a finite probability of non perfect score attainment. The fit of the data supported the hypothesis that question groupings are successively attained. A numerical taxonomy package, SNOB, was used to classify sub- samples of the students according to their question responses. Interviews with the students was used to help suggest characteristics of students in these classifications. Two longitudinal samples, one from Australia and one from New Zealand, were followed from years 9 to 11, and students in the Australian sample were interviewed at Year 12. The Rasch abilities of these students was used to test a two parameter model proposed by Keats in 1980 for cognitive development, No other test of this model has been reported in the literature. A new research tool - that of asking students to make up a mathematical problem that would be difficult for a friend to solve - was developed and explored. Use of interview techniques for probing students' methods and understanding was made in this and other sections of the research. Implications of the findings for the learning and teaching of mathematics were discussed.