An empirically derived hierarchy of intraconcept relationships : whole number division



Journal Title

Journal ISSN

Volume Title



The primary purpose of this investigation was to use a process involving intraconcept analysis and indirect validation procedures to generate and validate a hierarchy of mathematical relationships related to the traditional long division method. This study considered division with a one digit whole number divisor and a whole number dividend with at most three digits. Previous research supported the theory cf the hierarchical structure of knowledge. A sequence of subordinate tasks in a learning hierarchy should describe a program that will accomplish curriculum objectives. Therefore, the task seemed to be that researchers must validate a hierarchy in order to develop optimal instructional sequences. An instrument with K-R 21 reliability .95 consisting of 30 tasks based on the intraconcept analysis was constructed and administered to 744 subjects in grades 5, 7, and 9 in thirteen elementary and junior high schools of the Houston Independent School District. The analysis of data used the results from only the subjects who did not answer all tasks correctly or all tasks incorrectly (N=468). Item means yielded a linear ordering of the tasks based on the proportion of students answering each task correctly. From this ordering a nesting pattern was discerned based on nine factors related to the long division task using only the traditional method. There is a progressive nesting of the following variables in this order: [...]