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UBC Theses and Dissertations

A test of the validity of a concrete-abstract hierarchy of addition facts on kindergarten children Ward, Brian


The study attempts to validate a task-analytic hierarchy for the number facts of addition between 6 and 9 along a Concrete-Abstract continuum. Literature is reviewed in four areas: Task Analytic hierarchies in arithmetic; the; learning theory which is associated with task analytic hierarchies; the interaction of task analytic learning theory and developmental learning theory; and attempts to indicate behaviours which specify a definition of a concept of numbers which can be hierarchically arranged, researched and discussed. A Concrete-Abstract hierarchy is presented with seven Task Levels arranged in four Conceptual Levels designated Concrete, Semi-Concrete, Concrete-Abstract and Abstract. Questions are posed which consider the possibilities of validating both a seven-stage hierarchy and a four-stage hierarchy. The target population was kindergarten children between the ages of 5 years 0 months and 5 years 11 months in one school district in British Columbia. This entire kindergarten population was screened and the original 125 children reduced to 49 in order to conform with pretest Entering Behaviours of a developmental nature concerned with the concept of numbers. From these children seven subjects were chosen as subjects for a pilot test to investigate whether subjects would answer most of the questions on the test designed, and to standardize the testing instructions. Twenty-eight subjects were chosen for the main study: 14 boys and 14 girls. Twenty-one of the 26 number facts between 6 and 9 were used in the main study and the other 5 in the pilot study. A Greco-Latin Square was employed to design a balanced test which incorporated the seven levels of the hierarchy with seven sets of items. Each item set of three number facts was also balanced so that the level of difficulty of these sets was, according to established criteria, approximately equal. Prior to the testing session each subject was taught three items at each Task Level of the hypothesized hierarchy. This teaching was given to seven groups of children, each group composed of 2 boys and 2 girls. In order to decide the items taught at each level of the hierarchy, a row of a Latin Square was chosen at random and matched with a balanced set of items. When subjected to a Guttman Scalogram Analysis data supported the existence of a four-level hierarchy. Using a cut-off point of 2 out of 3 items correct a coefficient of reproducibility of 0.95 was achieved. Ceiling effects did occur, however, which made it difficult to substantiate the hypothesized differences between the Concrete and Semi-Concrete Levels. A possible cause advanced, is the pretest which eliminated 76 children from the population prior to beginning the study, thus producing a group of subjects who were most likely high in intelligence and intelligence-related behaviours.

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