# An empirically derived hierarchy of intraconcept relationships: rational number multiplication

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This study presented an intraconcept analysis of thirty-four selected tasks for rational number multiplication based on the performance of elementary and junior high school students. Specifically, this study focused on learner performance patterns following repeated exposure to multiplication of fractions. The two questions investigated were: 1. What is the performance on the set by students? 2. What hierarchical structure, if any, underlies the set of thirty-four tasks related to multiplication of rational numbers? The thirty-four tasks were constructed from five concepts â€” integer, unit fraction, non-unit fraction, improper fraction, and mixed numeral. Permutations were formed using the concepts two at a time to represent factors in a multiplication task. The case of an integer with an integer was omitted. The student sample used in this study consisted of 127 elementary and junior high school students of the Fort Bend Independent School District (Fort Bend County, Texas). Of the 127 subjects, 20 were fifth graders, 53 were sixth graders, and 54 were seventh graders. A test of the thirty-four tasks for rational number multiplication was administered to the subjects. In the present study an intraconcept analysis (Uprichard & Phillips, 1975,1976) was used to develop tasks. The procedures used in developing and generating hypothesized hierarchies are detailed below. 1. Rank tasks according to means. 2. Revise ranking by means by asking questions regarding concepts contained in each task. The result is an intraconcept analysis ordering of a hypothesized linear hierarchy. 3. The hypothesized linear hierarchy of Step 2 was examined using Walbesser ratios and did not meet the levels set by Phillips (1971). The rejection of a linear hierarchy was followed by a search for branched hierarchies. 4. A further examination of task means, concepts, and Walbesser ratios yielded the formation of clusters in the hierarchy developed in Step 2. Based on the observations branched hierarchies were hypothesized. 5. The branched hierarchies were examined using Walbesser ratios and fourteen empirically determined hierarchies resulted. The analysis was completed by inspection of means and altering the arrangement slightly to conform to the following questions regarding concepts contained in the tasks. Question 1: Is there a mixed numeral factor? Order from most difficult to least difficult (a) mixed numeral factor items, and (b) non-mixed numeral factor items. Question 2: Is there a mixed numeral product? Order from most difficult to least difficult (a) mixed numeral product items, and (b) non-mixed numeral product items. Question 3: Is there a mixed numeral factor in the right position of items 1(a) and 2(a). Order from most difficult to least difficult (a) mixed numeral factors in right position and (b) mixed numeral factors in left position. Question 4: Order items 3(a) and 3(b) from most difficult to least difficult integer, improper, non-unit, and unit fraction. Question 5: Is there an integer, improper, non-unit, or unit fraction as a factor? Order from most difficult to least difficult integer improper, non-unit, and unit fraction. Question 6: Are the factors of Question 5 in right position? Order from most difficult to least difficult integer right, integer left; improper right, improper left; non-unit right, non-unit left; and unit right, unit left. Question 7: Order the arrangement of Question 6 from most difficult to least difficult improper, non-unit, and unit fraction. Conclusions. The analysis of the data gathered for this study has led the researcher to formulate several conclusions. These conclusions are based on evidence obtained from the findings of this research and the researcherâ€™s interpertation of these findings. 1. Hierarchical relationships exist among the selected tasks for rational number multiplication. 2. The Walbesser technique and item analysis yielded a valid measure of intraconcept relationships.