A Knowledge Structure for the Arithmetic Mean: Relationships between Statistical Conceptualizations and Mathematical Concepts
This study examined cognitive relationships between the fairshare and centerofbalance conceptualizations of the arithmetic mean. It also hypothesized the use of these conceptualizations as blending spaces for the mathematical and statistical domains within a proposed knowledge structure for the arithmetic mean. Twentynine undergraduate liberal arts students completed pre/post verbal protocols with written solutions to arithmetic mean problems. The problems emphasized either the fairshare or centerofbalance conceptualization, or mathematical concepts related to the arithmetic mean. The participants were divided into three groups: those that received fairshare instruction, those that received centerofbalance instruction, and a control group. The data was analyzed using statistical methods, including contingency tables and ANCOVA, to investigate the effects fairshare and centerofbalance instruction had on knowledge of fairshare, centerofbalance, and mathematical concepts regarding the arithmetic mean. A qualitative analysis of the verbal protocols helped explain any statistically significant connection between the fairshare and centerofbalance conceptualizations, or between either conceptualization and mathematical concepts related to the arithmetic mean. Analysis of the data indicated participants increased their knowledge of the fairshare conceptualization after receiving instruction that was focused on centerofbalance. Similarly, participants increased their knowledge of the centerofbalance conceptualization after receiving instruction that was focused on fairshare. In either case, the concept, 'the sum of the deviations from the mean is zero,' was used to transfer knowledge between the conceptualizations. In addition, instruction in either the fairshare or centerofbalance conceptualization increased knowledge of the mathematical concepts related to the arithmetic mean. However, only specific mathematical concepts were impacted by each of the conceptualizations. The results suggest that both the fairshare and centerofbalance conceptualizations are pertinent to pedagogical decisions regarding the arithmetic mean. Furthermore, the concept, 'the sum of the deviations from the mean is zero,' is a viable cognitive connection between the fairshare and centerofbalance conceptualizations.
Year of publication: 
20090108


Authors:  Marnich, Mark A 
Other Persons:  Ellen Ansell (contributor) ; Gabriel Stylianides (contributor) ; Nancy Pfenning (contributor) ; Margaret S. Smith (contributor) ; James G. Greeno (contributor) 
Publisher: 
PIT 
Subject:  Instruction and Learning 
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