| This study examined cognitive relationships between the fair-share and center-of-balance 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.;Twenty-nine undergraduate liberal arts students completed pre/post verbal protocols with written solutions to arithmetic mean problems. The problems emphasized either the fair-share or center-of-balance conceptualization, or mathematical concepts related to the arithmetic mean. The participants were divided into three groups: those that received fair-share instruction, those that received center-of-balance instruction, and a control group.;The data was analyzed using statistical methods, including contingency tables and ANCOVA, to investigate the effects fair-share and center-of-balance instruction had on knowledge of fair-share, center-of-balance, and mathematical concepts regarding the arithmetic mean. A qualitative analysis of the verbal protocols helped explain any statistically significant connection between the fair-share and center-of-balance conceptualizations, or between either conceptualization and mathematical concepts related to the arithmetic mean.;Analysis of the data indicated participants increased their knowledge of the fair-share conceptualization after receiving instruction that was focused on center-of-balance. Similarly, participants increased their knowledge of the center-of-balance conceptualization after receiving instruction that was focused on fair-share. 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 fair-share or center-of-balance 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 fair-share and center-of-balance 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 fair-share and center-of-balance conceptualizations. |
