ROLE OF LINEAR REPRESENTATION OF LARGE MAGNITUDES ON UNDERSTANDING AND ESTIMATION

Abstract

Having a linear representation of magnitude across scales is essential in understanding many scientific concepts (Tretter, et al., 2006a) and is predictive of a range of mathematical achievement tests (Siegler & Booth, 2004). Despite the importance of understanding magnitude and scale, people have substantial difficulty comparing magnitudes outside of human perception (e.g., Jones, et al., 2008). The present work aims to examine the way people learn to represent and reason about large magnitudes through the development of two science of learning activities based on hierarchical alignment activity and corrective feedback. The hierarchical alignment activity utilizes several analogical reasoning principles: hierarchical alignment, progressive alignment, structural alignment, and multiple opportunities to make analogies. Study 1 examines the effectiveness of hierarchical alignment by contrasting it with a conventional activity that uses all the analogical reasoning principles described above except for hierarchical alignment. Study 2 examines a corrective feedback activity, based on the same analogical reasoning principles used in study 1, except, using corrective feedback instead of progressive alignment and hierarchical alignment. Thus, study 2 examines the necessity of hierarchical and progressive alignment. That both activities were successful in developing linear representations of geologic time (and for study 1, astronomical distances), suggests that multiple opportunities to make analogies through structural alignment are key components in developing analogies for learning magnitude. There appears to be an additive benefit of including hierarchical alignment (i.e., practice aligning magnitude relations across scales) in analogies for learning about magnitudes. Corrective feedback may also be a useful strategy in learning about scale information. Pedagogical implications are discussed. Both activities were based on the hypothesis that magnitudes at scales outside human perception are represented and reasoned about in the same way as magnitudes at human scales. The Category Adjustment Model (Huttenlocher, et al., 1988) suggests magnitude at human scales is stored as a hierarchical combination of metric and categorical information. People may use category boundaries to help make estimations in lieu of precise metric information. Variation in estimation, therefore, occurs because of imprecision of category boundaries (Shipley & Zacks, 2008; Zacks & Tversky, 2001). The current studies provided salient category boundaries to develop a more linear representation of magnitude. Thus, the effectiveness of the hierarchical alignment activity and the corrective feedback activity supports the hypothesis that people use hierarchically organized categorical information when making estimations across scales and across dimensions; and that providing people with more salient category boundary information improves estimation. Similarities and differences among temporal, spatial, and abstract line estimations are identified. Theoretical implications, including the potential application of the Category Adjustment Model to mental number lines, are discussed.