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. / Psychology

The ability to remember spatial locations is critical to human functioning, both in an evolutionary and an everyday sense. And yet, spatial memories and judgments often show systematic errors. Explanations for such errors have ranged from assumptions that memories are nonmetric, to the use of imperfect inferences, to the optimal combination of multiple sources of information. More recently, bias has been explained through the Category Adjustment Model - a Bayesian model in which fine-grained and categorical information are optimally combined (Huttenlocher, Hedges, & Duncan, 1991). However, experiments testing this model have largely used locations contained in simple geometric shapes. Use of this paradigm raises the issue of whether the results generalize to location memory in the complex natural world, as it should if it is to provide an over-arching framework for thinking about spatial memory. Here, this issue is addressed using a novel extension of the location memory paradigm that allows for testing of location memory in an everyday, 3D environment. The results support two predictions of the Category Adjustment Model - that memory for locations is biased toward central values, and that the magnitude of error increases with the retention interval. Future directions for testing the model in an increasingly ecologically valid manner are discussed. / Psychology