Participants in 2 experiments interacted with computer simulations designed to foster understanding of scientific principles governing complex adaptive systems. The quality of participants' transportable understanding was measured by the amount of transfer between 2 simulations governed by the same principle. The perceptual concreteness of the elements within the first simulation was manipulated. The elements either remained concrete throughout the simulation, remained idealized, or switched midway into the simulation from concrete to idealized or vice versa. Transfer was better when the appearance of the elements switched, consistent with theories predicting more general schemas when the schemas are multiply instantiated. The best transfer was observed when originally concrete elements became idealized. These results are interpreted in terms of tradeoffs between grounded, concrete construals of simulations and more abstract, transportable construals. Progressive idealization ("concreteness fading") allows originally grounded and interpretable principles to become less tied to specific contexts and hence more transferable.Cognitive psychologists and educators have often debated the merits of concrete versus idealized materials for fostering scientific understanding. Should chemical molecules be represented by detailed, shaded, and realistically illuminated balls or by simple ball-and-stick figures? Should a medical illustration of a pancreas include a meticulous rendering of the islets of Langerhans or convey in a more stylized manner the organ's general form? Our informal interviews with mycologists at the Royal Kew Gardens (personal communication, Brian Spooner and David Pegler, May 1998) indicate a schism between authors of mushroom field guides.

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Development in any domain is often characterized by increasingly abstract representations. Recent evidence in the domain of shape recognition provides one example; between 18 and 24 months children appear to build increasingly abstract representations of object shape [Smith, L. B. (2003). Learning to recognize objects. Psychological Science, 14, 244-250]. Abstraction is in part simplification because it requires the removal of irrelevant information. At the same time, part of generalization is ignoring irrelevant differences. The resulting prediction is this: simplification may enable generalization. Four experiments asked whether simple training instances could shortcut the process of abstraction and directly promote appropriate generalization. Toddlers were taught novel object categories with either simple or complex training exemplars. We found that children who learned with simple objects were able to generalize according to shape similarity, typically relevant for early object categories, better than those who learned with complex objects. Abstraction is the product of learning; using simplified - already abstracted instances - can short-cut that learning, leading to robust generalization.

Learning in educational settings emphasizes declarative and procedural knowledge. Studies of expertise, however, point to other crucial components of learning, especially improvements produced by experience in the extraction of information: perceptual learning. We suggest that such improvements characterize both simple sensory and complex cognitive, even symbolic, tasks through common processes of discovery and selection. We apply these ideas in the form of perceptual learning modules (PLMs) to mathematics learning. We tested three PLMs, each emphasizing different aspects of complex task performance, in middle and high school mathematics. In the MultiRep PLM, practice in matching function information across multiple representations improved students’ abilities to generate correct graphs and equations from word problems. In the Algebraic Transformations PLM, practice in seeing equation structure across transformations (but not solving equations) led to dramatic improvements in the speed of equation solving. In the Linear Measurement PLM, interactive trials involving extraction of information about units and lengths produced successful transfer to novel measurement problems and fraction problem solving. Taken together, these results suggest 1) that PL techniques have the potential to address crucial, neglected dimensions of learning, including discovery and fluent processing of relations, 2) PL effects apply even to complex tasks that involve symbolic processing, and 3) appropriately designed perceptual learning technology can produce rapid and enduring advances in learning.