The representation of physics problems in relation to the organization of physics knowledge is investigated in experts and novices. Four experiments examine the existence of problem categories as a basis for representation; differences in the categories used by experts and novices; differences in the knowledge associated with the categories; and features in the problems that contribute to problem categorization and representation. Results from sorting tasks and protocols reveal that experts and novices begin their problem representations with specifiably different problem categories, (...) and completion of the representations depends on the knowledge associated with the categories. For, the experts initially abstract physics principles to approach and solve a problem representation, whereas novices base their representation and approaches on the problem's literal features. (shrink)

This research investigates the development of analogy: In particular, we wish to study the development of systematicity in analogy. Systematicity refers to the mapping of systems of mutually constraining relations, such as causal chains or chains of implication. A preference for systematic mappings is a central aspect of analogical processing in adults (Gentner, 1980, 1983). This research asks two questions: Does systematicity make analogical mapping easier? And, if so, when, developmentally, do children become able to utilize systematicity?Children aged 5–7 and (...) 8–10 acted out stories with toy characters. Then they were asked to act out the same stories with new characters. Two variables were manipulated: systematicity, or the degree of explicit causal structure in the original stories, and the transparency of the object‐mappings. Transparency was manipulated by varying the similarity between the original characters and the corresponding new characters: it was included in order to vary the difficulty of the transfer task. If children can utilize systematicity, then their transfer accuracy should be greater for systematic stories.The results show: (1) As expected, transparency strongly influenced transfer accuracy (for both age groups, transfer accuracy dropped sharply as the object correspondences became less transparent); and (2) for the older group, there was also a strong effect of systematicity and an interaction between the two variables. Given a systematic story, 9‐year‐olds could transfer it accurately regardless of the transparency of the object correspondence. (shrink)

Although the field of perceptual learning has mostly been concerned with low- to middle-level changes to perceptual systems due to experience, we consider high-level perceptual changes that accompany learning in science and mathematics. In science, we explore the transfer of a scientific principle (competitive specialization) across superficially dissimilar pedagogical simulations. We argue that transfer occurs when students develop perceptual interpretations of an initial simulation and simply continue to use the same interpretational bias when interacting with a second simulation. In arithmetic (...) and algebraic reasoning, we find that proficiency in mathematics involves executing spatially explicit transformations to notational elements. People learn to attend mathematical operations in the order in which they should be executed, and the extent to which students employ their perceptual attention in this manner is positively correlated with their mathematical experience. For both science and mathematics, relatively sophisticated performance is achieved not by ignoring perceptual features in favor of deep conceptual features, but rather by adapting perceptual processing so as to conform with and support formally sanctioned responses. These “rigged-up perceptual systems” offer a promising approach to educational reform. (shrink)

People can be taught to manipulate symbols according to formal mathematical and logical rules. Cognitive scientists have traditionally viewed this capacity—the capacity for symbolic reasoning—as grounded in the ability to internally represent numbers, logical relationships, and mathematical rules in an abstract, amodal fashion. We present an alternative view, portraying symbolic reasoning as a special kind of embodied reasoning in which arithmetic and logical formulae, externally represented as notations, serve as targets for powerful perceptual and sensorimotor systems. Although symbolic reasoning often (...) conforms to abstract mathematical principles, it is typically implemented by perceptual and sensorimotor engagement with concrete environmental structures. (shrink)

This article describes the Adaptive Control of Thought–Rational (ACT–R) cognitive architecture (Anderson et al., 2004; Anderson & Lebiere, 1998) and its detailed application to the learning of algebraic symbol manipulation. The theory is applied to modeling the data from a study by Qin, Anderson, Silk, Stenger, & Carter (2004) in which children learn to solve linear equations and perfect their skills over a 6‐day period. Functional MRI data show that: (a) a motor region tracks the output of equation solutions, (b) (...) a prefrontal region tracks the retrieval of declarative information, (c) a parietal region tracks the transformation of mental representations of the equation, (d) an anterior cingulate region tracks the setting of goal information to control the information flow, and (e) a caudate region tracks the firing of productions in the ACT–R model. The article concludes with an architectural comparison of the competence children display in this task and the competence that monkeys have shown in tasks that require manipulations of sequences of elements. (shrink)

We present a new model of skilled performance in geometry proof problem solving called the Diagram Configuration model (DC). While previous models plan proofs in a step‐by‐step fashion, we observed that experts plan at a more abstract level: They focus on the key steps and skip the less important ones. DC models this abstract planning behavior by parsing geometry problem diagrams into perceptual chunks, called diagram configurations, which cue relevant schematic knowledge. We provide verbal protocol evidence that DC's schemas correspond (...) with the step‐skipping inferences experts make in their initial planning. We compare DC with other models of geometry expertise and then, in the final section, we discuss more general implications of our research. DC's reasoning has important similarities with Larkin's (1988) display‐based reasoning approach and Johnson‐Laird's (1983) mental model approach. DC's perceptually based schemas are a step towards a unified explanation of (1) experts' superior problem‐solving effectiveness, (2) experts' superior problem‐state memory, and (3) experts' ability, in certain domains, to solve relatively simple problems by pure forward inferencing. We also argue that the particular and efficient knowledge organization of DC challenges current theories of skill acquisition as it presents an end‐state of learning that is difficult to explain within such theories. Finally, we discuss the implications of DC for geometry instruction. (shrink)

Despite the accumulation of substantial cognitive science research relevant to education, there remains confusion and controversy in the application of research to educational practice. In support of a more systematic approach, we describe the Knowledge-Learning-Instruction (KLI) framework. KLI promotes the emergence of instructional principles of high potential for generality, while explicitly identifying constraints of and opportunities for detailed analysis of the knowledge students may acquire in courses. Drawing on research across domains of science, math, and language learning, we illustrate the (...) analyses of knowledge, learning, and instructional events that the KLI framework affords. We present a set of three coordinated taxonomies of knowledge, learning, and instruction. For example, we identify three broad classes of learning events (LEs): (a) memory and fluency processes, (b) induction and refinement processes, and (c) understanding and sense-making processes, and we show how these can lead to different knowledge changes and constraints on optimal instructional choices. (shrink)

This study investigated the roles of problem structure and strategy use in problem encoding. Fourth‐grade students solved and explained a set of typical addition problems (e.g., 5 + 4 + 9 + 5 = _) and mathematical equivalence problems (e.g., 4 + 3 + 6 = 4 + _ or 6 + 4 + 5 = _ + 5). Next, they completed an encoding task in which they reconstructed addition and equivalence problems after viewing each for 5 s. Equivalence problems (...) of the form 4 + 3 + 6 = 4 + _ overlap with a perceptual pattern found in traditional arithmetic problems (i.e., answer blank in final position), and students' encoding was poorest on problems of this type. Individual differences in encoding the equivalence problems were related to variations in strategy use. Some students solved blank‐final equivalence problems using the standard arithmetic strategy of performing all given operations on all given numbers. These students made more errors in encoding problem structure, but fewer errors in encoding the numbers, than did students who solved the problems using correct or other incorrect strategies. Moreover, students who expressed many strategies for solving the blank‐final equivalence problems made fewer errors in encoding problem structure, but more errors in encoding the numbers, than did students who expressed only a single strategy. Results highlight that encoding is intended to guide action and that prior experience can simultaneously facilitate and interfere with accurate encoding. (shrink)