This study investigated how participant’s specificity in shar-ing of information in collaborative problem solving was criti-cal to them reaching a successful shared perspective. We ana-lyzed participants’ communication strategies in a collaborativetask designed to make finding common ground challenging.We set out to better understand the difference between suc-cessful and unsuccessful collaborations by conducting a cog-nitive task analysis. From participants’ utterances, we inferredcognitive processes associated with repeating communicationmoves and coded those processes as if-then production rules.We thereby specified the communication strategies used duringinteractions and developed a production-rule model to explainwhether and how shared perspective developed or not. Ourcognitive task analysis indicated that although all collaboratingpairs described the objects they were seeing with a variety offeatures, the successful pairs were more specific in using com-binations of features. Quantitatively, we found significant cor-relations between frequency of combined feature statementsand success in sharing perspectives.

We have been refining a cognitive model, written in ACT-R, of student performance in early algebra problem solving. "Early algebra" refers to a class of problems and competencies at the boundary between arithmetic and algebra. Our empirical studies in this domain establish a striking contrast between students' difficulties with symbolic algebra and their relative success with certain kinds of "intuitive" algebraic reasoning. To better understand this contrast, we analyzed student solutions to identify the strategies and errors exhibited and then set out to account for this detailed process data with the utility-based choice mechanism of ACT-R. Our first model contained production mles for explicitly selecting strategies and for making certain systematic errors or bugs. It provided a good quantitative fit to student performance data (R2=.90), however, it had two quahtative shortcomings: 1) the productions for strategy selection appeared to serve no computational purpose and 2) the model systematically underpredicted the frequency of non-trivial errors on more complex problems. We created a new model in which explicit strategy selection was eliminated (strategic behavior is emergent) and in which failure to fire a production (an implicit, non-buggy error) is an option at every model choice point. Compared to the first model, this model achieved an equivalent quantitative fit with fewer productions and without the systematic deviations from the error data. We consider the implications of implicit strategies and errors for instruction.

In this paper, we review the literature on the relation between solving nonspecific goal problems and learning. Research has shown that reduced goal-specificity facilitates learning of rules and principles of the target domain. Researchers have accounted for this effect using a cognitive load theory (Sweller, 1988) and a dual space theory of problem solving (Vollmeyer, Bums, & Holyoak, 1996). Other researchers have shown that learning can be both facilitated by nonspecific as well as specific goals and account for their findings using goal appropriateness theory (Miller, Lehman. & Koedinger, 1997). W e judge each theoretical account by evaluating their consistencies with unified theories of cognition and other empirical data. We note the shortcomings of the each theory and incorporate elements of each to explain all the data.

Retrieval practice of information through testing has been shown to improve learning. So has studying examples. In this paper, we address inconsistencies in the literature concerning which of these two approaches is best. We test the hypothesis that learning depends on what is being learned; whereas practice emphasizes memorization, studying examples allows for selectivity of encoding, resulting in different information being learned. Accordingly, we predicted that practice will improve learning in situations that emphasize memorization (such as learning facts or simple associations), whereas studying examples will improve learning in situations where there are multiple pieces of information available and selectivity is necessary (such as when learning skills or procedures). We report evidence from a laboratory study using naturalistic materials showing results consistent with these predictions.

This paper is part of an effort to extend research on mathematical problem solving beyond the traditional focus on formal procedures (both in the classroom and in problem solving research). W e are beginning to investigate students' inductive discovery-oriented strategies and the interaction between these and formal deductive strategies. In contrast to typical classroom problems in math and science which demand the application of a learned formal procedure (e.g., prove X), we gave students more open-ended problems (e.g., is X true?) for which the formal deductive procedure is useful, but other, possibly informal or inductive, strategies are also potentially useful. The normative approach for solving these problems, in fact, requires the use of both a deductive strategy, which is definitive only when X is true, and an inductive search for examples, which is definitive only when X is not universally true. When presented with these problems we found that geometry students have some limited facility to perform the deductive strategy (though, less so in this context than when they are directly asked to vwite a proof) and use a degenerate version of the inductive strategy. Instead of considering multiple examples and looking for a counter-example, students tend to read off the conclusion from the single example (or model) we provided.

In current theories of skill acquisition it is quite common to assume that the input to learning mechanisms is a problem representation based on direct translations of problem instructions or simple inductions from problem solving examples. W e call such a problem representation an execution space because it is made up of operators corresponding to the external actions agents perform while executing problem solutions. Learning proceeds by modifications and combinations of these execution space operators. W e have built a model of geometry expertise based on verbal report evidence which contains operators which can be described as modifications (e.g., abstractions) and combinations (e.g., compositions) of execution operators. However, a number of points of evidence lead us to conclude that these operators were not derived from execution space operators. In contrast, it appears these operators derive from discoveries about the structure and properties of domain objects, particularly, perceptual properties. We have yet to develop a detailed and integrated theory of this "perceptual churJdng", but we present the exp)ert model is a challenge to ciurent dieories of skill acquisition.

A person's ability to translate a mathematical problem into symbols is an increasingly important skill as computational devices play an increasing role in academia and the workplace. Thus it is important to better understand this "symbolization" skill and how it develops. We are working toward a model of the acquisition of skill at symbolizing and scaffolding strategies for assisting that acquisition. We are using a difficulties factors assessment as an efficient methodology for identifying the critical cognitive factors that distinguish competent from less competent symbolizers. The current study indicates there is more to symbolizing than translating individual phrases into symbols and using long-term schematic knowledge to fill in implied information. In particular, students must be able to compose these individual translation operations into a complete symbolic sentence. We provide evidence that in contrast to many prior models of word problem solving which address story comprehension skills, a critical element of student competence is symbolic production skills.