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We analyze log-data generated by an experiment with Mathtutor, an intelligent tutoring system for fractions. The experiment compares the educational effectiveness of instruction with single and multiple graphical representations. We extract the error-making and hint-seeking behaviors of each student to characterize their learning strategy. Using an expectation-maximization approach, we cluster the students by their strategic profile. We find that a) experimental condition and learning outcome are clearly associated b) experimental condition and learning strategy are not, and c) almost all of (...) the association between experimental condition and learning outcome is found among students implementing just one of the learning strategies we identify. This class of students is characterized by relatively high rates of error as well as a marked reluctance to seek help. They also show the greatest educational gains from instruction with multiple rather than single representations. The behaviors that characterize this group illuminate the mechanism underlying the effectiveness of multiple representations and suggest strategies for tailoring instruction to individual students. Our methodology can be implemented in an on-line tutoring system to dynamically tailor individualized instruction. (shrink)

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Conceptual understanding of representations and fluency in using representations are important aspects of expertise. However, little is known about how these competencies interact: does representational understanding facilitate learning of fluency, or does fluency enhance learning of representational understanding? We analyze log data obtained from an experiment that investigates the effects of intelligent tutoring systems support for understanding and fluency in connection-making between fractions representations. The experiment shows that instructional support for both representational understanding and fluency are needed for students to (...) benefit from the ITS. In analyzing the ITS log data, we contrast the understanding-first hypothesis and the fluency-first hypothesis, testing whether errors made during the learning phase mediate the effect of experimental condition. Finding that a simple statistical model does not the fit data, we searched over all plausible causal path analysis models. Our results support the understanding-first hypothesis but not the fluency-first hypothesis. (shrink)

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Although learning from multiple representations has been shown to be effective in a variety of domains, little is known about the mechanisms by which it occurs. We analyzed log data on error-rate, hint-use, and time-spent obtained from two experiments with a Cognitive Tutor for fractions. The goal of the experiments was to compare learning from multiple graphical representations of fractions to learning from a single graphical representation. Finding that a simple statistical model did not fit data from either experiment, we (...) searched over all possible mediation models consistent with background knowledge, finding several that fit the data well. We also searched over alternative measures of student error-rate, hint-use, and time-spent to see if our data were better modeled with simple monotonic or u-shaped non-monotonic relationships. We found no evidence for non-monotonicity. No matter what measures we used, time-spent was irrelevant, and hint-use was only occasionally relevant. Although the total effect of multiple representations on learning was positive, they also had a negative effect on learning, mediated by a higher error-rate. Our evidence suggests that multiple representations increase error-rate, which in turn inhibits learning. The mechanisms by which multiple representations improve learning are as yet unmodeled. (shrink)

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