Annie Rafferty

Knowledge affects human deductive reasoning, but the mechanisms by which this occurs are not fully understood. Example knowledge—the ability to generate and categorise specific examples of general possibilities—is proposed to play a central role, and individual differences in such knowledge are proposed to contribute to differences in deductive reasoning. To test these hypotheses, four studies investigated the role of example knowledge in adults’ conditional reasoning about algebra. Individual differences in domain-specific knowledge predicted conditional reasoning about algebra when controlling for everyday and abstract conditional reasoning (Study 1), individuals spontaneously referred to examples during conditional reasoning about algebra (Study 2), individual differences in example knowledge predicted differences in conditional reasoning about algebra when controlling for general algebra knowledge and everyday conditional reasoning (Study 3), and training designed to increase example knowledge improved conditional reasoning about algebra (Study 4). The findings suggest that deductive reasoning in the domain of maths is in part a domain-specific skill, insofar as it depends on domain-specific knowledge, and that one cause of this knowledge-dependence is individuals’ reliance on examples during reasoning.

Math problem solving frequently involves choices among alternative strategies. Strategy choices, and effects of problem features on strategy choices, both vary among individuals. We propose that individual differences in strategy choices can be well characterized in terms of parametric variation in three types of influence: global bias, relevant feature effects, and irrelevant feature effects. We test this framework by applying it to children's strategy choices in fraction and decimal arithmetic. We describe a simple mathematical model of strategy choice in this domain that is based on a recent theory of arithmetic development and includes parameters representing the three types of influence above. We estimate these parameters in a sample of 120 fifth to ninth graders and find that all of them vary substantially among children. Further, we find that different parameters relate differently to other domain‐specific and domain‐general abilities, supporting the utility of distinguishing among the parameters and estimating them separately for individuals. We discuss implications of the results regarding the nature and origins of individual differences in strategy choice in fraction and decimal arithmetic and math more broadly.

Online educational technologies offer opportunities for providing individualized feedback and detailed profiles of students' skills. Yet many technologies for mathematics education assess students based only on the correctness of either their final answers or responses to individual steps. In contrast, examining the choices students make for how to solve the equation and the ways in which they might answer incorrectly offers the opportunity to obtain a more nuanced perspective of their algebra skills. To automatically make sense of step‐by‐step solutions, we propose a Bayesian inverse planning model for equation solving that computes an assessment of a learner's skills based on her pattern of errors in individual steps and her choices about what sequence of problem‐solving steps to take. Bayesian inverse planning builds on existing machine learning tools to create a generative model relating (mis)‐understandings to equation solving choices. Two behavioral experiments demonstrate that the model can interpret people's equation solving and that its assessments are consistent with those of experienced teachers. A third experiment uses this model to tailor guidance for learners based on individual differences in misunderstandings, closing the loop between assessing understanding, and using that assessment within an educational technology. Finally, because the bottleneck in applying inverse planning to a new domain is in creating the model of possible student misunderstandings, we show how to combine inverse planning with an existing production rule model to make inferences about student misunderstandings of fraction arithmetic.

Human and automated tutors attempt to choose pedagogical activities that will maximize student learning, informed by their estimates of the student's current knowledge. There has been substantial research on tracking and modeling student learning, but significantly less attention on how to plan teaching actions and how the assumed student model impacts the resulting plans. We frame the problem of optimally selecting teaching actions using a decision-theoretic approach and show how to formulate teaching as a partially observable Markov decision process planning problem. This framework makes it possible to explore how different assumptions about student learning and behavior should affect the selection of teaching actions. We consider how to apply this framework to concept learning problems, and we present approximate methods for finding optimal teaching actions, given the large state and action spaces that arise in teaching. Through simulations and behavioral experiments, we explore the consequences of choosing teacher actions under different assumed student models. In two concept-learning tasks, we show that this technique can accelerate learning relative to baseline performance.