Manipulation of physical models such as tangrams and tiles is a popular approach to teaching early mathematics concepts. This pedagogical approach is extended by new computational media, where mathematical entities such as equations and vectors can be virtually manipulated. The cognitive and neural mechanisms supporting such manipulation-based learning—particularly how actions generate new internal structures that support problem-solving—are not understood. We develop a model of the way manipulations generate internal traces embedding actions, and how these action-traces recombine during problem-solving. This model (...) is based on a study of two groups of sixth-grade students solving area problems. Before problem-solving, one group manipulated a tangram, the other group answered a descriptive test. Eye-movement trajectories during problem-solving were different between the groups. A second study showed that this difference required the tangram's geometrical structure, just manipulation was not enough. We propose a theoretical model accounting for these results, and discuss its implications. (shrink)

We present a computational model of multiple-object tracking that makes trial-level predictions about the allocation of visual attention and the effect of this allocation on observers' ability to track multiple objects simultaneously. This model follows the intuition that increased attention to a location increases the spatial resolution of its internal representation. Using a combination of empirical and computational experiments, we demonstrate the existence of a tight coupling between cognitive and perceptual resources in this task: Low-level tracking of objects generates bottom-up (...) predictions of error likelihood, and high-level attention allocation selectively reduces error probabilities in attended locations while increasing it at non-attended locations. Whereas earlier models of multiple-object tracking have predicted the big picture relationship between stimulus complexity and response accuracy, our approach makes accurate predictions of both the macro-scale effect of target number and velocity on tracking difficulty and micro-scale variations in difficulty across individual trials and targets arising from the idiosyncratic within-trial interactions of targets and distractors. (shrink)

The dissociation between statistical prediction and scientific explanation advanced by Bowers et al. for studies of vision using deep neural networks is also observed in several other domains of behavior research, and is in fact unavoidable when fitting large models such as deep nets and other supervised learners, with weak theoretical commitments, to restricted samples of highly stochastic behavioral phenomena.