Non-formal mechanisms in mathematical cognitive development: The case of arithmetic.

The idea that cognitive development involves a shift towards abstraction has a long history in psychology. One incarnation of this idea holds that development in the domain of mathematics involves a shift from non-formal mechanisms to formal rules and axioms. Contrary to this view, the present study provides evidence that reliance on non-formal mechanisms may actually increase with age. Participants - Dutch primary school children - evaluated three-term arithmetic expressions in which violation of formally correct order of evaluation led to errors, termed foil errors. Participants solved the problems as part of their regular mathematics practice through an online study platform, and data were collected from over 50,000 children representing approximately 10% of all primary schools in the Netherlands, suggesting that the results have high external validity. Foil errors were more common for problems in which formally lower-priority sub-expressions were spaced close together, and also for problems in which such sub-expressions were relatively easy to calculate. We interpret these effects as resulting from reliance on two non-formal mechanisms, perceptual grouping and opportunistic selection, to determine order of evaluation. Critically, these effects reliably increased with participants' grade level, suggesting that these mechanisms are not phased out but actually become more important over development, even when they cause systematic violations of formal rules. This conclusion presents a challenge for the shift towards abstraction view as a description of cognitive development in arithmetic. Implications of this result for educational practice are discussed.

Carving nature at its joints or cutting its effective loops? On the dangers of trying to disentangle intertwined mental processes.

Attention is often inextricably intertwined with perception, and it is deployed not only to spatial regions, but also to sensory dimensions, learned dimensions, and learned complex configurations. Firestone & Scholl's (F&S)'s tactic of isolating visual perceptual processes from attention and action has the negative consequence of neglecting interactions that are critically important for allowing people to perceive their world in efficient and useful ways.

Fitting perception in and to cognition.

Perceptual modules adapt at evolutionary, lifelong, and moment-to-moment temporal scales to better serve the informational needs of cognizers. Perceptual learning is a powerful way for an individual to become tuned to frequently recurring patterns in its specific local environment that are pertinent to its goals without requiring costly executive control resources to be deployed. Mechanisms like predictive coding, categorical perception, and action-informed vision allow our perceptual systems to interface well with cognition by generating perceptual outputs that are systematically guided by how they will be used. In classic conceptions of perceptual modules, people have access to the modules' outputs but no ability to adjust their internal workings. However, humans routinely and strategically alter their perceptual systems via training regimes that have predictable and specific outcomes. In fact, employing a combination of strategic and automatic devices for adapting perception is one of the most promising approaches to improving cognition.

The effect of explanation in simple binary decision tasks.

Many studies of explanation have focused on higher level tasks and on how explanations draw upon relevant prior knowledge, which then helps in understanding some event or observation. However, explanations may also affect performance in simple tasks even when they include no task-relevant information. In three experiments, we show that explanations adding no task-relevant information alter performance in a sequential binary decision task. Whereas people with no explanation for why two events occurred at different rates tended to predict each outcome in proportion to their probability of occurrence (to "probability match"), people with an explanation tended to predict the more likely event more often (to "overmatch," a better strategy). These results suggest a broader view of explanation, which includes a role in shaping simple tasks outside of higher level reasoning.

The education of perception.

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.