The RUPS Hypthesis (Goldstone, Landy, & Son, 2010) is this:
An important way to efficiently perform sophisticated cognitive tasks is to convert originally demanding, rule-based operations into into learned perceptual tasks.
Much of current American math education has been characterized as overly procedural with an emphasis on the appropriate use of symbolic rules. This, of course, leads to errors such as inappropriate use of rules (shown in the addition problem below). However, a more sophisticated mistake would be to make an error of magnitude. In the problem below, if a student had written an answer roughly greater than 1 but less than 1.5, we would consider that close in magnitude.
In collaboration with UCLA's Human Perception Lab and University of Pennsylvania's PENNlincs, several programs called Perceptual Learning Modules (PLMs) in order to foster perceptual learning have been developed and implemented in schools. By giving students experience with principled perceptual representations (such as number lines and magnitudes), students can develop perceptual expertise and intuitions for fractions and operations on fractions. This simultaneously gives students a strong foundation for thinking about numerical relationships but also starts to give meaning to previously arbitrary rules.
Rigged Up Perceptual Systems (RUPS) in Math