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…

Contemporary theories of learning postulate one or at most a small number of different learning mechanisms. However, people are capable of mastering a given task through qualitatively different learning paths such as learning by instruction and learning by doing. We hypothesize that the knowledge acquired through such alternative paths differs…