On some necessary conditions of learning

The purpose of this article is to identify some necessary conditions of learning. To learn something, the learner must discern what is to be learned (the object of learning). Discerning the object of learning amounts to discerning its critical aspects. To discern an aspect, the learner must experience potential alternatives, that is, variation in a dimension corresponding to that aspect, against the background of invariance in other aspects of the same object of learning. (One could not discern the color of things, for instance, if there was only one color.) The study results illustrate that what students learn in a sequence of lessons is indeed a function of the pattern of variation and invariance constituted in that sequence. All teachers make use of variation and invariance in their teaching, but this study shows that teachers informed by a systematic framework do it more systematically, with striking effects on their students' learning.