On the Structure of Attention in the Learning of Mathematics

What actually happens when something is learned? Something changes, but what? I suggest that what changes cognitively is the structure of attention (what is attended to and how, or in other words, sensitivity to notice); what changes emotionally is disposition, commitment and intention; what changes behaviourally is increased competence and facility. These changes constitute what we call learning. Here I look mostly at the cognitive. My analysis leads to a framework which has much in common with the van Hieles' observations, but which applies to fleeting moments rather than phases or levels. I also suggest ways in which this might influence the practice of teaching of mathematics. The notions of 'dimensions of possible variation' and 'range of permissible change' play an important role as devices for probing learner awareness as well as for structuring tasks, thinking about concepts, and enabling learners to extend ordinary tasks fruitfully for themselves.