Richard Skemp was a mathematician who later studied psychology1. He drew on both these disciplines to explain learning in mathematics. The main 'thrust' of his argument is that learners construct schemata to link what they already know with new learning. According to Skemp, mathematics involves an extensive hierarchy of concepts - we cannot form any particular concept until we have formed all the subsidiary ones upon which it is depends. Skemp also suggested that emotions play a dominant part in the way in which we learn.
Skemp suggested that there are two kinds of learning in mathematics:
Instrumental understanding2: a mechanical, rote or 'learn the rule/method/algorithm' kind of learning (which gives quicker results for the teacher in the short term), e.g. writing 10 would be understood as "This is how we write 10" in instrumental terms.
Relational understanding2: a more meaningful learning in which the pupil is able to understand the links and relationships which give mathematics its structure (which is more beneficial in the long term and aids motivation), e.g. writing 10 would be understood as "This is why we write 10 like this (in terms of place value)" in relational terms.
Both are deemed important for mathematics.
1See Skemp, R.R (1986) The Psychology of Learning Mathematics. 2nd Edition. London: Penguin Books.
2See Skemp, R.R (1977) Relational Understanding and Instrumental Understanding, Mathematics Teaching, 77: 20-6