In this chapter, the ideas of ‘set-before’ and ‘met-before’ are employed to provide a framework for the longer-term development of mathematical thinking. A set-before is ‘a mental structure that we are born with, which may take a little time to mature as our brains make connections in early life’. As such it is a fundamental structure that is shared by all. A met-before is ‘a mental structure we have now as a result of experiences we have met before’. This will depend partly on the nature of the mathematics, and so there will be communalities between different individuals, and partly on the individual’s cognitive growth, leading to widespread differences.

In Chapter 3 we saw how mathematical concepts arise through three forms of compression: categorization, encapsulation and definition. In this chapter we relate these developments through three major ‘set-befores’ based on our sensori-motor system and our use of language. These are the sensory power of recognition through perception, the motor power of repetition of actions to routinize into mathematical operations, and the use of language to describe increasingly sophisticated concepts.