Once children grasp the pattern that underlies cardinal number assignments, the way is open for a systematic and generalised concept of number that incorporates not only cardinal, but also ordinal and nominal contexts. In the present chapter, I show how these different contexts can be integrated into a model of our cognitive number domain. The questions that will lead our discussion are: how do children acquire the full range of number contexts? What are the components that make up a unified concept of number, and how do they come together?

Let me sketch a picture of the cognitive number domain as it arises from our results thus far. According to our view of numbers and number assignments, the basis of our numerical concepts is the representation of an infinite progression as provided by counting sequences, and its application in cardinal, ordinal, and nominal number assignments. The representation of these number assignments gives rise to the concepts of (a) numerical quantity and measure, (b) numerical rank, and (c) numerical label. These different numerical concepts constitute different subdomains of the cognitive number domain that are linked up with each other through their association with numerical tools. The acquisition of these concepts is supported by early, pre-numerical cognitive capacities that allow us to grasp the empirical properties we identify with numbers – cardinality, serial order, or identity.