From a Radical Constructivist (RC) perspective this thesis deals with children’s construction of early arithmetic learning as an evolving process through the cognitive system of self-regulation and self-organising. Thus the child’s learning must guide teaching. RC views early arithmetic as verbal and preceding the system of written arithmetic.

The purpose in my thesis is to build hypothetical models of the child’s conceptual progression, in the case of four fundamental rules of arithmetic, and thereby construct the ontogenesis of arithmetic. The material for this is documentations of longitudinal research of the child’s arithmetic carried out over a lot of years and with different children within the RC paradigm. Through analysing about 250 transcriptions from video recordings, included in my material, I construct what concept is lying behind the child’s counting activity, when it solves problems with its counting scheme. Following children’s progression longitudinally, I could make my hypothesis of the child’s knowledge viable.

In short, my results show that the child gives through five counting schemes qualitatively changeable meanings to numbers. It begins with perceptual and figurative collections in two pre-numerical counting schemes. In the three subsequent numerical schemes the significations of numbers are a numerical composite, an abstract composite unit and an iterable unit of one.

The models provide the discipline of special education a starting point for didactical decisions how to assess and influence the child’s early arithmetic learning. Research is needed to document the children’s progression in the course of schooling in order to elaborate or modify the models. This theoretical perspective makes both teaching and researching viable through longitudinal approaches. Further, it highlights the need to bring research into the practice of education.