ABSTRACT
This chapter begins with two questions. One was whether pre-school children do understand anything about mathematics before they go to school. The other was whether any mathematical understanding that they build up during this time does help them understand arithmetic that they are taught when they go to school. There may be a conceptual connection between two kinds of mathematical experiences, informal, preschool experiences and formal classroom ones, but this does not mean that the children themselves will make it. The barriers between informal and formal knowledge have been shown to be quite daunting for children and for adults in other contexts: people need to know whether children surmount them in this case. Young children, and particularly young children who are learning English numbers, make many mistakes when they count, and these mistakes are remarkably persistent. Children's early knowledge and use of number words is without doubt the result of patient coaching from other people.
