The results of an investigation of the development of children's knowledge of addition and subtraction concepts before they start school are detailed. The purpose of the study was to test the predictions of the three-stage model about the distinctions between the last two stages. Twenty-four children participated in the investigation. None of these children had started first grade. The results support the existence of the qualitative and quantitative stages. The data indicate that: (1) Before children begin school and are given formal instruction on addition and subtraction problems, they can use the magnitude of the difference between two arrays and two transformations to make predictions about final numerosities; (2) A method of examining the relationship between understanding addition and subtraction and number conservation now exists; and (3) The distinction between qualitative and quantitative understanding allows for children to solve complex problems without relying on estimator solutions. (MP)