The primary objective of this study was to provide an experimental model of children's representations of addition and subtraction concepts viewed as constructed schemes. How children with different counting schemes differ in their addition and subtraction concepts and how the types of problems children solve correlate with the addition and subtraction concepts were specifically explored. The 3-week study was conducted as a teaching experiment, with children's behavior observed and their mental processes probed in interviews, and in teaching episodes. Eight children in grades 1 and 2 were selected to reflect possible variations in counting, addition, and subtraction schemes. Four representations of addition concepts and six representations of subtraction concepts were found, with one or more specific schemes identified with each representation. The schemes were classified by developmental levels. Children who constructed higher level schemes also solved all kinds of addition and subtraction problems which involved larger numbers. Children's uses of their schemes reflected awareness of the difficulty of a problem and basic understanding of adding and subtracting. (MNS)