Through videotaped task-based clinical interviews, this study describes and explores the beginning of the process of acquiring the concept of negative numbers by a small number of young children. It focuses on the children's use of various external representations of signed numbers (including representations based on the number line and on signed cardinality) as indicative of internal representations that they are developing.
It also explores the connection between the models used to introduce negative numbers and the types of internal representations that children construct as a result of their experiences with these models.
Two second-grade classes comprised of 37 students all together were selected for the study. The students in these classes participated in two group activities, each introducing the concept of signed numbers in a different representational context designed to incorporate either the ordinal or signed cardinality aspect. These activities were conducted in different orders in the two classrooms. Each student was also a subject of three structured interviews: a preliminary interview, and a follow-up interview after each activity. All interviews and group activities were videotaped. Twelve students, six from each class, were selected for detailed analysis of their preliminary interviews. Three of these students were selected for detailed analysis of their follow-up interviews as well.
Inferences were drawn about each child's construction and use of internal representations based on the child's observed behavior, including words and gestures captured on videotape and the child's written work.
The study was designed to be exploratory and purely qualitative. The study (a) demonstrates that second graders can develop meaningful internal representations of signed numbers; (b) suggests that, at this early stage of development, the internal representation of signed numbers tends to have two components, the ordinal component and the signed cardinality component, and these components can develop independently of each other; (c) suggests a possibility of connections between the specifics of a child's internal representation of counting numbers and zero, and the processes through which internal representation of signed numbers develops; and (d) highlights some obstacles that can occur while introducing young children to signed numbers. For example, it offers evidence that the focus on procedural teaching of a subtraction algorithm can generate an obstacle in the process of developing an internal representation of signed numbers. The implications of this study include some suggestions about the timing and the means of introducing young children to negative numbers. In addition, the study poses questions for further research.