Children ’ s Number Sequences : An Explanation of Steffe ’ s Constructs and an Extrapolation to Rational Numbers of Arithmetic

In regarding young children as mathematicians we need to pay close attention to how children develop their mathematical thinking. It would be misleading to assume that young children think mathematically in the same way as adult mathematicians. Children have to develop their mathematical structures and their ways and means of operating mathematically. These structures and operations have to be constructed from the child's own activities. They cannot be given to the child " ready made. " One of the most fundamental mathematical structures that a child develops early on in life is that of a number sequence. The basic activity that leads to the construction of a number sequence is that of counting. The activity of counting, however, does not occur all at once for a child. Steffe, von Glasersfeld,. Richards and Cobb (1983) indicated in their research of young children's counting activities that early counting progresses through five distinct types of activity, from the counting of perceptual unit items to the counting of abstract unit items. They liken this progression to Piaget's concept of " object permanence " (p.117). A child's number sequence, however, is not a static structure. It also progresses through several developmental changes that are brought about through adaptations in the children's counting activities as they encounter more complex numerical situations. Following their work on children's counting types, Steffe and Cobb (1988) further developed the notion of children's abstract number sequences from their teaching experiments with first and second grade children. They described the development of three successive number sequences: the Initial Number Sequence (INS), the Tacitly Nested Number Sequence (TNS) and the Explicitly Nested Number Sequence (ENS). What follows are my attempts to clarify my own thinking about these hypothetical number sequences and an extrapolation from these sequences to a Generalized Number Sequence (GNS) and subsequently to the Rational Numbers of Arithmetic (RNA). While it is beyond the scope of this paper to attempt a thorough psychological discussion of learning, the following key psychological aspects of number sequences are important for making sense out of the developmental sequence presented in this paper. The following brief definitions are provided merely as an orientation for the reader rather than as a thorough explanation of terms. It is hoped that their meanings will become clearer through the examples given in the descriptions of the number sequences. Schemes Number Sequences are mental constructs-schemes. According to von Glasersfeld (1980) a …