Children’s Fractional Knowledge pp 27-47 | Cite as

# Operations That Produce Numerical Counting Schemes

## Abstract

The primary goal of this chapter is to present a model of important steps in children’s construction of their numerical counting schemes because the basic hypothesis that guides our work is that children’s fraction schemes can emerge as accommodations in their numerical counting schemes. I consider a *number sequence* to be the recognition template of a numerical counting scheme; that is, its assimilating structure. This way of thinking of a number sequence was basic in the formulation of the reorganization hypothesis. A number sequence is a discrete numerical structure; it is a sequence of arithmetical unit items that contain records of counting acts. At all stages of construction, children use their number sequences to provide meaning for number words. A number word such as “twenty-one,” say, can refer to a sequence of arithmetical unit items from “one” up to and including “twenty one.” It is the operations that children can perform using their number sequences that distinguish among distinct stages of the number sequences. In what follows, I explain the operations that produce two prenumerical counting schemes as well as three distinctly different number sequences and, hence, three distinctly different numerical counting schemes. I also explain discrete structures that precede number sequences in development that I refer to as perceptual and figurative lots. These lot structures are produced by the operation of categorizing discrete items together, where categorizing is based on reprocessing sensory-motor items of experience using an operation called unitizing. In categorizing, when reprocessing is coordinated with re-presenting discrete items of experience, recursive unitizing emerges. Recursive unitizing is that operation which produces arithmetical units and numerical structures. I start by presenting an attentional model of unitizing and the different levels of units that this operation produces.

### Keywords

Assimilation### References

- Ceccato S (1974) In the garden of choices. In: Smock CD, von Glasersfeld E (eds) Epistemology and education. Follow Through Publications, Athens, GA, pp 125–142Google Scholar
- Cooper RG Jr (1990) The role of mathematical transformations and practice in mathematical development. In: Steffe LP (ed) Epistemological foundations of mathematical experience. Springer, New York, pp 102–123Google Scholar
- Dörfler W (1996) Is the metaphor of mental object appropriate for a theory of learning mathematics? In: Steffe LP, Nesher P, Cobb P, Goldin GA, Greer B (eds) Theories of mathematical learning. Erlbaum, Mahwah, NJ, pp 467–476Google Scholar
- Inhelder B, Piaget J (1964) The early growth of logic in the child. W. W. Norton, New YorkGoogle Scholar
- Piaget J (1937) The construction of reality in the child. Basic Books, New YorkGoogle Scholar
- Steffe LP (1992) Schemes of action and operation involving composite units. Learn Indiv Differ 4:259–309CrossRefGoogle Scholar
- Steffe LP, Cobb P (with von Glasersfeld E) (1988) Construction of arithmetical meanings and strategies. Springer, New YorkGoogle Scholar
- Steffe LP, von Glasersfeld E, Richards J, Cobb P (1983) Children’s counting types: philosophy, theory, and application. Praeger Scientific, New YorkGoogle Scholar
- Thompson PW (1982) A theoretical framework for understanding young children’s concepts of numeration. Unpublished doctoral dissertation, The University of Georgia, Athens.Google Scholar
- von Glasersfeld E (1981) An attentional model for the conceptual construction of units and number. J Res Math Educ 12:83–94CrossRefGoogle Scholar
- von Glasersfeld E (1995a) Radical constructivism: a way of knowing and learning. Falmer Press, Washington, DCCrossRefGoogle Scholar