Children’s Fractional Knowledge pp 49-74 | Cite as

# Articulation of the Reorganization Hypothesis

## Abstract

When fractions are introduced in school mathematics, they are usually introduced in the context of continuous quantity. Number sequences are essentially excluded because, as quantitative schemes, they are thought to be relevant only in discrete quantitative situations. Even though I developed number sequences in Chap. 3 in the context of discrete quantity, I can see no principled reason to keep them separate from continuous quantity. Reserving number sequences for discrete quantity stands in opposition to the concept of the real number line in higher mathematics, and, in this chapter, I argue that it also stands in opposition to the development of quantitative schemes. In articulating the reorganization hypothesis, I establish that a composite unit of specific numerosity can be used to make a split in the way that Confrey (1994) explained. This involves more than simply indicating the possibility of transferring the operations involved in compounding discrete units together to splitting continuous units. I do a deeper developmental analysis of children’s quantitative schemes in which I explore whether the operations that produce discrete quantity and the operations that produce continuous quantity can be regarded as unifying quantitative operations. If so, these quantitative operations would justify the reorganization hypothesis.

## Keywords

Number Sequence Arithmetical Unit Segmented Unit Continuous Quantity Composite Unit## References

- Confrey J (1994) Splitting, similarity, and rate of change: a new approach to multiplication and exponential functions. In: Harel G, Confrey J (eds) The development of multiplicative reasoning in the learning of mathematics. State University of New York Press, Albany, NY, pp 291–330Google Scholar
- Davydov VV (1975) The psychological characteristics of the “prenumerical period” of mathematics instruction. In: Steffe LP (ed) Children’s capacity for learning mathematics. School Mathematics Study Group, StanfordGoogle Scholar
- Gelman R, Gallistel CR (1978) The child’s understanding of number. Harvard University Press, CambridgeGoogle Scholar
- Hunting RP, Sharpley CF (1991) Pre-fraction concepts of preschoolers. In: Hunting RP, Davis G (eds) Early fraction learning. Springer, New York, pp 9–26CrossRefGoogle Scholar
- Kagan VF (1963) Essays on geometry. Moscow University, MoscowGoogle Scholar
- Kieren T (1994) Multiple views of multiplicative structures. In: Harel G, Confrey J (eds) The development of multiplicative reasoning in the learning of mathematics. State University of New York Press, Albany, NY, pp 387–397Google Scholar
- Lamon SJ (1996) The development of unitizing: its role in children’s partitioning strategies. J Res Math Educ 27:170–193CrossRefGoogle Scholar
- Piaget J (1970) Genetic epistemology. Columbia University Press, New YorkGoogle Scholar
- Piaget J, Szeminska A (1952) Child’s conception of number. Routledge, LondonGoogle Scholar
- Piaget J, Inhelder B, Szeminska A (1960) The child’s conception of geometry. Basic Books, New YorkGoogle Scholar
- Schwartz J (1988) Intensive quantity and referent-transforming operations. In: Hiebert J, Behr M (eds) Number concepts and operations in the middle grades. National Council of Teachers of Mathematics, Reston, VA, pp 41–52Google Scholar
- Steffe LP (1991) Operations that generate quantity. Learn Indiv Differ 3:61–82CrossRefGoogle Scholar
- Steffe LP, Cobb P (with von Glasersfeld E) (1988) Construction of arithmetical meanings and strategies. Springer, New YorkGoogle Scholar
- von Glasersfeld E (1981) An attentional model for the conceptual construction of units and number. J Res Math Educ 12:83–94CrossRefGoogle Scholar