Operations that generate quantity

Abstract

Segmenting sensory experience into units is advanced as the fundamental operation that produces object concepts. How such unitizing activity might be produced by children through interactions with elements in their environments is suggested by an interpretation of several recent works on the construction of objects by infants. The arithmetical units that Piaget demonstrated as being essential for numerical reasoning are also shown to be the product of a specific kind of unitizing activity. The main thesis developed is that unitizing activity precedes as well as follows Piaget's specific kind and produces object concepts, pluralities, collections, numerical lots, and other kinds of composite wholes. In particular, "discrete" and "continuous" quantity are distinquished by the sensory material and articulation of unitizing activity. "Discrete" quantity is developed as a property of numerical lots and "continuous" quantity is developed as a property of continuous but segmented units. Piaget's gross, intensive, and extensive quantity are then interpreted in terms of these two types of quantity. Finally, an integrative view of quantitative reasoning is developed for mathematics education that includes counting and measuring.