Conceptual Structures for Multiunit Numbers: Implications for Learning and Teaching Multidigit Addition, Subtraction, and Place Value

Abstract

Multiunit numbers are whole numbers composed of one or more kinds of multiunits (collections of single units) and possibly some single units. Multiunit numbers are expressible by number words and by written number marks. This article identifies conceptual structures necessary for understanding the named-multiunit-value English system of number words, the unnamed-value positional system of written marks for multiunit numbers, and several different kinds of conceptual multiunits that give meaning to these two different systems. Conceptual components of multiunit addition and subtraction are described. The special difficulties caused by irregularities in the English words for multiunit numbers composed of tens and ones are discussed and compared with the much simpler learning task for children speaking a system of number words that names the tens and ones in a regular fashion (Asian systems based on Chinese). Literature concerning the poor performance of U.S. children on place-value tasks and on multidigit addition and subtraction is reviewed, and two special conceptual structures (sequence multiunit and concatenated single-digit structures) used by U.S. children are identified. The latter conceptual structure is associated with use of many different partially correct procedural rules for multidigit addition and subtraction that lead to characteristic errors made by U.S. children and that violate particular conceptual components of multiunit addition or subtraction. Classroom experiences that support children's construction of the requisite conceptual structures are discussed, with particular attention to the role of objects that display collectible multiunits. Limitations of current U.S. textbook treatments and curricular placement of place value and multidigit addition and subtraction are described, and possible alternative paths to children's