Developmental Processes and Stages in the Acquisition of Cardinality

Abstract

This is a study of the level of children'·s understanding of cardinality, focusing on the difference between a true cardinality response and the .application of a mechanically learned rule. The authors also evaluate and discuss the possible relationship between cardinality and counting., The 'subjects were two groups of 32 preschool children, ranging in age from 4 years 3 moriths to 6 years 3 months. Experimental methodology included two large sets of tests (elements-cardinal vs cardinal-elements), using both numbers and vowels with forward vs backward counting, and visual vs verbal presentation conditions. Results show that cardinality responses, are affected by both the direction and nature of the elements in the counting sequence. Scrutiny of errors committed in the various tests enables us to suggest six stages in the acquisition of cardinality. Although there appears to be a developmental dependency between counting and cardinality, this relationship is not significant in all cases.

How Spencer made number: First uses of the number words

“…By that time, his counting also was completely accurate-having been so for 2 months. This indicates that Spencer did not attribute special status to the last word in a count, as others have observed in previous research (Bermejo & Lago, 1990;Frye, Braisby, Lowe, Maroudas, & Nicholls, 1989;Fuson, 1988;Wynn, 1990;Wynn, 1992). Instead, he used the same response (i.e., ''two") that he used for similar queries in informal conversation and in spontaneous labeling.…”

“…They can match equivalent sets, label small numerosities, and use counting to determine cardinality (e.g., Bermejo, 1996;Bermejo & Lago, 1990;Fuson, 1988;Gelman & Gallistel, 1978;Huttenlocher, Jordan, & Levine, 1994;LeCorre & Carey, 2007;Mix, 1999;Mix, Sandhofer, & Baroody, 2005;Wynn, 1990). There have been different ideas about the origins of these skills-ideas that have generated abundant research on infancy and early childhood while largely ignoring toddlerhood (see Mix, Huttenlocher, & Levine, 2002, for a review).…”

Supporting children’s development of cardinality understanding

“…Moreover, when children use the ''counting on'' strategy, for instance, in solving an addition problem, one does not conclude that addition is a component of counting, because counting is only a means to achieve a goal (addition). We argue that the relationship between counting and cardinality is also similar to the relationship between means and goals, and between forms and functions (Bermejo & Lago, 1990;Saxe et al, 1987).…”

“…Finally, in the Saxe, Guberman, and Gearhart (1987) model, counting and cardinality appeared as two different phenomena, just as forms and functions are different; and Bermejo (1996) and Bermejo and Lago (1990) found empirically that counting and cardinality are different, as means and goals are different. Furthermore, Bermejo and Lago found that children could count correctly-assign number-words and objects on a one-to-one correspondence-without giving a correct response in cardinality, and vice versa-offering a correct response in cardinality without counting correctly.…”

“…Cardinality representation and understanding are important aspects of children's cognitive development between ages three and five (Bermejo, 1996;Bermejo & Lago, 1990, 1994Fuson, 1988;Gelman, Meck, & Merkin, 1986). Preschoolers are frequently faced with different situations related to cardinality, or the amount of items in a set (e.g., ''How many dogs are there in the picture?''…”