Young Children's Understanding of Counting and Cardinality

Abstract

4-year-old's knowledge of counting and cardinality--the last count word reached represents the numerosity of the set--was tested in 2 experiments. Experiment 1 investigated the nature of early cardinality responses by presenting different forms of the cardinality question before, after, and before and after the child counted. Both type and time of question has large effects. Experiment 2 examined whether children of this age could recognize errors in 4 counting procedures and whether they would reject a cardinality response arrived at through a mistaken counting procedure. The children were very good at recognizing a standard counting procedure as correct. They had only limited success at treating procedures that violated the stable order of count words or violated the one-one correspondence between count word and object as incorrect. They lacked an understanding of the order irrelevance in that they judged valid, nonstandard counting orders as incorrect. The children did not seem to link their evaluation of a cardinality response with their evaluation of the counting procedure used to reach that response. The results do not indicate that counting principles initially govern the child's acquisition of counting knowledge. They are consistent with the suggestion that early cardinality responses are last-word responses.

How counting represents number: What children must learn and when they learn it

“…For example, three-year-old children violate the one-one principle by skipping or double-counting items, or by using the same numeral twice in a count (Baroody & Price, 1983;Briars & Siegler, 1984;Frye, Braisby, Lowe, Maroudas, & Nicholls, 1989;Fuson, 1988;Miller, Smith, Zhu, & Zhang, 1995;Schaeffer et al, 1974;Wagner & Walters, 1982). Children also frequently violate the stable-order principle by producing different number-word sequences at different times (Baroody & Price, 1983;Frye et al, 1989;Fuson et al, 1982;Fuson, Secada, & Hall, 1983;Miller et al, 1995;Wagner & Walters, 1982). These findings have led many observers to conclude that the how-to-count principles, rather than being understood intuitively, are in fact learned gradually.…”

“…After the child had counted the chips, the line was immediately covered with a piece of cardboard and the child was asked how many chips were Studies using a How-Many task have found that many children respond incorrectly to the "how many" question even after they have counted the array correctly. When asked "how many," children often try to count the set again, or (if prevented from recounting) either make no response or guess some numeral other than the last numeral in the count sequence (Frye et al, 1989;Fuson, 1992;Markman, 1979;Rittle-Johnson & Siegler, 1998;Schaeffer et al, 1974;Wynn, 1990;Wynn, 1992). Moreover, some investigators have concluded that the How-Many task overestimates children's knowledge, because some children actually do repeat the last numeral used in counting without (apparently) understanding that it refers to the cardinal value of the set-this superficial understanding has been called a 'Last-Word Rule' to distinguish it from the cardinal principle (Frye et al, 1989;Fuson, 1988).…”

“…(This task is also called "Give-a-Number" by Wynn, 1990Wynn, , 1992 and others; "Give-me-X" by Frye et al, 1989; and "Make a set of N" by Fuson, 1988). In this task, the child is asked to create a set with a particular number of items.…”

“…Then, the Last-Word task was used to probe for a last-word rule as described by Fuson (1988) and Frye et al (1989). Finally, two new tasks (the Direction and Unit tasks) tested whether children knew that (a) adding an element to a set requires going forward in the numeral list to represent the number in the resulting set, while subtracting requires going backward (the Direction task) and (b) it is the next numeral that represents the set if 1 item has been added and the numeral after that if 2 have been added (the Unit task).…”

Why the verbal counting principles are constructed out of representations of small sets of individuals: A reply to Gallistel

“…It's in light of this evidence that we asserted that children's early counting is a routine that they learn without grasping its numerical significance. It is also the evidence that has led most researchers in the field to consider the debate between nativist and constructivist views of the acquisition of verbal counting to be settled in favor of the latter (Condry & Spelke, in press;Frye, Braisby, Lowe, Maroudas, & Nicholls, 1989;Fuson, 1988;Hurford, 1987;Le Corre et al, 2006;Schaeffer, Eggleston, & Scott, 1974;Siegler, 1991;Mix, Huttenlocher, & Levine, 2002;Wynn, 1992).…”