Blake's Development of the Number Words “One,” “Two,” and “Three”

Abstract

A mother tracked her preschooler's number word development daily from 18 to 49 months of age. Naturalistic observations were supplemented with observations during structured (Kumon) training and microgenetic testing. The boy's everyday use of “two” did not become highly reliable and selective for 10 months (at 28 months), emerged later than that of words representing less abstract concepts, and was used in a relatively abstract manner to describe various visible pairs of items. He quickly generalized “two” to partially visible collections and then those that were not visible. Highly reliable use of “one” and “two” appeared to develop simultaneously, before he started using a plural rule, and before he could put out two items upon request. Reliable and accurate use of number words in everyday situations, particularly child-initiated efforts, preceded such use in the contexts of the Kumon training and microgenetic testing, both of which involved adult-initiated tasks. Educational implications include underscoring differences among the first number words by contrasting, for instance, one with two, and pointing out non-examples of a number (“not two”) as well as a wide variety of examples, such as “two blocks, two hands, two socks, two airplanes.”

ACKNOWLEDGMENTS

This article is based on research supported by grant BCS-0111829 from the National Science Foundation (“The Developmental Bases of Number and Operation Sense”). The data analyses and write up also were supported, in part, by grant 200400033 from the Spencer Foundation (“Key Transitions in Preschoolers’ Number and Arithmetic Development”), grant R305A080479 (“Fostering Fluency with Basic Addition and Subtraction”) from the Institute of Education Science (U.S. Department of Education), and grant R01-HD051538–01 (“Computer-guided Comprehensive Mathematics Assessment for Young Children”) from the National Institutes of Health. Any opinions, findings, conclusions, or recommendations expressed are solely those of the authors and do not necessarily reflect the position, policy, or endorsement of the National Science Foundation, Spencer Foundation, Institute of Education Science, or National Institutes of Health. Portions of this case study were originally reported at the 2002 annual meeting of the American Educational Research Association and the 2003 biennial meeting of the Society for Research in Child Development.

Notes

Gallistel and Gelman (1992) hypothesize an innate, nonverbal counting system governed by the same counting principles as govern verbal counting. Moreover, this innate system involves a sequential list of nonverbal symbols that results in reasonably accurate cardinal representations of small numbers and permits nonverbal infants to discriminate among small even heterogeneous collections or across different modalities.

Other enumeration activities involved counting collections of objects, such as seven turtles in side-by-side rows of four and three, and the instructions to count the turtles. Other mathematical worksheets involved identifying the missing numeral in a counting sequence.

Note that Blake did not develop functional self-initiated one-to-one counting skill (use counting to quantify a collection) until 29 months—after recognition of “two” had become highly reliable—and, as with many newly learned skills, did so deliberately (e.g., started the counting with “one,” pointed with his fingers) for many months afterward. Functional one-to-one counting in adult-initiated situations did not begin for many months later. Consistent with most previous research (Klahr & Wallace, 1973; Sarnecka et al., 2007; Schaeffer et al., 1974; Sophian, 1992; Starkey & Cooper, 1995; von Glasersfeld, 1982; but cf. Fuson, 1988), the functional development of two in self-initiated situations particularly, then, was probably not aided by experiences with one-to-one counting. Moreover, many of the more abstract uses of two discussed in the next subsection would not lend themselves to one-to-one object counting.

For Issues #2 to #4, it is possible that the feedback Blake received in the Kumon training had a positive effect on his understanding and recognition of two. However, there are reasons to believe otherwise. Blake's progress in selectively using “two” in purposeful and engaging everyday situations ran well ahead of such progress in the sterile or purposeless Kumon context (as far as the child is concerned). Indeed, he never achieved functional use of “two” in the Kumon context and did not achieve even semi-functional use until 29 months, one month after achieving functional use in everyday situations. This evidence suggests that the training probably did not interest or motivate Blake. Given his relatively poor performance and apparent disinterest over the entire course of Kumon, it not clear that he attended carefully and benefitted significantly from the feedback provided in this training. In regard to Issue #5, the Kumon training did not involve counting out collections of items.