More precisely defining and measuring the order-irrelevance principle

R. Gelman and C. Gallistel (1978, Young Children's Understanding of Number, Cambridge, MA: Harvard Univ. Press) use two definitions of the order-irrelevance principle interchangeably: (1) count tags do not have to be assigned in a fixed order and (2) the order in which elements of a set are enumerated does not affect the cardinal designation of the set. A study involving 107 kindergarten and first grade children indicates that the two are actually distinct concepts. Apparently, a willingness to arbitrarily assign tags is a developmentally less sophisticated ability than--and hence does not necessarily imply--an ability to predict that differently ordered counts produce the same cardinal designation. Thus it appears that evidence of the second ability is necessary to infer a full understanding of the order-irrelevance principle. The first ability alone implies what might better be termed an "order-indifferent tagging scheme." Suggestions for measuring and further researching the order-irrelevance principle are discussed.     

Fostering At-Risk Kindergarten Children's Number Sense

A 9-month training experiment evaluated whether computer-assisted discovery learning of arithmetic regularities can facilitate kindergartners’ fluency with the easiest sums. After a pretest, kindergartners with at least one risk factor (n = 28) were randomly assigned to either a structured add-0/1 training condition, which focused on recognizing the n + 0/0 + n = n and the n + 1/1 + n = the number-after-n rules, or an active control group. Using pretest fluency as the covariate, ANCOVAs revealed that the structured add-0/1 group significantly outperformed the control group on both practiced and unpracticed (transfer) n + 0/0 + n and n + 1/1 + n items at the delayed posttest and had significantly larger gains in mathematics achievement. Key instructional implications include: Early intervention that targets discovering rules for adding with 0 and 1 and family-specific developmental prerequisites is feasible and more effective than typical classroom instruction in promoting fluency with such basic sums. Such rules may be a critically important bridge between informal and formal mathematics.     

The development of procedural knowledge: An alternative explanation for chronometric trends of mental arithmetic

In a review of the Chronometrie literature, M. H. Ashcraft (Developmental Review, 1982, 2, 213–236) concluded that the development of number fact efficiency is due to a shift from relying on procedural knowledge such as counting to relying on declarative knowledge (a stored network of facts). This model assumes that all procedural processes are slow or remain slow, which is probably not the case. An alternative account posits that the key change in number fact efficiency involves a shift from slow counting procedures to principled procedural knowledge. As rules, heuristics, and principles become more familiar and interconnected, their use, for example, in producing the number facts becomes more automatic. The use of such procedural knowledge would be cognitively more economical than storing individual facts in long-term memory. Finally, existing Chronometric data can readily be interpreted in terms of this alternative model.     

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

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.”     

Kindergartners' understanding of additive commutativity within the context of word problems

Baroody and Gannon (1984) proposed that children's understanding of additive commutativity progresses through several levels of understanding based on a unary view of addition (change meaning) before developing a "true" level of understanding based on a binary conception (part-whole meaning). Resnick (1992) implied that children have both a unary and a binary conception of additive commutativity from the earliest stages of development. Fifty-three 5- and 6-year-old (M = 6-0) kindergartners' unary and binary understanding of additive commutativity was investigated using performance on tasks involving change-add-to and part-part-whole word problems, respectively. The data were inconsistent with the predictions of both models and suggest three alternate theoretical explanations. Moreover, the data indicate that success on a task involving change-add-to problems may be a more rigorous test of understanding of additive commutativity than that involving part-part-whole problems.     

The construct and measurement of spontaneous attention to a number

The Spontaneous Attention to a Number (SAN) construct serves a different purpose than Hannula-Sormunen and colleagues’ Spontaneous Focus on Number (SFON) construct. As an extension of Eleanor J. Gibson’s differentiation theory, the premise of SAN is that children’s step-wise construction of small number concepts enables them to perceptually differentiate among increasingly larger numbers—to distinguish reliably between “oneness” or “twoness” and larger numbers, then between “threeness” and “larger numbers,” and eventually between “fourness” and larger numbers. In contrast, SFON refers to the tendency to attend to numbers in general—an attentional process that, unlike SAN, is separate from enumeration skill. Not surprisingly, then, although the prototype for both the SAN and SFON tasks is Nancy C. Jordan and colleagues’ non-directive nonverbal number task, the independent development of the SAN and SFON tasks resulted in key differences in how they are administered and scored and to whom they are administered.