Preschoolers' Understanding of Subtraction-Related Principles

Abstract

Little research has focused on an informal understanding of subtractive negation (e.g., 3 − 3 = 0) and subtractive identity (e.g., 3 − 0 = 3). Previous research indicates that preschoolers may have a fragile (i.e., unreliable or localized) understanding of the addition-subtraction inverse principle (e.g., 2 + 1 − 1 = 2). Recognition of a small collection's cardinal value and computational experience, particularly with subtractive negation, may play a key role in the construction of an understanding of inversion. Testing with eighty 3 to 7 year olds revealed that most children demonstrated a reliable and general understanding of subtractive negation and identity at 4 years of age. In contrast, such an understanding of the inverse principle was not achieved by most children until 6 years of age and was preceded by recognition of “two” and “three” and an understanding of subtractive negation and identity.

The research described was supported, in part, by a grant from the National Science Foundation (BCS-0111829) and the Spencer Foundation (200400033). The opinions expressed are solely those of the authors and do not necessarily reflect the position, policy, or endorsement of the National Science Foundation or the Spencer Foundation.

Notes

¹Bisanz et al. (this issue) thoughtfully observed that using the Binomial Theorem to analyze data on an individual basis with a small number of inversion trials might underestimate understanding of inversion. They argue that those in the process of constructing the concept might tend to apply the principle inconsistently and, thus, might be classified as “non-users.” This is a possibility, but its likelihood depends on whether a child otherwise tended to respond haphazardly.

On trials children did not apply their inverse knowledge because, say, they were unsure of the numbers in the transformation, failed to pay attention, or did not have a clear-cut understanding of inversion, many might resort to guessing (a random process). If so, the scoring system described would—except for extreme cases—most likely result in an inconsistent child being scored as at least marginally reliable, not as a non-user. Specifically, with three choices, as was done in the present study (see also Baroody & Lai, 2007; Lai et al., 2008), participants stood a 33.3% chance of choosing the correct answer on these trials. A child who failed to apply his or her inverse knowledge on one or two trials would be categorized as reliably or marginally successful and, thus, stood a 0% chance of being scored a “non-user.” If a child was perfectly inconsistent and failed to use his/her inverse knowledge on half of the six trials, then it is likely that the child would have been scored as at least “marginally reliable” (p = .7037, Binomial Theorem). In other words, the child would have to be extremely unlucky (choose incorrectly on all three guessed trials) to be scored as a “non-user” (only a 29.63% chance). Using the same logic, if a child was highly inconsistent (failed to use his/her inverse knowledge on four trials) or most inconsistent (failed to do so on five trials), there is still a 40% and 21% chance, respectively, that the child would have been scored as at least “marginally reliable.”

The previous analysis is not applicable if a child responds with one or more systematic errors. In an extensive error analysis of systematic errors, Baroody and Lai (2007) found that 8 of the 15 and 4 of the 11 unsuccessful 4 and 5 year olds, respectively, reliably (on at least 10 of 14 trials) used a response bias, such as equating any (qualitative as well as quantitative) as a change (consistently switching between “more” or “fewer”). In the present study, 7 of 16, 6 of 8, and 3 of 3 unsuccessful 3, 4, and 5 year olds, respectively, consistently responded “the same” or reliably stated “more.” This does not count children may have used several systematic incorrect strategies. The current scoring system, then, might underestimate the competence of inconsistent users of inversion in some cases. It is useful, therefore, to compare the conservative estimates of competence obtained with algebraic-reasoning tasks with the liberal estimates obtained with shortcut tasks to obtain converging evidence on emerging inverse competence or a balanced estimate of this development (Baroody & Lai, 2007).

²A test for the equality of partially overlapping frequencies effectively tests the hypothesis that if variable x is a necessary condition (developmental prerequisite) of variable y, then the frequency of participants who are successful on x but unsuccessful on y (light shaded cells in Table 4) should be significantly higher than the reverse (dark shaded cells in Table 4). (If the hypothesis is true and in the absence of measurement error, the dark shaded cells in Table 4 would be only cells with 0.)