Infants' understanding of numerical transformations

Abstract

This research explored infants' understanding of transformations that change the numerosity of a set. Do infants know how the insertion or deletion of a single object affects the relative numerosity of two sets? Four age groups (14‐, 18‐, 24‐ and 28‐month‐olds) were tested on problems involving sets whose numerosities were never greater than 2. The children were encouraged to find the set with more objects, after watching the experimenter cover up two sets and then transform one of them by performing either an insertion or a deletion. On the start‐equal problems, the two sets had the same number of objects until the transformation was performed, so that when an object was inserted the transformed set always had more and when an object was deleted the untransformed set always had more. On the start‐unequal problems, the initial numerosities of the two sets were varied so that these relations between the type of transformation that was performed and the relative numerosities of the two sets no longer held. One set initially had two objects in it and the other set had none. Insertions were performed on the smaller set, so that it still had less than the untransformed set; and deletions were performed on the larger set, so that it still had more than the untransformed set. Performance improved significantly across the four ages tested. However, every age group showed some understanding of the transformations. These results add to the growing body of evidence for an understanding of numerical relationships in infancy.

Can Young Infants Add and Subtract?

“…Furthermore, a number of studies have investigated the development of addition and subtraction in toddlers and young preschoolers using dependent measures such as reaching, object manipulation, and verbal response (e.g., Houdé, 1997;Huttenlocher et al, 1994;Sophian & Adams, 1987;Starkey, 1992;Vilette, 1996;Vilette & Mazouz, 1998). Young children's performance on simple addition and subtraction tasks with very small sets of concrete objects indicate that by their second year children know the ordinal effect of these operations, that addition yields more and subtraction yields less (e.g., Sophian & Adams, 1987;Starkey, 1992). The ability to calculate the exact results of simple addition and subtraction problems, however, develops gradually during early childhood (e.g., Huttenlocher et al, 1994;Starkey, 1992).…”

How we use what we learn in Math: An integrative account of the development of commutativity.

“…However, even though considerable interest in research on counting and addition principles emerged already in the 1980s and still continues (e.g., Baroody, 1984;Baroody & Gannon, 1984;Baroody et al, 1983;Briars & Siegler, 1984;Canobi, Reeve & Pattison, 1998, 2002, 2003Fuson, 1988;Gelman & Gallistel, 1978;Gelman & Meck, 1983;Resnick, 1992;Sophian & Adams, 1987;Starkey & Gelman, 1982), the central question has not been solved yet: How and when do children acquire integrated knowledge representations, in the sense of true formal arithmetic principles? For instance, Geary (2006) stated that it is not clear when children "explicitly understand commutativity as a formal arithmetical principle" (p. 791).…”

The Representations Underlying Infants' Choice of More: Object Files Versus Analog Magnitudes

“…Cooper reported that 14-to 16-month-olds successfully discriminated both same/different and more/less relationships, but did not present data on performance with particular numerosities. Finally, Sophian and Adams (1987) gave children of 14, 18, 24, and 28 months choices between arrays of varying numerosity. With 1-versus-2 comparisons, the oldest and the youngest age groups chose the array with the greater numerosity.…”