Children’s Understanding of the Counting Algorithm After Mastering Its Usage

• In Event: 3-226 - Number Knowledge Before, During, and After Children’s Mastery of Counting

Integrative Statement

Children learn to count by going through systematic stages of knowledge (learning the meaning of the first three or four number words before mastering how to count). While much research has focused on children’s knowledge before learning to count, far less is known about children’s knowledge after they master the counting algorithm. Specifically, are there stages of counting-knowledge after children can successfully count? Here we test this idea by looking at children’s understanding of abstract properties about the counting algorithm. Namely, when reapplying the counting algorithm will give the same answer, and how replacing number words with another ordered list continues to be a valid counting algorithm.

Study 1 tests whether counters recognize that, after counting a set, it is unnecessary to recount after one object is substituted for a new object. 50 CP-knowers (as determined by the Give-N task; mean age: 4.60 years) saw nine objects in a row and were asked to count them. Participants completed two (counterbalanced) test trials: an “ambiguous substitution” where the experimenter piled all objects and replaced one with an identical object, and a “last substitution” where the last object the child counted was replaced with an identical object. After each transformation, a puppet appeared, and the child was asked to tell the puppet how many objects there were. If children understand the invariance of the counting algorithm, then they should never recount. Overall, 54.5% of kids recounted in the last substitution trial, and 56.4% recounted in the ambiguous substitution trial. Participants either recounted in both trials (50.9%) or in neither trial (40%). The probability of recounting decreased with age (β=-7.99; p = .01, using a mixed-effects logistic regression). Altogether, this suggests that children who can count may not necessarily be aware of when counting will produce a different answer.

Study 2 tests whether counters understand that other ordered lists can be used to count. 54 CP-knowers (mean age: 4.82 years) saw a list of eight animals ordered by size from left to right, and were told that some people count using this list. The experimenter demonstrated by counting four objects (“ant, mouse, cat, pig”), and eight objects (“ant, mouse, …, elephant, giraffe”). Children completed three (counterbalanced) test trials (2vs3; 3vs6; and 6vs7 objects). In each trial, participants watched an agent count objects in two boxes. The boxes were visible, but their contents were not. The agent stated how many objects were in each box using the animal list and children were asked which box had more objects. Performance was above 79% on all trials, with the probability of answering correctly increasing with age (β=1.93; p<0.005). 68.25% of children performed at ceiling. Altogether, this suggests that children who understand the meaning of counting are mostly able to recognize that another ordered list can be used to count, although visual cues of size may help facilitate this understanding.

Collectively, these studies suggest that children continue to develop their understanding of the counting algorithm, even after they have learned to use it effectively.

Authors

• Madison Flowers, Yale University
  Presenting Author

• Lindsay Stoner, University of Pennsylvania
  Non-Presenting Author

• Julian Jara-Ettinger, Yale University
  Non-Presenting Author