Children use the solutions to non-symbolic arithmetic computations to balance unequal sets

• In Event: Poster Session 04
  In Poster Session: PS 04 Topic - Cognitive Processes

Abstract

Preschool children can intuitively perform basic arithmetic operations over non-symbolic representations of quantity. However, while non-symbolic computation bares some algorithmic similarity to symbolic arithmetic, the extent to which non-symbolic computation obeys similar functional rules to symbolic arithmetic is currently unknown. For instance, in symbolic arithmetic, the solutions to arithmetic operations are independent numerals that can serve as operands in new problems. We asked whether the approximate magnitude representations that arise from non-symbolic arithmetic operations can be used as inputs into new non-symbolic arithmetic operations.
In Experiment 1, we asked 30 4- to 6-year old children (mean: 5.4 years, range: 4.0-6.9; 18 girls) to concurrently solve two arithmetic problems with unknown addends (5+x=17 and 5+y=9) presented non-symbolically, with collections of real objects and a “magic cup” that acted as the unknown x and y. Solving each problem required representing the starting quantity before the cup adds, the final quantity after addition, and performing a difference computation to produce a third magnitude representation, the solution, without ever directly observing the quantities in the cups. In the Test trial, we showed children two sets containing different quantities of objects (8 and either 12 or 20), and asked children to choose which of two the solutions to add to the smaller set to make the sets “about the same” (Figure 1). 20/30 children (67%) chose the correct solution, not significantly above chance (binomial test, p=.10). However, further analysis showed that children who were older than the median age of the group performed above chance, suggesting potential development of this ability.
In Experiment 2, we tested a larger sample of children to investigate a) developmental change in the age range and b) whether the precision of children’s solutions impacts their ability to use those solutions in further computations. Forty-nine children (mean: 5.4 years, range: 4.1-6.9; 24 girls) completed a virtual version of Experiment 1. We also added a series of 8 post-test trials in which we asked children to compare their representations of the solutions to visible quantities, varying the ratio between solutions and visible quantities. Children chose the correct solution to balance the sets at rates significantly above chance (32/49, 65%, p =.04). Unlike in Experiment 1, children’s performance was not related to age (p=.42). Children’s combined responses in Experiments 1 and 2 were significantly above chance (52/79 correct, p=.007; Figure 2a). In the post-test trials, children discriminated the solutions from visible quantities at rates significantly above chance overall (M=77%, p<.001; Figure 2b). However, children’s patterns of responses on individual comparisons suggested that they may have overestimated the magnitude of both solutions, which may have made the balancing task more difficult.
Together, these results suggest that young children can solve two non-symbolic unknown-addend problems and may use the solutions of these problems as input into a balancing computation, suggesting that non-symbolic arithmetic may parallel at least one functional rule of symbolic arithmetic computation. However, there likely are constraints on this ability, including the precision with which the original computations are accomplished.

Authors

• Chen Cheng, Hong Kong University of Science and Technology
  Presenting Author

• Melissa M. Kibbe, Boston University
  Non-Presenting Author