Young children ‘solve for x’ using the Approximate Number System

Abstract

The Approximate Number System (ANS) supports basic arithmetic computation in early childhood, but it is unclear whether the ANS also supports the more complex computations introduced later in formal education. ‘Solving for x’ in addend-unknown problems is notoriously difficult for children, who often struggle with these types of problems well into high school. Here we asked whether 4–6-year-old children could solve for an unknown addend using the ANS. We presented problems either symbolically, using Arabic numerals or verbal number words, or non-symbolically, using collections of objects while preventing verbal counting. Across five experiments, children failed to identify the value of the unknown addend when problems were presented symbolically, but succeeded when problems were presented non-symbolically. Our results suggest that, well before formal exposure to unknown-addend problems, children appear to ‘solve for x’ in an intuitive way, using the ANS.

Numerical error monitoring

“…For instance, Gelman and Gallistel (1978) report that children often restart counting when they skip an object in the set or a number word. Another study shows that when math problems are framed nonsymbolically using two magic cups that add different numbers of items to the original set, children can infer from which of the two cups the new items came from (i.e., an addendunknown problem; Kibbe & Feigenson, 2015). In both of these situations, children demonstrated the basic ability to compare the expected outcome with the actual one and use that error information.…”

Individual differences in algebraic cognition: Relation to the approximate number and semantic memory systems

“…We extend the debate to algebra and demonstrate nuance in the relation between ANS acuity and fact memory and individual differences in this area (see also Kibbe & Feigenson, 2015). …”

“…The ANS is an inherent system for representing, comparing, and combining the magnitudes of collections of objects (see Feigenson, Dehaene, & Spelke, 2004; Geary, Berch, & Mann Koepke, 2015), and there is some evidence that poor acuity of this system contributes to difficulties in learning mathematics (Piazza et al, 2010), and to individual differences in mathematics achievement more generally (Chen & Li, 2014; Fazio, Bailey, Thompson, & Siegler, 2014; Kibbe & Feigenson, 2015; Libertus, Halberda, & Feigenson, 2011; Starr, Libertus, & Brannon, 2013). Other studies, however, suggest that children’s and adults’ formal mathematical competencies, whether or not they have learning difficulties, are largely independent of ANS acuity and that individual differences in mathematics achievement are more consistently related to the fluency of processing symbolic numerical and arithmetical information (e.g., Bugden & Ansari, 2011; De Smedt et al, 2011; De Smedt, Noël, Gilmore, & Ansari, 2013; Iuculano, Tang, Hall, & Butterworth, 2008; Rousselle & Noël, 2007) or to more basic processes, such as inhibitory control that influence performance on both ANS tasks and mathematics achievement tests (Fuhs & McNeil, 2013; Gilmore et al, 2013; but see Keller & Libertus, 2015).…”

Approximate Arithmetic Training Improves Informal Math Performance in Low Achieving Preschoolers

“…Previous research has demonstrated that children can solve math problems non-symbolically and approximately before they comprehend the same operations symbolically (Barth et al, 2005 ). With the ANS, young children can compare, add, subtract, multiply, and divide, and solve simple linear equations using sets of objects with ratio-dependent precision (Barth et al, 2006 ; McCrink and Spelke, 2010 , 2016 ; Kibbe and Feigenson, 2015 ). In contrast to these prodigious non-symbolic and approximate mathematical abilities, children must be explicitly taught how to solve the same symbolic mathematical problems effectively over years of formal schooling.…”