Symbolic arithmetic knowledge without instruction


Symbolic arithmetic is fundamental to science, technology and economics, but its acquisition by children typically requires years of effort, instruction and drill1,2. When adults perform mental arithmetic, they activate nonsymbolic, approximate number representations3,4, and their performance suffers if this nonsymbolic system is impaired5. Nonsymbolic number representations also allow adults, children, and even infants to add or subtract pairs of dot arrays and to compare the resulting sum or difference to a third array, provided that only approximate accuracy is required6,7,8,9,10. Here we report that young children, who have mastered verbal counting and are on the threshold of arithmetic instruction, can build on their nonsymbolic number system to perform symbolic addition and subtraction11,12,13,14,15. Children across a broad socio-economic spectrum solved symbolic problems involving approximate addition or subtraction of large numbers, both in a laboratory test and in a school setting. Aspects of symbolic arithmetic therefore lie within the reach of children who have learned no algorithms for manipulating numerical symbols. Our findings help to delimit the sources of children’s difficulties learning symbolic arithmetic, and they suggest ways to enhance children’s engagement with formal mathematics.

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Figure 1: Example problems of symbolic, approximate arithmetic.
Figure 2: Five-year-old children’s performance on approximate addition, subtraction and comparison problems.


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We thank C. Katz and R. Lizcano for help with data collection. This work was supported by a ROLE grant from the National Science Foundation (to E.S.S.).

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Correspondence to Camilla K. Gilmore or Elizabeth S. Spelke.

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Reprints and permissions information is available at The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains Supplementary Methods and Analyses, Supplementary Tables 1 – 5 and Supplementary Figures 1 – 4 with Legends. This document contains a list of problems used in the experiments, a description of the tests of school achievement and their relation to performance of symbolic approximate arithmetic, and a description of all the tests for guessing strategies. (PDF 727 kb)

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Gilmore, C., McCarthy, S. & Spelke, E. Symbolic arithmetic knowledge without instruction. Nature 447, 589–591 (2007).

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