 Award ID(s):
 1654089
 NSFPAR ID:
 10400281
 Date Published:
 Journal Name:
 Quarterly Journal of Experimental Psychology
 ISSN:
 17470218
 Page Range / eLocation ID:
 174702182211135
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6 to 9yearold children and college students to perform symbolic and nonsymbolic approximate division. Subjects were presented with nonsymbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and nonsymbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide nonsymbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate nonsymbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.more » « less

Prior work indicates that children have an untrained ability to approximately calculate using their approximate number system (ANS). For example, children can mentally double or halve a large array of discrete objects. Here, we asked whether children can perform a true multiplication operation, ﬂexibly attending to both the multiplier and multiplicand, prior to formal multiplication instruction. We presented 5 to 8yearolds with nonsymbolic multiplicands (dot arrays) or symbolic multiplicands (Arabic numerals) ranging from 2 to 12 and with nonsymbolic multipliers ranging from 2 to 8. Children compared each imagined product with a visible comparison quantity. Children performed with abovechance accuracy on both nonsymbolic and symbolic approximate multiplication, and their performance was dependent on the ratio between the imagined product and the comparison target. Children who could not solve any singledigit symbolic multiplication equations (e.g., 2 3) on a basic math test were nevertheless successful on both our approximate multiplication tasks, indicating that children have an intuitive sense of multiplication that emerges independent of formal instruction about symbolic multiplication. Nonsymbolic multiplication performance mediated the relation between children’s Weber fraction and symbolic math abilities, suggesting a pathway by which the ANS contributes to children’s emerging symbolic math competence. These ﬁndings may inform future educational interventions that allow children to use their basic arithmetic intuition as a scaffold to facilitate symbolic math learning.more » « less

Abstract Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic‐like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function‐like structure, like symbolic arithmetic. Children (
n = 74 4‐ to ‐8‐year‐olds in Experiment 1;n = 52 7‐ to 8‐year‐olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them “about the same.” We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge. 
Park and Brannon (2013, https://doi.org/10.1177/0956797613482944) found that practicing nonsymbolic approximate arithmetic increased performance on an objective numeracy task, specifically symbolic arithmetic. Manipulating objective numeracy would be useful for many researchers, particularly those who wish to investigate causal effects of objective numeracy on performance. Objective numeracy has been linked to performance in multiple areas, such as judgmentanddecisionmaking (JDM) competence, but most existing studies are correlational. Here, we expanded upon Park and Brannon’s method to experimentally manipulate objective numeracy and we investigated whether numeracy’s link with JDM performance was causal. Experimental participants drawn from a diverse internet sample trained on approximatearithmetic tasks whereas active control participants trained on a spatial workingmemory task. Numeracy training followed a 2 × 2 design: Experimental participants quickly estimated the sum of OR difference between presented numeric stimuli, using symbolic numbers (i.e., Arabic numbers) OR nonsymbolic numeric stimuli (i.e., dot arrays). We partially replicated Park and Brannon’s findings: The numeracy training improved objectivenumeracy performance more than control training, but this improvement was evidenced by performance on the Objective Numeracy Scale, not the symbolic arithmetic task. Subsequently, we found that experimental participants also perceived risks more consistently than active control participants, and this riskconsistency benefit was mediated by objective numeracy. These results provide the first known experimental evidence of a causal link between objective numeracy and the consistency of risk judgments.more » « less

Abstract Mathematical knowledge is constructed hierarchically from basic understanding of quantities and the symbols that denote them. Discrimination of numerical quantity in both symbolic and non‐symbolic formats has been linked to mathematical problem‐solving abilities. However, little is known of the extent to which overlap in quantity representations between symbolic and non‐symbolic formats is related to individual differences in numerical problem solving and whether this relation changes with different stages of development and skill acquisition. Here we investigate the association between neural representational similarity (NRS) across symbolic and non‐symbolic quantity discrimination and arithmetic problem‐solving skills in early and late developmental stages: elementary school children (ages 7–10 years) and adolescents and young adults (AYA, ages 14–21 years). In children, cross‐format NRS in distributed brain regions, including parietal and frontal cortices and the hippocampus, was positively correlated with arithmetic skills. In contrast, no brain region showed a significant association between cross‐format NRS and arithmetic skills in the AYA group. Our findings suggest that the relationship between symbolic‐non‐symbolic NRS and arithmetic skills depends on developmental stage. Taken together, our study provides evidence for both mapping and estrangement hypotheses in the context of numerical problem solving, albeit over different cognitive developmental stages.