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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 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic 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 non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically 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.
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Arithmetic operations without symbols are unimpaired in adults with math anxiety
This study characterises a previously unstudied facet of a major causal model of math anxiety. The model posits that impaired “basic number abilities” can lead to math anxiety, but what constitutes a basic number ability remains underdefined. Previous work has raised the idea that our perceptual ability to represent quantities approximately without using symbols constitutes one of the basic number abilities. Indeed, several recent studies tested how participants with math anxiety estimate and compare non-symbolic quantities. However, little is known about how participants with math anxiety perform arithmetic operations (addition and subtraction) on non-symbolic quantities. This is an important question because poor arithmetic performance on symbolic numbers is one of the primary signatures of high math anxiety. To test the question, we recruited 92 participants and asked them to complete a math anxiety survey, two measures of working memory, a timed symbolic arithmetic test, and a non-symbolic “approximate arithmetic” task. We hypothesised that if impaired ability to perform operations was a potential causal factor to math anxiety, we should see relationships between math anxiety and both symbolic and approximate arithmetic. However, if math anxiety relates to precise or symbolic representation, only a relationship between math anxiety and symbolic arithmetic should appear. Our results show no relationship between math anxiety and the ability to perform operations with approximate quantities, suggesting that difficulties performing perceptually based arithmetic operations do not constitute a basic number ability linked to math anxiety.
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- Award ID(s):
- 1654089
- PAR ID:
- 10400281
- Date Published:
- Journal Name:
- Quarterly Journal of Experimental Psychology
- ISSN:
- 1747-0218
- Page Range / eLocation ID:
- 174702182211135
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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 per-form a true multiplication operation, flexibly attending to both the multiplier and multiplicand, prior to formal multiplication instruc-tion. We presented 5- to 8-year-olds with nonsymbolic multipli-cands (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 vis-ible comparison quantity. Children performed with above-chance accuracy on both nonsymbolic and symbolic approximate multipli-cation, and their performance was dependent on the ratio between the imagined product and the comparison target. Children who could not solve any single-digit 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 chil-dren’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
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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.more » « less
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Park and Brannon (2013, https://doi.org/10.1177/0956797613482944) found that practicing non-symbolic 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 judgment-and-decision-making (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 approximate-arithmetic tasks whereas active control participants trained on a spatial working-memory 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 non-symbolic numeric stimuli (i.e., dot arrays). We partially replicated Park and Brannon’s findings: The numeracy training improved objective-numeracy 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 risk-consistency 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
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Objective There are many measures of health and/or financial literacy and/or numeracy, which vary widely in terms of length, content, and the extent to which numbers or math operations are involved. Although the literature is large, there is less considers what is known about math prediction and development, perhaps because the bulk of this literature is with adults. This is important because the literature on how math develops, what predicts it, and how to intervene with it, is very large (Cirino, 2022). To the extent that performance and prediction are similar, then information from the developmental literature of mathematics can be brought to bear with regard to health and financial numeracy. Here we assess math cognition variables (arithmetic concepts and number line estimation) alongside working memory, likely the most robust cognitive predictor of math, as well as sociodemographic covariates. We expect all predictors to relate to each type of outcome, though we expect reading to be more related to health and financial numeracy relative to symbolic math. Participants and Methods Participants were 238 young adults, diverse in language and race/ethnicity, enrolled in their first and entry-level college math class at either community college or university; approximately 30% were taking developmental coursework. For this study, participants were given three sets of analogous math problems: (a) pure symbolic; (b) health numeracy context; (c) financial numeracy context. Additional measures were of reading (KTEA-3 Reading Comprehension), math cognition (Arithmetic Concepts and Number Line Estimation), and complex span working memory (Symmetry Span and Reading Span). Correlations assessed relations, and multiple regression assessed prediction. Results All measures involving math correlated, though symbolic math less well than health and financial numeracy with one another. For symbolic math, math cognition and working memory together accounted for R2=56% variance, and all were unique predictors, with arithmetic concepts strongest (ηp2 = .19). For health numeracy, all predictors also accounted for R2=56% variance. Beyond symbolic math, math cognition and working memory were unique predictors (all p < .05); reading comprehension was not. The clearly strongest unique predictor was number line estimation (ηp2 = .06). For financial numeracy, all predictors accounted for R2=61% variance. Beyond symbolic math and reading comprehension, again math cognition and working memory were unique predictors (all p < .05), and again number line estimation was the strongest (ηp2 = .08). Results held with covariate control. Conclusions Math cognition and working memory are known important contributors to math skill. This study shows these to be equally important whether math is in a pure symbolic context, or a health and or financial context. This suggests that the utility of health and financial numeracy measures (and potentially the constructs themselves) needs to consider the underlying concomitants of math skill more generally, particularly as the extent to which numbers and/or specific math operations are used in such measures varies widely. Context is likely important, however, and future work will need to consider practical outcomes (e.g., risky health or financial behaviors and management) across a range of populations.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1177/17470218221113555
