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1. Schneider, R.M., & Barner, D. (2020). Children use one-to-one correspondence to establish equality after learning to count. 42nd Annual Meeting of the Cognitive Science Society.

   Schneider, R.M. (January 2020, 42nd Annual Meeting of the Cognitive Science Society.)

   null (Ed.)

   Humans make frequent and powerful use of external symbols to express number exactly, leading some to question whether exact number concepts are only available through the acquisition of symbolic number systems. Although prior work has addressed this longstanding debate on the relationship between language and thought in innumerate populations and seminumerate children, it has frequently produced conflicting results, leaving the origin of exact number concepts unclear. Here, we return to this question by replicating methods previously used to assess exact number knowledge in innumerate groups, such as the Piraha, with a large sample of semi- numerate US toddlers. We replicate previous findings from both innumerate cultures and developmental studies showing that numeracy is linked to the concept of exact number. However, we also find evidence that this knowledge is surprisingly fragile even amongst numerate children, suggesting that numeracy alone does not guarantee a full understanding of exactness. 

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2. Counting and the ontogenetic origins of exact equality

   https://doi.org/10.1016/j.cognition.2021.104952  

   Schneider, R. M. (January 2022, Cognition)

   Humans are unique in their capacity to both represent number exactly and to express these representations symbolically. This correlation has prompted debate regarding whether symbolic number systems are necessary to represent large exact number. Previous work addressing this question in innumerate adults and semi-numerate children has been limited by conflicting results and differing methodologies, and has not yielded a clear answer. We address this debate by adapting methods used with innumerate populations (a “set-matching” task) for 3- to 5-year-old US children at varying stages of symbolic number acquisition. In five studies we find that children’s ability to match sets exactly is related not simply to knowing the meanings of a few number words, but also to understanding how counting is used to generate sets (i.e., the cardinal principle). However, while children were more likely to match sets after acquiring the cardinal principle, they nevertheless demonstrated failures, compatible with the hypothesis that the ability to reason about exact equality emerges sometime later. These findings provide important data on the origin of exact number concepts, and point to knowledge of a counting system, rather than number language in general, as a key ingredient in the ability to reason about large exact number. 

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3. Math and health and financial numeracy in college students

   Cirino, PT; Boada, C; Salentine, C; Wall, J; vanTerheyden, S (February 2025, International Neuropsychological Society)

   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. 

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4. The causal impact of objective numeracy on judgments: Improving numeracy via symbolic and non-symbolic approximate arithmetic training yields more consistent risk judgments

   https://doi.org/10.5964/jnc.6925  

   Chesney, Dana L.; Shoots-Reinhard, Brittany; Peters, Ellen (January 2021, Journal of Numerical Cognition)

   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. 

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5. Money matters (especially if you are good at math): Numeracy, verbal intelligence, education, and income in satisfaction judgments

   https://doi.org/10.1371/journal.pone.0259331  

   Bjälkebring, Pär; Peters, Ellen (November 2021, PLOS ONE)

   Luciano, Michelle (Ed.)

   Objective numeracy, the ability to understand and use mathematical concepts, has been related to superior decisions and life outcomes. Unknown is whether it relates to greater satisfaction in life. We investigated numeracy’s relations with income satisfaction and overall life satisfaction in a diverse sample of 5,525 American adults. First, more numerate individuals had higher incomes; for every one point higher on the eight-item numeracy test, individuals reported $4,062 more in annual income, controlling for education and verbal intelligence. Combined, numeracy, education, and verbal intelligence explained 25% of the variance in income while Big-5 personality traits explained less than 4%. Further, the higher incomes associated with greater numeracy were related to more positive life evaluations (income and life satisfaction). Second, extant research also has indicated that the highly numerate compare numbers more than the less numerate. Consistent with numeracy-related income comparisons, numeracy moderated the relation between income and life evaluations, meaning that the same income was valued differently by those better and worse at math. Specifically, among those with lower incomes, the highly numerate were less satisfied than the less numerate; this effect reversed among those with higher incomes as if the highly numerate were aware of and made comparisons to others’ incomes. Further, no clear income satiation point was seen among those highest in numeracy, and satiation among the least numerate appeared to occur at a point below $50,000. Third, both education and verbal intelligence related to income evaluations in similar ways, and numeracy’s relations held when controlling for these other relations. Although causal claims cannot be made from cross-sectional data, these novel results indicate that numeracy may be an important factor underlying life evaluations and especially for evaluations concerning numbers such as incomes. Finally, this study adds to our understanding of education and intelligence effects in life satisfaction and happiness. 

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