Whether the meaning of symbolic numerals is represented by the approximate number system (ANS) is a subject of long-standing interest, with contradictory results reported across educated adults and developing children. To resolve past discrepancies, we investigated the time required to make judgments of numerical inequality across an expansive numerical spectrum (1-999) and range of ratios (1.125-4.5) in 217 children (ages 4-12) and 66 adults. Testing an experience-dependent framework, Study 1 evaluated whether the relative fluency of symbolic and non-symbolic processing changes with age according to the power law of practice. As predicted, the law provided an excellent fit to the data, with cognitive efficiency progressing through three range-dependent phases (prefluent, fluent, and overlearned) that tracked the ecological frequency of numbers. Study 2 evaluated cross-format switch costs, revealing a significant age-by-range interaction. Young children exhibited deeply negative switch costs for triple-digit numbers, with switch costs systematically rising with age. Rather than showing a number-wide overhaul, the data suggest that the time required to process symbolic magnitudes decreases as a non-linear function of experience.

Could the average estimate of a class of kindergartners be more accurate than any adult estimate? Normally, yes: a "wisdom of crowds" (WoC) effect would be observed if individual estimates were independent and unbiased, leading any crowd of children to outperform any individual adult. However, children's estimates have been proposed to be biased by logarithmic encoding of numerical value, which would undermine the WoC effect. Here, we examined the WoC effect to test whether estimates reflect bias (e.g., logarithmic encoding), noise (e.g., uncertainty), or both. In Study 1, 80 4- to 7-year-olds and 80 adults were asked to estimate the number of dots on a number line, and we found that WoC effects and effects of crowd size were stronger on 0-30 than 0-100 number-lines. In Study 2, 80 children and 80 adults completed a symbolic number-line task, and we found similar effects. In both studies, we also compared a crowd of less-advanced with individual more-advanced children. Again, the WoC effect was stronger on 0-30 than 0-100 number lines. Overall, results provide further evidence that development of numerical estimation involves reducing bias (especially for large numbers), not just reducing noise. Implications for theories of number representation are discussed.

Natural language is often depicted as the sine qua non of mathematical thinking, a view buttressed by findings of language-of-training effects among bilinguals. These findings, however, have been limited to studies of arithmetic. Here, we asked whether algebraic thinking differs. We trained Chinese-English bilinguals and English monolinguals to solve arithmetic and algebra problems in either Chinese or English and tested them on new and old problems in both languages. In Experiments 1 and 2, bilinguals solved arithmetic problems faster in their trained than untrained language, and old arithmetic problems were solved faster than new ones. However, both the language-of-training and novelty effect were reduced or eliminated when learning algebraic rules. Strikingly, when English monolinguals were given Chinese problems, they successfully learned to solve the algebraic-but not arithmetic-problems. Together, the findings suggest that-unlike rote arithmetic-algebraic rules need not be encoded in natural language.

Fractions are the gatekeepers to advanced mathematics but are difficult to learn. One powerful learning mechanism is analogy, which builds fraction understanding on a pre-existing foundation of integer knowledge. Indeed, a short intervention that aligned fractions and integers on number lines improved children’s estimates of fractions (Yu et al., 2022). The breadth and durability of such gains, however, are unknown, and analogies to other sources (such as percentages) may be equally powerful. To investigate this issue, we randomly assigned 109 fourth and fifth graders to one of three experimental conditions with different analogical sources (integers, percentages, or fractions) or a control condition. During training, children in the experimental conditions solved pairs of aligned fraction number line problems and proportionally-equivalent problems expressed in integers, percentages, or fractions (e.g., 3/8 on a 0–1 number line aligned with 3 on a 0–8 number line). Children in the control group solved fraction number-line problems sequentially. At pretest and a two-week delayed posttest, children completed a broad fraction knowledge battery, including estimation, comparison, categorization, ordering, and arithmetic. Results showed that aligning integers and fractions on number lines facilitated better estimation of fractional magnitudes, and the training effect transferred to novel fraction problems after two weeks. Similar gains were not observed for analogies using percentages. These findings highlight the importance of building new mathematical knowledge through analogies to familiar, similar sources.

Political partisanship might lead educated adults-even the highly numerate-to reason selectively about numbers that are relevant to and support their ideology ("motivated numeracy"). In this pre-registered study, we sought to examine the replicability of motivated numeracy, and investigate whether cognitive support (number lines) that improves the reasoning of children might also improve the reasoning of political partisans. To test this, we asked 1000 adults about their political ideology and asked them to interpret fictional data, in a table or number-line format, about ideology relevant (i.e., the effect of gun control on crime) or irrelevant (i.e., the effect of skin cream on rash) issues. We failed to replicate motivated numeracy when political identity was used but observed motivated numeracy when prior attitude was used. Moreover, data presented on number lines elicited 75 percent greater accuracy than data presented in tables, regardless of whether the information was ideology-relevant, or whether data supported, was neutral to, or contradicted participants' political outlooks. Findings imply that political partisans require cognitive support to be more objective about policy data.

The psychophysical function that best fits human data from number-line estimation is the subject of a lively, on-going debate with important theoretical and practical implications. We comprehensively reviewed articles which tested competing psychophysical functions and found systematic variablility in task design. To test whether one function could account for data across diverse tasks, we examined 158 children's and adults' estimates using two 2 x 2 designs, crossing symbol (symbolic, non-symbolic) and boundedness (bounded, unbounded) on free number-line tasks (Experiment 1) and crossing the same factors on anchored tasks (Experiment 2). This yielded eight varieties of number-line estimation: four old varieties for testing replicability and four new varieties for testing generalizability. Across the eight varieties, 88.84 % of participants provided estimates better fit by a mixed log-linear model than competing models, with weights of the logarithmic component (lambda) decreasing with age in each task. Unlike parameters of competing models, lambda on any given task significantly predicted lambda on the other 7 tasks, as well as predicting arithmetic skills. Results suggest that representations of numerical magnitude play the largest part in number-line estimation, and the "logarithmic-to-linear shift" provides the most accurate and generalizable description of how number-line estimation develops. (196 words)

Ordinal processing plays a fundamental role in both the representation and manipulation of symbolic numbers. As such, it is important to understand how children come to develop a sense of ordinality in the first place. The current study examines the role of the count-list in the development of ordinal knowledge through the investigation of two research questions: (1) Do K-1 children struggle to extend the notion of numerical order beyond the count-list, and if so (2) does this extension develop incrementally or manifest as a qualitative re-organization of how children recognize the ordinality of numerical sequences. Overall, we observed that although young children reliably identified adjacent ordered sequences (i.e., those that match the count-list; '2-3-4') as being in the correct ascending order, they performed significantly below chance on non-adjacent ordered trials (i.e., those that do not match the count-list but are in the correct order; '2-4-6') from the beginning of kindergarten to the end of first grade. Further, both qualitative and quantitative analyses supported the conclusion that the ability to extend notions of ordinality beyond the count-list emerged as a conceptual shift in ordinal understanding rather than through incremental improvements. These findings are the first to suggest that the ability to extend notions of ordinality beyond the count-list to include non-adjacent numbers is non-trivial and reflects a significant developmental hurdle that most children must overcome in order to develop a mature sense of ordinality.

When estimating the number of dots on a number line, logarithmicity of estimates typically increases with the range of numbers tested. This effect may reflect a logarithmic encoding of numbers (Dehaene et al, 2008). An alternative hypothesis is that numerosity is encoded linearly, but uncertainty about large numbers drives logarithmic compression (Pomé et al, 2021). We tested these two hypotheses by orthogonally manipulating number and entropy, an information-theoretic measure of uncertainty. In the experiment, 163 participants estimated the numerosity of a group of dots in four conditions. The range of numbers was either 0 to 30 (small number) or 0 to 100 (large number). Entropy was manipulated by using one (low entropy) or multiple (high entropy) dot colors. A multiple regression indicated logarithmicity increased with numerical range (b = .25, p < .01), but was not affected by perceptual entropy (b = .04, p = .66). This result suggests that objective uncertainty (perceptual entropy) does not drive logarithmic compression of estimates. Next, we investigated the effect of subjective uncertainty on estimates by analyzing variability of estimates. When estimates were analyzed trial-to-trial, variability of estimates increased with trial order (b = .72, p < .001), yet logarithmicity decreased with trial order (b = -.67, p < .001), such that they were negatively correlated (b = -.76, p < .001). Together, the study provides further evidence that numbers are encoded logarithmically, and neither objective uncertainty (perceptual entropy) nor subjective uncertainty (variability) drive logarithmic compression of estimates.

Author(s): Yu, Shuyuan; Opfer, John | Abstract: Political ideology leads educated adults–especially the highly numerate–to selectively reason about numbers that support their beliefs (“motivated numeracy”). We investigated whether supports that help children’s quantitative reasoning (number-lines) might also help political partisans. To test this, we asked 429 adults to interpret fictional data, in table or number-line format, about the effect of gun control on crime or the effect of a skin cream on rashes. We found data presented in number-line formats yielded greater accuracy than table formats controlling for numeracy skills (χ2 (1) = 21.88, p l .001), regardless of whether the true interpretation of data affirms, neutral to, or disaffirms participants’ political outlooks. Solving table problems after number-line problems yielded greater accuracy compared to solving table problems first (χ2 (1) = 4.78, p l .005), suggesting number-line practice is educational. Our research has important implications for communicating policy data and improving objectivity.

Clarke and Beck import certain assumptions about the nature of numbers. Although these are widespread within research on number cognition, they are highly contentious among philosophers of mathematics. In this commentary, we isolate and critically evaluate one core assumption: the identity thesis.

Chinese children routinely outperform American peers in standardized tests of mathematics knowledge. To examine mediators of this effect, 95 Chinese and US 5-year-olds completed a test of overall symbolic arithmetic, an IQ subtest, and three tests each of symbolic and non-symbolic numerical magnitude knowledge (magnitude comparison, approximate addition, and number-line estimation). Overall Chinese children performed better in symbolic arithmetic than US children, and all measures of IQ and number knowledge predicted overall symbolic arithmetic. Chinese children were more accurate than US peers in symbolic numerical magnitude comparison, symbolic approximate addition, and both symbolic and non-symbolic number-line estimation; Chinese and U.S. children did not differ in IQ and non-symbolic magnitude comparison and approximate addition. A substantial amount of the nationality difference in overall symbolic arithmetic was mediated by performance on the symbolic and number-line tests.

Perceptual judgments result from a dynamic process, but little is known about the dynamics of number-line estimation. A recent study proposed a computational model that combined a model of trial-to-trial changes with a model for the internal scaling of discrete numbers. Here, we tested a surprising prediction of the model-a situation in which children's estimates of numerosity would be better than those of adults. Consistent with the model simulations, task contexts led to a clear developmental reversal: children made more adult-like, linear estimates when to-be-estimated numbers were descending over trials (i.e., backward condition), whereas adults became more like children with logarithmic estimates when numbers were ascending (i.e., forward condition). In addition, adults' estimates were subject to inter-trial differences regardless of stimulus order. In contrast, children were not able to use the trial-to-trial dynamics unless stimuli varied systematically, indicating the limited cognitive capacity for dynamic updates. Together, the model adequately predicts both developmental and trial-to-trial changes in number-line tasks.

To characterize numerical representations, the number-line task asks participants to estimate the location of a given number on a line flanked with zero and an upper-bound number. An open question is whether estimates for symbolic numbers (e.g., Arabic numerals) and non-symbolic numbers (e.g., number of dots) rely on common processes with a common developmental pathway. To address this question, we explored whether well-established findings in symbolic number-line estimation generalize to non-symbolic number-line estimation. For exhaustive investigations without sacrificing data quality, we applied a novel Bayesian active learning algorithm, dubbed Gaussian process active learning (GPAL), that adaptively optimizes experimental designs. The results showed that the non-symbolic number estimation in participants of diverse ages (5–73 years old, n = 238) exhibited three characteristic features of symbolic number estimation.

Kim and Opfer (2017) found that number-line estimates increased approximately logarithmically with number when an upper bound (e.g., 100 or 1000) was explicitly marked (bounded condition) and when no upper bound was marked (unbounded condition). Using procedural suggestions from Cohen and Ray (2020), we examined whether this logarithmicity might come from restrictions on the response space provided. Consistent with our previous findings, logarithmicity was evident whether tasks were bounded or unbounded, with the degree of logarithmicity tied to the numerical value of the estimates rather than the response space per se. We also found a clear log-to-linear shift in numerical estimates. Results from Bayesian modeling supported the idea that unbounded tasks are qualitatively similar to bounded ones, but unbounded ones lead to greater logarithmicity. Our findings support the original findings of Kim and Opfer (2017) and extend their generality to more age groups and more varieties of number-line estimation. (PsycINFO Database Record (c) 2020 APA, all rights reserved).

Substantial evidence has suggested that reading and math are supported by executive processes (EP). However, to date little is known about which portion of the neural system underpinning domain-general executive skills works to support reading and math. In this study, we aimed to answer this question using fMRI via two complementary approaches. First, imaging data were acquired whilst a sample of 231 adolescents performed each of three separate tasks designed to assess reading comprehension, numerical magnitude estimation, and EP in working memory (WM), respectively. With careful task designs and conjunction analyses, we were able to isolate cross-domain brain activity specifically related to EP, as opposed to lower-level domain-general processes (e.g., visual processing). Second, the meta-analytic tool Neurosynth was used to independently identify brain regions involved reading, math, and EP. Using a combination of forward and reverse statistical inference and conjunction analyses, we again isolated brain regions specifically supporting domain-general EP. Results from both approaches yielded overlapping activation for reading, math, and EP in the left ventrolateral prefrontal cortex, left inferior frontal junction, and left precentral gyrus. This pattern suggests that posterior regions of the prefrontal cortex, rather than more central regions such as mid-DLPFC, play a leading role in supporting domain-general EP utilized by both reading and math.

Background The number line task assesses the ability to estimate numerical magnitudes. People vary greatly in this ability, and this variability has been previously associated with mathematical skills. However, the sources of individual differences in number line estimation and its association with mathematics are not fully understood. Aims This large-scale genetically sensitive study uses a twin design to estimate the magnitude of the effects of genes and environments on: (1) individual variation in number line estimation and (2) the covariation of number line estimation with mathematics. Samples We used over 3,000 8- to 16-year-old twins from the United States, Canada, the United Kingdom, and Russia, and a sample of 1,456 8- to 18-year-old singleton Russian students. Methods Twins were assessed on: (1) estimation of numerical magnitudes using a number line task and (2) two mathematics components: fluency and problem-solving. Results Results suggest that environments largely drive individual differences in number line estimation. Both genes and environments contribute to different extents to the number line estimation and mathematics correlation, depending on the sample and mathematics component. Conclusions Taken together, the results suggest that in more heterogeneous school settings, environments may be more important in driving variation in number line estimation and its association with mathematics, whereas in more homogeneous school settings, genetic effects drive the covariation between number line estimation and mathematics. These results are discussed in the light of development and educational settings.