From integers to fractions: The role of analogy in transfer and long-term learning.

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.

From integers to fractions: The role of analogy in developing a coherent understanding of proportional magnitude.

Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs. 3 of 4 apples), but have considerable difficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., "1/5" vs. "3/4"). Theoretically, reducing the gap between representations of implicit versus explicit proportions would improve understanding of fractions, but little is known about how the representations develop and interact with one another. To address this, we asked 177 third, fourth, and fifth graders (M = 9.85 years, 87 girls, 69% White, 19% low income) to estimate the position of proportionally equivalent integers and fractions on number lines (e.g., 3 on a 0-8 number line vs. 3/8 on a 0-1 number line, Study 1). With increasing age, children's estimates of implicit and explicit proportions became more coherent, such that a child's estimates of fractions on a 0-1 number-line was a linear function of the same child's estimates of equivalent integers. To further investigate whether preexisting integer knowledge can facilitate fraction learning through analogy, we assigned 100 third to fifth graders (M = 10.04 years, 55 girls, 76% White) to an Alignment condition, where children estimated fractions and integers on aligned number lines, or to a No Alignment condition (Study 2). Results showed that aligning integers and fractions on number lines facilitated a better understanding of fractional magnitudes, and increased children's fraction estimation accuracy to the level of college students'. Together, findings suggest that analogies can play an important role in building a coherent understanding of proportions. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

A number-line task with a Bayesian active learning algorithm provides insights into the development of non-symbolic number estimation.

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.

Dynamics Versus Development in Numerosity Estimation: A Computational Model Accurately Predicts a Developmental Reversal.

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.

Cognitive mediators of US-China differences in early symbolic arithmetic.

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.

Compression is evident in children's unbounded and bounded numerical estimation: Reply to Cohen and Ray (2020).

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).

The nature of the association between number line and mathematical performance: An international twin study.

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.

Dynamics and development in number-to-space mapping.

Young children's estimates of numerical magnitude increase approximately logarithmically with actual magnitude. The conventional interpretation of this finding is that children's estimates reflect an innate logarithmic encoding of number. A recent set of findings, however, suggests that logarithmic number-line estimates emerge via a dynamic encoding mechanism that is sensitive to previously encountered stimuli. Here we examine trial-to-trial changes in logarithmicity of numerosity estimates to test an alternative dynamic model (D-MLLM) with both a strong logarithmic component and a weak response to previous stimuli. In support of D-MLLM, first-trial numerosity estimates in both adults (Study 1, 2, 3, and 4) and children (Study 4) were strongly logarithmic, despite zero previous stimuli. Additionally, although numerosity of a previous trial affected adults' estimates, the influence of previous numbers always accompanied the logarithmic-to-linear shift predicted by D-MLLM. We conclude that a dynamic encoding mechanism is not necessary for compressive mapping, but sequential effects on response scaling are a possible source of linearity in adults' numerosity estimation.

A unified framework for bounded and unbounded numerical estimation.

Representations of numerical value have been assessed by using bounded (e.g., 0-1,000) and unbounded (e.g., 0-?) number-line tasks, with considerable debate regarding whether 1 or both tasks elicit unique cognitive strategies (e.g., addition or subtraction) and require unique cognitive models. To test this, we examined how well a mixed log-linear model accounted for 86 5- to 9-year-olds' estimates on bounded and unbounded number-line tasks and how well it predicted mathematical performance. Compared with mixtures of 4 alternative models, the mixed log-linear model better predicted 76% of individual children's estimates on bounded number lines and 100% of children's estimates on unbounded number lines. Furthermore, the distribution of estimates was fit better by a Bayesian log-linear model than by a Bayesian distributional model that depicted estimates as being anchored to varying number of reference points. Finally, estimates were generally more logarithmic on unbounded than bounded number lines, but logarithmicity scores on both tasks predicted addition and subtraction skills, whereas model parameters of alternative models failed to do so. Results suggest that the logarithmic-to-linear shift theory provides a simple, unified framework for numerical estimation with high descriptive adequacy and yields uniquely accurate predictions for children's early math proficiency. (PsycINFO Database Record

How not to develop a sense of number.

The authors rightly point to the theoretical importance of interactions of space and number through the life span, yet propose a theory with several weaknesses. In addition to proclaiming itself unfalsifiable, its stage-like format and emphasis on the role of selective attention are at odds with what is known about the development of spatial-numerical associations in infancy.

Learning Linear Spatial-Numeric Associations Improves Accuracy of Memory for Numbers.

Memory for numbers improves with age and experience. One potential source of improvement is a logarithmic-to-linear shift in children's representations of magnitude. To test this, Kindergartners and second graders estimated the location of numbers on number lines and recalled numbers presented in vignettes (Study 1). Accuracy at number-line estimation predicted memory accuracy on a numerical recall task after controlling for the effect of age and ability to approximately order magnitudes (mapper status). To test more directly whether linear numeric magnitude representations caused improvements in memory, half of children were given feedback on their number-line estimates (Study 2). As expected, learning linear representations was again linked to memory for numerical information even after controlling for age and mapper status. These results suggest that linear representations of numerical magnitude may be a causal factor in development of numeric recall accuracy.

Free versus anchored numerical estimation: A unified approach.

Children's number-line estimation has produced a lively debate about representational change, supported by apparently incompatible data regarding descriptive adequacy of logarithmic (Opfer, Siegler, & Young, 2011) and cyclic power models (Slusser, Santiago, & Barth, 2013). To test whether methodological differences might explain discrepant findings, we created a fully crossed 2×2 design and assigned 96 children to one of four cells. In the design, we crossed anchoring (free, anchored) and sampling (over-, even-), which were candidate factors to explain discrepant findings. In three conditions (free/over-sampling, free/even-sampling, and anchored/over-sampling), the majority of children provided estimates better fit by the logarithmic than cyclic power function. In the last condition (anchored/even-sampling), the reverse was found. Results suggest that logarithmically-compressed numerical estimates do not depend on sampling, that the fit of cyclic power functions to children's estimates is likely an effect of anchors, and that a mixed log/linear model provides a useful model for both free and anchored numerical estimation.

The value of numbers in economic rewards.

Previous work has identified a distributed network of neural systems involved in appraising the value of rewards, such as when winning $100 versus $1. These studies, however, confounded monetary value and the number used to represent it, which leads to the possibility that some elements in the network may be specialized for processing numeric rather than monetary value. To test this hypothesis, we manipulated numeric magnitude and units to construct a range of economic rewards for simple decisions (e.g., 1¢, $1, 100¢, $100). Consistent with previous research in numerical cognition, results showed that blood-oxygen-level-dependent (BOLD) activity in intraparietal sulcus was correlated with changes in numeric magnitude, independent of monetary value, whereas activity in orbitofrontal cortex was correlated with monetary value, independent of numeric magnitude. Finally, region-of-interest analyses revealed that the BOLD response to numeric magnitude, but not monetary value, described a compressive function. Together, these findings highlight the importance of numerical cognition for understanding how the brain processes monetary rewards.

Learning without representational change: development of numerical estimation in individuals with Williams syndrome.

Experience engenders learning, but not all learning involves representational change. In this paper, we provide a dramatic case study of the distinction between learning and representational change. Specifically, we examined long- and short-term changes in representations of numeric magnitudes by asking individuals with Williams syndrome (WS) and typically developing (TD) children to estimate the position of numbers on a number line. As with TD children, accuracy of WS children's numerical estimates improved with age (Experiment 1) and feedback (Experiment 2). Both long- and short-term changes in estimates of WS individuals, however, followed an atypical developmental trajectory: as TD children gained in age and experience, increases in accuracy were accompanied by a logarithmic-to-linear shift in estimates of numerical magnitudes, whereas in WS individuals, accuracy increased but logarithmic estimation patterns persisted well into adulthood and after extensive training. These findings suggest that development of numerical estimation in WS is both arrested and atypical.

Children are not like older adults: a diffusion model analysis of developmental changes in speeded responses.

Children (n = 130; M(age) = 8.51-15.68 years) and college-aged adults (n = 72; M(age) = 20.50 years) completed numerosity discrimination and lexical decision tasks. Children produced longer response times (RTs) than adults. R. Ratcliff's (1978) diffusion model, which divides processing into components (e.g., quality of evidence, decision criteria settings, nondecision time), was fit to the accuracy and RT distribution data. Differences in all components were responsible for slowing in children in these tasks. Children extract lower quality evidence than college-aged adults, unlike older adults who extract a similar quality of evidence as college-aged adults. Thus, processing components responsible for changes in RTs at the beginning of the life span are somewhat different from those responsible for changes occurring with healthy aging.

The powers of noise-fitting: reply to Barth and Paladino.

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude.

How 15 hundred is like 15 cherries: effect of progressive alignment on representational changes in numerical cognition.

How does understanding the decimal system change with age and experience? Second, third, sixth graders, and adults (Experiment 1: N = 96, mean ages = 7.9, 9.23, 12.06, and 19.96 years, respectively) made number line estimates across 3 scales (0-1,000, 0-10,000, and 0-100,000). Generation of linear estimates increased with age but decreased with numerical scale. Therefore, the authors hypothesized highlighting commonalities between small and large scales (15:100::1500:10000) might prompt children to generalize their linear representations to ever-larger scales. Experiment 2 assigned second graders (N = 46, mean age = 7.78 years) to experimental groups differing in how commonalities of small and large numerical scales were highlighted. Only children experiencing progressive alignment of small and large scales successfully produced linear estimates on increasingly larger scales, suggesting analogies between numeric scales elicit broad generalization of linear representations.

Early development of spatial-numeric associations: evidence from spatial and quantitative performance of preschoolers.

Numeric magnitudes often bias adults' spatial performance. Partly because the direction of this bias (left-to-right versus right-to-left) is culture-specific, it has been assumed that the orientation of spatial-numeric associations is a late development, tied to reading practice or schooling. Challenging this assumption, we found that preschoolers expected numbers to be ordered from left-to-right when they searched for objects in numbered containers, when they counted, and (to a lesser extent) when they added and subtracted. Further, preschoolers who lacked these biases demonstrated more immature, logarithmic representations of numeric value than preschoolers who exhibited the directional bias, suggesting that spatial-numeric associations aid magnitude representations for symbols denoting increasingly large numbers.

What makes relational reasoning smart? Revisiting the perceptual-to-relational shift in the development of generalization.

Development of reasoning is often depicted as involving increasing use of relational similarities and decreasing use of perceptual similarities ('the perceptual-to-relational shift'). We argue that this shift is a special case of a broader developmental trend: increasing sensitivity to the predictive accuracy of different similarity types. To test this hypothesis, we asked participants (3-, 4-, 5-year-olds and adults) to generalize novel information on two types of problems--offspring problems, where relational matches yield accurate generalizations, and prey problems, where perceptual matches yield accurate generalizations. On offspring problems, we replicated prior findings of increasing relational matches with age. However, we observed decreasing relational matches on prey problems. Provided feedback on their responses, 3-year-olds showed the same trend. Findings suggest that the relational shift commonly observed in categorization and analogical reasoning may reflect a general increase in children's sensitivity to cue validity rather than an overall preference to generalize over perceptual similarity.