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.

Unwarranted philosophical assumptions in research on ANS.

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.

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).

Left posterior prefrontal regions support domain-general executive processes needed for both reading and math.

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.

Characterizing and decomposing the neural correlates of individual differences in reading ability among adolescents with task-based fMRI.

To better characterize the neural correlates of the full spectrum of reading ability, this fMRI study examined how variations in reading ability correlate with task-based brain activity during reading among a large community sample of adolescents (N = 234). In addition, complimentary approaches taking advantage of empirical as well as independent meta-analytic information were employed to isolate neural substrates of domain-general executive processes that are predictive of reading ability. Age-related differences in brain activity were also examined. Better reading was associated with increased activation in left anterior and inferior temporal regions and parts of orbitofrontal cortex, along with reduced activation in the thalamus and left frontal eye field (FEF). Converging evidence suggests that FEF activity corresponds to executive processes during reading. In contrast, activity in temporal regions is likely to reflect cognitive processes specific to reading. Older adolescents also demonstrated increased activation in an orbitofrontal region that overlaps with the aforementioned age-independent, reading-related regions, along with reduced activity in parietal and occipital regions. These results suggest that comparedto poor readers, proficient readers benefit from efficient reading-specific processes and require less executive effort, implemented via the FEF, during a reading comprehension task.

Linear Spatial-Numeric Associations Aid Memory for Single Numbers.

Memory for numbers improves with age. One source of this improvement may be learning linear spatial-numeric associations, but previous evidence for this hypothesis likely confounded memory span with quality of numerical magnitude representations and failed to distinguish spatial-numeric mappings from other numeric abilities, such as counting or number word-cardinality mapping. To obviate the influence of memory span on numerical memory, we examined 39 3- to 5-year-olds' ability to recall one spontaneously produced number (1-20) after a delay, and the relation between numeric recall (controlling for non-numeric recall) and quality of mapping between symbolic and non-symbolic quantities using number-line estimation, give-a-number estimation, and counting tasks. Consistent with previous reports, mapping of numerals to space, to discrete quantities, and to numbers in memory displayed a logarithmic-to-linear shift. Also, linearity of spatial-numeric mapping correlated strongly with multiple measures of numeric recall (percent correct and percent absolute error), even when controlling for age and non-numeric memory. Results suggest that linear spatial-numeric mappings may aid memory for number over and above children's other numeric skills.

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.

Nuclear IHC enumeration: A digital phantom to evaluate the performance of automated algorithms in digital pathology.

Automatic and accurate detection of positive and negative nuclei from images of immunostained tissue biopsies is critical to the success of digital pathology. The evaluation of most nuclei detection algorithms relies on manually generated ground truth prepared by pathologists, which is unfortunately time-consuming and suffers from inter-pathologist variability. In this work, we developed a digital immunohistochemistry (IHC) phantom that can be used for evaluating computer algorithms for enumeration of IHC positive cells. Our phantom development consists of two main steps, 1) extraction of the individual as well as nuclei clumps of both positive and negative nuclei from real WSI images, and 2) systematic placement of the extracted nuclei clumps on an image canvas. The resulting images are visually similar to the original tissue images. We created a set of 42 images with different concentrations of positive and negative nuclei. These images were evaluated by four board certified pathologists in the task of estimating the ratio of positive to total number of nuclei. The resulting concordance correlation coefficients (CCC) between the pathologist and the true ratio range from 0.86 to 0.95 (point estimates). The same ratio was also computed by an automated computer algorithm, which yielded a CCC value of 0.99. Reading the phantom data with known ground truth, the human readers show substantial variability and lower average performance than the computer algorithm in terms of CCC. This shows the limitation of using a human reader panel to establish a reference standard for the evaluation of computer algorithms, thereby highlighting the usefulness of the phantom developed in this work. Using our phantom images, we further developed a function that can approximate the true ratio from the area of the positive and negative nuclei, hence avoiding the need to detect individual nuclei. The predicted ratios of 10 held-out images using the function (trained on 32 images) are within ±2.68% of the true ratio. Moreover, we also report the evaluation of a computerized image analysis method on the synthetic tissue dataset.

A field guide for teaching evolution in the social sciences.

The theory of evolution by natural selection has begun to revolutionize our understanding of perception, cognition, language, social behavior, and cultural practices. Despite the centrality of evolutionary theory to the social sciences, many students, teachers, and even scientists struggle to understand how natural selection works. Our goal is to provide a field guide for social scientists on teaching evolution, based on research in cognitive psychology, developmental psychology, and education. We synthesize what is known about the psychological obstacles to understanding evolution, methods for assessing evolution understanding, and pedagogical strategies for improving evolution understanding. We review what is known about teaching evolution about nonhuman species and then explore implications of these findings for the teaching of evolution about humans. By leveraging our knowledge of how to teach evolution in general, we hope to motivate and equip social scientists to begin teaching evolution in the context of their own field.

Number sense and mathematics: Which, when and how?

Individual differences in number sense correlate with mathematical ability and performance, although the presence and strength of this relationship differs across studies. Inconsistencies in the literature may stem from heterogeneity of number sense and mathematical ability constructs. Sample characteristics may also play a role as changes in the relationship between number sense and mathematics may differ across development and cultural contexts. In this study, 4,984 16-year-old students were assessed on estimation ability, one aspect of number sense. Estimation was measured using 2 different tasks: number line and dot-comparison. Using cognitive and achievement data previously collected from these students at ages 7, 9, 10, 12, and 14, the study explored for which of the measures and when in development these links are observed, and how strong these links are and how much these links are moderated by other cognitive abilities. The 2 number sense measures correlated modestly with each other (r = .22), but moderately with mathematics at age 16. Both measures were also associated with earlier mathematics; but this association was uneven across development and was moderated by other cognitive abilities. (PsycINFO Database Record

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.