Clarissa A. Thompson

An Integrated Theory of Whole Number and Fractions Development.

This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds indicate that, as with whole numbers, accuracy of fraction magnitude representations is closely related to both fractions arithmetic proficiency and overall mathematics achievement test scores, that fraction magnitude representations account for substantial variance in mathematics achievement test scores beyond that explained by fraction arithmetic proficiency, and that developing effective strategies plays a key role in improved knowledge of fractions. Theoretical and instructional implications are discussed.

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.

Linear Numerical-Magnitude Representations Aid Children’s Memory for Numbers

We investigated the relation between children’s numerical-magnitude representations and their memory for numbers. Results of three experiments indicated that the more linear children’s magnitude representations were, the more closely their memory of the numbers approximated the numbers presented. This relation was present for preschoolers and second graders, for children from low-income and middle-income backgrounds, for the ranges 0 through 20 and 0 through 1,000, and for four different tasks (categorization and number-line, measurement, and numerosity estimation) measuring numerical-magnitude representations. Other types of numerical knowledge—numeral identification and counting—were unrelated to recall of the same numerical information. The results also indicated that children’s representations vary from trial to trial with the numbers they need to represent and remember and that general strategy-choice mechanisms may operate in selection of numerical representations, as in other domains.

Costs and benefits of representational change: Effects of context on age and sex differences in symbolic magnitude estimation

Studies have reported high correlations in accuracy across estimation contexts, robust transfer of estimation training to novel numerical contexts, and adults drawing mistaken analogies between numerical and fractional values. We hypothesized that these disparate findings may reflect the benefits and costs of learning linear representations of numerical magnitude. Specifically, children learn that their default logarithmic representations are inappropriate for many numerical tasks, leading them to adopt more appropriate linear representations despite linear representations being inappropriate for estimating fractional magnitude. In Experiment 1, this hypothesis accurately predicted a developmental shift from logarithmic to linear estimates of numerical magnitude and a negative correlation between accuracy of numerical and fractional magnitude estimates (r=-.80). In Experiment 2, training that improved numerical estimates also led to poorer fractional magnitude estimates. Finally, both before and after training that eliminated age differences in estimation accuracy, complementary sex differences were observed across the two estimation contexts.

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. Ratcliffs (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.

Modeling individual differences in response time and accuracy in numeracy

In the study of numeracy, some hypotheses have been based on response time (RT) as a dependent variable and some on accuracy, and considerable controversy has arisen about the presence or absence of correlations between RT and accuracy, between RT or accuracy and individual differences like IQ and math ability, and between various numeracy tasks. In this article, we show that an integration of the two dependent variables is required, which we accomplish with a theory-based model of decision making. We report data from four tasks: numerosity discrimination, number discrimination, memory for two-digit numbers, and memory for three-digit numbers. Accuracy correlated across tasks, as did RTs. However, the negative correlations that might be expected between RT and accuracy were not obtained; if a subject was accurate, it did not mean that they were fast (and vice versa). When the diffusion decision-making model was applied to the data (Ratcliff, 1978), we found significant correlations across the tasks between the quality of the numeracy information (drift rate) driving the decision process and between the speed/accuracy criterion settings, suggesting that similar numeracy skills and similar speed-accuracy settings are involved in the four tasks. In the model, accuracy is related to drift rate and RT is related to speed-accuracy criteria, but drift rate and criteria are not related to each other across subjects. This provides a theoretical basis for understanding why negative correlations were not obtained between accuracy and RT. We also manipulated criteria by instructing subjects to maximize either speed or accuracy, but still found correlations between the criteria settings between and within tasks, suggesting that the settings may represent an individual trait that can be modulated but not equated across subjects. Our results demonstrate that a decision-making model may provide a way to reconcile inconsistent and sometimes contradictory results in numeracy research.

Free versus anchored numerical estimation: A unified approach

Childrens 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 childrens 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.

Numerical landmarks are useful—except when they’re not

Placing landmarks on number lines, such as marking each tenth on a 0-1 line with a hatch mark and the corresponding decimal, has been recommended as a useful tool for improving childrens number sense. Four experiments indicated that some landmarks do have beneficial effects, others have harmful effects, and yet others have no effects on representations of common fractions (N/M). The effects of the landmarks were seen not only on the number line task where they appeared but also on a subsequent magnitude comparison task and on correlations with mathematics achievement tests. Landmarks appeared to exert their effects through the encodings and strategies that they promoted. Theoretical and educational implications are discussed.

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.

Children’s mental representation when comparing fractions with common numerators

Researchers debate whether one represents the magnitude of a fraction according to its real numerical value or just the discrete numerosity of its numerator or denominator. The present study examined three effects based on the notion that people possess a mental number line to explore how children represent fractions when they compare fractions with common numerators. Specifically, the effect of the spatial numerical association of response codes (SNARC), the distance effect and the size effect in representing fractions were examined in a sample of 72 sixth graders, who successfully solved the fraction comparison task with a real number (.2) or a fraction (1/5) as the reference. Results showed that in the fraction-reference group (1/5 as the reference), there was a significant reverse SNARC effect and a distance effect between the denominators of the target fractions and the reference fraction; in the real number-reference group, the three effects were also observed. These results revealed that both groups used the mental number line to represent fractions and did not represent their real numerical values but rather the discrete numerosities of denominators when comparing fractions with common numerators. It seems that the way people represent fractions may depend on their strategy choices.