The mental representation of numerical fractions: real or integer?

Numerical fractions are commonly used to express ratios and proportions (i.e., real numbers), but little is known about how they are mentally represented and processed by skilled adults. Four experiments employed comparison tasks to investigate the distance effect and the effect of the spatial numerical association of response codes (SNARC) for fractions. Results showed that fractions were processed componentially and that the real numerical value of the fraction was not accessed, indicating that processing the fraction's magnitude is not automatic. In contrast, responses were influenced by the numerical magnitude of the components and reflected the simple comparison between numerators, denominators, and reference, depending on the strategy adopted. Strategies were used even by highly skilled participants and were flexibly adapted to the specific experimental context. In line with results on the whole number bias in children, these findings suggest that the understanding of fractions is rooted in the ability to represent discrete numerosities (i.e., integers) rather than real numbers and that the well-known difficulties of children in mastering fractions are circumvented by skilled adults through a flexible use of strategies based on the integer components.     

Language supports for children's understanding of numerical fractions: cross-national comparisons.

This study examined Croatian, Korean, and U.S. children's knowledge of numerical fractions prior to school instruction. The research is part of an ongoing project examining the influence of language characteristics on mathematical thinking and performance. The part-whole quantitative relation denoted by numerical fractions may be easier to understand in East Asian languages like Korean. In these languages, the concept of fractional parts is embedded in the mathematics terms used for fractions. The results from this study suggest that the Korean vocabulary of fractions may influence the meaning 6- to 7-year-old children ascribe to numerical fractions and that this results in children being able to associate numerical fractions with corresponding pictorial representations prior to formal instruction.     

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.     

Comparative judgment of numerosity and numerical magnitude: attention preempts automaticity

It is commonly believed that humans are unable to ignore the meanings of numerical symbols, even when these meanings are irrelevant to the task at hand. In 5 experiments, the authors tested the notion of automatic activation of numerical magnitude by asking participants to compare, while timed, pairs of numerical arrays on either numerosity or numerical value. Garner and Stroop effects were used to gauge the degree of interactive processing. The results showed that both effects were sensitive to the discriminability of values along the constituent dimensions, to the number of stimulus values used, and to practice and motivation. Notably, Stroop and Garner effects were eliminated under several conditions. These findings are incompatible with claims of obligatory activation of meaning in numerical processing, and they cast doubt on theories positing automatic processing of semantic information for alphanumerical symbols.     

The mental representation of the magnitude of symbolic and nonsymbolic ratios in adults

This study mainly investigated the specificity of the processing of fraction magnitudes. Adults performed a magnitude-estimation task on fractions, the ratios of collections of dots, and the ratios of surface areas. Their performance on fractions was directly compared with that on nonsymbolic ratios. At odds with the hypothesis that the symbolic notation impedes the processing of the ratio magnitudes, the estimates were less variable and more accurate for fractions than for nonsymbolic ratios. This indicates that the symbolic notation activated a more precise mental representation than did the nonsymbolic ratios. This study also showed, for both fractions and the ratios of dot collections, that the larger the components the less precise the mental representation of the magnitude of the ratio. This effect suggests that the mental representation of the magnitude of the ratio was activated from the mental representation of the magnitude of the components and the processing of their numerical relation (indirect access). Finally, because most previous studies of fractions have used a numerical comparison task, we tested whether the mental representation of magnitude activated in the fraction-estimation task could also underlie performance in the fraction-comparison task. The subjective distance between the fractions to be compared was computed from the mean and the variability of the estimates. This distance was the best predictor of the time taken to compare the fractions, suggesting that the same approximate mental representation of the magnitude was activated in both tasks.     

The mental representation of the magnitude of symbolic and nonsymbolic ratios in adults

This study mainly investigated the specificity of the processing of fraction magnitudes. Adults performed a magnitude-estimation task on fractions, the ratios of collections of dots, and the ratios of surface areas. Their performance on fractions was directly compared with that on nonsymbolic ratios. At odds with the hypothesis that the symbolic notation impedes the processing of the ratio magnitudes, the estimates were less variable and more accurate for fractions than for nonsymbolic ratios. This indicates that the symbolic notation activated a more precise mental representation than did the nonsymbolic ratios. This study also showed, for both fractions and the ratios of dot collections, that the larger the components the less precise the mental representation of the magnitude of the ratio. This effect suggests that the mental representation of the magnitude of the ratio was activated from the mental representation of the magnitude of the components and the processing of their numerical relation (indirect access). Finally, because most previous studies of fractions have used a numerical comparison task, we tested whether the mental representation of magnitude activated in the fraction-estimation task could also underlie performance in the fraction-comparison task. The subjective distance between the fractions to be compared was computed from the mean and the variability of the estimates. This distance was the best predictor of the time taken to compare the fractions, suggesting that the same approximate mental representation of the magnitude was activated in both tasks.     

Understanding the real value of fractions and decimals

Understanding fractions and decimals is difficult because whole numbers are the most frequently and earliest experienced type of number, and learners must avoid conceptualizing fractions and decimals in terms of their whole-number components (the "whole-number bias"). We explored the understanding of fractions, decimals, two-digit integers, and money in adults and 10-year-olds using two number line tasks: marking the line to indicate the target number, and estimating the numerical value of a mark on the line. Results were very similar for decimals, integers, and money in both tasks for both groups, demonstrating that the linear representation previously shown for integers is also evident for decimals already by the age of 10. Fractions seem to be "task dependent" so that when asked to place a fractional value on a line, both adults and children displayed a linear representation, while this pattern did not occur in the reverse task.     

小学生分数比较中的加工模式:基于反应时和口语报告的研究

选取北京和四川两地53名小学六年级学生分别完成同分子、同分母与异分子异分母三类分数比较任务,收集被试口语报告的策略作为直接证据,并以分数大小和分数距离对反应时的回归分析结果作为间接证据,共同探究被试在分数比较任务中的加工模式,结果发现,（1）在三类分数比较中,被试均采用成分加工模式而非整体加工模式;（2）口语报告的策略与反应时回归分析的结果并不完全吻合,从侧面证明了原有研究方法的不稳定性。     

Understanding the real value of fractions and decimals

Understanding fractions and decimals is difficult because whole numbers are the most frequently and earliest experienced type of number, and learners must avoid conceptualizing fractions and decimals in terms of their whole-number components (the “whole-number bias”). We explored the understanding of fractions, decimals, two-digit integers, and money in adults and 10-year-olds using two number line tasks: marking the line to indicate the target number, and estimating the numerical value of a mark on the line. Results were very similar for decimals, integers, and money in both tasks for both groups, demonstrating that the linear representation previously shown for integers is also evident for decimals already by the age of 10. Fractions seem to be “task dependent” so that when asked to place a fractional value on a line, both adults and children displayed a linear representation, while this pattern did not occur in the reverse task.     

Sequential analysis of the numerical Stroop effect reveals response suppression

Automatic processing of irrelevant stimulus dimensions has been demonstrated in a variety of tasks. Previous studies have shown that conflict between relevant and irrelevant dimensions can be reduced when a feature of the irrelevant dimension is repeated. The specific level at which the automatic process is suppressed (e.g., perceptual repetition, response repetition), however, is less understood. In the current experiment we used the numerical Stroop paradigm, in which the processing of irrelevant numerical values of 2 digits interferes with the processing of their physical size, to pinpoint the precise level of the suppression. Using a sequential analysis, we dissociated perceptual repetition from response repetition of the relevant and irrelevant dimension. Our analyses of reaction times, error rates, and diffusion modeling revealed that the congruity effect is significantly reduced or even absent when the response sequence of the irrelevant dimension, rather than the numerical value or the physical size, is repeated. These results suggest that automatic activation of the irrelevant dimension is suppressed at the response level. The current results shed light on the level of interaction between numerical magnitude and physical size as well as the effect of variability of responses and stimuli on automatic processing.     

Fractions We Cannot Ignore: The Nonsymbolic Ratio Congruity Effect

Although many researchers theorize that primitive numerosity processing abilities may lay the foundation for whole number concepts, other classes of numbers, like fractions, are sometimes assumed to be inaccessible to primitive architectures. This research presents evidence that the automatic processing of nonsymbolic magnitudes affects processing of symbolic fractions. Participants completed modified Stroop tasks in which they selected the larger of two symbolic fractions while the ratios of the fonts in which the fractions were printed and the overall sizes of the compared fractions were manipulated as irrelevant dimensions. Participants were slower and less accurate when nonsymbolic dimensions of printed fractions were incongruent with the symbolic comparison decision. Results indicated a robust basic sensitivity to nonsymbolic ratios that exceeds prior conceptions about the accessibility of fraction values. Results also indicated a congruity effect for overall fraction size, contrary to findings of prior research. These findings have implications for extending theory about the nature of human number sense and mathematical cognition more generally.     

Are 1/2 and 0.5 represented in the same way?

Adults' processing of unit and decimal fractions was investigated using the numerical comparison task. When unit fractions were compared to integers, the pattern of distance effect found suggests that they were perceived to be on the same mental number line as integers; however, their representation was undifferentiated, as they were perceived to have the same magnitude. This was found both with simultaneous and with sequential presentation. When decimal fractions were compared to integers, the pattern of results suggests that they were also represented on the same mental number line with integers, but their representation was differentiated. Possible explanations for the different patterns found for unit and decimal fractions are discussed. Moreover, compatibility between the magnitude of the whole fraction and that of its components relative to the compared integer affected performance in the case of decimal fractions and unit fractions presented simultaneously, but not in the case of unit fractions presented sequentially. This suggests that sequential processing reduces the components representation of fractions and the whole number bias.     

The representation of negative numbers: Exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.     

The representation of negative numbers: exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.     

Attention and preattention in theories of automaticity.

The theoretical relation between preattentive processes and automatic processes is different in different approaches to attention and automaticity. In the modal view, automatic and preattentive processes are one and the same; automatic processing is preattentive. In recent views that treat automaticity as a memory phenomenon, automatic processing is postattentive. These views are described and evidence for them is discussed. Two experiments are reported that test whether the training that makes processing automatic also makes it preattentive. The data suggest a dissociation between automatic and preattentive processes that is more consistent with the memory view of automaticity than with the modal view.     

Laterality briefed: laterality modulates performance in a numerosity-congruity task

It is widely agreed that irrelevant numerical values are automatically activated. However, automatic and intentional activations may give rise to different numerical representations. We examined processing of symbolic and non-symbolic (i.e., numerosity) representations asking whether they differ in automatic and intentional processing. Participants were presented with two-dimensional displays containing repetitions of a digit and were asked to report, in different blocks, whether the digit or numerosity was smaller or larger than 5. Incongruent trials differed either in laterality between the relevant and irrelevant dimensions (i.e., the location of both dimensions in reference to the midpoint 5) or in numerical distance between dimensions. Congruency affected performance regardless of symbolic or non-symbolic presentation. For incongruent trials, laterality (not distance) affected performance, again regardless of presentation. This implies that automaticity does not mean similar processing of relevant and irrelevant dimensions. Specifically, the relevant dimension is processed elaborately whereas the irrelevant dimension is processed crudely.     

Individual differences in children's mathematical competence are related to the intentional but not automatic processing of Arabic numerals

In recent years, there has been an increasing focus on the role played by basic numerical magnitude processing in the typical and atypical development of mathematical skills. In this context, tasks measuring both the intentional and automatic processing of numerical magnitude have been employed to characterize how children’s representation and processing of numerical magnitude changes over developmental time. To date, however, there has been little effort to differentiate between different measures of ‘number sense’. The aim of the present study was to examine the relationship between automatic and intentional measures of magnitude processing as well as their relationships to individual differences in children’s mathematical achievement. A group of 119 children in 1st and 2nd grade were tested on the physical size congruity paradigm (automatic processing) as well as the number comparison paradigm to measure the ratio effect (intentional processing). The results reveal that measures of intentional and automatic processing are uncorrelated with one another, suggesting that these tasks tap into different levels of numerical magnitude processing in children. Furthermore, while children’s performance on the number comparison paradigm was found to correlate with their mathematical achievement scores, no such correlations could be obtained for any of the measures typically derived from the physical size congruity task. These findings therefore suggest that different tasks measuring ‘number sense’ tap into different levels of numerical magnitude representation that may be unrelated to one another and have differential predictive power for individual differences in mathematical achievement.     

Automaticity of two-digit numbers

Automatic processing of 2-digit numbers was demonstrated using the size congruency effect (SiCE). The SiCE indicates the processing of the irrelevant (numerical) dimension when 2 digits differing both numerically and physically are compared on the relevant (physical) dimension. The SiCE was affected by the compatibility between unit and decade digits but was unaffected by the global magnitude of the numbers. Together these results suggest automatic processing of the magnitudes of the components of the 2-digit numbers but not of whole numbers. Finally, the SiCE was affected more by the magnitude of the decade digits compared with the unit digits, indicating that the syntactic roles of the digits were represented. The implications of these results for understanding the numerical representations are discussed.     

Indexing the approximate number system

Much recent research attention has focused on understanding individual differences in the approximate number system, a cognitive system believed to underlie human mathematical competence. To date researchers have used four main indices of ANS acuity, and have typically assumed that they measure similar properties. Here we report a study which questions this assumption. We demonstrate that the numerical ratio effect has poor test–retest reliability and that it does not relate to either Weber fractions or accuracy on nonsymbolic comparison tasks. Furthermore, we show that Weber fractions follow a strongly skewed distribution and that they have lower test–retest reliability than a simple accuracy measure. We conclude by arguing that in the future researchers interested in indexing individual differences in ANS acuity should use accuracy figures, not Weber fractions or numerical ratio effects.     

Length illusions in conventional and single-wing Müller-Lyer stimuli

Warren and Bashford (1977) reported that eliminating one of the wing components from the conventional (i.e., two-wing) Müller-Lyer figures had no appreciable effect on the magnitude of the acute-angle (contraction) illusion but substantially reduced the magnitude of the obtuse-angle (expansion) illusion. In addition, they found that whereas the contractionary effects of the acute-angle components tended to be confined to the region of the shaft adjacent to the angles, the expansionary effects of the obtuse-angle components were more uniformly distributed across the shaft. Since these findings challenge many theories of the Müller-Lyer illusion, the purpose of the present investigation was to evaluate further Warren and Bashford's work with four experiments. Experiments 1 and 2 assessed length illusion magnitudes by requiring subjects to adjust either the length of a plain comparison line to match the length of the Müller-Lyer test figures (Experiment 1) or the length of comparison Müller-Lyer figures to match the length of plain test lines (Experiment 2). Experiments 3 and 4 used a bisection task to assess whether the illusory effects of the angle components are confined mainly to regions of the shaft adjacent to the angles. Consistent with most theories of the Müller-Lyer illusion, eliminating one of the wing components reduced both forms of the Müller-Lyer length illusion to a similar extent. In addition, the acute- and obtuse-angle forms yielded similar patterns of bisection errors, with substantial errors for regions of the shaft adjacent to the angles and negligible errors for regions of the shaft distant from the angles.