Rational numbers: componential versus holistic representation of fractions in a magnitude comparison task

No power: exponential expressions are not processed automatically as such

Little is known about the mental representation of exponential expressions. The present study examined the automatic processing of exponential expressions under the framework of multi-digit numbers, specifically asking which component of the expression (i.e., the base/power) is more salient during this type of processing. In a series of three experiments, participants performed a physical size comparison task. They were presented with pairs of exponential expressions that appeared in frames that differed in their physical sizes. Participants were instructed to ignore the stimuli within the frames and choose the larger frame. In all experiments, the pairs of exponential expressions varied in the numerical values of their base and/or power component. We manipulated the compatibility between the base and the power components, as well as their physical sizes to create a standard versus nonstandard syntax of exponential expressions. Experiments 1 and 3 demonstrate that the physically larger component drives the size congruity effect, which is typically the base but was manipulated here in some cases to be the power. Moreover, Experiments 2 and 3 revealed similar patterns, even when manipulating the compatibility between base and power components. Our findings support componential processing of exponents by demonstrating that participants were drawn to the physically larger component, even though in exponential expressions, the power, which is physically smaller, has the greater mathematical contribution. Thus, revealing that the syntactic structure of an exponential expression is not processed automatically. We discuss these results with regard to multi-digit numbers research.

Linking inhibitory control to math achievement via comparison of conflicting decimal numbers

The relationship between executive functions (EF) and academic achievement is well-established, but leveraging this insight to improve educational outcomes remains elusive. Here, we propose a framework for relating the role of specific EF on specific precursor skills that support later academic learning. Starting from the premise that executive functions contribute to general math skills both directly - supporting the execution of problem solving strategies - and indirectly - supporting the acquisition of precursor mathematical content, we hypothesize that the contribution of domain-general EF capacities to precursor skills that support later learning can help explain relations between EF and overall math skills. We test this hypothesis by examining whether the contribution of inhibitory control on general math knowledge can be explained by inhibition's contribution to processing rational number pairs that conflict with individual's prior whole number knowledge. In 97 college students (79 female, age = 20.58 years), we collected three measures of EF: working memory (backwards spatial span), inhibition (color-word Stroop) and cognitive flexibility (task switching), and timed and untimed standardized measures of math achievement. Our target precursor skill was a decimals comparison task where correct responses were inconsistent with prior whole number knowledge (e.g., 0.27 vs. 0.9). Participants performed worse on these trials relative to the consistent decimals pairs (e.g., 0.2 vs. 0.87). Individual differences in the Stroop task predicted performance on inconsistent decimal comparisons, which in turn predicted general math achievement. With respect to relating inhibitory control to math achievement, Stroop performance was an independent predictor of achievement after accounting for age, working memory and cognitive flexibility, but decimal performance mediated this relationship. Finally, we found inconsistent decimals performance mediated the relationship of inhibition with rational number performance, but not other advanced mathematical concepts. These results pinpoint the specific contribution of inhibitory control to rational number understanding, and more broadly are consistent with the hypothesis that acquisition of foundational mathematical content can explain the relationships between executive functions and academic outcomes, making them promising targets for intervention.

The processing of prices across numerical formats

We evaluated whether the format in which prices are presented determines the processing of their magnitude. A price comparison task was used in which two-digit prices with Arabic digits, written number words and auditory number words were presented in the euro currency. Prices were number-monetary category (NMC) compatible (49 euros - 36 cents) when the number and monetary category of one price were larger than those of the other (49 > 36, euros > cents); or NMC incompatible (49 cents - 36 euros) when the number of one price was larger but the monetary category smaller than those of the other (49 > 36, cents < euros). In addition, there were unit-decade (UD) compatible prices when the decade and unit of one price were larger than those of the other (49 euros - 36 cents, 4 > 3, 9 > 6); and UD incompatible prices when the decade of one price was larger but the unit smaller than those of the other (46 euros - 39 cents, 4 > 3, 6 < 9). The results showed NMC compatibility effects in all numerical formats. However, the UD compatibility effect was not found in any numerical format. The results are discussed within the hybrid model of multisymbolic magnitude processing.

Taking the Relational Structure of Fractions Seriously: Relational Reasoning Predicts Fraction Knowledge in Elementary School Children

Understanding and using symbolic fractions in mathematics is critical for access to advanced STEM concepts. However, children and adults consistently struggle with fractions. Here, we take a novel perspective on symbolic fractions, considering them within the framework of relational structures in cognitive psychology, such as those studied in analogy research. We tested the hypothesis that relational reasoning ability is important for reasoning about fractions by examining the relation between scores on a domain-general test of relational reasoning (TORR Jr.) and a test of fraction knowledge consisting of various types of fraction problems in 194 second grade and 145 fifth grade students. We found that relational reasoning was a significant predictor of fractions knowledge, even when controlling for non-verbal IQ and fractions magnitude processing for both grades. The effects of relational reasoning also remained significant when controlling for overall mathematics knowledge and skill for second graders but was attenuated for fifth graders. These findings suggest that this important subdomain of mathematical cognition is integrally tied to relational reasoning and opens the possibility that instruction targeting relational reasoning may prove to be a viable avenue for improving children's fractions skills.

A Study on Congruency Effects and Numerical Distance in Fraction Comparison by Expert Undergraduate Students

School mathematics comprises a diversity of concepts whose cognitive complexity is still poorly understood, a chief example being fractions. These are typically taught in middle school, but many students fail to master them, and misconceptions frequently persist into adulthood. In this study, we investigate fraction comparison, a task that taps into both conceptual and procedural knowledge of fractions, by looking at performance of highly mathematically skilled young adults. Fifty-seven Chilean engineering undergraduate students answered a computerized fraction comparison task, while their answers and response times were recorded. Task items were selected according to a number of mathematically and/or cognitively relevant characteristics: (a) whether the fractions to be compared shared a common component, (b) the numerical distance between fractions, and (c) the applicability of two strategies to answer successfully: a congruency strategy (a fraction is larger if it has larger natural number components than another) and gap thinking (a fraction is larger if it is missing fewer pieces than another to complete the whole). In line with previous research, our data indicated that the congruency strategy is inadequate to describe participants’ performance, as congruent items turned out to be more difficult than incongruent ones when fractions had no common component. Although we hypothesized that this lower performance for congruent items would be explained by the use of gap thinking, this turned out not to be the case: evidence was insufficient to show that the applicability of the gap thinking strategy modulated either participants’ accuracy rates or response times (although individual-level data suggest that there is an effect for response times). When fractions shared a common component, instead, our data display a more complex pattern that expected: an advantage for congruent items is present in the first experimental block but fades as the experiment progresses. Numerical distance had an effect in fraction comparison that was statistically significant for items without common components only. Altogether, our results from experts’ reasoning reveal nuances in the fraction comparison task with respect to previous studies and contribute to future models of reasoning in this task.

Symbolic fractions elicit an analog magnitude representation in school-age children

A fundamental question about fractions is whether they are grounded in an abstract nonsymbolic magnitude code similar to that postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving nonsymbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second- and fifth-grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, nonsymbolic ratios, and mixed symbolic-nonsymbolic pairs. This paradigm allowed us to test three key questions: (a) whether children show an analog magnitude code for rational numbers, (b) whether that code is compatible with mental representations of symbolic fractions, and (c) how formal education with fractions affects the symbolic-nonsymbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth graders' reaction times and error rates showed classic distance and notation effects. Nonsymbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions but had a smaller advantage for nonsymbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance and that fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a nonsymbolic ratio magnitude code and that symbolic fractions can be translated into that magnitude code.

Inhibiting the Whole Number Bias in a Fraction Comparison Task: An Event-Related Potential Study

Introduction

People often use heuristics derived from natural number tasks to solve fraction comparison tasks. For instance, one may falsely consider a fraction with a larger natural number to be the larger in magnitude, as in the case of 1/5 vs 1/4. We hypothesized that inhibitory control was needed to overcome this type of bias.

Methods

To test the hypothesis, Event-related potentials (ERP) were collected when participants were conducting fraction comparison tasks that designed with the negative priming paradigm. Twenty-eight adult participants performed three types of fraction comparison tasks: congruent items, incongruent items and neutral items.

Results

We found a negative priming effect in terms of response time. Consistently, ERP results demonstrated larger N1 and N2 amplitudes and a smaller P3 amplitude in the test trial than in the control trial.

Conclusion

These findings indicated that adults still need to inhibit the “larger natural number-larger fraction” misleading strategy when solving fraction comparison tasks with common components.

No calculation necessary: Accessing magnitude through decimals and fractions

Research on how humans understand the relative magnitude of symbolic fractions presents a unique case of the symbol-grounding problem with numbers. Specifically, how do people access a holistic sense of rational number magnitude from decimal fractions (e.g. 0.125) and common fractions (e.g. 1/8)? Researchers have previously suggested that people cannot directly access magnitude information from common fraction notation, but instead must use a form of calculation to access this meaning. Questions remain regarding the nature of calculation and whether a division-like conversion to decimals is a necessary process that permits access to fraction magnitudes. To test whether calculation is necessary to access fractions magnitudes, we carried out a series of six parallel experiments in which we examined how adults access the magnitude of rational numbers (decimals and common fractions) under varying task demands. We asked adult participants to indicate which of two fractions was larger in three different conditions: decimal-decimal, fraction-fraction, and mixed decimal-fraction pairs. Across experiments, we manipulated two aspects of the task demands. 1) Response windows were limited to 1, 2 or 5 s, and 2) participants either did or did not have to identify when the two stimuli were the same magnitude (catch trials). Participants were able to successfully complete the task even at a response window of 1 s and showed evidence of holistic magnitude processing. These results indicate that calculation strategies with fractions are not necessary for accessing a sense of a fractions meaning but are strategic routes to magnitude that participants may use when granted sufficient time. We suggest that rapid magnitude processing with fractions and decimals may occur by mapping symbolic components onto common amodal mental representations of rational numbers.

Magnitude processing of symbolic and non-symbolic proportions: an fMRI study

Background

Recent research indicates that processing proportion magnitude is associated with activation in the intraparietal sulcus. Thus, brain areas associated with the processing of numbers (i.e., absolute magnitude) were activated during processing symbolic fractions as well as non-symbolic proportions. Here, we investigated systematically the cognitive processing of symbolic (e.g., fractions and decimals) and non-symbolic proportions (e.g., dot patterns and pie charts) in a two-stage procedure. First, we investigated relative magnitude-related activations of proportion processing. Second, we evaluated whether symbolic and non-symbolic proportions share common neural substrates.

Methods

We conducted an fMRI study using magnitude comparison tasks with symbolic and non-symbolic proportions, respectively. As an indicator for magnitude-related processing of proportions, the distance effect was evaluated.

Results

A conjunction analysis indicated joint activation of specific occipito-parietal areas including right intraparietal sulcus (IPS) during proportion magnitude processing. More specifically, results indicate that the IPS, which is commonly associated with absolute magnitude processing, is involved in processing relative magnitude information as well, irrespective of symbolic or non-symbolic presentation format. However, we also found distinct activation patterns for the magnitude processing of the different presentation formats.

Conclusion

Our findings suggest that processing for the separate presentation formats is not only associated with magnitude manipulations in the IPS, but also increasing demands on executive functions and strategy use associated with frontal brain regions as well as visual attention and encoding in occipital regions. Thus, the magnitude processing of proportions may not exclusively reflect processing of number magnitude information but also rather domain-general processes.

Using propensity score matching to construct experimental stimuli

Propensity score matching is widely used in various fields of research, including psychology, medicine, education, and sociology. It is usually applied to find a matched control group for a treatment group. In the present article, we suggest that propensity score matching might also be used to construct item sets matched for different parameters. We constructed stimuli to illustrate the use of propensity score matching in item construction for the exemplary cases of numerical cognition research and reading research. In particular, we provide a step-by-step approach, using the statistics software R, for how to apply propensity score matching for constructing matched stimuli. This approach involves deciding on a population of stimuli, determining and calculating the covariates, and finally applying the propensity-matching method to find a set of items matched to another predefined set. Thereby, we were able to construct well-matched item sets for both examples. Hence, we conclude that the propensity-score-matching method is useful for constructing matched stimuli. Further cases of application are discussed.

Mental representation of fractions: It all depends on whether they are common or uncommon

This study examined whether common and uncommon fractions are mentally represented differently and whether common ones are used in accessing the magnitudes of uncommon ones. In Experiments 1 and 2, college education majors, most of whom were female, Caucasian, and in their early 20s, made comparisons involving common and uncommon fractions. In Experiment 3, participants were presented with comparison tasks involving uncommon fractions and asked to describe the strategies which they used in making such comparisons. Analysis of reaction times and error rates support the hypothesis that for common fractions, it is their holistic real value, rather than their individual components, that gets represented. For uncommon fractions, the access of their magnitudes is a process of retrieving and using a known common one having a similar value. Such results suggest that the development of the cognisance of the magnitudes of fractions may be principally a matter of common ones only and that learners' handling of uncommon fractions may be greatly facilitated through instructions on matching them with common ones having a similar value.

The fractions SNARC revisited: Processing fractions on a consistent mental number line

The ability to understand fractions is key to establishing a solid foundation in mathematics, yet children and adults struggle to comprehend them. Previous studies have suggested that these struggles emerge because people fail to process fraction magnitude holistically on the mental number line (MNL), focusing instead on fraction components. Subsequent studies have produced evidence for default holistic processing but examined only magnitude processing, not spatial representations. We explored the spatial representations of fractions on the MNL in a series of three experiments. Experiment 1 replicated Bonato et al.; 30 naïve undergraduates compared unit fractions (1/1-1/9) to 1/5, resulting in a reverse SNARC (Spatial-Numerical Association of Response Codes) effect. Experiment 2 countered potential strategic biases induced by the limited set of fractions used by Bonato et al. by expanding the stimulus set to include all irreducible, single-digit proper fractions and asked participants to compare them against 1/2. We observed a classic SNARC effect, completely reversing the pattern from Experiment 1. Together, Experiments 1 and 2 demonstrate that stimulus properties dramatically impact spatial representations of fractions. In Experiment 3, we demonstrated within-subjects reliability of the SNARC effect across both a fractions and whole number comparison task. Our results suggest that adults can indeed process fraction magnitudes holistically, and that their spatial representations occur on a consistent MNL for both whole numbers and fractions.

Developmental changes in the whole number bias

Many students' knowledge of fractions is adversely affected by whole number bias, the tendency to focus on the separate whole number components (numerator and denominator) of a fraction rather than on the fraction's magnitude (ratio of numerator to denominator). Although whole number bias appears early in the fraction learning process and under speeded conditions persists into adulthood, even among mathematicians, little is known about its development. Performance with equivalent fractions indicated that between fourth and eighth grade, whole number bias decreased, and reliance on fraction magnitudes increased. These trends were present on both fraction magnitude comparison and number line estimation. However, analyses of individual children's performance indicated that a substantial minority of fourth graders did not show whole number bias and that a substantial minority of eighth graders did show it. Implications of the findings for development of understanding of fraction equivalence and for theories of numerical development are discussed.

Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions

Why might it be (at least sometimes) beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse primes for the equation that immediately followed it (e.g., 4 × 3/4 = 3 followed by 3 × 8/6 = 4). Students with relatively high math ability showed relational priming (speeded solution times to the second of two successive relationally related fraction equations) both with and without high perceptual similarity (Experiment 2). Students with relatively low math ability also showed priming, but only when the structurally inverse equation pairs were supported by high perceptual similarity between numbers (e.g., 4 × 3/4 = 3 followed by 3 × 4/3 = 4). Several additional experiments established boundary conditions on relational priming with fractions. These findings are interpreted in terms of componential processing of fractions in a relational multiplication context that takes advantage of their inherent connections to a multiplicative schema for whole numbers.

Physical similarity or numerical representation counts in same-different, numerical comparison, physical comparison, and priming tasks?

Recent studies have highlighted the fact that some tasks used to study symbolic number representations are confounded by judgments about physical similarity. Here, we investigated whether the contribution of physical similarity and numerical representation differed in the often-used symbolic same-different, numerical comparison, physical comparison, and priming tasks. Experiment 1 showed that subjective physical similarity was the best predictor of participants' performance in the same-different task, regardless of simultaneous or sequential presentation. Furthermore, the contribution of subjective physical similarity was larger in a simultaneous presentation than in a sequential presentation. Experiment 2 showed that only numerical representation was involved in numerical comparison. Experiment 3 showed that both subjective physical similarity and numerical representation contributed to participants' physical comparison performance. Finally, only numerical representation contributed to participants' performance in a priming task as revealed by Experiment 4. Taken together, the contribution of physical similarity and numerical representation depends on task demands. Performance primarily seems to rely on numerical properties in tasks that require explicit quantitative comparison judgments (physical or numerical), while physical stimulus properties exert an effect in the same-different task.

Proportional Reasoning

Abstract. Comparison of ratios is difficult for children and adults. We studied the role of salience and congruity in comparison of ratios using reaction time and functional magnetic resonance imaging (fMRI). Participants were asked to decide which of two mixtures of red and white paint drops (presented in Arabic numerals) was darker. In congruent trials the mixture with more red drops was darker and in incongruent trials it was lighter. Half of the trials were red salience (more red than white drops in both mixtures) and half of them were white salience. Interaction between congruity and salience was observed. Behaviorally, accuracy was higher and reaction time of correct responses (RTC) was shorter in congruent red salience and incongruent white salience conditions. For these conditions higher activation in a fronto-parietal numerical network was observed in fMRI. These findings suggest that automatic processing of natural numbers supports or suppresses the comparison of ratios as a function of congruity and salience.

Delaware Longitudinal Study of Fraction Learning: Implications for Helping Children With Mathematics Difficulties

The goal of the present article is to synthesize findings to date from the Delaware Longitudinal Study of Fraction Learning. The study followed a large cohort of children ( N = 536) between Grades 3 and 6. The findings showed that many students, especially those with diagnosed learning disabilities, made minimal growth in fraction knowledge and that some showed only a basic grasp of the meaning of a fraction even after several years of instruction. Children with low growth in fraction knowledge during the intermediate grades were much more likely to fail to meet state standards on a broad mathematics measure at the end of Grade 6. Although a range of general and mathematics-specific competencies predicted fraction outcomes, the ability to estimate numerical magnitudes on a number line was a uniquely important marker of fraction success. Many children with mathematics difficulties have deep-seated problems related to whole number magnitude representations that are complicated by the introduction of fractions into the curriculum. Implications for helping students with mathematics difficulties are discussed.

Response trajectories capture the continuous dynamics of the size congruity effect

In a comparison task involving numbers, the size congruity effect refers to the general finding that responses are usually faster when there is a match between numerical size and physical size (e.g., 2-8) than when there is a mismatch (e.g., 2-8). In the present study, we used computer mouse tracking to test two competing models of the size congruity effect: an early interaction model, where interference occurs at an early representational stage, and a late interaction model, where interference occurs as dynamic competition between response options. In three experiments, we found that the curvature of responses for incongruent trials was greater than for congruent trials. In Experiment 2 we showed that this curvature effect was reliably modulated by the numerical distance between the two stimulus numbers, with large distance pairs exhibiting a larger curvature effect than small distance pairs. In Experiment 3 we demonstrated that the congruity effects persist into response execution. These findings indicate that incongruities between numerical and physical sizes are carried throughout the response process and result from competition between parallel and partially active response options, lending further support to a late interaction model of the size congruity effect.

Response trajectories reveal the temporal dynamics of fraction representations

Previous studies on mental arithmetic with fractions have been equivocal with respect to the nature of mental representations that are formed with fractions. It is not clear from present evidence whether fractions form perceptual primitives independent from components or whether component magnitudes must be processed in addition to the holistic magnitude. In the present study, we attempt to resolve this issue by using computer mouse-tracking. We analyzed the dynamics of participants' hand movements as they compared presented fractions to 1/2. We found that before settling to the correct answer, hand trajectories showed competitive influences of component magnitude and overall fraction magnitude, but the influence of components happened much earlier. These data support the idea that in fraction comparison, component magnitudes and holistic magnitude are processed together in a continuous, competitive manner.

Eye movements reflect and shape strategies in fraction comparison

The comparison of fractions is a difficult task that can often be facilitated by separately comparing components (numerators and denominators) of the fractions—that is, by applying so-called component-based strategies. The usefulness of such strategies depends on the type of fraction pair to be compared. We investigated the temporal organization and the flexibility of strategy deployment in fraction comparison by evaluating sequences of eye movements in 20 young adults. We found that component-based strategies could account for the response times and the overall number of fixations observed for the different fraction pairs. The analysis of eye movement sequences showed that the initial eye movements in a trial were characterized by stereotypical scanning patterns indicative of an exploratory phase that served to establish the kind of fraction pair presented. Eye movements that followed this phase adapted to the particular type of fraction pair and indicated the deployment of specific comparison strategies. These results demonstrate that participants employ eye movements systematically to support strategy use in fraction comparison. Participants showed a remarkable flexibility to adapt to the most efficient strategy on a trial-by-trial basis. Our results confirm the value of eye movement measurements in the exploration of strategic adaptation in complex tasks.

Common magnitude representation of fractions and decimals is task dependent

Although several studies have compared the representation of fractions and decimals, no study has investigated whether fractions and decimals, as two types of rational numbers, share a common representation of magnitude. The current study aimed to answer the question of whether fractions and decimals share a common representation of magnitude and whether the answer is influenced by task paradigms. We included two different number pairs, which were presented sequentially: fraction-decimal mixed pairs and decimal-fraction mixed pairs in all four experiments. Results showed that when the mixed pairs were very close numerically with the distance 0.1 or 0.3, there was a significant distance effect in the comparison task but not in the matching task. However, when the mixed pairs were further apart numerically with the distance 0.3 or 1.3, the distance effect appeared in the matching task regardless of the specific stimuli. We conclude that magnitudes of fractions and decimals can be represented in a common manner, but how they are represented is dependent on the given task. Fractions and decimals could be translated into a common representation of magnitude in the numerical comparison task. In the numerical matching task, fractions and decimals also shared a common representation. However, both of them were represented coarsely, leading to a weak distance effect. Specifically, fractions and decimals produced a significant distance effect only when the numerical distance was larger.

The componential processing of fractions in adults and children: effects of stimuli variability and contextual interference

Recent studies have indicated that people have a strong tendency to compare fractions based on constituent numerators or denominators. This is called componential processing. This study explored whether componential processing was preferred in tasks involving high stimuli variability and high contextual interference, when fractions could be compared based either on the holistic values of fractions or on their denominators. Here, stimuli variability referred to the fact that fractions were not monotonous but diversiform. Contextual interference referred to the fact that the processing of fractions was interfered by other stimuli. To our ends, three tasks were used. In Task 1, participants compared a standard fraction 1/5 to unit fractions. This task was used as a low stimuli variability and low contextual interference task. In Task 2 stimuli variability was increased by mixing unit and non-unit fractions. In Task 3, high contextual interference was created by incorporating decimals into fractions. The RT results showed that the processing patterns of fractions were very similar for adults and children. In task 1 and task 3, only componential processing was utilzied. In contrast, both holistic processing and componential processing were utilized in task 2. These results suggest that, if individuals are presented with the opportunity to perform componential processing, both adults and children will tend to do so, even if they are faced with high variability of fractions or high contextual interference.

Early predictors of middle school fraction knowledge

Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first grade predicted knowledge of fraction arithmetic in middle school, controlling for whole number magnitude knowledge in first grade and the other control variables. In contrast, neither type of early whole number knowledge uniquely predicted middle school reading achievement. We discuss the implications of these findings for theories of numerical development and for improving mathematics learning.

Unlimited capacity parallel quantity comparison of multiple integers

Research has shown that integer comparison is quick and efficient. This efficiency may be a function of the structure of the integer comparison system. The present study tests whether integers are compared with an unlimited capacity system or a limited capacity system. We tested these models using a visual search task with time delimitation. The data from Experiments 1 and 2 indicate that integers are encoded, identified, and compared within an unlimited capacity system. The data from Experiment 3 indicate that 2nd-order magnitude comparisons are processed with a highly efficient limited capacity system.

Adaptive processing of fractions--evidence from eye-tracking

Recent evidence indicated that fraction pair type determined whether a particular fraction is processed holistically, componentially or in a hybrid manner. Going beyond previous studies, we investigated how participants adapt their processing of fractions not only to fraction type, but also to experimental context. To examine adaptation in fraction processing, we recorded participants' eye-fixation behaviour in a fraction magnitude comparison task. Participants' eye fixation behaviour indicated componential processing of fraction pairs with common components for which the decision-relevant components are easy to identify. Importantly, we observed that fraction processing was adapted to experimental context: Evidence for componential processing was stronger, when experimental context allowed valid expectations about which components are decision-relevant. Taken together, we conclude that fraction processing is adaptive beyond the comparison of different fraction types, because participants continuously adjust to the experimental context in which fractions are processed.

Long-distance neural synchrony correlates with processing strategies to compare fractions

Adults use different processing strategies to work with fractions. Depending on task requirements, they may analyze the fraction components separately (componential processing strategy, CPS) or consider the fraction as a whole (holistic processing strategy, HPS). It is so far unknown what is the brain coordination dynamics underlying these types of fraction processing strategies. To elucidate this issue, we analyzed oscillatory brain activity during a fraction comparison task, presenting pairs of fractions either with or without common components. Results show that CPS induces a left frontal-parietal alpha phase desynchronization after the onset of fraction pairs, while HPS induces an increase of phase synchrony on theta and gamma bands, over frontal and central-parietal sites, respectively. Additionally, the HPS evokes more negative ERPs around 400 ms over the right frontal scalp than the CPS. This ERP activity correlates with the increase of Theta phase synchrony. Our results reveal the emergence of different functional neural networks depending on the kind of cognitive strategy used for processing fractions.

The processing of symbolic and nonsymbolic ratios in school-age children

This study tested the processing of ratios of natural numbers in school-age children. Nine- and eleven-year-olds were presented collections made up of orange and grey dots (i.e., nonsymbolic format) and fractions (i.e., symbolic format). They were asked to estimate ratios between the number of orange dots and the total number of dots and fractions by producing an equivalent ratio of surface areas (filling up a virtual glass). First, we tested whether symbolic notation of ratios affects their processing by directly comparing performance on fractions with that on dot sets. Second, we investigated whether children’s estimates of nonsymbolic ratios of natural numbers relied at least in part on ratios of surface areas by contrasting a condition in which the ratio of surface areas occupied by dots covaried with the ratio of natural numbers and a condition in which this ratio of surface areas was kept constant across ratios of natural numbers. The results showed that symbolic notation did not really have a negative impact on performance among 9-year-olds, while it led to more accurate estimates in 11-year-olds. Furthermore, in dot conditions, children’s estimates increased consistently with ratios between the number of orange dots and the total number of dots even when the ratio of surface areas was kept constant but were less accurate in that condition than when the ratio of surface areas covaried with the ratio of natural numbers. In summary, these results indicate that mental magnitude representation is more accurate when it is activated from symbolic ratios in children as young as 11 years old and that school-age children rely at least in part on ratios of surface areas to process nonsymbolic ratios of natural numbers when given the opportunity to do so.

The development of the mental representations of the magnitude of fractions

We investigated the development of the mental representation of the magnitude of fractions during the initial stages of fraction learning in grade 5, 6 and 7 children as well as in adults. We examined the activation of global fraction magnitude in a numerical comparison task and a matching task. There were global distance effects in the comparison task, but not in the matching task. This suggests that the activation of the global magnitude representation of fractions is not automatic in all tasks involving magnitude judgments. The slope of the global distance effect increased during early fraction learning and declined by adulthood, demonstrating that the development of the fraction global distance effect differs from that of the integer distance effect.

An ERP study of the processing of common and decimal fractions: how different they are

This study explored event-related potential (ERP) correlates of common fractions (1/5) and decimal fractions (0.2). Thirteen subjects performed a numerical magnitude matching task under two conditions. In the common fraction condition, a nonsymbolic fraction was asked to be judged whether its magnitude matched the magnitude of a common fraction; in the decimal fraction condition, a nonsymbolic fraction was asked to be matched with a decimal fraction. Behavioral results showed significant main effects of condition and numerical distance, but no significant interaction of condition and numerical distance. Electrophysiological data showed that when nonsymbolic fractions were compared to common fractions, they displayed larger N1 and P3 amplitudes than when they were compared to decimal fractions. This finding suggested that the visual identification for nonsymbolic fractions was different under the two conditions, which was not due to perceptual differences but to task demands. For symbolic fractions, the condition effect was observed in the N1 and P3 components, revealing stimulus-specific visual identification processing. The effect of numerical distance as an index of numerical magnitude representation was observed in the P2, N3 and P3 components under the two conditions. However, the topography of the distance effect was different under the two conditions, suggesting stimulus specific semantic processing of common fractions and decimal fractions.

The mental representations of fractions: adults' same-different judgments

Two experiments examined whether the processing of the magnitude of fractions is global or componential. Previously, some authors concluded that adults process the numerators and denominators of fractions separately and do not access the global magnitude of fractions. Conversely, others reported evidence suggesting that the global magnitude of fractions is accessed. We hypothesized that in a fraction matching task, participants automatically extract the magnitude of the components but that the activation of the global magnitude of the whole fraction is only optional or strategic. Participants carried out same/different judgment tasks. Two different tasks were used: a physical matching task and a numerical matching task. Pairs of fractions were presented either simultaneously or sequentially. Results showed that participants only accessed the representation of the global magnitude of fractions in the numerical matching task. The mode of stimulus presentation did not affect the processing of fractions. The present study allows a deeper understanding of the conditions in which the magnitude of fractions is mentally represented by using matching tasks and two different modes of presentation.

Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) versus low mathematics achievement

Fractions pose significant challenges for many children, but for some children those challenges persist into high school. Here we administered a fractions magnitude comparison test to 122 children, from Grades 4 to 8, to test whether their knowledge of fractions typically learned early in the sequence of formal math instruction (e.g., fractions equivalent to one-half, fraction pairs with common denominators) differentiates those with mathematics learning disability (MLD) versus low achievement (LA) or typical achievement (TA) in mathematics and whether long-term learning trajectories of this knowledge also differentiate these groups. We confirmed that although fourth graders with TA (n=93) were more accurate in evaluating "one-half" fractions than in evaluating "non-half" fractions (until they reached ceiling performance levels on both types of fractions), children with MLD (n=11) did not show a one-half advantage until Grade 7 and did not reach ceiling performance even by Grade 8. Both the MLD and LA groups had early difficulties with fractions, but by Grade 5 the LA group approached performance levels of the TA group and deviated from the MLD group. All groups showed a visual model advantage over Arabic number representation of fractions, but this advantage was short-lived for the TA group (because ceiling level was achieved across formats), whereas it was slightly more persistent for the LA group and persisted through Grade 8 for children with MLD. Thus, difficulties with fractions persist through Grade 8 for many students, but the nature and trajectories of those difficulties vary across children with math difficulties (MLD or LA).

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.

On the relativity of relative frequencies

The most prominent models of numerical representation posit that numerical symbols are converted into a single internal, abstract representation prior to estimation and comparison processing. Here, we (1) provide a mathematical analysis of the predictions of the abstract-representation hypothesis, assuming the validity of the analog-representation hypothesis, (2) run a simulation to assess the patterns of data that result from our mathematical analysis, and (3) conduct two experiments to test the predictions of our model, using relative frequencies as inputs. We assess relative frequencies in a typical numerical distance task, whereby participants are presented with two relative frequencies and asked to identify the one that represents the larger quantity. Our data reveal that relative frequencies' numerical representations (1) are analog and (2) are scale-specific (i.e., nonabstract).

The role of numerosity in processing nonsymbolic proportions

The difficulty in processing fractions seems to be related to the interference between the whole-number value of the numerator and the denominator and the real value of the fraction. Here we assess whether the reported problems with symbolic fractions extend to the nonsymbolic domain, by presenting fractions as arrays of black and white dots representing the two operands. Participants were asked to compare a target array with a reference array in two separate tasks using the same stimuli: a numerosity task comparing just the number of white dots in the two arrays; and a proportion task comparing the proportion of black and white dots. The proportion task yielded lower accuracy and slower response, confirming that even with nonsymbolic stimuli accessing proportional information is relatively difficult. However, using a congruity manipulation in which the greater numerosity of white dots could co-occur with a lower proportion of them, and vice versa, it was found that both task-irrelevant dimensions would interfere with the task-relevant dimension suggesting that both numerosity and proportion information was automatically accessed. The results indicate that the magnitude of fractions can be automatically and holistically processed in the nonsymbolic domain.

Relating magnitudes: the brain's code for proportions

Whereas much is known about how we categorize and reason based on absolute quantity, data exploring ratios of quantities, as in proportions and fractions, are comparatively sparse. Until recently, it remained elusive whether these two representations of number are connected, how proportions are implemented by neurons and how language shapes this code. New data derived with complementary methods and from different model systems now shed light on the mechanisms of magnitude ratio representations. A coding scheme for proportions has emerged that is remarkably reminiscent of the representation of absolute number. These novel findings suggest a sense for ratios that grants the brain automatic access to proportions independently of language and the format of presentation.

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.

When meaningful components interrupt the processing of the whole: the case of fractions

Numerical fractions are composed of a numerator and a denominator that are natural numbers. These components influence processing of the fraction. This study was conducted to test whether eliminating the fractional components would result in the processing of fractions as unique numerical entities. Participants that learned to relate fractional values to arbitrary figures in a training task showed automatic processing of the numerical values of the new figures. The processing of fractions written in regular form improved following training, but did not show automatic processing. The results suggest that eliminating the influence of the fractional components allowed individual fractions to be represented in long-term memory.

Fractions but not negative numbers are represented on the mental number line

The present study is the first to directly compare numerical representations of positive numbers, negative numbers and unit fractions. The results show that negative numbers and unit fractions were not represented in the same way. Distance effects were found when positive numbers were compared with fractions but not when they were compared with negative numbers, thus suggesting that unit fractions but not negative numbers were represented on the number line with positive numbers. As indicated by the semantic congruity effect, negative numbers were perceived to be small, positive numbers were perceived as large, while unit fractions were perceived neither as large nor small. Comparisons between negative numbers were faster than between unit fractions, possibly due to the smaller differences between the holistic magnitudes of the unit fractions. Finally, comparing unit fractions to 1 was faster than comparing them to 0, consistent with the idea that unit fractions are perceived as entities smaller than 1 (Kallai & Tzelgov, 2009). The results are consistent with the idea of a mental division between numbers that represent a quantity (positive numbers and unit fractions) and those that do not (negative numbers).