The processing and representation of fractions within the brain: an fMRI investigation

Cognitive Effort during Visuospatial Problem Solving in Physical Real World, on Computer Screen, and in Virtual Reality

Spatial cognition plays a crucial role in academic achievement, particularly in science, technology, engineering, and mathematics (STEM) domains. Immersive virtual environments (VRs) have the growing potential to reduce cognitive load and improve spatial reasoning. However, traditional methods struggle to assess the mental effort required for visuospatial processes due to the difficulty in verbalizing actions and other limitations in self-reported evaluations. In this neuroergonomics study, we aimed to capture the neural activity associated with cognitive workload during visuospatial tasks and evaluate the impact of the visualization medium on visuospatial task performance. We utilized functional near-infrared spectroscopy (fNIRS) wearable neuroimaging to assess cognitive effort during spatial-reasoning-based problem-solving and compared a VR, a computer screen, and a physical real-world task presentation. Our results reveal a higher neural efficiency in the prefrontal cortex (PFC) during 3D geometry puzzles in VR settings compared to the settings in the physical world and on the computer screen. VR appears to reduce the visuospatial task load by facilitating spatial visualization and providing visual cues. This makes it a valuable tool for spatial cognition training, especially for beginners. Additionally, our multimodal approach allows for progressively increasing task complexity, maintaining a challenge throughout training. This study underscores the potential of VR in developing spatial skills and highlights the value of comparing brain data and human interaction across different training settings.

Why fractions are difficult? Modeling optimal and sub-optimal integration strategies of numerators and denominators by educated adults

Many children and educated adults experience difficulties in understanding and manipulating fractions. In this study, we argue that a major cause of this challenge is rooted in the need to integrate information from two separate informational sources (i.e., denominator and numerator) according to a normative arithmetic rule (i.e., division). We contend that in some tasks, the correct arithmetic rule is replaced by an inadequate (sub-optimal) operation (e.g., multiplication), which leads to inaccurate representation of fractions. We tested this conjecture by applying two rigorous models of information integration : (a) functional measurement (Experiments 1-3) and (b) conjoint measurement (Experiment 4-5) to data from number-to-line and comparative judgment tasks. These allowed us to compare participants' integration strategies with that of an ideal-observer model. Functional measurement analyses on data from the number-to-line task, revealed that participants could represent the global magnitude of proper and improper fractions quite accurately and combine the fractions' components according to an ideal-observer model. However, conjoint measurement analyses on data from the comparative judgment task, showed that most participants combined these fractions' components according to a sub-optimal (saturated) observer model, that is inconsistent with an ideal-observer (additive) model. These results support the view that educated adults are capable of extracting multiple types of representations of fractions depending on the task at-hand. These representations can be either accurate and conform with normative arithmetic or approximated and inconsistent with normative arithmetic. The latter may lead to the observed difficulties people experience with fractions.

Competing numerical magnitude codes in decimal comparison: Whole number and rational number distance both impact performance

A critical difference between decimal and whole numbers is that among whole numbers the number of digits provides reliable information about the size of the number, e.g., double-digit numbers are larger than single-digit numbers. However, for decimals, fewer digits can sometimes denote a larger number (i.e., 0.8 > 0.27). Accordingly, children and adults perform worse when comparing such Inconsistent decimal pairs relative to Consistent pairs, where the larger number also has more digits (i.e., 0.87 > 0.2). Two explanations have been posited for this effect. The string length congruity account proposes that participants compare each position in the place value system, and they additionally compare the number of digits. The semantic interference account suggests that participants additionally activate the whole number referents of numbers - the numbers unadorned with decimal points (e.g., 8 < 27) - and compare these. The semantic interference account uniquely predicts that for Inconsistent problems with the same actual rational distance, those with larger whole number distances should be harder, e.g., 0.9 vs. 0.81 should be harder than 0.3 vs. 0.21 because 9 < < 81 whereas 3 < 21. Here we test this prediction in two experiments with college students (Study 1: n = 58 participants, Study 2: n = 78). Across both, we find a main effect of consistency, demonstrating string length effects, and also that whole number distance interferes with processing conflicting decimals, demonstrating semantic interference effects. Evidence for both effects supports the semantic interference account, highlighting that decimal comparison difficulties arise from multiple competing numerical codes. Finally, for accuracy we found no relationship between whole number distance sensitivity and math achievement, indicating that whole number magnitude interference affects participants similarly across the spectrum of math achievement.

Comparison of visual quantities in untrained neural networks

The ability to compare quantities of visual objects with two distinct measures, proportion and difference, is observed even in newborn animals. However, how this function originates in the brain, even before visual experience, remains unknown. Here, we propose a model in which neuronal tuning for quantity comparisons can arise spontaneously in completely untrained neural circuits. Using a biologically inspired model neural network, we find that single units selective to proportions and differences between visual quantities emerge in randomly initialized feedforward wirings and that they enable the network to perform quantity comparison tasks. Notably, we find that two distinct tunings to proportion and difference originate from a random summation of monotonic, nonlinear neural activities and that a slight difference in the nonlinear response function determines the type of measure. Our results suggest that visual quantity comparisons are primitive types of functions that can emerge spontaneously before learning in young brains.

Middle-schoolers' misconceptions in discretized nonsymbolic proportional reasoning explain fraction biases better than their continuous reasoning: Evidence from correlation and cluster analyses

Early emerging nonsymbolic proportional skills have been posited as a foundational ability for later fraction learning. A positive relation between nonsymbolic and symbolic proportional reasoning has been reported, as well as successful nonsymbolic training and intervention programs enhancing fraction magnitude skills. However, little is known about the mechanisms underlying this relationship. Of particular interest are nonsymbolic representations, which can be in continuous formats that may emphasize proportional relations and in discretized formats that may prompt erroneous whole-number strategies and hamper access to fraction magnitudes. We assessed the proportional comparison skills of 159 middle-school students (mean age = 12.54 years, 43% females, 55% males, 2% other or prefer not to say) across three types of representations: (a) continuous, unsegmented bars, (b) discretized, segmented bars that allowed counting strategies, and (c) symbolic fractions. Using both correlational and cluster approaches, we also examined their relations to symbolic fraction comparison ability. Within each stimulus type, we varied proportional distance, and in the discretized and symbolic stimuli, we also manipulated whole-number congruency. We found that fraction distance across all formats modulated middle-schoolers' performance; however, whole-number information affected discretized and symbolic comparison performance. Further, continuous and discretized nonsymbolic performance was related to fraction comparison ability; however, discretized skills explained variance above and beyond the contributions of continuous skills. Finally, our cluster analyses revealed three nonsymbolic comparison profiles: students who chose the bars with the largest number of segments (whole-number bias), chance-level performers, and high performers. Crucially, students with a whole-number bias profile showed this bias in their fraction skills and failed to show any symbolic distance modulation. Together, our results indicate that the relation between nonsymbolic and symbolic proportional skills may be determined by the (mis)conceptions based on discretized representations, rather than understandings of proportional magnitudes, suggesting that interventions focusing on competence with discretized representations may show dividends for fraction understanding.

Fraction mapping and fraction comparison skills among grade 4 Chinese students: An error analysis

BACKGROUND

Mapping fraction symbols to magnitudes is easier for students to master than comparing fraction magnitudes. Fraction mapping assesses students' understanding of part-whole interpretations of fractions; fractions represent the parts of an object or set of objects. Fraction comparison assesses students' understanding of measurement interpretations of fractions; a fraction is a single numerical quantity, not a combination of two whole numbers.

AIM

To examine and compare the types of errors made by emergent fraction learners on fraction mapping and comparison tasks.

SAMPLE

Grade 4 Chinese students (N = 1036; 577 boys; M_age  = 9.9 years).

METHOD

We examined performance and identified errors on fraction mapping and comparison tasks. For mapping, students converted pictorial representations into fraction notation. For comparison, they chose the larger of two symbolic fractions.

RESULTS

Consistent with curriculum expectations, most students successfully mapped pictorial representations to fraction notation. In contrast, few students were able to accurately compare fraction magnitudes. Within each task, students' errors were consistent across trials, suggesting that they applied systematic but incorrect procedures. However, errors were not consistent between tasks and the correlation between mapping and comparison performance was weak.

CONCLUSION

Emergent fraction learners can acquire part-whole knowledge of fractions without acquiring measurement interpretations of fractions. Moreover, misconceptions about different interpretations of fractions need not overlap. Awareness of the types of errors that students make can assist educators in identifying misconceptions early so that students do not build their fraction knowledge on erroneous beliefs.

Neural Dissociations between Magnitude Processing of Fractions and Decimals

Fraction and decimal magnitude processing are crucial for mathematic achievement. Previous neuroimaging results showed that fraction and decimal processing activated both overlapping and distinct neural substrates, but temporal dissociations between fraction and decimal processing remained unknown. This event-related potential (ERP) study explored differences in neural activities between magnitude processing of fractions and decimals, by examining the notation effect (fraction vs. decimal) and distance effect (far vs. close) on early components of P1/N1, P2 and N2. Results showed that decimals elicited larger N1 and smaller P1 than fractions at the parietal region. Fractions demonstrated the significant distance effect on fronto-central P2 while decimals showed the distance effect on left anterior N2. ERP results reflect distinct processing of identification and semantic access stages between fractions and decimals. Identification is located at the visual-related region with enhanced perception acuity and identification efficiency for decimals. Semantic access activates the fronto-central region associated with elaborative magnitude manipulation for fractions, while semantic access reflects automatic phonological retrieval for decimals. Our findings disintegrate the magnitude processing of fractions and decimals from identification to magnitude processing. It reveals that temporal discrepancies between fraction and decimal magnitude processing appear as early as post-stimulus 100 ms.

Neurofunctional plasticity in fraction learning: An fMRI training study

BACKGROUND

Fractions are known to be difficult for children and adults. Behavioral studies suggest that magnitude processing of fractions can be improved via number line estimation (NLE) trainings, but little is known about the neural correlates of fraction learning.

METHOD

To examine the neuro-cognitive foundations of fraction learning, behavioral performance and neural correlates were measured before and after a five-day NLE training.

RESULTS

In all evaluation tasks behavioral performance increased after training. We observed a fronto-parietal network associated with number magnitude processing to be recruited in all tasks as indicated by a numerical distance effect. For symbolic fractions, the distance effect on intraparietal activation was only observed after training.

CONCLUSION

The absence of a distance effect of symbolic fractions before the training could indicate an initially less automatic access to their overall magnitude. NLE training facilitates processing of overall fraction magnitude as indicated by the distance effect in neural activation.

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.

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.

Processing symbolic and non-symbolic proportions: Domain-specific numerical and domain-general processes in intraparietal cortex

Previous studies on the processing of fractions and proportions focused mainly on the processing of their overall magnitude information in the intraparietal sulcus (IPS). However, the IPS is also associated with domain-general cognitive functions beyond processing overall magnitude, which may nevertheless be involved in operating on magnitude information of proportions. To pursue this issue, the present study aimed at investigating whether there is a shared neural correlate for proportion processing in the intraparietal cortex beyond overall magnitude processing and how part-whole relations are processed on the neural level. Across four presentation formats (i.e., fractions, decimals, dot patterns, and pie charts) we observed a shared neural substrate in bilateral inferior parietal cortex, slightly anterior and inferior to IPS areas recently found for overall magnitude proportion processing. Nevertheless, when evaluating the neural correlates of part-whole processing (i.e., contrasting fractions, dot patterns, and pie charts vs. decimals), we found wide-spread activation in fronto-parietal brain areas. These results indicate involvement of domain-general cognitive processes in part-whole processing beyond processing the overall magnitude of proportions. The dissociation between proportions involving part-whole relations and decimals was further substantiated by a representational similarity analysis, which revealed common neural processing for fractions, pie charts, and dot patterns, possibly representing their bipartite part-whole structure. In contrast, decimals seemed to be processed differently on the neural level, possibly reflecting missing processes of actual proportion calculation in decimals.

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.

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.

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.

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.

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.

The Impact of Density and Ratio on Object-Ensemble Representation in Human Anterior-Medial Ventral Visual Cortex

Behavioral research has demonstrated that observers can extract summary statistics from ensembles of multiple objects. We recently showed that a region of anterior-medial ventral visual cortex, overlapping largely with the scene-sensitive parahippocampal place area (PPA), participates in object-ensemble representation. Here we investigated the encoding of ensemble density in this brain region using fMRI-adaptation. In Experiment 1, we varied density by changing the spacing between objects and found no sensitivity in PPA to such density changes. Thus, density may not be encoded in PPA, possibly because object spacing is not perceived as an intrinsic ensemble property. In Experiment 2, we varied relative density by changing the ratio of 2 types of objects comprising an ensemble, and observed significant sensitivity in PPA to such ratio change. Although colorful ensembles were shown in Experiment 2, Experiment 3 demonstrated that sensitivity to object ratio change was not driven mainly by a change in the ratio of colors. Thus, while anterior-medial ventral visual cortex is insensitive to density (object spacing) changes, it does code relative density (object ratio) within an ensemble. Object-ensemble processing in this region may thus depend on high-level visual information, such as object ratio, rather than low-level information, such as spacing/spatial frequency.

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.

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).

Health numeracy and cognitive decline in advanced age

The study aimed at investigating health numeracy in cognitively well performing healthy participants aged from 50 to 95 years as well as in participants with cognitive impairment, but no dementia (CIND). In cognitively well performing participants (n = 401), demographic variables and cognitive abilities (executive functions, reading comprehension, mental calculation, vocabulary) were associated with health numeracy. Older age, lower education, female gender as well as lower cognitive functions predicted low health numeracy. The effect of older age was partly mediated by executive functions and calculation abilities. Participants with CIND (n = 51) performed significantly lower than healthy controls in health numeracy. The findings suggest that cognitively well performing old individuals have difficulties in understanding health-related numerical information. The risk of misunderstanding health-related numerical information is increased in persons with CIND. As these population groups are frequently involved in health care decisions, particular attention has to be paid to providing numerical information in comprehensible form.

Fractions: the new frontier for theories of numerical development

Recent research on fractions has broadened and deepened theories of numerical development. Learning about fractions requires children to recognize that many properties of whole numbers are not true of numbers in general and also to recognize that the one property that unites all real numbers is that they possess magnitudes that can be ordered on number lines. The difficulty of attaining this understanding makes the acquisition of knowledge about fractions an important issue educationally, as well as theoretically. This article examines the neural underpinnings of fraction understanding, developmental and individual differences in that understanding, and interventions that improve the understanding. Accurate representation of fraction magnitudes emerges as crucial both to conceptual understanding of fractions and to fraction arithmetic.

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).

Holistic Representation of Unit Fractions

The present study examined the processing of unit fractions and the extent to which it is affected by context. Using a numerical comparison task we found evidence for a holistic representation of unit fractions when the immediate context of the fractions was emphasized, that is when the stimuli set included in addition to the unit fractions also the numbers 0 and 1. The holistic representation was indicated by the semantic congruity effect for comparisons of pairs of fractions and by the distance effect in comparisons of a fraction and 0 and 1. Consistent with previous results (Bonato, Fabbri, Umilta, & Zorzi, 2007) there was no evidence for a holistic representation of unit fractions when the stimulus set included only fractions. These findings suggest that fraction processing is context-dependent. Finally, the present results are discussed in the context of processing other complex numbers beyond the first decade.

Rapid Communication: 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.

Comparison of quantities: core and format-dependent regions as revealed by fMRI

The perception and handling of numbers is central to education. Numerous imaging studies have focused on how quantities are encoded in the brain. Yet, only a few studies have touched upon number mining: the ability to extract the magnitude encoded in a visual stimulus. This article aims to characterize how analogue (i.e., disks and dots) and symbolic (i.e., positive and negative integers) formats influence number mining and the representation of quantities. Sixteen adult volunteers completed a comparison task while we recorded the blood oxygen level-dependent response using functional magnetic resonance imaging. The results revealed that a restricted set of specific subdivisions in the right intraparietal sulcus is activated in all conditions. With respect to magnitude assessment, the results show that 1) analogue stimuli are predominantly processed in the right hemisphere and that 2) symbolic stimuli encompass the analogue system and further recruit areas in the left hemisphere. Crucially, we found that polarity is encoded independently from magnitude. We refine the triple-code model by integrating our findings.

Representations of visual proportions in the primate posterior parietal and prefrontal cortices

The primate prefrontal (PFC) and posterior parietal cortices (PPC) have been shown to be cardinal structures in processing abstract absolute magnitudes, such as numerosity or length. The neuronal representation of quantity relations, however, remained largely elusive. Recent functional imaging studies in humans showed that blood flow changes systematically both in the PFC and the PPC as a function of relational distance between proportions. We investigated the response properties of single neurons in the lateral PFC and the inferior parietal lobule (IPL, area 7) in rhesus monkeys performing a lengths-proportion-discrimination task. Neurons in both areas shared many characteristics and showed peaked tuning functions with preferred proportions. However, a significantly higher percentage of neurons coding proportions was found in the PFC compared with the IPL. In agreement with human studies, our study shows that proportions are represented in the fronto-parietal network that has already been implicated for absolute magnitude processing.

Mental Representations in Fraction Comparison

In this study, we investigated the mental representations used in a fraction comparison task. Adults were asked to quickly and accurately pick the larger of two fractions presented on a computer screen and provide trial-by-trial reports of the types of strategies they used. We found that adults used a variety of strategies to compare fractions, ranging among just knowing the answer, using holistic knowledge of fractions to determine the answer, and using component-based procedures such as cross multiplication. Across all strategy types, regression analyses identified that reaction times were significantly predicted by numerical distance between fractions, indicating that the participants used a magnitude-based representation to compare the fraction magnitudes. In addition, a variant of the problem-size effect (e.g., Ashcraft, 1992) appeared, whereby reaction times were significantly predicted by the average cross product of the two fractions. This effect was primarily found for component-based strategies, indicating a role for strategy choice in the formation of mental representations of fractions.