Nonsymbolic Ratios Research Articles

Circling around number: People can accurately extract numeric values from circle area ratios.

It has long been known that people have the ability to estimate numerical quantities without counting. A standard account is that people develop a sense of the size of symbolic numbers by learning to map symbolic numbers (e.g., 6) to their corresponding numerosities (e.g. :::) and concomitant approximate magnitude system (ANS) representations. However, we here demonstrate that adults are capable of extracting fractional numerical quantities from non-symbolic visual ratios (i.e., labeling a ratio of two circle areas with the appropriate symbolic fraction). Not only were adult participants able to perform this task, but they were remarkably accurate: linear regressions on median estimates yielded slopes near 1, and accounted for 97% of the variability. Participants also performed at least as well on line-estimation and ratio-estimation tasks using non-numeric circular stimuli as they did in earlier experiments using non-symbolic numerosities, which are frequently considered to be numeric stimuli. We discuss results as consistent with accounts suggesting that non-symbolic ratios have the potential to act as a reliable and stable ground for symbolic number, even when composed of non-numeric stimuli.

Non-symbolic Ratio Reasoning in Kindergarteners: Underlying Unidimensional Heuristics and Relations With Math Abilities.

Although it is thought that young children focus on the magnitude of the target dimension across ratio sets during binary comparison of ratios, it is unknown whether this is the default approach to ratio reasoning, or if such approach varies across representation formats (discrete entities and continuous amounts) that naturally afford different opportunities to process the dimensions in each ratio set. In the current study, 132 kindergarteners (Mage = 68 months, SD = 3.5, range = 62–75 months) performed binary comparisons of ratios with discrete and continuous representations. Results from a linear mixed model revealed that children followed an additive strategy to ratio reasoning—i.e., they focused on the magnitude of the target dimension across ratio sets as well as on the absolute magnitude of the ratio set. This approach did not vary substantially across representation formats. Results also showed an association between ratio reasoning and children’s math problem-solving abilities; children with better math abilities performed better on ratio reasoning tasks and processed additional dimensions across ratio sets. Findings are discussed in terms of the processes that underlie ratio reasoning and add to the extant debate on whether true ratio reasoning is observed in young children.

Revisiting and refining relations between nonsymbolic ratio processing and symbolic math achievement

In their 2016 Psych Science article, Matthews, Lewis and Hubbard (2016, https://doi.org/10.1177/0956797615617799) leveled a challenge against the prevailing theory that fractions—as opposed to whole numbers—are incompatible with humans’ primitive nonsymbolic number sense. Their ratio processing system (RPS) account holds that humans possess a primitive system that confers the ability to process nonysmbolic ratio magnitudes. Perhaps the most striking finding from Matthews et al. was that ratio processing ability predicted symbolic fractions knowledge and algebraic competence. The purpose of the current study was to replicate Matthews et al.’s novel results and to extend the study by including a control measure of fluid intelligence and an additional nonsymbolic magnitude format as predictors of multiple symbolic math outcomes. Ninety-nine college students completed three comparison tasks deciding which of two nonsymbolic ratios was numerically larger along with three simple magnitude comparison tasks in corresponding formats that served as controls. The formats included were lines, circles, and dots. We found that RPS acuity predicted fractions knowledge for three university math placement exam subtests when controlling for simple magnitude acuities and inhibitory control. However, this predictive power of the RPS measure appeared to stem primarily from acuity of the line-ratio format, and that predictive power was attenuated with the inclusion of fluid intelligence. These findings may help refine theories positing the RPS as a domain-specific foundation for building fractional knowledge and related higher mathematics.

Math anxiety differentially impairs symbolic, but not nonsymbolic, fraction skills across development.

Although important for the acquisition of later math skills, fractions are notoriously difficult. Previous studies have shown that higher math anxiety (MA) is associated with lower performance in symbolic fraction tasks in adults and have suggested that MA may negatively impact the acquisition of fractions in children. However, the effects of MA on fraction skills in school-aged children remain underexplored. We, therefore, investigated the impact of MA on the performance of younger (second and third graders) and older (fifth and sixth graders) children in math fluency (MF), written calculation, fraction knowledge (FK), and symbolic fraction and nonsymbolic ratio processing. On the basis of our prior work suggesting a perceptual foundation for fraction processing, we predicted that symbolic, but not nonsymbolic, math skills (especially fractions) would be impaired by MA. As predicted, higher MA was associated with lower performance in general mathematics achievement and symbolic fraction tasks, but nonsymbolic ratio processing was not affected by MA in either age group. Furthermore, working memory capacity partially mediated the effects of MA on general mathematics achievement, FK, and symbolic fraction processing. These results suggest that understanding the bidirectional interactions between MA and fractions may be important for helping children acquire these critical skills.

First and Second Graders Successfully Reason About Ratios With Both Dot Arrays and Arabic Numerals

Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio reasoning in both formats and improved with age. Children who engaged in the nonsymbolic task first performed better on the symbolic task compared to children tested in the reverse order, suggesting that nonsymbolic ratio reasoning may function as a scaffold for symbolic ratio reasoning. Nonsymbolic ratio reasoning mediated the relation between children's numerosity comparison performance and symbolic mathematics performance in the domain of probabilities, but numerosity comparison performance explained significant unique variance in general numeration skills.

Non-symbolic and symbolic number and the approximate number system.

The distinction between non-symbolic and symbolic number is poorly addressed by the authors despite being relevant in numerical cognition, and even more important in light of the proposal that the approximate number system (ANS) represents rational numbers. Although evidence on non-symbolic number and ratios fits with ANS representations, the case for symbolic number and rational numbers is still open.

The ratio processing system and its role in fraction understanding: Evidence from a match-to-sample task in children and adults with and without dyscalculia.

It has been hypothesised that the human neurocognitive architecture may include a perceptual ratio processing system (RPS) that supports symbolic fraction understanding. In the present study, we aimed to provide further evidence for the existence of the RPS by exploring whether individuals with a range of math skills are indeed perceptually sensitive to non-symbolic ratio magnitudes. We also aimed to test to what extent the RPS may underlie symbolic fraction processing in those individuals. In a match-to-sample task, typical adults, elementary school children, and adults with dyscalculia were asked to match a non-symbolic ratio (i.e., target) to one of two non-symbolic ratios (i.e., the match and distractor). We found that all groups of participants were sensitive to the ratio between the match and the distractor, suggesting a common reliance on the RPS. This ratio sensitivity was also observed in another group of typical adults who had to choose which of two symbolic fractions match a non-symbolic ratio, indicating that the RPS may also contribute to symbolic fraction understanding. However, no ratio dependence was observed when participants had to choose which of two symbolic fractions match another symbolic fraction, suggesting that reliance on the RPS in symbolic fraction processing is limited and may not support exact fraction processing.

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.

The Relational SNARC: Spatial Representation of Nonsymbolic Ratios.

Recent research in numerical cognition has begun to systematically detail the ability of humans and nonhuman animals to perceive the magnitudes of nonsymbolic ratios. These relationally defined analogs to rational numbers offer new potential insights into the nature of human numerical processing. However, research into their similarities with and connections to symbolic numbers remains in its infancy. The current research aims to further explore these similarities by investigating whether the magnitudes of nonsymbolic ratios are associated with space just as symbolic numbers are. In two experiments, we found that responses were faster on the left for smaller nonsymbolic ratio magnitudes and faster on the right for larger nonsymbolic ratio magnitudes. These results further elucidate the nature of nonsymbolic ratio processing, extending the literature of spatial-numerical associations to nonsymbolic relative magnitudes. We discuss potential implications of these findings for theories of human magnitude processing in general and how this general processing relates to numerical processing.

The roles of place-value understanding and non-symbolic ratio processing system in symbolic rational number processing.

While it has been widely demonstrated that children's and adolescents' understanding of rational number plays an important role in their mathematics achievement, we have limited knowledge about the cognitive correlates of this understanding. The current study aimed at examining whether children's non-symbolic ratio processing and their understanding of place-value structure of whole numbers play a role in their understanding of fractions and decimals and whether their roles are different for fractions versus decimal understanding. A sample of 124 fourth graders was tested. Participants were tested on their symbolic rational number processing, non-symbolic ratio processing, place-value understanding of whole numbers, mathematics achievement, as well as a series of domain-general and domain-specific cognitive skills related to symbolic rational number processing. The findings suggest that, while the understanding of place value of whole numbers significantly predicted the understanding of both fractions and decimals, non-symbolic ratio processing specifically predicted the understanding of fractions, but not decimals. The findings highlight the roles of place-value understanding and non-symbolic ratio processing in the acquisition of symbolic rational numbers.

Ratios and differences in perceptual comparison: A reexamination of Torgerson’s conjecture

How do we compare stimuli that vary in magnitude? According to a well-known conjecture by Torgerson (1961), observers perceive only a single relation between stimuli, that is either a ratio or difference, but which one cannot be determined empirically. Previous research has used direct scaling procedures in which observers have judged ratios and differences numerically, but with mixed results. We used a novel behavioral task in which observers learned to produce non-symbolic ratios and differences by feedback and without explicit instruction. In two sets of experiments, observers viewed pairs of stimuli that varied in brightness, number of dots, or circle areas, and responded by clicking along a bar. Feedback was provided based on either the stimulus ratios or differences and the response location. Observers produced ratios and differences accurately, with average individual correlations of r=.94 and .95 across experiments, respectively. Regressions showed that responding was controlled jointly by ratios and differences, with the untrained relation predicting significant variance in approximately half of individual cases. Results were further supported by rank-order analyses, non-metric scaling analyses, and Monte Carlo simulations. Although Torgerson’s conjecture was originally proposed in explicit tasks in which observers were required to use their mathematical knowledge, our results show that the perceptual system automatically computes both differences and ratios when comparing stimuli in a non-symbolic, implicit learning task. Thus Torgerson’s conjecture, as applied to our task, is fundamentally false. Evidence for two operations rather than one suggests that the perceptual system might represent elements of an algebraic field, which could support adaptive behavior such as spatial navigation that appears to be computationally complex.

From continuous magnitudes to symbolic numbers: The centrality of ratio.

Leibovich et al.'s theory neither accounts for the deep connections between whole numbers and other classes of number nor provides a potential mechanism for mapping continuous magnitudes to symbolic numbers. We argue that focusing on non-symbolic ratio processing abilities can furnish a more expansive account of numerical cognition that remedies these shortcomings.

Fractions We Cannot Ignore: The Nonsymbolic Ratio Congruity Effect.

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

Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance

What basic capacities lay the foundation for advanced numerical cognition? Are there basic nonsymbolic abilities that support the understanding of advanced numerical concepts, such as fractions? To date, most theories have posited that previously identified core numerical systems, such as the approximate number system (ANS), are ill-suited for learning fraction concepts. However, recent research in developmental psychology and neuroscience has revealed a ratio-processing system (RPS) that is sensitive to magnitudes of nonsymbolic ratios and may be ideally suited for supporting fraction concepts. We provide evidence for this hypothesis by showing that individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university entrance exam in algebra. We suggest that the nonsymbolic RPS may support symbolic fraction understanding much as the ANS supports whole-number concepts. Thus, even abstract mathematical concepts, such as fractions, may be grounded not only in higher-order logic and language, but also in basic nonsymbolic processing abilities.

Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes

This study presents evidence that humans have intuitive, perceptually based access to the abstract fraction magnitudes instantiated by nonsymbolic ratio stimuli. Moreover, it shows these perceptually accessed magnitudes can be easily compared with symbolically represented fractions. In cross-format comparisons, participants picked the larger of two ratios. Ratios were presented either symbolically as fractions or nonsymbolically as paired dot arrays or as paired circles. Response patterns were consistent with participants comparing specific analog fractional magnitudes independently of the particular formats in which they were presented. These results pose a challenge to accounts that argue human cognitive architecture is ill-suited for processing fractions. Instead, it seems that humans can process nonsymbolic ratio magnitudes via perceptual routes and without recourse to conscious symbolic algorithms, analogous to the processing of whole number magnitudes. These findings have important implications for theories regarding the nature of human number sense – they imply that fractions may in some sense be natural numbers, too.

Rethinking the implications of numerical ratio effects for understanding the development of representational precision and numerical processing across formats.

Numerical ratio effects are a hallmark of numerical comparison tasks. Moreover, ratio effects have been used to draw strong conclusions about the nature of numerical representations, how these representations develop, and the degree to which they generalize across stimulus formats. Here, we compute ratio effects for 1,719 children from Grades K-6 for each individual separately by computing not just the average ratio effect for each person, but also the variability and statistical magnitude (effect-size) of their ratio effect. We find that individuals' ratio effect-sizes in fact increase over development, calling into question the view that decreasing ratio effects over development indicate increasing representational precision. Our data also strongly caution against the use of ratio effects in inferring the nature of symbolic number representation. While 75% of children showed a statistically significant ratio effect for nonsymbolic comparisons, only 30% did so for symbolic comparisons. Furthermore, whether a child's nonsymbolic ratio effect was significant did not predict whether the same was true of their symbolic ratio effect. These results undercut the notions (a) that individuals' ratio effects are indicative of representational precision in symbolic numbers, and (b) that a common process generates ratio effects in symbolic and nonsymbolic formats. Finally, for both formats, it was the variability of an individual child's ratio effect (not its slope or even effect-size) that correlated with arithmetic ability. Taken together, these results call into question many of the long-held tenets regarding the interpretation of ratio effects-especially with respect to symbolic numbers.

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 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.

Tuning to non‐symbolic proportions in the human frontoparietal cortex

Humans share with many species a non-verbal system to estimate absolute quantity. This sense of number has been linked to the activity of quantity-selective neurons that respond maximally to preferred numerosities. With functional magnetic resonance imaging adaptation, we now show that populations of neurons in the human parietal and frontal cortex are also capable of encoding quantity ratios, or proportions, using the same non-verbal analog code as for absolute number. Following adaptation to visually presented constant proportions (specified by the ratio of line lengths or numerosities), we introduced novel relative magnitudes to examine the tuning characteristics of the population of stimulated neurons. In bilateral parietal and frontal cortex we found that blood oxygenation level-dependent signal recovery from adaptation was a function of numerical distance between the deviant proportion and the adaptation stimulus. The strongest effects were observed in the cortex surrounding the anterior intraparietal sulcus, a region considered pivotal for the processing of absolute magnitudes. Overall, there was substantial overlap of frontoparietal structures representing whole numbers and proportions. The identification of tuning to non-symbolic ratio stimuli, irrespective of notation, adds to the magnitude system a remarkable level of sophistication by demonstrating automatic access to a composite, derived quantitative measure. Our results argue that abstract concepts of both absolute and relative number are deeply rooted in the primate brain as fundamental determinants of higher-level numerical cognition.