A Two-Minute Paper-and-Pencil Test of Symbolic and Nonsymbolic Numerical Magnitude Processing Explains Variability in Primary School Children's Arithmetic Competence

Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children’s ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children’s performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children’s ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1–3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics.     

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.     

Numerical magnitude in the human parietal lobe; tests of representational generality and domain specificity

Behavioral evidence suggests that human adults have a single system for representing the numerical magnitude of both symbolic numbers (e.g., Arabic digits) and nonsymbolic number stimuli (e.g., dot arrays). Brain imaging studies have implicated a specific parietal region in symbolic number processing, leading to the influential hypothesis that this region is the locus of a dedicated, domain-specific number system. Here we evaluated a prediction of this hypothesis, that this region should be activated not only by symbolic but also nonsymbolic number processing. Using nonsymbolic stimuli, we tested for higher parietal activations for number than for nonnumber comparison tasks (experiment 1), fMRI adaptation for numerosity repetition (experiment 2), and greater fMRI increases with increasing task difficulty for number than nonnumber tasks (experiment 3). None of these predictions were supported by the data, posing a serious challenge to the hypothesis that a single, domain-specific parietal region underlies both symbolic and nonsymbolic number representation.     

Multi-representation of symbolic and nonsymbolic numerical magnitude in Chinese number processing

Numerical information can be conveyed by either symbolic or nonsymbolic representation. Some symbolic numerals can also be identified as nonsymbolic quantities defined by the number of lines (e.g., I, II, III in Roman and -, =, ≡ in Japanese Kanji and Chinese). Here we report that such multi-representation of magnitude can facilitate the processing of these numerals under certain circumstances. In a magnitude comparison task judging 1 to 9 (except 5) Chinese and Arabic numerals presented at the foveal (at the center) or parafoveal (3° left or right of the center) location, multi-representational small-value Chinese numerals showed a processing advantage over single-representational Arabic numerals and large-value Chinese numerals only in the parafoveal condition, demonstrated by lower error rates and faster reaction times. Further event-related potential (ERP) analysis showed that such a processing advantage was not reflected by traditional ERP components identified in previous studies of number processing, such as N1 or P2p. Instead, the difference was found much later in a N400 component between 300-550 msec over parietal regions, suggesting that those behavioral differences may not be due to early processing of visual identification, but later processing of subitizing or accessing mental number line when lacking attentional resources. These results suggest that there could be three stages of number processing represented separately by the N1, P2p and N400 ERP components. In addition, numerical information can be represented simultaneously by both symbolic and nonsymbolic systems, which will facilitate number processing in certain situations.     

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.     

Children’s Mapping between Non-Symbolic and Symbolic Numerical Magnitudes and Its Association with Timed and Untimed Tests of Mathematics Achievement

The ability to map between non-symbolic numerical magnitudes and Arabic numerals has been put forward as a key factor in children’s mathematical development. This mapping ability has been mainly examined indirectly by looking at children’s performance on a symbolic magnitude comparison task. The present study investigated mapping in a more direct way by using a task in which children had to choose which of two choice quantities (Arabic digits or dot arrays) matched the target quantity (dot array or Arabic digit), thereby focusing on small quantities ranging from 1 to 9. We aimed to determine the development of mapping over time and its relation to mathematics achievement. Participants were 36 first graders (M = 6 years 8 months) and 46 third graders (M = 8 years 8 months) who all completed mapping tasks, symbolic and non-symbolic magnitude comparison tasks and standardized timed and untimed tests of mathematics achievement. Findings revealed that children are able to map between non-symbolic and symbolic representations and that this mapping ability develops over time. Moreover, we found that children’s mapping ability is related to timed and untimed measures of mathematics achievement, over and above the variance accounted for by their numerical magnitude comparison skills.     

Impact of high mathematics education on the number sense

In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS.     

The number domain- can we count on parietal cortex?

Does the primate brain contain a dedicated and localized neural circuitry for processing generic numerical information? The human parietal cortex, particularly the intraparietal sulcus (IPS), has long been implicated in processing symbolic (verbal) number information. If the IPS is indeed the site of generic numerical processing, however, its neurons should also encode nonsymbolic numerosity information. Two recent studies by Shuman and Kanwisher and by Piazza et al. published in this issue of Neuron tested this assumption...with quite different results.     

A magnitude code common to numerosities and number symbols in human intraparietal cortex

Activation of the horizontal segment of the intraparietal sulcus (hIPS) has been observed in various number-processing tasks, whether numbers were conveyed by symbolic numerals (digits, number words) or by nonsymbolic displays (dot patterns). This suggests an abstract coding of numerical magnitude. Here, we critically tested this hypothesis using fMRI adaptation to demonstrate notation-independent coding of numerical quantity in the hIPS. Once subjects were adapted either to dot patterns or to Arabic digits, activation in the hIPS and in frontal regions recovered in a distance-dependent fashion whenever a new number was presented, irrespective of notation changes. This remained unchanged when analyzing the hIPS peaks from an independent localizer scan of mental calculation. These results suggest an abstract coding of approximate number common to dots, digits, and number words. They support the idea that symbols acquire meaning by linking neural populations coding symbol shapes to those holding nonsymbolic representations of quantities.     

Symbolic Numerical Magnitude Processing Is as Important to Arithmetic as Phonological Awareness Is to Reading

In this article, we tested, using a 1-year longitudinal design, whether symbolic numerical magnitude processing or children’s numerical representation of Arabic digits, is as important to arithmetic as phonological awareness is to reading. Children completed measures of symbolic comparison, phonological awareness, arithmetic, reading at the start of third grade and the latter two were retested at the start of fourth grade. Cross-sectional and longitudinal correlations indicated that symbolic comparison was a powerful domain-specific predictor of arithmetic and that phonological awareness was a unique predictor of reading. Crucially, the strength of these independent associations was not significantly different. This indicates that symbolic numerical magnitude processing is as important to arithmetic development as phonological awareness is to reading and suggests that symbolic numerical magnitude processing is a good candidate for screening children at risk for developing mathematical difficulties.     

Fostering Formal Commutativity Knowledge with Approximate Arithmetic

How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural and conceptual knowledge of a more abstract arithmetic principle. Experiment 1 tested first graders who had not been instructed about commutativity in school yet. Approximate calculation with symbolic quantities positively influenced the use of commutativity-based shortcuts in formal arithmetic. We replicated this finding with older first graders (Experiment 2) and third graders (Experiment 3). Despite the positive effect of approximation on the spontaneous application of commutativity-based shortcuts in arithmetic problems, we found no comparable impact on the application of conceptual knowledge of the commutativity principle. Overall, our results show that the usage of a specific arithmetic principle can benefit from approximation. However, the findings also suggest that the correct use of certain procedures does not always imply conceptual understanding. Rather, the conceptual understanding of commutativity seems to lag behind procedural proficiency during elementary school.     

Functional imaging of numerical processing in adults and 4-y-old children

Adult humans, infants, pre-school children, and non-human animals appear to share a system of approximate numerical processing for non-symbolic stimuli such as arrays of dots or sequences of tones. Behavioral studies of adult humans implicate a link between these non-symbolic numerical abilities and symbolic numerical processing (e.g., similar distance effects in accuracy and reaction-time for arrays of dots and Arabic numerals). However, neuroimaging studies have remained inconclusive on the neural basis of this link. The intraparietal sulcus (IPS) is known to respond selectively to symbolic numerical stimuli such as Arabic numerals. Recent studies, however, have arrived at conflicting conclusions regarding the role of the IPS in processing non-symbolic, numerosity arrays in adulthood, and very little is known about the brain basis of numerical processing early in development. Addressing the question of whether there is an early-developing neural basis for abstract numerical processing is essential for understanding the cognitive origins of our uniquely human capacity for math and science. Using functional magnetic resonance imaging (fMRI) at 4-Tesla and an event-related fMRI adaptation paradigm, we found that adults showed a greater IPS response to visual arrays that deviated from standard stimuli in their number of elements, than to stimuli that deviated in local element shape. These results support previous claims that there is a neurophysiological link between non-symbolic and symbolic numerical processing in adulthood. In parallel, we tested 4-y-old children with the same fMRI adaptation paradigm as adults to determine whether the neural locus of non-symbolic numerical activity in adults shows continuity in function over development. We found that the IPS responded to numerical deviants similarly in 4-y-old children and adults. To our knowledge, this is the first evidence that the neural locus of adult numerical cognition takes form early in development, prior to sophisticated symbolic numerical experience. More broadly, this is also, to our knowledge, the first cognitive fMRI study to test healthy children as young as 4 y, providing new insights into the neurophysiology of human cognitive development.     

Processing ordinality and quantity: the case of developmental dyscalculia

In contrast to quantity processing, up to date, the nature of ordinality has received little attention from researchers despite the fact that both quantity and ordinality are embodied in numerical information. Here we ask if there are two separate core systems that lie at the foundations of numerical cognition: (1) the traditionally and well accepted numerical magnitude system but also (2) core system for representing ordinal information. We report two novel experiments of ordinal processing that explored the relation between ordinal and numerical information processing in typically developing adults and adults with developmental dyscalculia (DD). Participants made "ordered" or "non-ordered" judgments about 3 groups of dots (non-symbolic numerical stimuli; in Experiment 1) and 3 numbers (symbolic task: Experiment 2). In contrast to previous findings and arguments about quantity deficit in DD participants, when quantity and ordinality are dissociated (as in the current tasks), DD participants exhibited a normal ratio effect in the non-symbolic ordinal task. They did not show, however, the ordinality effect. Ordinality effect in DD appeared only when area and density were randomized, but only in the descending direction. In the symbolic task, the ordinality effect was modulated by ratio and direction in both groups. These findings suggest that there might be two separate cognitive representations of ordinal and quantity information and that linguistic knowledge may facilitate estimation of ordinal information.     

THE EFFECT OF 40 HOURS OF CONSTANT WAKEFULNESS ON NUMBER COMPARISON PERFORMANCE

We investigated the effects of sleep loss and circadian rhythm on number comparison performance. Magnitude comparison of single-digits is robustly characterized by a distance effect: Close numbers (e.g., 5 versus 6) produce longer reaction times than numbers further apart (e.g., 2 versus 8). This distance effect is assumed to reflect the difficulty of a comparison process based on an analogous representation of general magnitude. Twelve male participants were required to stay awake for 40?h in a quasi-constant-routine protocol. Response speed and accuracy deteriorated between 00:00 and 06:00?h but recovered afterwards during the next day, indicating a circadian rhythm of elementary cognitive function (i.e., attention and speed of mental processing). The symbolic distance effect, however, did not increase during the nighttime, indicating that neither cumulative sleep loss nor the circadian clock prolongs numerical comparison processes. The present findings provide first evidence for a relative insensitivity of symbolic magnitude processing against the temporal variation in energy state. (Author correspondence: michael.steinborn@uni-tuebingen.de)     

Childhood Overweight and Academic Performance: National Study of Kindergartners and First‐Graders

Objective: To examine the association between children's overweight status in kindergarten and their academic achievement in kindergarten and first grade. Research Methods and Procedures: The data analyzed consisted of 11, 192 first time kindergartners from the Early Childhood Longitudinal Study, a nationally representative sample of kindergartners in the U.S. in 1998. Multivariate regression techniques were used to estimate the independent association of overweight status with children's math and reading standardized test scores in kindergarten and grade 1. We controlled for socioeconomic status, parent‐child interaction, birth weight, physical activity, and television watching. Results: Overweight children had significantly lower math and reading test scores compared with nonoverweight children in kindergarten. Both groups were gaining similarly on math and reading test scores, resulting in significantly lower test scores among overweight children at the end of grade 1. However, these differences, except for boys’ math scores at baseline (difference = 1.22 points, p = 0.001), became insignificant after including socioeconomic and behavioral variables, indicating that overweight is a marker but not a causal factor. Race/ethnicity and mother's education were stronger predictors of test score gains or levels than overweight status. Discussion: Significant differences in test scores by overweight status at the beginning of kindergarten and the end of grade 1 can be explained by other individual characteristics, including parental education and the home environment. However, overweight is more easily observable by other students compared with socioeconomic characteristics, and its significant (unadjusted) association with worse academic performance can contribute to the stigma of overweight as early as the first years of elementary school.     

Magnitude Representations in Williams Syndrome: Differential Acuity in Time,Space and Number Processing

For some authors, the human sensitivity to numerosities would be grounded in our ability to process non-numerical magnitudes. In the present study, the developmental relationships between non numerical and numerical magnitude processing are examined in people with Williams syndrome (WS), a genetic disorder known to associate visuo-spatial and math learning disabilities. Twenty patients with WS and 40 typically developing children matched on verbal or non-verbal abilities were administered three comparison tasks in which they had to compare numerosities, lengths or durations. Participants with WS showed lower acuity (manifested by a higher Weber fraction) than their verbal matched peers when processing numerical and spatial but not temporal magnitudes, indicating that they do not present a domain-general dysfunction of all magnitude processing. Conversely, they do not differ from non-verbal matched participants in any of the three tasks. Finally, correlational analyses revealed that non-numerical and numerical acuity indexes were both related to the first mathematical acquisitions but not with later arithmetical skills.     

Preschoolers' Dot Enumeration Abilities Are Markers of Their Arithmetic Competence

The abilities to enumerate small sets of items (e.g., dots) and to compare magnitudes are claimed to be indexes of core numerical competences that scaffold early math development. Insofar as this is correct, these abilities may be diagnostic markers of math competence in preschoolers. However, unlike magnitude comparison abilities, little research has examined preschoolers'' ability to enumerate small sets, or its significance for emerging math abilities; which is surprising since dot enumeration is a marker of school-aged children''s math competence. It is nevertheless possible that general cognitive functions (working memory, response inhibition in particular) are associated with preschoolers'' math abilities and underlie nascent dot enumeration abilities. We investigated whether preschoolers'' dot enumeration abilities predict their non-verbal arithmetic ability, over and above the influence of working memory and response inhibition. Two measures of dot enumeration ability were examined—inverse efficiency and paradigm specific (response time profiles) measures—to determine which has the better diagnostic utility as a marker of math competence. Seventy-eight 42-to-57 month-olds completed dot enumeration, working memory, response inhibition, and non-verbal addition and subtraction tasks. Dot enumeration efficiency predicted arithmetic ability over and above the influence of general cognitive functions. While dot enumeration efficiency was a better predictor of arithmetic ability than paradigm specific response time profiles; the response time profile displaying the smallest subitizing range and steepest subitizing slope, also displayed poor addition abilities, suggesting a weak subitizing profile may have diagnostic significance in preschoolers. Overall, the findings support the claim that dot enumeration abilities and general cognitive functions are markers of preschoolers'' math ability.     

Strategy Choice Mediates the Link between Auditory Processing and Spelling

Relations among linguistic auditory processing, nonlinguistic auditory processing, spelling ability, and spelling strategy choice were examined. Sixty-three undergraduate students completed measures of auditory processing (one involving distinguishing similar tones, one involving distinguishing similar phonemes, and one involving selecting appropriate spellings for individual phonemes). Participants also completed a modified version of a standardized spelling test, and a secondary spelling test with retrospective strategy reports. Once testing was completed, participants were divided into phonological versus nonphonological spellers on the basis of the number of words they spelled using phonological strategies only. Results indicated a) moderate to strong positive correlations among the different auditory processing tasks in terms of reaction time, but not accuracy levels, and b) weak to moderate positive correlations between measures of linguistic auditory processing (phoneme distinction and phoneme spelling choice in the presence of foils) and spelling ability for phonological spellers, but not for nonphonological spellers. These results suggest a possible explanation for past contradictory research on auditory processing and spelling, which has been divided in terms of whether or not disabled spellers seemed to have poorer auditory processing than did typically developing spellers, and suggest implications for teaching spelling to children with good versus poor auditory processing abilities.     

The ontogeny of visual–motor memory and its importance in handwriting and reading: a developing construct

Humans have evolved a remarkable ability to remember visual shapes and use these representations to generate motor activity (from Palaeolithic cave drawings through Jiahu symbols to cursive handwriting). The term visual–motor memory (VMM) describes this psychological ability, which must have conveyed an evolutionary advantage and remains critically important to humans (e.g. when learning to write). Surprisingly, little empirical investigation of this unique human ability exists—almost certainly because of the technological difficulties involved in measuring VMM. We deployed a novel technique for measuring this construct in 87 children (6–11 years old, 44 females). Children drew novel shapes presented briefly on a tablet laptop screen, drawing their responses from memory on the screen using a digitizer stylus. Sophisticated algorithms (using point-registration techniques) objectively quantified the accuracy of the children''s reproductions. VMM improved with age and performance decreased with shape complexity, indicating that the measure captured meaningful developmental changes. The relationship between VMM and scores on nationally standardized writing assessments were explored with the results showing a clear relationship between these measures, even after controlling for age. Moreover, a relationship between VMM and the nationally standardized reading test was mediated via writing ability, suggesting VMM''s wider importance within language development.     

Microbiological evaluation of a biological safety cabinet modified for bedding disposal

A biological safety cabinet modified for bedding disposal was tested to determine the cabinet's ability to protect operators and experiments from aerosol exposure during routine microbiological and cage cleaning procedures. Stringent test conditions were provided by modifications of standardized protocols in addition to simulated cage dumping procedures, both of which utilized bacterial aerosols as challenges. Results of standardized test procedures (with no operator present) indicated good performance in protecting both operators and experiments. Procedures involving the dumping (by an operator) of contaminated bedding within the unit showed that the cabinet was able to contain 99.96% or greater of the total particles generated.