Numerical and nonnumerical estimation in children with and without mathematical learning disabilities

There are currently multiple explanations for mathematical learning disabilities (MLD). The present study focused on those assuming that MLD are due to a basic numerical deficit affecting the ability to represent and to manipulate number magnitude ( Butterworth, 1999 , 2005 ; A. J. Wilson & Dehaene, 2007 ) and/or to access that number magnitude representation from numerical symbols ( Rousselle & No?l, 2007 ). The present study provides an original contribution to this issue by testing MLD children (carefully selected on the basis of preserved abilities in other domains) on numerical estimation tasks with contrasting symbolic (Arabic numerals) and nonsymbolic (collection of dots) numbers used as input or output. MLD children performed consistently less accurately than control children on all the estimation tasks. However, MLD children were even weaker when the task involved the mapping between symbolic and nonsymbolic numbers than when the task required a mapping between two nonsymbolic numerical formats. Moreover, in the estimation of nonsymbolic numerosities, MLD children relied more than control children on perceptual cues such as the cumulative area of the dots. Finally, the task requiring a mapping from a nonsymbolic format to a symbolic format was the best predictor of MLD. In order to explain these present results, as well as those reported in the literature, we propose that the impoverished number magnitude representation of MLD children may arise from an initial mapping deficit between number symbols and that magnitude representation.     

Improving children's mental rotation accuracy with computer game playing

The authors investigated the relation between mental rotation (MR) and computer game-playing experience. Third grade boys (n = 24) and girls (n = 23) completed a 2-dimensional MR test before and after playing computer games (during 11 separate 30-min sessions), which either involved the use of MR skills (the experimental group) or did not involve the use of MR skills (the control group). The experimental group outperformed the control group on the MR posttest but not on the pretest. Boys outperformed girls on the pretest but not on the posttest. Children whose initial MR performance was low improved after playing computer games that entailed MR skills. The findings imply that computer-based instructional activities can be used in schools to enhance children's spatial abilities.     

Sensori-motor spatial training of number magnitude representation

An adequately developed spatial representation of number magnitude is associated with children's general arithmetic achievement. Therefore, a new spatial-numerical training program for kindergarten children was developed in which presentation and response were associated with a congruent spatial numerical representation. In particular, children responded by a full-body spatial movement on a digital dance mat in a magnitude comparison task. This spatial-numerical training was more effective than a non-spatial control training in enhancing children's performance on a number line estimation task and a subtest of a standardized mathematical achievement battery (TEDI-MATH). A mediation analysis suggested that these improvements were driven by an improvement of children's mental number line representation and not only by unspecific factors such as attention or motivation. These results suggest a benefit of spatial numerical associations. Rather than being a merely associated covariate, they work as an independently manipulated variable which is functional for numerical development.     

Reducing the gap in numerical knowledge between low- and middle-income preschoolers

We compared the learning from playing a linear number board game of preschoolers from middle-income backgrounds to the learning of preschoolers from low-income backgrounds. Playing this game produced greater learning by both groups than engaging in other numerical activities for the same amount of time. The benefits were present on number line estimation, magnitude comparison, numeral identification, and arithmetic learning. Children with less initial knowledge generally learned more, and children from low-income backgrounds learned at least as much, and on several measures more, than preschoolers from middle-income backgrounds with comparable initial knowledge. The findings suggest a class of intervention that might be especially effective for reducing the gap between low-income and middle-income children's knowledge when they enter school.     

The relation between spatial skill and early number knowledge: The role of the linear number line

[Correction Notice: An Erratum for this article was reported in Vol 48(5) of Developmental Psychology (see record 2012-11771-001). The grey boxes around the faces in Figure 2 are missing. The correct version is presented in the erratum.] Spatial skill is highly related to success in math and science (e.g., Casey, Nuttall, Pezaris, & Benbow, 1995). However, little work has investigated the cognitive pathways by which the relation between spatial skill and math achievement emerges. We hypothesized that spatial skill plays a crucial role in the development of numerical reasoning by helping children to create a spatially meaningful, powerful numerical representation-the linear number line. In turn, a strong linear number representation improves other aspects of numerical knowledge such as arithmetic estimation. We tested this hypothesis using 2 longitudinal data sets. First, we found that children's spatial skill (i.e., mental transformation ability) at the beginning of 1st and 2nd grades predicted improvement in linear number line knowledge over the course of the school year. Second, we found that children's spatial skill at age 5 years predicted their performance on an approximate symbolic calculation task at age 8 and that this relation was mediated by children's linear number line knowledge at age 6. The results are consistent with the hypothesis that spatial skill can improve children's development of numerical knowledge by helping them to acquire a linear spatial representation of numbers. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties

This study examined numerical magnitude processing in first graders with severe and mild forms of mathematical difficulties, children with mathematics learning disabilities (MLD) and children with low achievement (LA) in mathematics, respectively. In total, 20 children with MLD, 21 children with LA, and 41 regular achievers completed a numerical magnitude comparison task and an approximate addition task, which were presented in a symbolic and a nonsymbolic (dot arrays) format. Children with MLD and LA were impaired on tasks that involved the access of numerical magnitude information from symbolic representations, with the LA children showing a less severe performance pattern than children with MLD. They showed no deficits in accessing magnitude from underlying nonsymbolic magnitude representations. Our findings indicate that this performance pattern occurs in children from first grade onward and generalizes beyond numerical magnitude comparison tasks. These findings shed light on the types of intervention that may help children who struggle with learning mathematics.     

"The relation between spatial skill and early number knowledge: The role of the linear number line": Correction to Gunderson et al. (2012)

Reports an error in "The Relation Between Spatial Skill and Early Number Knowledge: The Role of the Linear Number Line" by Elizabeth A. Gunderson, Gerardo Ramirez, Sian L. Beilock and Susan C. Levine (Developmental Psychology, Advanced Online Publication, Mar 5, 2012, np). The grey boxes around the faces in Figure 2 are missing. The correct version is presented in the erratum. (The following abstract of the original article appeared in record 2012-05400-001.) Spatial skill is highly related to success in math and science (e.g., Casey, Nuttall, Pezaris, & Benbow, 1995). However, little work has investigated the cognitive pathways by which the relation between spatial skill and math achievement emerges. We hypothesized that spatial skill plays a crucial role in the development of numerical reasoning by helping children to create a spatially meaningful, powerful numerical representation-the linear number line. In turn, a strong linear number representation improves other aspects of numerical knowledge such as arithmetic estimation. We tested this hypothesis using 2 longitudinal data sets. First, we found that children's spatial skill (i.e., mental transformation ability) at the beginning of 1st and 2nd grades predicted improvement in linear number line knowledge over the course of the school year. Second, we found that children's spatial skill at age 5 years predicted their performance on an approximate symbolic calculation task at age 8 and that this relation was mediated by children's linear number line knowledge at age 6. The results are consistent with the hypothesis that spatial skill can improve children's development of numerical knowledge by helping them to acquire a linear spatial representation of numbers. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Implicit response-irrelevant number information triggers the SNARC effect: Evidence using a neural overlap paradigm

There is evidence from the SNARC (spatial–numerical association of response codes) effect and NDE (numerical distance effect) that number activates spatial representations. Most of this evidence comes from tasks with explicit reference to number, whether through presentation of Arabic digits (SNARC) or through magnitude decisions to nonsymbolic representations (NDE). Here, we report four studies that use the neural overlap paradigm developed by Fias, Lauwereyns, and Lammertyn (2001) to examine whether the presentation of implicit and task-irrelevant numerosity information (nonsymbolic arrays and auditory numbers) is enough to activate a spatial representation of number. Participants were presented with either numerosity arrays (1–9 circles or triangles) to which they made colour (Experiment 1) or orientation (Experiment 2) judgements, or auditory numbers coupled with an on-screen stimulus to which they made a colour (Experiment 3) or orientation (Experiment 4) judgement. SNARC effects were observed only for the orientation tasks. Following the logic of Fias et al., we argue that this SNARC effect occurs as a result of overlap in parietal processing for number and orientation judgements irrespective of modality. Furthermore, we found stronger SNARC effects in the small number range (1–4) than in the larger number range (6–9) for both nonsymbolic displays and auditory numbers. These results suggest that quantity is extracted (and interferes with responses in the orientation task) but this is not exact for the entire number range. We discuss a number of alternative models and mechanisms of numerical processing that may account for such effects.     

The association between children's numerical magnitude processing and mental multi-digit subtraction

Children apply various strategies to mentally solve multi-digit subtraction problems and the efficient use of some of them may depend more or less on numerical magnitude processing. For example, the indirect addition strategy (solving 72–67 as “how much do I have to add up to 67 to get 72?”), which is particularly efficient when the two given numbers are close to each other, requires to determine the proximity of these two numbers, a process that may depend on numerical magnitude processing. In the present study, children completed a numerical magnitude comparison task and a number line estimation task, both in a symbolic and nonsymbolic format, to measure their numerical magnitude processing. We administered a multi-digit subtraction task, in which half of the items were specifically designed to elicit indirect addition. Partial correlational analyses, controlling for intellectual ability and motor speed, revealed significant associations between numerical magnitude processing and mental multi-digit subtraction. Additional analyses indicated that numerical magnitude processing was particularly important for those items for which the use of indirect addition is expected to be most efficient. Although this association was observed for both symbolic and nonsymbolic tasks, the strongest associations were found for the symbolic format, and they seemed to be more prominent on numerical magnitude comparison than on number line estimation.     

Playing linear numerical board games promotes low-income children's numerical development

The numerical knowledge of children from low-income backgrounds trails behind that of peers from middle-income backgrounds even before the children enter school. This gap may reflect differing prior experience with informal numerical activities, such as numerical board games. Experiment 1 indicated that the numerical magnitude knowledge of preschoolers from low-income families lagged behind that of peers from more affluent backgrounds. Experiment 2 indicated that playing a simple numerical board game for four 15-minute sessions eliminated the differences in numerical estimation proficiency. Playing games that substituted colors for numbers did not have this effect. Thus, playing numerical board games offers an inexpensive means for reducing the gap in numerical knowledge that separates less and more affluent children when they begin school.     

Squeezing, striking, and vocalizing: Is number representation fundamentally spatial?

Numbers are fundamental entities in mathematics, but their cognitive bases are unclear. Abundant research points to linear space as a natural grounding for number representation. But, is number representation fundamentally spatial? We disentangle number representation from standard number-to-line reporting methods, and compare numerical estimations in educated participants using line-reporting with three nonspatial reporting conditions (squeezing, bell-striking, and vocalizing). All three cases of nonspatial-reporting consistently reproduced well-established results obtained with number-line methods. Furthermore, unlike line-reporting—and congruent with the psychophysical Weber–Fechner law—nonspatial reporting systematically produced logarithmic mappings for all nonsymbolic stimuli. Strikingly, linear mappings were obtained exclusively in conditions with culturally mediated elements (e.g., words). These results suggest that number representation is not fundamentally spatial, but builds on a deeper magnitude sense that manifests spatially and nonspatially mediated by magnitude, stimulus modality, and reporting condition. Number-to-space mappings—although ubiquitous in the modern world—do not seem to be rooted directly in brain evolution but have been culturally privileged and enhanced.     

Implicit response-irrelevant number information triggers the SNARC effect: Evidence using a neural overlap paradigm

There is evidence from the SNARC (spatial-numerical association of response codes) effect and NDE (numerical distance effect) that number activates spatial representations. Most of this evidence comes from tasks with explicit reference to number, whether through presentation of Arabic digits (SNARC) or through magnitude decisions to nonsymbolic representations (NDE). Here, we report four studies that use the neural overlap paradigm developed by Fias, Lauwereyns, and Lammertyn (2001) to examine whether the presentation of implicit and task-irrelevant numerosity information (nonsymbolic arrays and auditory numbers) is enough to activate a spatial representation of number. Participants were presented with either numerosity arrays (1-9 circles or triangles) to which they made colour (Experiment 1) or orientation (Experiment 2) judgements, or auditory numbers coupled with an on-screen stimulus to which they made a colour (Experiment 3) or orientation (Experiment 4) judgement. SNARC effects were observed only for the orientation tasks. Following the logic of Fias et al., we argue that this SNARC effect occurs as a result of overlap in parietal processing for number and orientation judgements irrespective of modality. Furthermore, we found stronger SNARC effects in the small number range (1-4) than in the larger number range (6-9) for both nonsymbolic displays and auditory numbers. These results suggest that quantity is extracted (and interferes with responses in the orientation task) but this is not exact for the entire number range. We discuss a number of alternative models and mechanisms of numerical processing that may account for such effects.     

Improving Children's Mental Rotation Accuracy With Computer Game Playing

The authors investigated the relation between mental rotation (MR) and computer game-playing experience. Third grade boys (n = 24) and girls (n = 23) completed a 2-dimensional MR test before and after playing computer games (during 11 separate 30-min sessions), which either involved the use of MR skills (the experimental group) or did not involve the use of MR skills (the control group). The experimental group outperformed the control group on the MR posttest but not on the pretest. Boys outperformed girls on the pretest but not on the posttest. Children whose initial MR performance was low improved after playing computer games that entailed MR skills. The findings imply that computer-based instructional activities can be used in schools to enhance children's spatial abilities.     

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.     

Sex differences in number line estimation: The role of numerical estimation

Sex differences in mathematical performance have frequently been examined over the last decades indicating an advantage for males especially when numerical problems cannot be solved by (classroom‐)learnt strategies and/or estimation. Even in basic numerical tasks such as number line estimation, males were found to outperform females – with sex differences argued to emerge from different solution strategies applied by males and females. We evaluated the latter using two versions of the number line estimation task: a bounded and an unbounded task version. Assuming that women tend more strongly to apply known procedures, we expected them to be at a particular disadvantage in the unbounded number line estimation task, which is less prone to be solved by specific strategies such as proportion judgement but requires numerical estimation. Results confirmed more pronounced sex differences for unbounded number line estimation with males performing significantly more accurately in this task version. This further adds to recent evidence suggesting that estimation performance in the bounded task version may reflect solution strategies rather than numerical estimation. Additionally, it indicates that sex differences regarding the spatial representation of number magnitude may not be universal, but associated with spatial–numerical estimations in particular.     

Differences between literates and illiterates on symbolic but not nonsymbolic numerical magnitude processing

The study of numerical magnitude processing provides a unique opportunity to examine interactions between phylogenetically ancient systems of semantic representations and those that are the product of enculturation. While nonsymbolic representations of numerical magnitude are processed similarly by humans and nonhuman animals, symbolic representations of numerical magnitude (e.g., Hindu–Arabic numerals) are culturally invented symbols that are uniquely human. Here, we report a comparison of symbolic and nonsymbolic numerical magnitude processing in two groups of participants who differ substantially in their level of literacy. In this study, level of literacy is used as an index of level of school-based numeracy skill. The data from these groups demonstrate that while the processing of nonsymbolic numerical magnitude (numerical distance effect) is unaffected by an individual’s level of literacy, the processing of Hindu–Arabic numerals differs between literate and illiterate individuals who live in a literature culture and have limited symbolic recognition skills. These findings reveal that nonsymbolic numerical magnitude processing is unaffected by enculturation, while the processing of numerical symbols is modulated by literacy.     

Working memory and number line representations in single-digit addition: Approximate versus exact,nonsymbolic versus symbolic

How do kindergarteners solve different single-digit addition problem formats? We administered problems that differed solely on the basis of two dimensions: response type (approximate or exact), and stimulus type (nonsymbolic, i.e., dots, or symbolic, i.e., Arabic numbers). We examined how performance differs across these dimensions, and which cognitive mechanism (mental model, transcoding, or phonological storage) underlies performance in each problem format with respect to working memory (WM) resources and mental number line representations. As expected, nonsymbolic problem formats were easier than symbolic ones. The visuospatial sketchpad was the primary predictor of nonsymbolic addition. Symbolic problem formats were harder because they either required the storage and manipulation of quantitative symbols phonologically or taxed more WM resources than their nonsymbolic counterparts. In symbolic addition, WM and mental number line results showed that when an approximate response was needed, children transcoded the information to the nonsymbolic code. When an exact response was needed, however, they phonologically stored numerical information in the symbolic code. Lastly, we found that more accurate symbolic mental number line representations were related to better performance in exact addition problem formats, not the approximate ones. This study extends our understanding of the cognitive processes underlying children's simple addition skills.     

The powers of noise-fitting: reply to Barth and Paladino

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude.     

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

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

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

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