Non-symbolic arithmetic in adults and young children

Barth, Hilary; Mont, Kristen La; Lipton, Jennifer S.; Dehaene, Stanislas; Kanwisher, Nancy; Spelke, Elizabeth S.

doi:10.1016/j.cognition.2004.09.011

Cited by 331 publications

(311 citation statements)

References 35 publications

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“…Our first experiment replicates previous findings of nonsymbolic numerical abilities in preschool children (e.g., Barth et al, 2003Barth et al, , 2005Barth et al, , 2006Brannon, 2002;McCrink & Wynn, 2004), and it extends those findings in two ways. First, performance in our studies on problems of the form x + y − z provides evidence that preschool children can perform two successive operations of addition and subtraction on large approximate numerosities presented in nonsymbolic form (as arrays of dots).…”

Section: Discussionsupporting

confidence: 88%

“…Moreover, adults with and without formal education, preschool children, and infants can perform approximate additions and subtractions on non-symbolic stimuli (Pica et al, 2004;McCrink & Wynn, 2004;Barth, La Mont, Lipton & Spelke, 2005;Barth, La Mont, Lipton, Dehaene, Kanwisher & Spelke, 2006). In an experiment that is a direct precursor to the present studies, 5-year-old children were presented with computer-animated events in which an array of blue dots appeared and moved into a box, and then a second set of blue dots moved into the box.…”

Section: Author Manuscriptmentioning

confidence: 99%

“…Cordes, Gelman, Gallistel, & Whalen, 2001;van Oeffelen & Vos, 1982). These non-symbolic representations are imprecise, they are subject to a ratio limit on discriminability, and they have been found in educated adults (Barth, Kanwisher & Spelke, 2003;Whalen, Gallistel & Gelman, 1999), preschool children , adults in an indigenous Amazonian community lacking any formal education (Pica, Lemer, Izard & Dehaene, 2004), pre-verbal infants (Brannon, 2002Xu & Spelke, 2000) and non-human animals (Meck & Church, 1983).Moreover, adults with and without formal education, preschool children, and infants can perform approximate additions and subtractions on non-symbolic stimuli (Pica et al, 2004;McCrink & Wynn, 2004;Barth, La Mont, Lipton & Spelke, 2005;Barth, La Mont, Lipton, Dehaene, Kanwisher & Spelke, 2006). In an experiment that is a direct precursor to the present studies, 5-year-old children were presented with computer-animated events in which an array of blue dots appeared and moved into a box, and then a second set of blue dots moved into the box.…”

mentioning

confidence: 99%

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Children’s understanding of the relationship between addition and subtraction

Gilmore

Spelke

2008

Cognition

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In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12 + 9 − 9 yields 12. Here, we investigate whether preschool children's approximate number knowledge nevertheless supports understanding of this relationship. Five-yearold children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation than those that did not, when problems were presented in either non-symbolic or symbolic form. In contrast they showed no advantage for problems involving an inverse transformation when exact arithmetic was involved. Prior to formal schooling, children therefore show generalized understanding of at least one logical principle of arithmetic. The teaching of mathematics may be enhanced by building on this understanding.To a large degree, mathematics is the discovery and use of general, abstract principles that make hard problems easy. The inverse relationship between addition and subtraction is a case in point. Problems of the form x + y − z = ? are intractable for those who lack knowledge of specific arithmetic facts (e.g., what is x + y?), and they require two successive calculations for those who possess the relevant knowledge. In contrast, problems of the form x + y − y = ? can immediately be solved, without arithmetic fact knowledge or calculation, by anyone who understands the logical relationship between addition and subtraction. The present research explores the origins of this understanding in children on the threshold of formal instruction in arithmetic.Previous research suggests that children's understanding of this relationship develops over many years of instruction in elementary mathematics. Children who have received arithmetic instruction perform more accurately on inverse problems of the form x + y − y than on matched problems of the form x + y − z (e.g. Bisanz & LeFevre, 1990;Bryant, Christie & Rendu, 1999;Gilmore & Bryant, 2006, Gilmore 2006Rasmussen, Ho & Bisanz, 2003;Siegler & Stern, 1998;Stern 1992), but they appear to learn about this principle in a piecemeal fashion. For example, children may recognize that subtracting 4 cancels the operation of adding 4, but they fail to recognize inversion as a general principle that can be applied to all numbers (Bisanz & LeFevre, 1990). Furthermore, these studies all involved children who were already receiving formal instruction in arithmetic, and thus the roots of this understanding are unclear. Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that ...

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Section: Discussionsupporting

confidence: 88%

Section: Author Manuscriptmentioning

confidence: 99%

mentioning

confidence: 99%

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Children’s understanding of the relationship between addition and subtraction

Gilmore

Spelke

2008

Cognition

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“…As shown in Figure 3, performance on these tasks was much less accurate than the numerical magnitude comparison tasks (Figure 2), which is consistent with previous research (Barth et al, 2006;Gilmore et al, 2007).…”

Section: Approximate Additionsupporting

confidence: 91%

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

Smedt

Gilmore

2011

Journal of Experimental Child Psychology

253

185

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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. Twenty children with MLD, twenty-one children with LA and forty-one regular achievers completed a numerical magnitude comparison task and an approximate addition task, which were presented in a symbolic and a non-symbolic (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 on and generalizes beyond numerical magnitude comparison tasks. These findings shed light on the types of interventions that may help children who struggle with learning mathematics.

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“…The standard deviation sd n 5 ASSESSING THE APPROXIMATE NUMBER SYSTEM for a given numerosity n should thus be proportional to n, multiplied by w: sd n = w × n, which is equivalent to w = sd n / n, that is, the definition of CV. Then, when assessed on the basis of the psychophysical model (Pica et al, 2004;Barth, La Mont, Lipton, Dehaene, Kanwisher, & Spelke, 2006), w and CV evaluate the exact same characteristic, based on the scalar variability of mental number representations. Actually, measures of w and CV in Western adult samples converge on similar figures, around 0.15 (w: Inglis & Gilmore, 2013, experiment 1;Lyons & Beilock, 2011;Pica et al, 2004;CV: Castronovo & Göbel, 2012, experiments 3 & 4;Frank, Fedorenko, Lai, Saxe, & Gibson, 2012;.…”

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confidence: 99%

Assessing the Approximate Number System: no relation between numerical comparison and estimation tasks

Guillaumé

Gevers

2015

Psychological Research

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Whether our general numerical skills and the mathematical knowledge that we acquire at school are entwined is a debated issue, which many researchers are still striving to investigate.The findings reported in the literature are actually inconsistent; some studies emphasised the existence of a relationship between the acuity of the Approximate Number System (ANS) and arithmetic competence, while some others did not observe any significant correlation. One potential explanation of the discrepancy might stem from the evaluation of the ANS itself. In the present study, we correlated two measures used to index ANS acuity with arithmetic performance. These measures were the Weber fraction (w), computed from a numerical comparison task and the Coefficient of Variation (CV), computed from a numerical estimation task. Arithmetic performance correlated with estimation CV but not with comparison w. We further investigated the meaning of this result by taking the relationship between w and CV into account. We expected a tight relation as both these measures are believed to assess ANS acuity. Crucially however, w and CV did not correlate with each other. Moreover, the value of w was modulated by the congruity of the relation between numerical magnitude and nonnumerical visual cues, potentially accounting for the lack of correlation between the measures. Our findings thus challenge the overuse of w to assess ANS acuity and more generally put into question the relevance of correlating this measure with arithmetic without any deeper understanding of what they are really indexing.Abstract word count: 243 words.

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Non-symbolic arithmetic in adults and young children

Cited by 331 publications

References 35 publications

Children’s understanding of the relationship between addition and subtraction

Children’s understanding of the relationship between addition and subtraction

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

Assessing the Approximate Number System: no relation between numerical comparison and estimation tasks

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