Exact and Approximate Arithmetic in an Amazonian Indigene Group

Pica, Pierre; Lemer, Cathy; Izard, Véronique; Dehaene, Stanislas

doi:10.1126/science.1102085

Cited by 1,110 publications

(987 citation statements)

References 22 publications

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“…Such a reliance on the count list matches Carey (2009)'s suggestion that it provides a key component of bootstrapping, a placeholder structure, which guides the critical induction. Indeed, cultures without number words lack the capacity to encode representations of exact cardinalities (Frank, Everett, Fedorenko, & Gibson, 2008;Gordon, 2004;Pica, Lemer, Izard, & Dehaene, 2004) 19 , although this does not prevent them from exactly matching large cardinalities (Frank et al, 2008).…”

Section: The Count List May Play a Crucial Role The Cp-transitionmentioning

confidence: 99%

Bootstrapping in a language of thought: A formal model of numerical concept learning

Piantadosi¹,

Tenenbaum²,

2012

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In acquiring number words, children exhibit a qualitative leap in which they transition from understanding a few number words, to possessing a recursive system of interrelated numerical concepts. We present a computational framework for understanding this inductive leap as the consequence of statistical inference over a sufficiently powerful representational system. We provide an implemented model that is powerful enough to learn number word meanings and other related conceptual systems from naturalistic data. The model shows that bootstrapping can be made computationally and philosophically well-founded as a theory of number learning. Our approach demonstrates how learners may combine core cognitive operations to build sophisticated representations during the course of development, and how this process explains observed developmental patterns in number word learning.

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Section: The Count List May Play a Crucial Role The Cp-transitionmentioning

confidence: 99%

Bootstrapping in a language of thought: A formal model of numerical concept learning

Piantadosi¹,

Tenenbaum²,

2012

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“…For example, if the magnitude of the ratio is large, for example, four dots versus eight dots (1:2 ratio), responses tend to be fast and precise, which indicates that a large ratio makes the comparison easy. If the magnitude of the ratio is small, for example, 15 dots versus 16 dots (15:16 ratio), responses tend to be slower than in the easy ratio condition, and the accuracy is typically lower (Barth, et al., 2003; Cordes, Gelman, Gallistel, & Whalen, 2001; Pica, Lemer, Izard, & Dehaene, 2004), indicating harder comparison. Converging evidence from developmental and comparative studies as well as studies with people whose languages do not have number words shows ratio‐dependent performance on nonsymbolic number comparison tasks suggesting a key feature of the ANS: independence from language (Cantlon, Brannon, Carter, & Pelphrey, 2006; Izard, Sann, Spelke, & Streri, 2009; Libertus and Brannon, 2009; Lipton & Spelke, 2003; Nieder, 2009; Pica et al., 2004; Xu & Spelke, 2000).…”

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The integration between nonsymbolic and symbolic numbers: Evidence from an EEG study

et al. 2018

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IntroductionAdults can represent numerical information in nonsymbolic and symbolic formats and flexibly switch between the two. While some studies suggest a strong link between the two number representation systems (e.g., Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004 Neuron, 44(3), 547), other studies show evidence against the strong‐link hypothesis (e.g., Lyons, Ansari, & Beilock, 2012 Journal of Experimental Psychology: General, 141(4), 635). This inconsistency could arise from the relation between task demands and the closeness of the link between the two number systems.MethodsWe used a passive viewing task and event‐related potentials (ERP) to examine the temporal dynamics of the implicit integration between the nonsymbolic and symbolic systems. We focused on two ERP components over posterior scalp sites that were found to be sensitive to numerical distances and ratio differences in both numerical formats: a negative component that peaks around 170 ms poststimulus (N1) and a positive component that peaks around 200 ms poststimulus (P2p). We examined adults' (n = 55) ERPs when they were passively viewing simultaneously presented dot quantities and Arabic numerals (i.e., nonsymbolic and symbolic numerical information) in the double‐digit range. For each stimulus, the nonsymbolic and symbolic content either matched or mismatched in number. We also asked each participant to estimate dot quantities in a separate behavioral task and observed that they tended to underestimate the actual dot quantities, suggesting a need to adjust the match between nonsymbolic and symbolic information to reflect the perceived quantity of the nonsymbolic information.ResultsUsing this adjustment, participants showed greater N1 and P2p amplitudes when perceived dot quantities matched Arabic numerals than when there was a mismatch. However, no differences were found between the unadjusted match and mismatch conditions.ConclusionOur findings suggest that adults rapidly integrate nonsymbolic and symbolic formats of double‐digit numbers, but evidence of such integration is best observed when the perceived (rather than veridical) dot quantity is considered.

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“…However, contrary to the estimation task, the direct computation of the CV is not possible in such numerical comparison tasks; for this reason, most authors instead relied on the computation of another index of ANS acuity: the Weber fraction (w). The Weber fraction is generally considered to quantify the precision or discriminability of mental number representations (Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004;Pica, Lemer, Izard, & Dehaene, 2004).…”

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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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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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Exact and Approximate Arithmetic in an Amazonian Indigene Group

Cited by 1,110 publications

References 22 publications

Bootstrapping in a language of thought: A formal model of numerical concept learning

Bootstrapping in a language of thought: A formal model of numerical concept learning

The integration between nonsymbolic and symbolic numbers: Evidence from an EEG study

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

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