Judgement of discrete and continuous quantity in adults: Number counts!

Three experiments involving a Stroop-like paradigm were conducted. In Experiment 1, adults received a number comparison task in which large sets of dots, orthogonally varying along a discrete dimension (number of dots) and a continuous dimension (cumulative area), were presented. Incongruent trials were processed more slowly and with less accuracy than congruent trials, suggesting that continuous dimensions such as cumulative area are automatically processed and integrated during a discrete quantity judgement task. Experiment 2, in which adults were asked to perform area comparison on the same stimuli, revealed the reciprocal interference from number on the continuous quantity judgements. Experiment 3, in which participants received both the number and area comparison tasks, confirmed the results of Experiments 1 and 2. Contrasting with earlier statements, the results support the view that number acts as a more salient cue than continuous dimensions in adults. Furthermore, the individual predisposition to automatically access approximate number representations was found to correlate significantly with adults' exact arithmetical skills.     

Operator and operand preview effects in simple addition and multiplication: A comparison of Canadian and Chinese adults

Using an arithmetic-based retrieval-induced forgetting (RIF) paradigm, researchers have found evidence that participants with very high arithmetic proficiency (Chinese adults), but not less-skilled participants (Canadian adults), solved some simple additions (e.g. 3 + 2) using fast procedural skills. Here we sought converging evidence for this using the operator-priming paradigm. Previous research testing simple addition and multiplication found that a 150-ms preview of the operator (+ or ×) facilitated only addition performance. This was taken as evidence that addition, but not multiplication, was solved by procedural algorithms that could be primed by presentation of the plus sign. In the present study, Chinese and Canadian adults (N = 144) were tested in the operator-priming paradigm but, in contrast to the RIF results, there was little evidence that operator-priming effects differed between the groups and robust operator priming was observed in both addition and multiplication. Thus, the operator preview results did not reinforce the results of previous research but the experiment revealed robust group differences in operand preview effects: For the Chinese, but not the Canadians, a preview of the numerical operands produced much greater facilitation for multiplication than addition. The fact that CN obtained a mean 103-ms gain for multiplication from the 150-ms preview of the operands strongly suggests that multiplication was their default operation in this paradigm. This result adds a potentially important new phenomenon to the behavioural distinctions between Chinese and North American adults' arithmetic skills.     

Children's Understanding of the Natural Numbers’ Structure

When young children attempt to locate numbers along a number line, they show logarithmic (or other compressive) placement. For example, the distance between “5” and “10” is larger than the distance between “75” and “80.” This has often been explained by assuming that children have a logarithmically scaled mental representation of number (e.g., Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, 2010 ; Siegler & Opfer, 2003 ). However, several investigators have questioned this argument (e.g., Barth & Paladino, 2011 ; Cantlon, Cordes, Libertus, & Brannon, 2009 ; Cohen & Blanc‐Goldhammer, 2011 ). We show here that children prefer linear number lines over logarithmic lines when they do not have to deal with the meanings of individual numerals (i.e., number symbols, such as “5” or “80”). In Experiments 1 and 2, when 5‐ and 6‐year‐olds choose between number lines in a forced‐choice task, they prefer linear to logarithmic and exponential displays. However, this preference does not persist when Experiment 3 presents the same lines without reference to numbers, and children simply choose which line they like best. In Experiments 4 and 5, children position beads on a number line to indicate how the integers 1–100 are arranged. The bead placement of 4‐ and 5‐year‐olds is better fit by a linear than by a logarithmic model. We argue that previous results from the number‐line task may depend on strategies specific to the task.     

Children's representation of symbolic and nonsymbolic magnitude examined with the priming paradigm

How people process and represent magnitude has often been studied using number comparison tasks. From the results of these tasks, a comparison distance effect (CDE) is generated, showing that it is easier to discriminate two numbers that are numerically further apart (e.g., 2 and 8) compared with numerically closer numbers (e.g., 6 and 8). However, it has been suggested that the CDE reflects decisional processes rather than magnitude representation. In this study, therefore, we investigated the development of symbolic and nonsymbolic number processes in kindergartners and first, second, and sixth graders using the priming paradigm. This task has been shown to measure magnitude and not decisional processes. Our findings revealed that a priming distance effect (PDE) is already present in kindergartners and that it remains stable across development. This suggests that formal schooling does not affect magnitude representation. No differences were found between the symbolic and nonsymbolic PDE, indicating that both notations are processed with comparable precision. Finally, a poorer performance on a standardized mathematics test seemed to be associated with a smaller PDE for both notations, possibly suggesting that children with lower mathematics scores have a less precise coding of magnitude. This supports the defective number module hypothesis, which assumes an impairment of number sense.     

Number versus extent in newborns’ spontaneous preference for collections of dots

This study investigated processing of number and extent in newborns. Using visual preference, we showed that newborns discriminated between small sets of dot collections relying solely on implicit numerical information when non-numerical continuous variables were strictly controlled (Experiment 1), and solely on continuous information when numerical variables were controlled (Experiment 2). When number and extent were pitted against each other (Experiment 3), newborns showed no visual preference, suggesting that the two variables play comparable roles in attracting newborns’ visual attention. In contrast to reports of dominance of continuous variables, these findings suggest that multiple dimensions attract newborns’ attention and guide their visual exploration.     

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like , is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self‐reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task‐specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.     

情绪加工的性别差异及神经机制

已有研究表明, 情绪加工存在显著性别差异。这主要表现为女性人群具有情绪识别优势, 更好的情绪记忆能力与更强的负性情绪易感性。此外, 情绪加工的性别差异也表现为情绪调节过程的不同：相比男性, 女性更善于抑制情绪行为, 却较难通过认知策略调节负性情绪。情绪加工的性别差异有重要生理基础, 与情绪脑结构, 荷尔蒙水平的男女差异有关。同时, 情绪加工的性别差异同女性更高的情绪障碍易感性有着密切联系：不善于调节负性情绪, 及更强的消极情绪易感性可能是女性更易患情绪障碍的重要原因。从情绪加工角度探讨情绪障碍的性别差异及其原因, 对于情绪障碍的预防和治疗具有重要意义。     

Beyond quantity: Individual differences in working memory and the ordinal understanding of numerical symbols

In two different contexts, we examined the hypothesis that individual differences in working memory (WM) capacity are related to the tendency to infer complex, ordinal relationships between numerical symbols. In Experiment 1, we assessed whether this tendency arises in a learning context that involves mapping novel symbols to quantities by training adult participants to associate dot-quantities with novel symbols, the overall relative order of which had to be inferred. Performance was best for participants who were higher in WM capacity (HWMs). HWMs also learned ordinal information about the symbols that lower WM individuals (LWMs) did not. In Experiment 2, we examined whether WM relates to performance when participants are explicitly instructed to make numerical order judgments about highly enculturated numerical symbols by having participants indicate whether sets of three Arabic numerals were in increasing order. All participants responded faster when sequential sets (3-4-5) were in order than when they were not. However, only HWMs responded faster when non-sequential, patterned sets (1-3-5) were in order, suggesting they were accessing ordinal associations that LWMs were not. Taken together, these experiments indicate that WM capacity plays a key role in extending symbolic number representations beyond their quantity referents to include symbol-symbol ordinal associations, both in a learning context and in terms of explicitly accessing ordinal relationships in highly enculturated stimuli.     

语言加工性别差异的神经机制

神经成像技术为语言加工的性别差异研究提供了新的手段和证据。研究者们从语言加工的词汇、句子、篇章理解和语言学习、再认和记忆等层面进行了大量的男女性脑功能差异研究。同时, 语言加工的性别差异也从脑结构–生理基础差异方面进行了探讨, 主要包括了灰质分布差异、胼胝体大小差异和荷尔蒙的影响。然而, 语言加工是否有性别差异还存在争议, 不同的研究者从加工时间要求、陈述性记忆差异和方法学考虑等方面对分歧做出了解释。     

Processing style and person recognition: Exploring the face inversion effect

It has frequently been reported that recognition performance is impaired when faces are presented in an inverted rather than upright orientation, a phenomenon termed the face inversion effect (FIE). Extending previous work on this topic, the current investigation explored whether individual differences in global precedence—the propensity to process nonfacial stimuli in a configural manner—impacts memory for faces. Based on performance on the Navon letter-classification task, two experimental groups were created that differed in relative global precedence (i.e., strong global precedence [SGP] and weak global precedence [WGP]). In a subsequent face-recognition task, results revealed that while both groups demonstrated a reliable FIE, this effect was attenuated among participants displaying WGP. These findings suggest that individual differences in general processing style modulate face recognition.     

Flexible and unique representations of two-digit decimals

We examined the representation of two-digit decimals through studying distance and compatibility effects in magnitude comparison tasks in four experiments. Using number pairs with different leftmost digits, we found both the second digit distance effect and compatibility effect with two-digit integers but only the second digit distance effect with two-digit pure decimals. This suggests that both integers and pure decimals are processed in a compositional manner. In contrast, neither the second digit distance effect nor the compatibility effect was observed in two-digit mixed decimals, thereby showing no evidence for compositional processing of two-digit mixed decimals. However, when the relevance of the rightmost digit processing was increased by adding some decimals pairs with the same leftmost digits, both pure and mixed decimals produced the compatibility effect. Overall, results suggest that the processing of decimals is flexible and depends on the relevance of unique digit positions. This processing mode is different from integer analysis in that two-digit mixed decimals demonstrate parallel compositional processing only when the rightmost digit is relevant. Findings suggest that people probably do not represent decimals by simply ignoring the decimal point and converting them to natural numbers.     

Low working memory capacity impedes both efficiency and learning of number transcoding in children

This study aimed to evaluate the impact of individual differences in working memory capacity on number transcoding. A recently proposed model, ADAPT (a developmental asemantic procedural transcoding model), accounts for the development of number transcoding from verbal form to Arabic form by two mechanisms: the learning of new production rules that enlarge the range of numbers a child can transcode and the increase of the mental lexicon. The working memory capacity of 7-year-olds was evaluated along with their ability to transcode one- to four-digit numbers. As ADAPT predicts, the rate of transcoding errors increased when more production rules were required and when children had low working memory capacity, with these two factors interacting. Moreover, qualitative analysis of the errors produced by high- and low-span children showed that the latter have a developmental delay in the acquisition of the production rules.     

Extrafoveal Processing in Categorical Search for Geometric Shapes: General Tendencies and Individual Variations

The paper addresses the capabilities and limitations of extrafoveal processing during a categorical visual search. Previous research has established that a target could be identified from the very first or without any saccade, suggesting that extrafoveal perception is necessarily involved. However, the limits in complexity defining the processed information are still not clear. We performed four experiments with a gradual increase of stimuli complexity to determine the role of extrafoveal processing in searching for the categorically defined geometric shape. The series of experiments demonstrated a significant role of extrafoveal processing while searching for simple two-dimensional shapes and its gradual decrease in a condition with more complicated three-dimensional shapes. The factors of objects’ spatial orientation and distractor homogeneity significantly influenced both reaction time and the number of saccades required to identify a categorically defined target. An analysis of the individual p-value distributions revealed pronounced individual differences in using extrafoveal analysis and allowed examination of the performance of each particular participant. The condition with the forced prohibition of eye movements enabled us to investigate the efficacy of covert attention in the condition with complicated shapes. Our results indicate that both foveal and extrafoveal processing are simultaneously involved during a categorical search, and the specificity of their interaction is determined by the spatial orientation of objects, type of distractors, the prohibition to use overt attention, and individual characteristics of the participants.     

The predictive value of numerical magnitude comparison for individual differences in mathematics achievement

Although it has been proposed that the ability to compare numerical magnitudes is related to mathematics achievement, it is not clear whether this ability predicts individual differences in later mathematics achievement. The current study addressed this question in typically developing children by means of a longitudinal design that examined the relationship between a number comparison task assessed at the start of formal schooling (mean age = 6 years 4 months) and a general mathematics achievement test administered 1 year later. Our findings provide longitudinal evidence that the size of the individual’s distance effect, calculated on the basis of reaction times, was predictively related to mathematics achievement. Regression analyses showed that this association was independent of age, intellectual ability, and speed of number identification.     

Changes in the prevalence of alpha activity associated with the repetition,performance and magnitude of arithmetical calculations

Summary Changes in prevalence of alpha activity (defined as from 8 to 13 c.p.s.) immediately before and during the solution of twenty arithmetical tasks (multiplications and additions in Hard and Easy categories) by 61 normal subjects, with closed eyes, have been measured above a preset threshold voltage over noise level by a percent time alpha computer. An average prevalence of alpha activity of 64.75% before calculation reduced to an average prevalence of 49.01% during calculations. The prevalence of alpha activity during calculation was significantly increased by repetition of similar arithmetical tasks whereas the prevalence of alpha activity before calculation declined from the beginning to the end of the experiment, thus showing an opposite trend. Prevalence of alpha activity was not related to duration of calculations, but, in contrast, low prevalences of alpha activity during calculation were significantly associated with more numerous errors, particularly in additions. It was found that as the information content of a task increased so the prevalence of alpha activity during its solution declined, especially in additions and Easy multiplications. Calculation times and errors were strongly correlated with information content. These results are evaluated in terms of their possible behavioural significance and of the neurophysiological mechanisms underlying attenuation of alpha activity, although the precise theoretical context in which they should be viewed; whether they should be related to habituation, learning, fatigue or solely calculation, may require further clarification.

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Zusammenfassung Veränderungen in der Dominanz (Prävalenz) der Alpha-Aktivität (als 8–13 c.p.s. definiert) wurden kurz vor und während der Lösung von 20 arithmetischen Aufgaben (Multiplikation und Addition in Leichten und Schweren Serien) bei 61 normalen Personen, welche die Augen geschlossen hielten, studiert. Die Messung wurde bei einer festgesetzten (preset) Schwellenspannung über dem Geräuschniveau mit einem Alpha-Zeit-Computer ausgeführt. Dabei ergab sich eine Reduzierung der Alpha-Aktivität von 64,75% vor dem Rechnen auf 49,01% während des Rechnens. Die Alpha-Dominanz während des Rechnens wurde statistisch durch Wiederholung von ähnlichen Aufgaben erhöht; im Gegensatz dazu ging die Alpha-Dominanz vor dem Rechnen progressiv vom Anfang bis zum Ende des Experiments zurück.Die Alpha-Dominanz hatte keine Beziehung zur Dauer des Rechnens; andererseits waren niedrige Frequenzen von Alpha-Aktivität während des Rechnens mit Häufigkeit von Fehlern, vor allem beim Addieren, verbunden. Wenn sich der Informations-inhalt einer Aufgabe erhöhte, so verminderte sich die Alpha-Dominanz während ihrer Lösung, vor allem beim Addieren und bei leichten Multiplikationen. Sowohl Rechen-Zeit wie Fehler waren stark mit dem Informations-inhalt korreliert.Diese Ergebnisse werden hinsichtlich ihrer möglichen Beziehungen zum psychischen Verhaltens-Zustand (Behaviour) und zu den mit Abschwächung der Alpha-Aktivität verbundenen neuro-physiologischen Prozessen analysiert. Eine Klärung des präzisen theoretischen Zusammenhanges, in dem die Resultate gesehen werden sollten, sowie ihre Beziehungen zu Gewöhnung, Lernen, Ermüdung oder zum Rechnen allein, erfordert weiteres Studium.

     

Children’s representation of symbolic magnitude: The development of the priming distance effect

The comparison distance effect (CDE), whereby discriminating between two numbers that are far apart is easier than discriminating between two numbers that are close, has been considered as an important indicator of how people represent magnitudes internally. However, the underlying mechanism of this CDE is still unclear. We tried to shed further light on how people represent magnitudes by using priming. Adults have been shown to exhibit a priming distance effect (PDE), whereby numbers are processed faster when they are preceded by a close number than when they are preceded by a more distant number. Surprisingly, there are no studies available that have investigated this effect in children. The current study examined this effect in typically developing first, third, and fifth graders and in adults. Our findings revealed that the PDE already occurs in first graders and remains stable across development. This study also documents the usefulness of number priming in children, making it an interesting tool for future research.     

Number bias for the discrimination of large visual sets in infancy

This brief report attempts to resolve the claim that infants preferentially attend to continuous variables over number [e.g. Psychol. Sci. 10 (1999) 408; Cognit. Psychol.44 (2002) 33] with the finding that when continuous variables are controlled, infants as young as 6-months of age discriminate large numerical values [e.g. Psychol. Sci. 14 (2003) 396; Cognition 89 (2003) B15; Cognition 74 (2000) B1]. In two parallel experiments, we compare 6-month-old infants' ability to discriminate number and ignore continuous variables with their ability to form a representation of a cumulative surface area and ignore number. We find that infants discriminate a 2-fold change in number but fail to discriminate a 2-fold change in cumulative surface area. The results point to a more complicated relationship between discrete and continuous dimensions than implied by previous literature.     

First Language Attrition Induces Changes in Online Morphosyntactic Processing and Re‐Analysis: An ERP Study of Number Agreement in Complex Italian Sentences

First language (L1) attrition in adulthood offers new insight on neuroplasticity and the role of language experience in shaping neurocognitive responses to language. Attriters are multilinguals for whom advancing L2 proficiency comes at the cost of the L1, as they experience a shift in exposure and dominance (e.g., due to immigration). To date, the neurocognitive mechanisms underlying L1 attrition are largely unexplored. Using event‐related potentials (ERPs), we examined L1‐Italian grammatical processing in 24 attriters and 30 Italian native‐controls. We assessed whether (a) attriters differed from non‐attriting native speakers in their online detection and re‐analysis/repair of number agreement violations, and whether (b) differences in processing were modulated by L1‐proficiency. To test both local and non‐local agreement violations, we manipulated agreement between three inflected constituents and examined ERP responses on two of these (subject, verb , modifier ). Our findings revealed group differences in amplitude, scalp distribution, and duration of LAN/N400 + P600 effects. We discuss these differences as reflecting influence of attriters’ L2‐English, as well as shallower online sentence repair processes than in non‐attriting native speakers. ERP responses were also predicted by L1‐Italian proficiency scores, with smaller N400/P600 amplitudes in lower proficiency individuals. Proficiency only modulated P600 amplitude between 650 and 900 ms, whereas the late P600 (beyond 900 ms) depended on group membership and amount of L1 exposure within attriters. Our study is the first to show qualitative and quantitative differences in ERP responses in attriters compared to non‐attriting native speakers. Our results also emphasize that proficiency predicts language processing profiles, even in native‐speakers, and that the P600 should not be considered a monolithic component.     

The role of numerical and non-numerical cues in nonsymbolic number processing: Evidence from the line bisection task

In line bisection tasks, adults and children bisect towards the numerically larger of two nonsymbolic numerosities [de Hevia, M. D., & Spelke, E. S. (2009 de Hevia, M. D., & Spelke, E. S. (2009). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198–207. doi:10.1016/j.cognition.2008.11.003[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198–207. doi:10.1016/j.cognition.2008.11.003]. However, it is not clear whether this effect is driven by number itself or rather by visual cues such as subtended area [Gebuis, T., & Gevers, W. (2011 Gebuis, T., & Gevers, W. (2011). Numerosities and space: Indeed a cognitive illusion! A reply to de Hevia and Spelke (2009). Cognition, 121, 248–252. doi:10.1016/j.cognition.2010.09.008[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Numbers and space: Indeed a cognitive illusion! A reply to de Hevia and Spelke (2009 de Hevia, M. D., & Spelke, E. S. (2009). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198–207. doi:10.1016/j.cognition.2008.11.003[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Cognition, 121, 248–252. doi:10.1016/j.cognition.2010.09.008]. Furthermore, this effect has only been demonstrated with flanking displays of two and nine items. Here, we report three studies that examined whether this “spatial bias” effect occurs across a range of absolute and ratio numerosity differences; in particular, we examined whether the bias would occur when both flankers were outside the subitizing range. Additionally, we manipulated the subtended area of the stimulus and the aggregate surface area to assess the influence of visual cues. We found that the spatial bias effect occurred for a range of flanking numerosities and for ratios of 3:5 and 5:6 when subtended area was not controlled (Experiment 1). However, when subtended area and aggregate surface area were held constant, the biasing effect was reversed such that participants bisected towards the flanker with fewer items (Experiment 2). Moreover, when flankers were identical, participants bisected towards the flanker with larger subtended area or larger aggregate surface area (Experiments 2 and 3). On the basis of these studies, we conclude that the spatial bias effect for nonsymbolic numerosities is primarily driven by visual cues.     

Intellectual growth in children as a function of domain specific and domain general working memory subgroups

This study examined whether children's growth on measures of fluid (Raven Colored Progressive Matrices) and crystallized (reading and math achievement) intelligence was attributable to domain-specific or domain-general functions of working memory (WM). A sample of 290 elementary school children was tested on measures of intelligence across three testing waves. Two methods, a hierarchical factor model predicting latent growth and a double dissociation design comparing children divided into high and low performers, tested whether general and/or specific components accounted for performance on intelligence measures. A general domain model for the total sample provided the best fit for latent growth on all measures. Further, the lack of a significant WM subgroup × domain interaction suggested that a general WM system underlies the individual differences on measures of fluid and crystallized intelligence. Overall, the findings support a domain-general view of WM capacity on intelligence measures in children, but also suggest that domain specific storage systems may come into play on isolated measures.