Number Processing and Calculation in Brazilian Children Aged 7-12 Years

Numerical cognition is based on two components - number processing and calculation. Its development is influenced by biological, cognitive, educational, and cultural factors. The objectives of the present study were to: i) assess number processing and calculation in Brazilian children aged 7-12 years from public schools using the Zareki-R (Battery of neuropsychological tests for number processing and calculation in children, Revised; von Aster & Dellatolas, 2006) in order to obtain normative data for Portuguese speakers; ii) identify how environment, age, and gender influences the development of these mathematical skills; iii) investigate the construct validity of the Zareki-R by the contrast with the Arithmetic subtest of WISC-III. The sample included 172 children, both genders, divided in two groups: urban (N= 119) and rural (N= 53) assessed by the Zareki-R. Rural children presented lower scores in one aspect of number processing; children aged 7-8 years demonstrated an inferior global score than older; boys presented a superior performance in both number processing and calculation. Construct validity of Zareki-R was demonstrated by high to moderate correlations with Arithmetic subtest of WISC-III. The Zareki-R therefore is a suitable instrument to assess the development of mathematical skills, which is influenced by factors such as environment, age, and gender.

Self-motion perception influences number processing: evidence from a parity task

We investigated the role of horizontal body motion on the processing of numbers. We hypothesized that leftward self-motion leads to shifts in spatial attention and therefore facilitates the processing of small numbers, and vice versa, we expected that rightward self-motion facilitates the processing of large numbers. Participants were displaced by means of a motion platform during a parity judgment task. We found a systematic influence of self-motion direction on number processing, suggesting that the processing of numbers is intertwined with the processing of self-motion perception. The results differed from known spatial numerical compatibility effects in that self-motion exerted a differential influence on inner and outer numbers of the given interval. The results highlight the involvement of sensory body motion information in higher-order spatial cognition.

Discalculia do desenvolvimento: avaliação da representação numérica pela ZAREKI-R

O presente estudo investigou aspectos da representação numérica (processamento numérico e cálculo) e memória operacional de crianças com transtornos de aprendizagem. Participaram 30 crianças de idade entre 9 e 10 anos, ambos os gêneros, divididas em dois grupos: sem dificuldade em aritmética (SDA; N=11) e com dificuldade em aritmética (CDA; N=19), avaliadas pela ZAREKI-R, Matrizes Coloridas de Raven, o Blocos de Corsi e o BCPR. Crianças CDA exibiram escores levemente mais baixos que as SDA quanto ao nível intelectual e nos Blocos de Corsi. Na ZAREKI-R apresentaram prejuízo nos subtestes ditado de números, cálculo mental, problemas aritméticos e total. Crianças CDA apresentaram déficits específicos em memória operacional visuoespacial e comprometimento em processamento numérico e cálculo, compatível com discalculia do desenvolvimento.

O efeito do treino musical sobre a capacidade da memória operacional e da cognição numérica de crianças com discalculia do desenvolvimento

Pós-graduação em Psicologia do Desenvolvimento e Aprendizagem - FC

Development of numerical cognition among Brazilian school-aged children

The Numerical Cognition is influenced by biological, cognitive, educational, and cultural factors and entails the following systems: Number Sense (NS) represents the innate ability to recognize, compare, add, and subtract small quantities, without the need of counting; Number Production (NP) which includes reading, writing and counting numbers or objects; Number Comprehension (NC), i.e., the understanding the nature of the numerical symbols and their number, and the calculation (CA). The aims of the present study were to: i) assess theoretical constructs (NS, NC, NP and CA) in children from public schools from 1 st -to 6 th - grades; and ii) investigate their relationship with schooling and working memory. The sample included 162 children, both genders, of 7-to 12-years-old that studied in public school from 1 st -to 6 th -grades, which participated in the normative study of Zareki-R (Battery of neuropsychological tests for number processing and calculation in children, Revised; von Aster & Dellatolas, 2006). Children of 1 st and 2 nd grades demonstrated an inferior global score in NC, NP and CA. There were no genderrelated differences. The results indicated that the contribution of NS domain in Zareki-R performance is low in comparison to the other three domains, which are dependent on school-related arithmetic skills.

Cognição Numérica de crianças pré-escolares brasileiras pela ZAREKI-K

The Numerical Cognition is infl uenced by biological, cognitive, educational and cultural factors. It consists of a primary system, called Number Sense that would be innate and universal, also of secondary systems: the Calculation, implied to perform mathematical operations by means of symbols or words and Number Processing, which is divided into two components, Number Comprehension, related with the understanding of numerical symbols and Number Production, which includes reading, writing and coun-ting numbers. However, studies that show the development of these functions in children of preschool age are scarce. Therefore, aims of this study were to investigate numerical cognition in preschool Brazilian children to demonstrate the construct validity of the ZAREKI-K (A Neuropsychological Battery for the Assessment of Treatment of Numbers and Calculation for preschool children). The participants were 42 children of both genders, who attended public elementary schools; the children were evaluated by this battery and WISC-III. The results indicated signifi cant differences associated with age which children of 6 years had better scores on subtests related to Number Production, Calculation and Number Comprehension, as well moderate and high correlations between some subtests of both instruments, demonstrating the construct validity of the battery. In conclusion, preliminary normative data were obtained for ZAREKI-K. The analyses suggested that it is a promising tool for the assessment of numerical cognition in preschool children.Keywords: Mathematics, number, preschoolers, working memory, Developmental Dyscalculia.

Componenti spaziali della rappresentazione cognitiva della grandezza del numero

The humans process the numbers in a similar way to animals. There are countless studies in which similar performance between animals and humans (adults and/or children) are reported. Three models have been developed to explain the cognitive mechanisms underlying the number processing. The triple-code model (Dehaene, 1992) posits an mental number line as preferred way to represent magnitude. The mental number line has three particular effects: the distance, the magnitude and the SNARC effects. The SNARC effect shows a spatial association between number and space representations. In other words, the small numbers are related to left space while large numbers are related to right space. Recently a vertical SNARC effect has been found (Ito & Hatta, 2004; Schwarz & Keus, 2004), reflecting a space-related bottom-to-up representation of numbers. The magnitude representations horizontally and vertically could influence the subject performance in explicit and implicit digit tasks. The goal of this research project aimed to investigate the spatial components of number representation using different experimental designs and tasks. The experiment 1 focused on horizontal and vertical number representations in a within- and between-subjects designs in a parity and magnitude comparative tasks, presenting positive or negative Arabic digits (1-9 without 5). The experiment 1A replied the SNARC and distance effects in both spatial arrangements. The experiment 1B showed an horizontal reversed SNARC effect in both tasks while a vertical reversed SNARC effect was found only in comparative task. In the experiment 1C two groups of subjects performed both tasks in two different instruction-responding hand assignments with positive numbers. The results did not show any significant differences between two assignments, even if the vertical number line seemed to be more flexible respect to horizontal one. On the whole the experiment 1 seemed to demonstrate a contextual (i.e. task set) influences of the nature of the SNARC effect. The experiment 2 focused on the effect of horizontal and vertical number representations on spatial biases in a paper-and-pencil bisecting tasks. In the experiment 2A the participants were requested to bisect physical and number (2 or 9) lines horizontally and vertically. The findings demonstrated that digit 9 strings tended to generate a more rightward bias comparing with digit 2 strings horizontally. However in vertical condition the digit 2 strings generated a more upperward bias respect to digit 9 strings, suggesting a top-to-bottom number line. In the experiment 2B the participants were asked to bisect lines flanked by numbers (i.e. 1 or 7) in four spatial arrangements: horizontal, vertical, right-diagonal and left-diagonal lines. Four number conditions were created according to congruent or incongruent number line representation: 1-1, 1-7, 7-1 and 7-7. The main results showed a more reliable rightward bias in horizontal congruent condition (1-7) respect to incongruent condition (7-1). Vertically the incongruent condition (1-7) determined a significant bias towards bottom side of line respect to congruent condition (7-1). The experiment 2 suggested a more rigid horizontal number line while in vertical condition the number representation could be more flexible. In the experiment 3 we adopted the materials of experiment 2B in order to find a number line effect on temporal (motor) performance. The participants were presented horizontal, vertical, rightdiagonal and left-diagonal lines flanked by the same digits (i.e. 1-1 or 7-7) or by different digits (i.e. 1-7 or 7-1). The digits were spatially congruent or incongruent with their respective hypothesized mental representations. Participants were instructed to touch the lines either close to the large digit, or close to the small digit, or to bisected the lines. Number processing influenced movement execution more than movement planning. Number congruency influenced spatial biases mostly along the horizontal but also along the vertical dimension. These results support a two-dimensional magnitude representation. Finally, the experiment 4 addressed the visuo-spatial manipulation of number representations for accessing and retrieval arithmetic facts. The participants were requested to perform a number-matching and an addition verification tasks. The findings showed an interference effect between sum-nodes and neutral-nodes only with an horizontal presentation of digit-cues, in number-matching tasks. In the addition verification task, the performance was similar for horizontal and vertical presentations of arithmetic problems. In conclusion the data seemed to show an automatic activation of horizontal number line also used to retrieval arithmetic facts. The horizontal number line seemed to be more rigid and the preferred way to order number from left-to-right. A possible explanation could be the left-to-right direction for reading and writing. The vertical number line seemed to be more flexible and more dependent from the tasks, reflecting perhaps several example in the environment representing numbers either from bottom-to-top or from top-to-bottom. However the bottom-to-top number line seemed to be activated by explicit task demands.

Counting is a spatial process: evidence from eye movements

Spatial-numerical associations (small numbers-left/lower space and large numbers-right/upper space) are regularly found in simple number categorization tasks. These associations were taken as evidence for a spatially oriented mental number line. However, the role of spatial-numerical associations during more complex number processing, such as counting or mental arithmetic is less clear. Here, we investigated whether counting is associated with a movement along the mental number line. Participants counted aloud upward or downward in steps of 3 for 45 s while looking at a blank screen. Gaze position during upward counting shifted rightward and upward, while the pattern for downward counting was less clear. Our results, therefore, confirm the hypothesis of a movement along the mental number line for addition. We conclude that space is not only used to represent number magnitudes but also to actively operate on numbers in more complex tasks such as counting, and that the eyes reflect this spatial mental operation.

Mathematical skills: intergenerational features and relationships with cognitive and linguistic abilities

This thesis aimed to investigate the cognitive underpinnings of math skills, with particular reference to cognitive, and linguistic markers, core mechanisms of number processing and environmental variables. In particular, the issue of intergenerational transmission of math skills has been deepened, comparing parents’ and children’s basic and formal math abilities. This pattern of relationships amongst these has been considered in two different age ranges, preschool and primary school children. In the first chapter, a general introduction on mathematical skills is offered, with a description of some seminal works up to recent studies and latest findings. The first chapter concludes with a review of studies about the influence of environmental variables. In particular, a review of studies about home numeracy and intergenerational transmission is examined. The first study analyzed the relationship between mathematical skills of children attending primary school and those of their mothers. The objective of this study was to understand the influence of mothers' math abilities on those of their children. In the second study, the relationship between parents’ and children numerical processing has been examined in a sample of preschool children. The goal was to understand how mathematical skills of parents were relevant for the development of the numerical skills of children, taking into account children’s cognitive and linguistic skills as well as the role of home numeracy. The third study had the objective of investigating whether the verbal and nonverbal cognitive skills presumed to underlie arithmetic are also related to reading. Primary school children were administered measures of reading and arithmetic to understand the relationships between these two abilities and testing for possible shared cognitive markers. Finally, in the general discussion a summary of main findings across the study is presented, together with clinical and theoretical implications.

Random number generation as an index of controlled processing

Random number generation (RNG) is a functionally complex process that is highly controlled and therefore dependent on Baddeley's central executive. This study addresses this issue by investigating whether key predictions from this framework are compatible with empirical data. In Experiment 1, the effect of increasing task demands by increasing the rate of the paced generation was comprehensively examined. As expected, faster rates affected performance negatively because central resources were increasingly depleted. Next, the effects of participants' exposure were manipulated in Experiment 2 by providing increasing amounts of practice on the task. There was no improvement over 10 practice trials, suggesting that the high level of strategic control required by the task was constant and not amenable to any automatization gain with repeated exposure. Together, the results demonstrate that RNG performance is a highly controlled and demanding process sensitive to additional demands on central resources (Experiment 1) and is unaffected by repeated performance or practice (Experiment 2). These features render the easily administered RNG task an ideal and robust index of executive function that is highly suitable for repeated clinical use.

The processing of number and gender agreement in Spanish: an event related potential investigation of the effects of structural distance

Previous research suggests that the processing of agreement is affected by the distance between the agreeing elements. However, the unique contribution of structural distance (number of intervening syntactic phrases) to the processing of agreement remains an open question, since previous investigations do not tease apart structural and linear distance (number of intervening words). We used event related potentials (ERPs) to examine the extent to which structural distance impacts the processing of Spanish number and gender agreement. Violations were realized both within the phrase and across the phrase. Across both levels of structural distance, linear distance was kept constant, as was the syntactic category of the agreeing elements. Number and gender agreement violations elicited a robust P600 between 400 and 900ms, a component associated with morphosyntactic processing. No amplitude differences were observed between number and gender violations, suggesting that the two features are processed similarly at the brain level. Within-phrase agreement yielded more positive waveforms than across-phrase agreement, both for agreement violations and for grammatical sentences (no agreement by distance interaction). These effects can be interpreted as evidence that structural distance impacts the establishment of agreement overall, consistent with sentence processing models which predict that hierarchical structure impacts the processing of syntactic dependencies. However, due to the lack of an agreement by distance interaction, the possibility cannot be ruled out that these effects are driven by differences in syntactic predictability between the within-phrase and across-phrase configurations, notably the fact that the syntactic category of the critical word was more predictable in the within-phrase conditions.

Examining second language development using event-related potentials: a cross-sectional study on the processing of gender and number agreement

This cross-sectional study examines the role of L1-L2 differences and structural distance in the processing of gender and number agreement by English-speaking learners of Spanish at three different levels of proficiency. Preliminary results show that differences between the L1 and L2 impact L2 development, as sensitivity to gender agreement violations, as opposed to number agreement violations, emerges only in learners at advanced levels of proficiency. Results also show that the establishment of agreement dependencies is impacted by the structural distance between the agreeing elements for native speakers and for learners at intermediate and advanced levels of proficiency but not for low proficiency. The overall pattern of results suggests that the linguistic factors examined here impact development but do not constrain ultimate attainment; for advanced learners, results suggest that second language processing is qualitatively similar to native processing.

The impact of number mismatch and passives on the real-time processing of relative clauses

Language processing plays a crucial role in language development, providing the ability to assign structural representations to input strings (e.g., Fodor, 1998). In this paper we aim at contributing to the study of children's processing routines, examining the operations underlying the auditory processing of relative clauses in children compared to adults. English-speaking children (6–8;11) and adults participated in the study, which employed a self-paced listening task with a final comprehension question. The aim was to determine (i) the role of number agreement in object relative clauses in which the subject and object NPs differ in terms of number properties, and (ii) the role of verb morphology (active vs. passive) in subject relative clauses. Even though children's off-line accuracy was not always comparable to that of adults, analyses of reaction times results support the view that children have the same structural processing reflexes observed in adults.