Vitor Haase

Mental calculation is thought to be tightly related to visuospatial abilities. One of the strongest evidence for this link is the widely replicated operational momentum (OM) effect: the tendency to overestimate the result of additions and to underestimate the result of subtractions. Although the OM effect has been found in both infants and adults, no study has directly investigated its developmental trajectory until now. However, to fully understand the cognitive mechanisms lying at the core of the OM effect it is important to investigate its developmental dynamics. In the present study, we investigated the development of the OM effect in a group of 162 children from 8 to 12 years old. Participants had to select among five response alternatives the correct result of approximate addition and subtraction problems. Response alternatives were simultaneously presented on the screen at different locations. While no effect was observed for the youngest age group, children aged 9 and older showed a clear OM effect. Interestingly, the OM effect monotonically increased with age. The increase of the OM effect was accompanied by an increase in overall accuracy. That is, while younger children made more and non-systematic errors, older children made less but systematic errors. This monotonous increase of the OM effect with age is not predicted by the compression account (i.e., linear calculation performed on a compressed code). The attentional shift account, however, provides a possible explanation of these results based on the functional relationship between visuospatial attention and mental calculation and on the influence of formal schooling. We propose that the acquisition of arithmetical skills could reinforce the systematic reliance on the spatial mental number line and attentional mechanisms that control the displacement along this metric. Our results provide a step in the understanding of the mechanisms underlying approximate calculation and an important empirical constraint for current accounts on the origin of the OM effect.

Math anxiety is a relatively frequent phenomenon often related to low mathematics achievement and dyscalculia. In the present study, the German and the Brazilian versions of the Mathematics Anxiety Questionnaire (MAQ) were examined. The two-dimensional structure originally reported for the German MAQ, that includes both affective and cognitive components of math anxiety was reproduced in the Brazilian version. Moreover, mathematics anxiety also was found to increase with age in both populations and was particularly associated with basic numeric competencies and more complex arithmetics. The present results suggest that mathematics anxiety as measured by the MAQ presents the same internal structure in culturally very different populations.

Introduction: Congenital amusia is a developmental disorder associated with deficits in pitch height discrimination or in integrating pitch sequences into melodies. This quasi-experimental pilot study investigated whether there is an association between pitch and numerical processing deficits in congenital amusia. Since pitch height discrimination is considered a form of magnitude processing, we investigated whether individuals with amusia present an impairment in numerical magnitude processing, which would reflect damage to a generalized magnitude system. Alternatively, we investigated whether the numerical processing deficit would reflect a disconnection between non-symbolic and symbolic number representations. Method: This study was conducted with 11 adult individuals with congenital amusia and a control comparison group of 6 typically developing individuals. Participants performed nonsymbolic and symbolic magnitude comparisons and number line tasks. Results were available from previous testing using the Montreal Battery of Evaluation of Amusia (MBEA) and a pitch change detection task (PCD). Results: Compared to the controls, individuals with amusia exhibited no significant differences in their performance on both the number line and the nonsymbolic magnitude tasks. Nevertheless, they showed significantly worse performance on the symbolic magnitude task. Moreover, individuals with congenital amusia, who presented worse performance in the Meter subtest, also presented less precise nonsymbolic numerical representation. Conclusions: The relationship between meter and nonsymbolic numerical discrimination could indicate a general ratio processing deficit. The finding of preserved nonsymbolic numerical magnitude discrimination and mental number line representations, with impaired symbolic number processing, in individuals with congenital amusia indicates that (a) pitch height and numerical magnitude processing may not share common neural representations, and (b) in addition to pitch processing, individuals with amusia may present a deficit in accessing nonsym-bolic numerical representations from symbolic representations. The symbolic access deficit could reflect a widespread impairment in the establishment of cortico-cortical connections between association areas.

Although progress has been made, the cognitive, biological and, particularly, the genetic underpinnings of math learning difficulties (MD) remain largely unknown. This difficulty stems from the heterogeneity of MD and from the large contribution of environmental factors to its etiology. Understanding endophenotypes, e.g., the role of the Approximate Number System (ANS), may help understanding the nature of MD. MD associated with ANS impairments has been described in some genetic conditions, e.g., 22q11.2 deletion syndrome (22q11.2DS or Velocardiofacial syndrome, VCFS). Recently, a girl with MD was identified in a school population screening. She has a new syndrome resulting from a microdeletion in 22q11.2 (LCR22-4 to LCR22-5), a region adjacent to but not overlapping with region 22q11.2 (LCR22-2 to LCR22-4), typically deleted in VCFS. Here, we describe her cognitive-neuropsychological and numerical-cognitive profiles. The girl was assessed twice, at 8 and 11 years. Her numerical-cognitive performance at both times was compared to demographically similar girls with normal intelligence in a single-case, quasi-experimental study. Neuropsychological assessment was normal, except for relatively minor impairments in executive functions. She presented severe and persistent difficulties in the simplest single-digit calculations. Difficulties in commutative operations improved from the first to the second assessment. Difficulties in subtraction persisted and were severe. No difficulties were observed in Arabic number writing. Difficulties in single-digit calculation co-occurred with basic numerical processing impairments in symbolic and non-symbolic (single-digit comparison, dot sets size comparison and estimation) tasks. Her difficulties suggest ANS impairment. No difficulties were detected in visuospatial/visuoconstructional and in phonological processing tasks. The main contributions of the present study are: (a) this is the first characterization of the neuropsychological phenotype in 22q11.2DS (LCR22-4 to LCR22.5) with normal intelligence; (b) mild forms of specific genetic conditions contribute to persistent MD

The approximate number system (ANS) has been consistently found to be associated with math achievement. However, little is known about the interactions between the different instantiations of the ANS and in how many ways they are related to exact calculation. In a cross-sectional design, we investigated the relationship between three measures of ANS acuity (non-symbolic comparison, non-symbolic estimation and non-symbolic addition), their cross-sectional trajectories and specific contributions to exact calculation. Children with mathematical difficulties (MD) and typically achieving (TA) controls attending the first six years of formal schooling participated in the study. The MD group exhibited impairments in multiple instantiations of the ANS compared to their TA peers. The ANS acuity measured by all three tasks positively correlated with age in TA children, while no correlation was found between non-symbolic comparison and age in the MD group. The measures of ANS acuity significantly correlated with each other, reflecting at least in part a common numerosity code. Crucially, we found that non-symbolic estimation partially and non-symbolic addition fully mediated the effects of non-symbolic comparison in exact calculation.

OBJECTIVES: The present study established norms for the spelling and arithmetic subtests of the School Achievement Test (Teste do Desempenho Escolar [TDE]) in two Brazilian cities located in the state of Minas Gerais and compared the results with those obtained from the original normative sample. METHODOLOGY: A stratified proportional sample of 1,034 students from Belo Horizonte and Mariana, from the 1 st to 6 th grades, was selected. The participants were assessed by the spelling and arithmetic subtests of the TDE. RESULTS: Significant differences were found between the results from Minas Gerais and Rio Grande do Sul, with moderate to high effect sizes. Significant differences were found in percentiles and classification parameters. The educational performance of the children from Minas Gerais was generally classified as less than expected (i.e., inferior) when the original norms were used as a classification parameter. DISCUSSION: Considering the high variability of educational data in different Brazilian regions, using norms for educational assessment based on only one Brazilian region is inappropriate.

Number transcoding (e.g., writing 29 when hearing "twenty-nine") is one of the most basic numerical abilities required in daily life and is paramount for mathematics achievement. The aim of this study is to investigate psychometric properties of an Arabic number-writing task and its capacity to identify children with mathematics difficulties. We assessed 786 children (55% girls) from first to fourth grades, who were classified as children with mathematics difficulties (n ¼ 103) or controls (n ¼ 683). Although error rates were low, the task presented adequate internal consistency (0.91). Analyses revealed effective diagnostic accuracy in first and second school grades (specificity equals to 0.67 and 0.76 respectively, and sensitivity equals to 0.70 and 0.88 respectively). Moreover, items tapping the understanding of place-value syntax were the most sensitive to mathematics achievement. Overall, we propose that number transcoding is a useful tool for the assessment of mathematics abilities in early elementary school.

Reading and spelling performance have a significant correlation with number transcoding, which is the ability to establish a relationship between the verbal and Arabic representations of numbers, when a conversion of numerical symbols from one notation to the other is necessary. The aim of the present study is to reveal shared and non-shared mechanisms involved in reading and writing of words and Arabic numerals in Brazilian school-aged children. One hundred and seventy-two children from second to fourth grades were evaluated. All of them had normal intelligence. We conducted a series of hierarchical regression models using scores on word spelling and reading single words and Arabic numerals, as dependent variables. As predictor variables we investigated intelligence, the phonological and visuospatial components of working memory (WM) and phonemic awareness. All of the writing and reading tasks (single word spelling and reading as well as number reading and number writing) were significantly correlated to each other. In the regression models, phonological WM was specifically associated to word reading. Phonemic awareness was the only cognitive variable that systematically predicted all of the school skills investigated, both numerical and word tasks. This suggests that phonemic awareness is a modular cognitive ability shared by several school tasks and might be an important factor associated to the comorbidity between dyslexia and dyscalculia.

Although verbal and numerical abilities have a well-established interaction, the impact of phonological processing on numeric abilities remains elusive. The aim of this study is to investigate the role of phonemic awareness in number processing and to explore its association with other functions such as working memory and magnitude processing. One hundred seventy-two children in 2nd grade to 4th grade were evaluated in terms of their intelligence, number transcoding, phonemic awareness, verbal and visuospatial working memory and number sense (non-symbolic magnitude comparison) performance. All of the children had normal intelligence. Among these measurements of magnitude processing, working memory and phonemic awareness, only the last was retained in regression and path models predicting transcoding ability. Phonemic awareness mediated the influence of verbal working memory on number transcoding. The evidence suggests that phonemic awareness significantly affects number transcoding. Such an association is robust and should be considered in cognitive models of both dyslexia and dyscalculia.

Transcoding between numerical systems is one of the most basic abilities acquired by children during their early school years. One important topic that requires further exploration is how mathematics proficiency can affect number transcoding. The aim of the current study was to investigate transcoding abilities (i.e., reading Arabic numerals and writing dictation) in Brazilian children with and without mathematics difficulties, focusing on different school grades. We observed that children with learning difficulties in mathematics demonstrated lower achievement in number trans-coding in both early and middle elementary school. In early elementary school, difficulties were observed in both the basic numerical lexicon and the management of numerical syntax. In middle elementary school, difficulties appeared mainly in the transcoding of more complex numbers. An error analysis revealed that the children with mathematics difficulties struggled mainly with the acquisition of transcoding rules. Although we confirmed 0022-0965/$-see front matter Ó

Physiological changes caused by COMT have been associated with different information processing profiles. The aim of this study is to investigate, by means of a literature review, if the COMT polymorphic types influence the performance of children and adolescents in working memory tasks. We investigated articles that compared behavioral measures of working memory between COMT genotypes. Fourteen articles were selected to be part of this review. Although only a minority of selected studies presented differences in working memory between polymorphisms, we may not affirm that there is not such difference since there is no consistency in the analysis of intervening variables related to the COMT influence on working memory. Futures studies should have better control of these variables.

Math anxiety (MA) is a phobic reaction to math activities, potentially impairing math achievement. Higher frequency of MA in females is explainable by the interaction between genetic and environmental factors. The molecular-genetic basis of MA has not been investigated. The COMT Val158Met polymorphism, which affects dopamine levels in the prefrontal cortex, has been associated with anxiety manifestations. The valine allele is associated with lower, and the methionine allele with higher, dopamine availability. In the present study, the effects of sex and COMT Val158Met genotypes on MA were investigated: 389 school children aged 7-12 years were assessed for intelligence, numerical estimation, arithmetic achievement and MA and genotyped for COMT Val158Met polymorphism. The Math Anxiety Questionnaire (MAQ) was used to assess the cognitive and affective components of MA. All genotype groups of boys and girls were comparable regarding genotype frequency, age, school grade, numerical estimation, and arithmetic abilities. We compared the results of all possible genetic models: codominance (Val/Val vs. Val/Met vs. Met/Met), heterosis (Val/Met vs. Val/Val plus Met/Met), valine dominance (Val/Val plus Val/Met vs. Met/Met), and methionine dominance (Met/Met plus Val/Met vs. Val/Val). Models were compared using AIC and AIC weights. No significant differences between girls and boys and no effects of the COMT Val158Met polymorphism on numerical estimation and arithmetic achievement were observed. Sex by genotype effects were significant for intelligence and MA. Intelligence scores were higher in Met/Met girls than in girls with at least one valine allele (valine dominance model).

Mathematics learning difficulties are a highly comorbid and heterogeneous set of disorders linked to several dissociable mechanisms and endophenotypes. Two of these endophenotypes consist of primary deficits in number sense and verbal numerical representations. However, currently acknowledged endophenotypes are underspecified regarding the role of automatic vs. controlled information processing, and their description should be complemented. Two children with specific deficits in number sense and verbal numerical representations and normal or above-normal intelligence and preserved visuospatial cognition illustrate this point. Child H.V. exhibited deficits in number sense and fact retrieval. Child G.A. presented severe deficits in orally presented problems and transcoding tasks. A partial confirmation of the two endophenotypes that relate to the number sense and verbal processing was obtained, but a much more clear differentiation between the deficits presented by H.V. and G.A. can be reached by looking at differential impairments in modes of processing. H.V. is notably competent in the use of controlled processing but has problems with more automatic processes, such as nonsymbolic magnitude processing, speeded counting and fact retrieval. In contrast, G.A. can retrieve facts and process nonsymbolic magnitudes but exhibits severe impairment in recruiting executive functions and the concentration that is necessary to accomplish transcoding tasks and word problem solving. These results indicate that typical endophenotypes might be insufficient to describe accurately the deficits that are observed in children with mathematics learning abilities. However, by incorporating domain-specificity and modes of processing into the assessment of the endophenotypes, individual deficit profiles can be much more accurately described. This process calls for further specification of the endophenotypes in mathematics learning difficulties.

Mathematics anxiety has been associated to performance in school mathematics. The association between math anxiety and psychosocial competencies as well as their specific contribution to explain school mathematics performance are still unclear. In the present study, the impact of sociodemographic factors, psychosocial competencies, and math anxiety on mathematics and spelling performance was examined in school children with and without mathematics difficulties. The specific contributions of psychosocial competencies (i.e., general anxiety and attentional deficits with hyperactivity) and math anxiety (i.e., self-assessment in mathematics) to school mathematics performance were found to be statistically independent from each other. Moreover, psychosocial competencies-but not math anxiety-were related also to spelling performance. These results suggest that psychosocial competencies are more related to general mechanisms of emotional regulation and emotional response towards academic performance, while mathematics anxiety is related to the specific cognitive aspect of self-assessment in mathematics.

The aim of the present article is to review contributions from neuroscience to a better understanding of mathematical learning processes. A review of the theoretical frames and methodological assumptions from the developmental cognitive neuropsychology is firstly introduced. Then we analyzed the evidences from the neuroscientific investigation of typical and atypical development of mathematical abilities and the implications of those evidences in the context of mathematics education. Finally we conclude that the interactions of neuroscience and mathematics education may benefit all students, by thinking of the cognitive mechanism underlying the development of mathematical abilities.

Catechol-O-methyltransferase (COMT) is an enzyme that is particularly important for the metabolism of dopamine. Functional polymorphisms of COMT have been implicated in working memory and numerical cognition. This is an exploratory study that aims at investigating associations between COMT polymorphisms, working memory, and numerical cognition. Elementary school children from 2th to 6th grades were divided into two groups according to their COMT val158met polymorphism [homozygous for valine allele (n = 61) vs. heterozygous plus methionine homozygous children or met+ group (n = 94)]. Both groups were matched for age and intelligence. Working memory was assessed through digit span and Corsi blocks. Symbolic numerical processing was assessed through transcoding and single-digit word problem tasks. Non-symbolic magnitude comparison and estimation tasks were used to assess number sense. Between-group differences were found in symbolic and non-symbolic numerical tasks, but not in working memory tasks. Children in the met+ group showed better performance in all numerical tasks while val homozygous children presented slower development of non-symbolic magnitude representations. These results suggest COMT-related dopaminergic modulation may be related not only to working memory, as found in previous studies, but also to the development of magnitude processing and magnitude representations.

Magnitude processing is one of the most central cognitive mechanisms that underlie persistent mathematics difficulties. No consensus has yet been reached about whether these difficulties can be predominantly attributed to deficits in symbolic or nonsymbolic magnitude processing. To investigate this issue, we assessed symbolic and nonsymbolic magnitude representations in children with low or typical achievement in school mathematics. Response latencies and the distance effect were comparable between groups in both symbolic and nonsymbolic tasks. The results indicated that both typical and low achievers were able to access magnitude representation via symbolic and nonsymbolic processing. However, low achievers presented higher error rates than typical achievers, especially in the nonsymbolic task. Furthermore, measures of nonsymbolic magnitude explained individual differences in school mathematics better than measures of symbolic magnitude when considering all of the children together. When examining the groups separately, symbolic magnitude representation explained differences in school mathematics in low achievers but not in typical achievers. These results suggest that symbolic magnitude is more relevant to solving arithmetic problems when mathematics achievement is particularly low. In contrast, individual differences in nonsymbolic processing appear to be related to mathematics achievement in a more general manner.

Math learning disability (MLD) is a heterogeneous condition characterized by severe and persistent difficulties in learning math, including difficulties in learning multiplication facts. Objective: In this article, we compared the responses of two MLD children to multiplication facts training. Methods: One of the children was a 9 year-old girl (HV) who presented mild math difficulties associated with lower accuracy of the Approximate Number System (ANS). The other was an 11 year-old boy (GA) who presented severe math difficulties related to impaired phonological processing due to developmental dyslexia. Both children underwent an intervention for multiplication, comprising conceptual instructions and retrieval practice of the times table. Results: HV's accuracy and response speed improved consistently on both training tasks, while GA's accuracy improved on the Simple Calculation Task only. Error analyses indicated that, after training, HV produced fewer errors of the type "close miss", and GA produced less omission but more operand errors. Conclusion: We argue that these differences between their responses to the training tasks were caused by differences in the mechanisms underlying their math difficulties. These results support the notion that individual specificities regarding math disabilities should be taken into account during preparation of training interventions. HeterogeneidAde dAs dificuldAdes nA AprendizAgem dA mAtemáticA e suAs implicAções pArA intervenções específicAs nAs HAbilidAdes de multiplicAção resumo. O transtorno de aprendizagem da matemática (MLD) é uma condição heterogênea caracterizada por dificuldades acentuadas e persistentes na aprendizagem da matemática, incluindo déficits na aprendizagem dos fatos de multiplicação. Objetivo: No presente artigo, nos comparamos a resposta de duas crianças com MLD em uma intervenção da multiplicação. Métodos: Uma das crianças, HV, sexo feminino, 9 anos de idade, apresentava dificuldades menos acentuadas na matemática, associadas a um déficit no Sistema Numérico Aproximado (ANS). A outra criança, GA, sexo masculino, 11 anos de idade, apresentava dificuldades na matemática mais acentuadas associadas a um comprometimento no processamento fonológico devido a um quadro de Dislexia do Desenvolvimento. Ambas as crianças passaram por um programa de intervenção para a multiplicação, o qual se consistia em instruções conceituais e práticas de memorização da tabuada. Resultados: HV apresentou uma melhora consistente na acurácia e tempo de resposta nas duas medidas de desfecho, enquanto, GA apresentou uma melhora apenas na Tarefa de Cálculos Simples. Análises nos tipos de erros evidenciam que, após a intervenção, HV cometeu menos "erros de aproximação", ao passo que, GA cometeu menos erros por omissão, mas mais erros de operando. Conclusão: Nossa hipótese é de que as diferenças na resposta a intervenção dos participantes estão relacionadas a mecanismos subjacentes distintos à aprendizagem da matemática. Esses resultados reforçam a necessidade de que as especificidades nas dificuldades na matemática de cada paciente sejam levadas em consideração durante o planejamento das intervenções. Palavras-chave: multiplicação, intervenção, transtornos de aprendizagem, discalculia.