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 (! = 119) and rural (! = 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.
Keywords: Zareki-R, mathematical skills, children, neuropsychological assessment, arithmetic.

La cognición numérica se basa en dos componentes: el procesamiento numérico y el cálculo. Su desarrollo está influenciado por factores
biológicos, cognitivos, educativos, y culturales. Los objetivos del presente trabajo fueron: a) evaluar el procesamiento numérico y el cálculo
en niños brasileños de entre 7-12 años de escuelas públicas utilizando el Zareki-R (batería de pruebas neuropsicológicas para el
procesamiento numérico y el cálculo en niños, revisada por von Aster y Dellatolas, 2006) con el fin de obtener datos normativos para
los hablantes de portugués, b) determinar cómo el medio ambiente, la edad y el género influyen en el desarrollo de estas habilidades
matemáticas, y c) investigar la validez de constructo del Zareki-R en contraste con el subtest de Aritmética WISC-III. La muestra incluyó
a 172 niños evaluados por el Zareki-R, niños de ambos sexos, divididos en dos grupos: urbano (N = 119) y rural (N = 53). Los niños de
origen rural presentaron puntuaciones más bajas en un aspecto de procesamiento numérico, los niños de 7-8 años demostraron una
puntuación inferior global que los mayores; los varones presentaron un rendimiento superior tanto en el procesamiento numérico como
en el cálculo. La validez del constructo del Zareki-R fue demostrada por las correlaciones de alta a moderada con el subtest de Aritmética
WISC-III. El Zareki-R por lo tanto, es un instrumento adecuado para evaluar el desarrollo de habilidades matemáticas, que está influenciado
por factores como el medio ambiente, la edad y el género.
Palabras clave: Zareki-R, habilidades matemáticas, niños, evaluación neuropsicológica, aritmética.

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

Flávia Heloísa Dos Santos1, Paulo Adilson Da Silva1, Fabiana Silva Ribeiro1, Ana Luiza Ribeiro
Pereira Dias1, Michele Cândida Frigério1, Georges Dellatolas2, Michael von Aster3

1Universidade Estadual Paulista (Brazil)
2Institut National de La Santé et de la Recherche Médicale (France)

3University of Zurich (Switzerland)

The Spanish Journal of Psychology Copyright 2012 by The Spanish Journal of Psychology
2012, Vol. 15, No. 2, 513-525 ISSN 1138-7416
http://dx.doi.org/10.5209/rev_SJOP.2012.v15.n2.38862

The study was developed at government schools in Assis and Ourinhos Cities, São Paulo State, Brazil. Results presented at the
Experimental Psychology Society; EPS London Meeting. London: University College London, 2008.

We especially thank the parents and children for their participation and acknowledge the cooperation of the government schools at Assis
and Ourinhos cities in São Paulo State. This project was supported by the International Cooperation Agreement between Fundação de Amparo
à Pesquisa do Estado de São Paulo (FAPESP) and Institut National de La Santé et de la Recherche Médicale (INSERM) for cooperation
between French and Brazilian researchers, case no 04/11.067-0. Financial support from FAPESP was also obtained for the granted students:
Juliana Molina no 05/00595-8, Bruna Paschoalini no 05/00594-1, Rosana Satiko Kikuchi no 08/54971-2; Michele Cândida Frigério no 05/00593-
5; Ana Luiza Ribeiro Pereira Dias nº 2005/00592-9; Fabiana Silva Ribeiro no 08/54970-2; Paulo Adilson da Silva nº 05/60375-1.

Correspondence concerning this article should be addressed to Flávia Heloísa Dos Santos. Universidade Estadual Paulista, UNESP/Assis.
Laboratório de Neuropsicologia. Programa de Pós-Graduação em Psicologia. Departamento de Psicologia Experimental e do Trabalho.
Avenida Dom Antônio 2100 - cep 19806-900, Assis – SP, (Brazil). Phone: + 55-18-33025902. Fax: + 55-18-33025804; E-mail:
flaviahs@assis.unesp.br

513

http://dx.doi.org/10.5209/rev_SJOP.2012.v15.n2.38862
mailto:flaviahs@assis.unesp.br


SANTOS, SILVA, RIBEIRO, DIAS, FRIGÉRIO, DELLATOLAS, AND VON ASTER

Quantitative skills such as numerosity comprehension,
ordinality, counting, and simple arithmetic develop during
childhood (Cantlon, Platt, & Brannon, 2009; Geary, 1995,
2000). In the primary school, children learn the base-10
system, different numerical representations; for example,
transcoding from verbal code (twenty three) into Arabic
code (23), and basic arithmetic (Geary, 2000). In the high
school, the complexity of these processes increases and
includes procedures of multiple steps, such as multiplication
and division (Dehaene, 1997; Geary, Frensch, & Wiley,
1993; O’Hare, 1999; Shalev, Manor, Amir, & Gross-Tsur,
1993).

Gelman and Galistel (1978) assumed five counting
principles listed below, three of them related to procedures
rules, one related to the concept of countable items, and
the last one is a combination of the four principles. The
one-to-one-correspondance emphasizes the discrimination
of a specific counting code for each object (Word, or Arabic
or other form); the child must coordinate two processes:
first each item to be counted must be transferred from the
original category to the counting category (division), once
counted the item must be annulated in order to not be
counted twice (tagging). A typical strategy is pointing the
items (visual code) while counting loudly (verbal code).
The stable-order. Counting involves more than the ability
to assign random tags to the items in an array. The counting
marks must be organized in a stable manner, repeated in
certain order. The cardinal principle. The last word in a
sequence of numbers in general represents the total amount
and the numerosity of the set. The comprehension of this
concept depends on the principle of one-to-one, in other
words, the principle of the stable order of correspondence
develops after the child learn to tag items already counted
and to name a set of items. The abstraction. The child must
understand that the counting is independent of the object
characteristics (toys, actions, sounds etc). The order-
irrelevance. The child must learn that direction of counting
is irrelevant (from left to the right and vice versa). The
regular use of this principle emerges around 4-5 years old.

The calculation refers to the mathematical operations
such as addition, subtraction, multiplication, and division;
which require operational words (more, less, times, divide)
and symbols (+, –, × or ÷), the retrieval of them and other
arithmetic facts, and the arithmetic calculation procedure.
On the other hand, number processing refers to
comprehension of the numeric symbols associated with
quantities, and number production in reading, writing and
counting of numbers (McCloskey, Caramazza, & Basili,
1985). The processing of the mathematical operations in
practice depends on: a) the availability of cognitive number
representations – visual Arabic, verbal, and analogue
magnitude – (triple-code-model; Dehaene, 1997), that interact
dynamically without the need of association between the
numbers, and the quantities themselves (Dehaene & Cohen,
2000); b) numerical skills that are based on a variety of

modality-specific representations (e.g., visuospatial and
verbal-auditory codes) in diverse number-processing tasks
supporting the specific-integrated (encoding-complex) view
of number processing (Campbell & Clark, 1992).

According to Dehaene (1997), the number sense
represents the innate ability to recognize, compare, add,
and subtract small quantities, without the need of counting,
in a mental number line, which is a spatially oriented
representation of the quantities, that increases with age and
experience, and account to large numerosities. The number
sense capacity plays a fundamental role in mathematical
competence (Dehaene, 2001; Iuculano, Tang, Hall, &
Butterworth, 2008). von Aster and Shalev (2007) described
a four-step Developmental Model of Numerical Cognition,
which considers that transitions from the pre-verbal period
to adulthood, in both number skills and number sense,
would culminate in the mental number line development.
However, these changes demands more than an intact
number system, they require experience, plasticity, and
development of other skills such as visual imagery,
language, and working memory.

On the other hand, socio-cultural, pedagogical, and
linguistic factors may affect both number processing and
calculation (Dellatolas, von Aster, Willardino-Braga, Meier,
& Deloche, 2000; Gross-Tsur, Manor, & Shalev, 1996; Hein,
Bzufka, & Neumärker, 2000; Koumoula et al., 2004). In
contrast, studies showed that working memory is culture free
of socioeconomic status and environmental aspects (Engel,
Santos, & Gathercole, 2008; Koumoula et al., 2004; Santos
& Bueno, 2003; Santos, Mello, Bueno, & Dellatolas, 2005).

Considering the complexity of the number processing
and calculation cognitive systems, it is necessary to assess
different aspects of the numerical domains separately, as
for differential diagnosis and planning appropriate
intervention strategies (von Aster, 2000). For this purpose,
the ZAREKI (in German: !europsychologische Testbatterie
fûr ZAhlenarbeitung und REtchnen bei KIndern; von Aster,
2001), or NUCALC for the acronym in English
(!europsychological Test Battery for �umber Processing
and Calculation in Children), was developed. The
preliminary version of this instrument was the EC301
battery (Batterie Standardisée D’evaluation du Calcul et
du Traitement des !ombres) developed to assess numerical
skills in adults (Dellatolas, Deloche, Basso, & Claros-
Salinas, 2001; Deloche, 1995; Deloche et al., 1995).

The ZAREKI was adapted for use in different countries
and languages, for instance, German, French, and Greek
(Dellatolas et al., 2000; Koumoula et al., 2004; von Aster,
2001; von Aster & Dellatolas, 2006). Since secondary
quantitative abilities are influenced by educational systems,
which change across countries and generations (Dellatolas
et al., 2000; Geary, 2000), it is important to assess
mathematical skills using an equivalent instrument in
different languages, which benefit cross-cultural studies.
Many studies were accomplished with the ZAREKI:

514



NUMERICAL COGNITION IN BRAZILIAN CHILDREN

assessment of German and French children (von Aster,
Deloche, Dellatolas, & Meier, 1997), illiterates (Deloche,
Souza, Willadino-Braga, & Dellatolas, 1999), and patients
with unilateral brain injury (Deloche, Dellatolas, Vendrell,
& Bergego, 1996; Deloche & Willmes, 2000), etc.

Functional analysis indicated that the theoretical
construct of the ZAREKI are oriented by Dehaene’s Triple
Code Model (Dehaene & Cohen, 2000; von Aster, 2000)
and is consistent with the four-step Developmental Model
of Numerical Cognition (von Aster & Shalev, 2007). The
Figure 1 illustrates the relationship between the subtests of
the ZAREKI with the three modules of representation
(Verbal, Analog and Arabic codes) in contrast to the different
combinations of the processing procedures overlapping
them (input, internal transcoding, and output). The three
modules are autonomous, interconnected and activated
according to the particular needs of a given task and
constitute a system for number processing and calculation.
In this sense, according to the triple-code model, abilities
such as approximation and number comparison depends on
the analogue module, whereas abilities such as counting
(in operations such as addition and subtraction, including
the arithmetical fact retrieval) depends on the verbal module.
Multi-digit operations and parity judgments rely on the
visual Arabic module number form, in that numbers are
represented by their Arabic code (von Aster, 2000).
Moreover, the literature indicated that ZAREKI clearly
differentiates and specifies the profile of mathematical skills
in children being suitable for the diagnosis of
Developmental Dyscalculia (DD) (von Aster, 2000; Shalev
et al., 1993).

DD, also named as Specific Disorder of Arithmetical
Skills (International Classification of Diseases, 10th Edition
[ICD-10], OMS, 1993) or Mathematics Disorder (Diagnostic
and Statistical Manual of Mental Disorders, 4th Edition, Text
Revision [DSM-IV-TR], (APA, 2002), constitutes a specific
difficulty in quantitative processing, which is expressed by
difficulties to accomplish elementary operations such as
addition, subtraction, multiplication, and division, not due
to inefficient teaching or global intellectual disability (F81.
2, World Health Organization, 2005); and its diagnosis
requires that these operations are assessed objectively by
standardized tests (315.1, APA, 2002). The aetiology of DD
is multi-factorial and includes genetic predisposition,
neurological disturbances, environmental, and psychological
factors (Shalev, 2004).

Hein et al. (2000) investigated the prevalence of
arithmetic impairment in urban and rural children using the
ZAREKI; they found a similar prevalence (around 6%)
between simple arithmetic and reading and spelling
disorders. In their study children with DD showed lower
scores than their counterparts; besides, the discrepancy
between the mean scores and standard deviation suggested
that boys performed better than girls. Koumoula et al.
(2004) assessed Greek children with the same instrument,

and identified that children from rural areas and low
socioeconomic level presented inferior performance than
the urban children in mental calculation, reading and writing
of numbers, oral comparison of numbers, contextual
estimation of quantities and problem solving. In Koumoula
et al. (2004) the prevalence of children performing
significantly worse in arithmetical skill was higher in rural
than in urban students.

von Aster (2000) found differences associated with age
and schooling on performance of German children.
Moreover, boys performed better than girls in tasks of
subtraction, oral comparison, and arithmetic problems; the
gender difference was also observed by Rosselli, Ardila,
Matute, and Inozemtseva (2009) in mental calculation and
solving arithmetic problem tasks, these authors argued that
boys presented a more efficient retrieval of arithmetic facts.
von Aster (1994) hypothesised that girls seems to be more
susceptible to anxiety and depression, which may affect
their performance in maths exam. This assumption was
supported by the study of Krinzinger and Kaufmann (2006)
in that girls reported higher levels of math anxiety and/or
more negative attitudes towards math than boys.

In a cross-cultural study of Brazilian, French and Swiss
children, aged 7 to 10 years Brazilian children were from
central and suburbs of Brasília city, recruited in public and
private schools (Dellatolas et al., 2000). The authors
obtained an age effect on the Score A, which is formed by
the core tasks of the battery: two involving number
transcription, the two tasks of number comparison, and the
two tasks of calculation. French children from 7 to 8 years
old obtained better scores than children from other
nationalities; however, the groups showed equivalent
performance at age 10. In regards to the Brazilian sample,
children from the suburbs presented lower scores than
children from central areas in specific tasks: at ages 7-8 in
written comparison, dictation and reading of numbers; at
ages 8-9 in mental calculation and arithmetic problem; at
ages 9-10 in mental calculation and contextual estimation.
Boys presented better performance than girls in both oral
and written comparison of numbers. However, this study
had an irregular distribution of children by age range in
different schooling grades, for instance a child at age 9
to10 could study at 2nd or 3rd grade, and this can be a
confounding factor in terms of schooling achievement.
Besides, the socioeconomic status was estimated according
to the city regions (central versus suburb) instead of being
objectively measured, which makes the discrimination of
environment and socioeconomic aspects difficult.

Other Brazilian neuropsychological studies in
mathematical skills remain rare (Dias et al., 2009). Correa
and Meireles (2000) observed a progressive development
in the capacity to establish relationship of inverse order
between divisor and quotient in children aged 5-7 years.
Haydu, Costa, and Pullin (2006) demonstrated that the
presentation in different forms of arithmetic problems

515



SANTOS, SILVA, RIBEIRO, DIAS, FRIGÉRIO, DELLATOLAS, AND VON ASTER516

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facilitates the addition operation. Raad, Pimentel, and
Almeida (2008) did not find differences on arithmetic
performance of boys and girls from the 2nd grade in two
public schools of Aracaju city in Sergipe State. Oliveira
and Tourinho (2001) showed better performance for
problems solving by the end of the 1st grade in comparison
with the beginning of it. These studies are useful
contributions, however, they assessed particular aspects of
mathematical skills in restrict age bands, using different
instruments, and, therefore, it is difficult to generalize the
results in terms of mathematical development, and the
factors that interfere in its functioning.

More recently a revised version of the Neuropsycho-
logical test battery for Number Processing and Calculation
in Children, recognized as Zareki-R was released (von Aster
& Dellatolas, 2006). However, studies with the revised
version still sparce (von Aster & Dellatolas, 2006; Rotzer
et al., 2009), and the present work is the first normative
study of the Zareki-R into Portuguese language. Silva and
Santos (2011) assessed Brazilian children with learning
disabilities and remarkable difficulties in arithmetic using
the Zareki-R. These children presented a global score one
and a half standard-deviation below children with learning
difficulties in reading and writing, and were impaired in
both number processing (dictation of numbers) and
calculation (mental calculation and arithmetic problem
solving). Their impairment was even worse - two standard
deviation below - when compared with typically developing
children, which is compatible with the first criterion of ICD-
10 for Developmental Dyscalculia (Silva & Santos, 2009).
Rotzer et al. (2009) also used Zareki-R and found similar
results. Moreover, Silva and Santos (2011) observed
significantly positive correlations between mental calculation
and problem solving subtests and visuospatial and
phonological working memory tasks. Santos and Silva (2008)
obtained a high positive correlation between the Zareki-R
total score (r = .73, p < .05) and the subtest “arithmetic” of
the SAT, School Achievement Test (TDE, Teste de
Desempenho Escolar; Stein, 1994), which indicates the
construct validity of the Zareki-R.

The objectives of the present study were to: a) assess
number processing and mental calculation in Brazilian
children aged 7-12 years from public schools using the
Zareki-R in order to obtain normative data for Portuguese
speakers; b) identify how environment, age and gender
factors influence the development of these mathematical
skills; c) investigate the validity of the construct of the
Zareki-R by the contrast with the Arithmetic subtest of
WISC-III. The authors hypothesize that rural children might
present a slight inferior performance than urban in both
number processing and mental calculation, therefore, a
global difference in mathematical skills. Besides, we
suppose that older children will perform better than
younger, and boys will perform better than girls in mental
calculation.

Method

Prior to testing, informed written consent for both expe-
riments was obtained from the children’s parents. It was
explained to each child that the experiment could be dis-
continued at any time. The Ethics Committee of UNESP, São
Paulo State University approved the study, case nº 0095/2005.

Participants

Participants were 172 children (86 boys) aged 7-12 years.
In regards to gender the children were distributed across
age bands: twelve boys at age 7, ten boys at age 8, eighteen
boys at age 9, twenty-five boys at age 10, eleven boys at
age 11 and ten boys at age 12. The children were raised in
urban area of Assis and Ourinhos cities in São Paulo State
(! = 119) or in rural area from the same region (! = 53).
The children were recruited in government schools from 1st

to 6th grades. The schools were selected according to three
aspects: a) being public schools, b) attending children at the
target age bands, and c) attending also children raised in
rural areas. It is important to mention that the rural children
of this sample lived in the rural environment but studied in
the same urban schools than their counterparts.

All of the children were Brazilian nationals, native
Portuguese speakers, and non-bilingual. The age bands and
schooling grades were equivalent in all cases. The inclusion
criterion was normal intellectual level: 50 children of the
sample were assessed by the Raven’s Coloured Progressive
Matrices (percentile > 24 and < 75; Angelini, Alves,
Custódio, Duarte, & Duarte, 1999) and 122 children were
assessed by the Wechsler Intelligence Scale for Children -
WISC III (IQ >80 and <120; Wechsler, 2002). The
intellectual level was assessed in order identify potential
outliers; participants were recruited for two different projects,
which explain the use of Raven’s matrices for some of them.

None of the children was identified as presenting learning
disabilities, emotional disturbances, motor difficulties, speech
or hearing impairments, or neurological or psychiatric diagnosis,
based on parent and teacher reports. The socioeconomic status
(SES) was assessed by the Brazilian Association of Marketing
Research Institutes Scale, which stratifies in five classes from
A/richest to E/poorest (ABIPEME; Almeida & Wickerhauser,
1991). Difference in SES between the two groups was found,
in that urban childrenM = 54.89, (SD = 18.44) obtained higher
scores (t = 5.61; p < .001) than rural children M = 38.45, (SD
= 15.16), these mean scores represent the opposite extremes
boundaries of middle class (38-58 points). Sociodemographic
information is presented on Table 1.

Material

The Zareki-R (von Aster & Dellatolas, 2006) assesses
the number representation in children considering cognitive
abilities that are prerequisites for the acquisition of

NUMERICAL COGNITION IN BRAZILIAN CHILDREN 517



arithmetic skills. Zareki-R has been originally published in
German and French language versions (von Aster, Weinhold
Zulauf, & Horn, 2006; von Aster & Dellatolas, 2006;
respectively) with its respective standardization. As for the
French version, von Aster and Dellatolas (2006) found
significant and high correlations between Zareki-R scores
and the mathematic part of the Test of Scholar Achievement
in different grades (TAS, Test d’Acquisitions Scolaires
Francais et Mathematiques; Lepez & Riquier, 1997) reported
by children’s teachers.

The Zareki-R has 11 subtests specialized in mathematical
skills, nine of them assess number processing (i-ix) and two
assess calculation (x-xi): i) Counting dots (CD): Children
have to enumerate different set of dots. The scoring system
considers the number of results. Qualitatively, it is
considered: production of the usual sequence of spoken
numbers, oral and manual synchronization while pointing
and counting dots, visual control for discrimination of the
dots already counted and remaining items, transcoding the
information from spoken verbal number to Arabic digit form
(the number of items, i.e., the k-value is 6) ; ii) Counting
backwards (CB): the children must count the dots in

backward the sequences: from 23 to 1 and from 67 to 54
(Items k = 2); iii) Dictation of numbers (D!): the child is
asked to write, in Arabic numerals, eight numbers presented
orally (e.g. 14), (Items k = 8); iv) Reading numbers (R!):
the child is asked to read loudly eight numbers written in
Arabic numerals, such as 15 and 6485, (Items k = 8); v)
Positioning numbers on an analogue scale (P!): the scales
are vertical lines represented with a “0” at the bottom and
a “100” at the top, which are transacted by 4 small
horizontal lines at different locations. The child is asked to
point to the horizontal which corresponds to an Arabic
numeral, aurally or visually presented by the examiner
(Items k = 12); vi) Oral comparison (OC): Eight pairs of
numbers are presented aurally (e.g., 34601 and 9678) and
the child must judge which one is larger (Items k = 8); vii)
Perceptive estimation (PE): The child has to give orally an
estimative, without counting, in of the quantities of items
shown in a picture, visually presented by 5 seconds, for
example the number of balls in the picture (the precise
answer is 57 balls), (Items k = 8); viii) Contextual estimation
(CE): The child must judge sentences in terms of coherence
between quantities and context, for instance: “ten leaves on
a tree” is ‘little’, ‘median’ or a ‘lot’?, (Items k = 10); ix)
Written comparison (WC): Pairs of numbers in Arabic
numeral form are presented visually, for instance, 1007 and
1070, and the child must judge which one is larger, (Items
k = 10); x) Mental calculation (MC): Eight additions, eight
subtractions, and six multiplications are presented aurally
(e.g.: 5 + 8; 14 - 6, and 3 x 2), (Items k = 22); xi) Problem
solving (PS): The child has to solve six numerical problems
of increasing complexity. The first story problem is: ‘Peter
has 12 marbles; He gave 5 to his friend Anne; how many
marbles Peter has now?’, (Items k = 6). The battery also
includes a subtest to measure phonological working memory,
however, it is not included in the total score: xii) Memory
of Digits (MD): This requires the forward (FDS) and
backward (BDS) repetition of digit sequences of increasing
length, three sequences of each length were performed. In
the FDS the child repeats the digits in the same order as
the experimenter spoke them, (Items k = 24). In the BDS,
the subject repeats the digits in reverse order. Additionally,
as presented by Dellatolas et al. (2000), the Score A is

SANTOS, SILVA, RIBEIRO, DIAS, FRIGÉRIO, DELLATOLAS, AND VON ASTER518

Table 1
Sample description by groups

Total Urban Rural
! % ! % ! %

Age (years)
Age 7 30 17.44 23 19.33 7 13.21
Age 8 20 11.63 11 9.24 9 16.98
Age 9 38 22.09 29 24.37 9 16.98
Age 10 44 25.58 33 27.73 11 20.75
Age 11 21 12.21 13 10.92 8 15.09
Age 12 19 11.05 10 8.40 9 16.98
Gender
Boys 86 50 60 50.42 26 49.06
Girls 86 50 59 49.58 27 50.94
Total 172 100 119 100 53 100

Legend. ! = number of participants; % = percentage of cases

Table 2
Sociodemographic aspects by groups

Total Urban Rural
! M (SD) ! M (SD) ! M (SD)

SES 160 49.45 (19.03) 107 54.89 (18.44) 53 38.45 (15.16) *
WISC-III– IQV 122 106.56 (11.75) 69 109.65 (10.89) 53 102.46 (11.68) *
CPM# 50 66.50 (20.01) 50 —

(*) p < .05; (#) urban children from Ourinhos city aged 9 to 10 years. Legend. ! = number of participants; M = mean; SD = standard
deviation; SES = socioeconomic status; WISC-III = Wechsler Intelligence Scale for Children; IQV = Verbal Intelligence Quotient;
CPM = Raven’s Coloured Progressive Matrices.



calculated by the sum of the six following subtests of
ZAREKI-R: dictation of number, reading numbers, mental
calculation, problem solving, oral comparison and written
comparison.

Procedures

The children were assessed with a larger battery of
neuropsychological tests, involving short and long term-
memory for visuospatial and verbal stimuli, which will not
be reported in this article. The neuropsychological assessment
was carried out individually at child’s school in a quiet
room. The Zareki-R was administered in a single session
and engaged 30-minute in average; the subtests order was
not fixed, but verbal tasks (MD, DN, CB, MC, OC, CE,
and PS) were alternated with visuomotor e perceptive ones
(CD, RN, WR, PE, and PN).

Statistical analyses

The analyses were carried out with STATISTICA, version
5.0 (Statsoft, 1995). Three sets of analyses were performed.
In the first, multivariate analysis of variance (MANOVA)
were carried out for Zareki-R subtests. In this case, the
between-subject factors were 2 groups (urban and rural),

while the within-subjects factors were the scores in each
subtest of Zareki-R, except the subtest Memory of Digits.
The t-test was carried out for the Total score, Score A, and
Memory of Digits of Zareki-R, with the same between-
subject factors. Due to differences between groups on Zareki-
R subtests, the subsequent analyses were conducted with
group as covariant.

In the second set of analyses, multivariate analysis of
covariance (MANCOVA) - group as a covariant - were
carried out, between-subject factors were 6 ages (from 7
to 12 years) and 2 genders (male and female) and the same
within-subject factors were used to investigate the factors
that determine the main effects obtained. Analyses of
covariance (ANCOVA) – group as a covariant – were
performed for the Total score, Score A, and Memory of
Digits of Zareki-R, with the same within- and between-
subject factors. In the first and second sets of analyses the
Scheffé post hoc test was used with a significant alpha level
of p ≤ .05.

In the third set, Pearson’s product moment correlations
between scores in Zareki-R and Arithmetic of WISC-III,
and Memory of Digits of Zareki-R were carried out. For
the discussion, we adopted moderate to high correlations
(r > .40) as minimum score. To assess the effect size of
group and gender effect was used Cohen’s d calculated

NUMERICAL COGNITION IN BRAZILIAN CHILDREN 519

Table 3
ZAREKI-R performance in Brazilian children by groups and gender

Urban (!=119) Rural (!=53)
p Cohen d

Boys (N=86) Girls (N=86)
p Cohen d

M (SD) M (SD) M (SD) M (SD)

CD 3.49 (.82) 3.58 (.72) .45 –.11 3.53 (.79) 3.50 (.79) .77 .03
CB 3.26 (1.09) 2.96 (1.18) .10 .26 3.31 (.99) 3.02 (1.24) .08 .25

DN 13.26 (4.11) 12.42 (4.27) .21 .20 13.42 (3.85) 12.58 (4.44) .18 .20
RN 14.16 (3.79) 13.83 (3.93) .60 .08 14.48 (3.57) 13.64 (4.03) .15 .22

PN 16.46 (5.07) 14.99 (5.31) .08 .28 16.68 (5.13) 15.33 (5.17) .08 .26
OC 14.08 (2.57) 13.25 (3.32) .07 .27 14.48 (2.17) 13.16 (3.26) < .01* .47

PE 6.08 (2.22) 6.40 (2.37) .40 –.13 6.38 (2.26) 5.98 (2.26) .24 .17
CE 12.72 (4.67) 14.04 (4.89) .09 –.27 12.84 (5.01) 13.42 (4.52) .42 –.12

WC 18.97 (2.06) 18.23 (2.62) .04* .31 19 (1.98) 18.49 (2.51) .13 .22
MC 29.88 (11.28) 26.51 (11.54) .07 .29 30.48 (11.37) 27.21 (11.33) .05* .28

PS 7.33 (3.64) 6.58 (3.62) .21 .20 7.99 (3.59) 6.21 (3.48) < .01* .50
Total 139.56 (31.85) 132.76 (33.56) .20 .20 142.58 (30.97) 132.35 (33.25) .03* .31

Score A 97.68 (24.02) 90.79 (25.01) .08 .28 99.82 (23.36) 91.29 (24.94) < .01* .35
MD 25.05 (6.83) 20.22 (5.19) < .01 .79 23.53 (6.93) 23.60 (6.59) .89 –.01

(*) p ≤ .05. Gender effect by ANCOVA, group as covariant. Legend. Zareki-R = Neuropsychological Tests Battery of for Number
Processing and Mental Calculation in children, Revised; != number of participants; M= mean; SD= standard deviation; CD = Counting
dots; CB = Counting backwards; DN = Dictation of numbers; RN = Reading numbers; PN = Positioning numbers on an analogue scale;
OC = Oral comparison; PE = Perceptive estimation; CE = Contextual estimation; WC = Written comparison; MC = Mental calculation;
PS = Problem solving. Score A= is calculated by the sum of the six following subtests of Zareki-R: dictation of number, reading numbers,
mental calculation, problem solving, oral comparison and written comparison. MD= Memory of Digits.



through the free software (Devilly, 2005); small, median,
and large effects are respectively correspondent to effect
sizes of .20, .50, and .80 (Rice & Harris, 2005).

Finally, considering that working memory could be a
possible factor explaining the results, using the same within-
and between-factors of first and second sets we carried out
exploratory analyses with Memory of Digits as covariant,
and the results were similar to the analyses reported below
in that group was the covariant.

Results

Tables 3 and 4 present the results obtained with the
Zareki-R by groups, gender and age, respectively. In general,
urban children had higher scores than rural in the subtest
Written comparison and Memory of Digits, the performance
increased with age, and boys had higher scores than girls
in Mental calculation, Oral comparison, Problem solving,
and Total score.

Group effect. A 2 (group) x 11 (subtest) MANOVA
was performed. A group effect was observed [R(11,160)
= 2.11; p = .02]. Urban children were better than rural in
Written comparison (p = .04). No effect of group was
observed by ANOVA in the Total Score (t = 1.27; p = .63)
and Score A of Zareki-R (t = 1.71; p = .70). There was

group effect in the subtest Memory of Digits of Zareki-R
(t = 4.58; p = .02), in that urban children performed better
than rural.

Gender effect. A 2 (gender) x 11 (subtest) MANCOVA
was performed, group as a covariant, and revealed a gender
effect [R(11,159) = 2.43; p < .01]. Boys performed better
than girls in Mental calculation (p = .05), Oral comparison
(p < .01), and Problem solving (p = .11). ANCOVAs –
group as a covariant – indicated gender effect to the Total
score [F(1, 169) = 4.30; p = .03] and Score A [F(1, 169)
= 5.32; p < .02]; boys performed better than girls, but not
to Memory of Digits [F(1, 169) = .01; p = .89].

Age effect. A 6 (age) x 11 (subtest) MANCOVA was
performed, group as a covariant, and revealed an age effect
[R (55,721) = 5.47; p < .001]. ANCOVA – group as a
covariant – observed an age effect to the Total Score [F(5,
165) = 58.57; p < .001], and Score A [F(5, 165) = 63.94;
p < .001]. ANCOVA – group as a covariant – also observed
an age effect to Memory of Digits [F(5, 165) = 2.33; p =
.04], but the post-hoc test did not discriminated the
differences. Except for Counting dots and Perceptive
estimation subtests, the age effect was observed in all tasks.
7-year-old children performed worse than all other ages for
the other subtests and Score A, except for Oral comparison,
in that 7- and 8-year-old children performed worse than 9-
to 12-year-old children. 8-year-old children performed worse

SANTOS, SILVA, RIBEIRO, DIAS, FRIGÉRIO, DELLATOLAS, AND VON ASTER520

Table 4
ZAREKI-R performance in Brazilian children by age

Age 7 (! = 30) Age 8 (! = 20) Age 9 (! = 38) Age 10 (! = 44) Age 11 (! = 21) Age 12 (! = 19)
M (SD) M (SD) M (SD) M (SD) M (SD) M (SD)

CD 3.40 (.89) 3.50 (.83) 3.26 (.89) 3.68 (.64) 3.67 (.66) 3.68 (.75)
CBa 1.97 (1.43) 3.25 (1.21) 3.21 (.99) 3.55 (.76) 3.57 (.60) 3.58 (.51)
DNb 6.07 (3.81) 11.65 (3.76) 14.45 (2.42) 15.23 (1.18) 15.05 (1.36) 15.05 (1.03)
RNb 7.67 (4.61) 13.30 (3.33) 15.53 (1.25) 15.80 (.67) 16.00 (.00) 15.84 (.37)
PNa 10.38 (5.96) 15.53 (5.33) 15.61 (4.46) 18.41 (2.97) 17.86 (4.01) 18.58 (3.01)
OCd 10.67 (3.49) 12.15 (3.80) 15.00 (1.41) 14.52 (1.64) 15.00 (1.30) 15.26 (1.37)
PE 5.20 (2.20) 5.40 (2.44) 6.42 (2.42) 6.73 (2.12) 6.33 (1.98) 6.63 (2.03)
CEa 9.13 (3.81) 13.40 (4.45) 12.68 (4.73) 14.14 (4.89) 14.48 (4.29) 16.21 (2.49)
WCa 16.33 (3.07) 18.40 (2.87) 19.26 (1.08) 19.41 (1.70) 19.52 (0.87) 19.47 (1.47)
MCc 12.47 (9.37) 25.15 (11.82) 31.26 (8.00) 34.80 (6.72) 34.43 (6.93) 33.79 (6.09)
PSc 2.77 (3.33) 5.50 (3.86) 7.53 (3.13) 8.66 (2.51) 8.62 (2.01) 9.47 (1.81)
Totalc 85.51 (27.46) 127.17 (31.70) 144.21 (18.38) 154.91 (14.25) 154.52 (12.85) 157.58 (11.81)
Score Ab 55.96 (21.26) 86.10 (24.12) 103.02 (11.95) 108.40 (10.58) 108.61 (8.72) 108.89 (8.96)
MD 21.46 (6.14) 23.90 (8.39) 23.00 (6.25) 24.63 (6.96) 22.95 (7.00) 25.89 (5.39)

Legend. Age effect by MANCOVA, group as covariant; Scheffé post-hoc: (a) 7 < 8-12; (b) 7 < 8 < 9-12; (c) 7 < 8 and 8 < 10-12; (d)
7-8 < 9-12; p ≤ .05 in all cases. Zareki-R = Neuropsychological Tests Battery of for Number Processing and Mental Calculation in
children, Revised; != number of participants; M= mean; SD= standard deviation; CD = Counting dots; CB = Counting backwards; DN
= Dictation of numbers; RN = Reading numbers; PN = Positioning numbers on an analogue scale; OC = Oral comparison; PE =
Perceptive estimation; CE = Contextual estimation; WC = Written comparison; MC = Mental calculation; PS = Problem solving; MD=
Memory of Digits. Score A= is calculated by the sum of the six following subtests of Zareki-R: dictation of number, reading numbers,
mental calculation, problem solving, oral comparison and written comparison.



than 9- to 12-year-old children in Dictation of numbers,
Reading numbers, and Score A, and worse than 10- to 12-
year-old children in Mental calculation, Problem solving
and Total score.

Exploratory analysis with gender and age as covariants
altogether in MANCOVA and ANCOVAs for the total score,
Memory of Digits, Zareki-R subtests, and for Score A
indicated that there were no interactions between these
factors.

Correlation between Zareki-R and Arithmetic
of WISC-III or Memory of Digits

The correlation matrix was calculated considering the
four-step developmental model of number acquisition (von
Aster & Shalev, 2007) which states that number line
development is associated with working memory (subtest
memory of dots). Moreover, the subtest Arithmetic of WISC-
III was used as a gold standard for the assessment of
mathematical skills. The results revealed significant and
positive relationship between all the subtests of Zareki-R
and the subtest Arithmetic of WISC-III. Moderate
correlations were observed to the following subtests:
Counting backwards (r = .49), Dictation of numbers (r =
.49), Mental calculation (r = .49), Reading numbers (r =
.49), Oral comparison (r = .41), Problem solving (r = .58),
Total score (r = .65), and Score A (r = .62). There were
significant and positive correlation between the Memory of
Digits and the others subtest of Zareki-R, except for
Counting Dots (p = .10) and Perceptive estimation (p = .87),
but none of them were over .4. See Table 5.

Discussion

The present study investigated the mathematical skills
such as calculation and number processing of Brazilian
children aged 7 to 12 years from public schools in two
cities of São Paulo State and identified some factors that
influenced the development of these abilities. The
performance was mainly age and gender-related, and
indicated minimal environmental influence.

As for the environmental factor, rural children achieved
lower score than urban children only in two tasks (Written
comparison and Memory of Digits), with preserved
calculation; whereas in the study of Koumoula et al. (2004)
rural children obtained lowers scores in seven subtests of
ZAREKI (von Aster, 2001), including Written comparison,
but not Memory of Digits. In previous studies it was
suggested that a low socioeconomic level and educational
environment could determine such differences (Dellatolas
et al., 2000; Koumoula et al., 2004). In the present study,
socioeconomic status was objectively assessed, rather than
inferred by the environment factor, and the statistical
analysis indicated that rural children in fact had low
socioeconomic scores than urban children. In spite of it,
the rural group showed a slight low score restricted to one
aspect of number processing; and, even though, the effect
size showed that this difference had small magnitude. It
means that, despite of this socioeconomic discrepancy both
groups performed Zareki-R almost similarly. Since all rural
children studied in the same schools as the urban children,
therefore obtain the same educational stimulation, we
consider that the pedagogy method is a more specific

NUMERICAL COGNITION IN BRAZILIAN CHILDREN 521

Table 5
Correlation between Zareki-R and Arithmetic of WISC-III

Arithmetic of WISC-III MD of Zareki-R
(! = 1 22) (! = 172)

Counting dots .18 .12
Counting backwards .49* .35
Dictation of numbers .49* .35
Reading numbers .49* .27
Positioning numbers on an analogue scale .33 .33
Oral comparison .41* .21
Perceptive estimation .22 –.01
Contextual estimation .36 .25
Written comparison .38 .32
Mental calculation .62* .36
Problem solving .58* .38
Total score .64* .39
Score A .62* .38

(*) Moderate and high correlations r > .40; p < .05. Legend. Zareki-R = Neuropsychological test battery for Number Processing and
Calculation in Children, Revised; WISC-III = Wechsler Intelligence Scale for Children; MD= Memory of Digits; Zareki-R =
Neuropsychological Tests Battery of for Number Processing and Mental Calculation in children, Revised.



determinant of the arithmetic performance than the
socioeconomic status.

The performance was age-related, even controlling
statistically the environmental factor, in that children aged
7 and 8 years obtained lower scores in Zareki-R in
comparison to older children, except in two subtests:
counting dots and perceptive estimation. This result was
expected since counting dots is an ability related to the core
verbal system of the four-step Developmental Model of
Numerical Cognition, which develops at the preschool age
(von Aster & Shalev, 2007), which is consistent with
Gelman and Gallistel (1978) principles, that the counting
competence allows children to compare quantities and solve
arithmetic using counting strategies as reasoning tools.
Besides, older children have more facility to perform the
battery, at least for items influenced by schooling; for
example, there was ceiling effect in Reading numbers
subtest at age 11.

Dellatolas et al. (2000), Koumoula et al. (2004), and
von Aster and Shalev (2007) showed that many domains
of number processing and calculation are dependent on
schooling achievement (Score A) and have an improvement
age-related since the symbolic number representation system
(verbal and Arabic) and the ordinal magnitude system
(mental number line) are learnt during primary school.
Therefore, mathematic skills seem to be dependent of both
brain maturation (von Aster, 2000) and schooling (von Aster
& Shalev, 2007).

Boys performed better than girls at Zareki-R total score,
in calculation (Mental calculation and Problem solving),
and in one subtest of number processing (Oral comparison),
in all cases the effect was confirmed controlling the
environmental factor by the ANCOVA. However, the effect
size showed that the gender difference had small magnitude
for Mental calculation and Oral comparison and median
magnitude for the Problem solving subtest. We consider
that the differences were number representation-related
rather than associated with general cognitive ability because
all children had average scores on intelligence measure,
phonological short-term span or other neuropsychological
tests. Previous studies have found lowers scores for girls
in calculation (Hein et al., 2000; Rosselli et al., 2009; von
Aster, 2000) and in number comparison (Dellatolas et al.,
2000; von Aster, 2000). Rosselli et al. (2009) attributed the
gender effect to more efficient arithmetical fact retrieval
for the boys. In our study boys and girls performed similarly
all memory tasks; so, it is unlikely that long-term memory
(results not reported in the present study) could explain the
gender contrast. Although, we did not assessed mood or
stress in our sample, Krinzinger and Kaufmann (2006)
considered that the low performance of girls in arithmetic
could be explained by a higher level of math anxiety and
to negative attitudes toward math, what is consistent with
von Aster (1994) that associated these lower scores of girls
with emotional aspects.

In the present study urban children performed the subtest
Memory of Digits of Zareki-R better than rural children.
Many studies showed that digit span task is free from
culture aspects, such as environment and socioeconomic
status (Engel et al., 2009; Koumoula et al., 2004; Santos
& Bueno, 2003; Santos et al., 2005). A possible explanation
for this incongruence is the scoring system of Zareki-R, in
that higher score in Memory of Digits cannot be interpreted
necessarily as higher span on phonological working
memory, instead higher scores are associated with the
consistence of correct response, because the child may have
a span, for instance of four items, but failure to reproduce
the three trials and consequently obtain less points than a
child with the same span but that answered correctly all
three trials. Moreover, the Memory of Digits presents a
composite score with the sum of forward and backward
trials. So, these particularities may be an artefact. In the
other hand, the correlations between Memory of Digits and
Zareki-R’s subtests were low statistically significant (.1 to
.3) and corroborate with previous studies associating
arithmetic skills and working memory (Gathercole &
Alloway, 2004; Gathercole, Alloway, Willis, & Adams,
2006; Geary, 2000; Koumoula et al., 2004; Siegel & Ryan,
1989; Silva & Santos, 2011; Swanson, 2006), which is also
consistent with the four-step Developmental Model of
Numerical Cognition (von Aster & Shalev, 2007).

In addition, it was found moderate to high positive
correlations (r > .40) between Zareki-R and the subtest
Arithmetic of the WISC-III, which demonstrated the Zareki-
R construct validity; however, in the Arithmetic WISC-III
subtest, different components of numerical cognition are
in concert, thus, it is a good measure for screening but not
accurate for diagnosis. Santos and Silva (2008) showed that
children who presented inferior scores in the Arithmetic
subtest of the School Achievement Test (SAT; Stein, 1994),
which assess only calculation according to the schooling
grade expectation, were impaired in both aspects of
mathematical skills - calculation and number processing
when assessed by the Zareki-R. Therefore, the Zareki-R
seems to be more comprehensive to assess mathematical
skills than SAT and WISC-III, and seems to be an important
instrument for the diagnosis of Developmental Dyscalculia.

The limitation of the present study is that the sample
included children from two cities of the Sao Paulo State
(Assis and Ourinhos), which restrict in part the
generalization of the results, however, the results were not
completely discrepant of the previous normative studies
with ZAREKI and Zareki-R in general aspects (Dellatolas
et al., 2000; Hein et al., 2000; Koumoula et al., 2004; von
Aster, 2000; Rotzer et al., 2009). A present goal from the
authors is to investigate if the current profile is equivalent
in different regions of the country and include children at
age 6 because recently Brazilian law anticipated the
beginning of the 1st grade of the primary school officially
from age 7 to age 6; whereas for younger children, in

SANTOS, SILVA, RIBEIRO, DIAS, FRIGÉRIO, DELLATOLAS, AND VON ASTER522



preschool level, seems to be more suitable to assess
mathematical skills by the Zareki-K (Weinhold-Zulauf,
Schweiter, & von Aster, 2003), Kindergarten adapted version
in Portuguese by Santos, Paschoalini, and Molina (2006).

In conclusion, the performance of the Brazilian children
at Zareki-R was slightly related to environmental, and mainly
influenced by age, and gender factors. In disagreement with
our previous hypothesis, rural children presented lower
performance only in one number processing aspect. It is
important to hallmark that the socioeconomic factor did not
seems to be main determinant of this difference, since the
pedagogy method produced similar performance between
rural and urban children. On the other hand, as it was
expected, children from 7 to 8 years old showed a global
inferior scores in comparison to older children; besides, boys
performed better than girls in calculation and in one aspect
of number processing. These results are important to the
development of the cognitive rehabilitation strategies.

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Received June 21, 2010
Revision received June 16, 2011

Accepted July 13, 2011

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