Introduction

Low arithmetic achievement or LA (^{Mazzocco, 2007}) tends to be transitory and mainly determined by environmental factors (^{Mazzocco & Räsänen, 2013}; ^{Räsänen, Salminen, Wilson, Aunio, & Dehaene, 2009}). It is a common condition in the school environment and may be worsened, if there is 'continuous progression in primary school', a measure applied in the Brazilian education system where children can progress in certain grades without having achieved the planned aims (^{Menezes & Santos, 2001}).

Importantly, LA can mask the presence of mathematical learning disability, which on the contrary is inherent. For instance, ^{Silva and Santos (2011}) and ^{Arias-Rodríguez, Nascimento, and Santos (2017}) researched several aspects of numerical cognition and working memory in children with an LA profile, who showed deficits in number processing, calculation and visuospatial working memory. However, deficits of some children in the sample (57.14%) were compatible with Developmental Dyscalculia (DD), since elementary performance of addition, subtraction, multiplication and division was at least 1.5 standard deviation below that expected for age and corresponded to a delay of two years of school grade taken; and this difficulty was not linked to inadequate teaching or intellectual disability neither to sensorial deficits (^{ICD-11, WHO, 2018}; ^{DSM-V, APA, 2013}). In fact, the persistence and severity of equivalent deficits are criteria for the diagnosis of DD (^{Kaufmann et al., 2013}).

The International Consensus indicated that when deficits in math are attributed exclusively to numerical cognition dysfunctions in the absence of comorbidities with other disorders and of impairments in other cognitive abilities, they should be classified as primary DD (^{Kaufmann & von Aster, 2012}; ^{Kaufmann et al., 2013}). However, children with deficits in numerical cognition linked with non-numerical cognitive dysfunctions, e.g. in working memory (^{Pina, Fuentes, Castillo & Diamantopoulou, 2014}) and those with comorbid neurodevelopmental disorders should be classified as secondary DD (^{Kaufmann et al., 2013}).

^{Devine, Soltész, Nobes, Goswami, and Szűcs (2013}), based on 17 epidemiological studies indicated that prevalence of DD is around 5-6%, their survey carried out with 1004 British children showed an equal presence of the disorder in boys and girls, depending on the inclusion criteria (^{Devine, et al., 2013}). The study by ^{Bastos, Cecato, and Martins (2016}) analysed the prevalence of DD in Brazilian primary school children (n = 226) and concluded that 7.8% met disorder criteria. Results analysis showed a higher prevalence among male children from schools in more vulnerable neighbourhoods, from parents with little schooling. Such sample characteristics suggest the presence of children with an LA profile among those with DD (^{Kaufmann et al., 2013}; ^{Mazzocco & Räsänen, 2013}).

Children with DD are vulnerable to risks such as persistent learning difficulties, school abandonment and emotional instability (^{Snyder & Bambara, 1997}), which can permanently impair their emotional functioning, schooling and future professional career (^{Herrera & Maldonado, 2002}). These children may acquire a negative attitude towards the subject, often characterized by a specific anxiety to mathematics (^{Krinzinger, Kaufmann, & Wilmes, 2009}; ^{Ramírez, Chang, Maloney, Levine, & Beilock, 2016}; ^{Rubinsten & Tannock, 2010}). Therefore, in these cases remediation is a sustained challenge.

Computerized training for numerical cognition in children with DD currently seems effective both for number sense (^{Wilson, Revkin, Cohen, & Dehaene, 2006}) and for the mental number line (^{Kucian et al., 2011}). Studies indicate remediation alternatives in the field of neuroscience (Gracia-Bafalluy & Escolano-Pérez, 2014); Numerical cognition tutorial (^{Iuculano et al., 2015}), although these interventions are generally individual. However, music can stimulate mathematics learning without using routine school activities, thus being more ecological and inclusive (^{Katsh & Merle-Fishman, 1985}). According to ^{Eugênio, Escalda and Lemos (2012}), musical learning always provides mathematical challenges, since the basis of musical theory and sound acoustics follow mathematical calculations.

The experience of musicalisation, for example, showed beneficial effects toward working memory. ^{Ribeiro and Santos (2012}) assessed children of 9 and 10 years of age, beginners and advanced in a musicalisation programme for one hour per week, compared to a control group that did not perform any regular and monitored musical activity. The evaluation of the working memory showed that children more exposed to the music programme presented better scores than beginners and no-exposed participants in both short-term memory tasks and working memory. This effect was seen in verbal and visuospatial modality tasks. This is due to the fact that music, language, reading and mathematics share some acoustic properties such as pitch, rhythm and timbre, so, stimulated by music, children can learn these concepts faster (^{Anvari, Trainor, Woodside, & Levy, 2002}; ^{Ribeiro & Santos, 2015}).

^{Koelsch and Siebel (2005}) pointed out that musical training produces a cognitive effect because it activates complex brain functions such as: acoustic analysis, auditory memory, transformation of interval relationships, syntax and musical semantics, as well as activation and representation of actions. In addition, contact with music elicits emotions, which stimulate the subjective sensations, autonomic nervous, hormonal and immune systems, bringing a feeling of well-being (^{Koelsch and Siebel (2005}).

Prospective studies observed different cognitive effects depending on the musical training characteristics. For 'instrumental musical training', one requiring regular keyboard or piano practice, benefits were found in the processing of verbal and visual working memory (^{Ho, Cheung, & Chan, 2003}), reasoning (^{Moreno et al., 2011}; ^{Schellenberg & Mankarious, 2012}) and visuospatial skills (^{Costa-Giomi, 1999}; ^{Rauscher & Zupan, 2000}). In 'non-instrumental musical training', where musical experiences with voice and perception of sounds predominate, benefits derived from the musical experience were observed in skills such as abstract reasoning, numerical cognition and working memory (^{Anvari et al., 2002}; Rauscher et al., 1997; ^{Ribeiro & Santos, 2012}; ^{Ribeiro & Santos, 2015}; ²⁰¹⁷; ^{Silva, Baldin, & Santos, 2017}). Students taking part in non-instrumental musical training linked to the learning of abstract representation and geometry, outperformed peers with little or no musical experience in tasks involving numerical and visuospatial mastery and improved performance in mathematics.

Other evidence suggests that music training can be a tool in the remediation of learning disorders in mathematics (^{Ribeiro & Santos, 2017}) or its prevention (^{Silva, Baldin, & Santos, 2017}). ^{Esteki (2013}) evaluated the effect of musical stimulation for the neurocognitive structuring of 13 girls, between 7 and 9 years old, diagnosed with DD. Following the 16 musical sessions, there were changes in the activities of different regions of the brain, particularly the parietal and frontal lobes, together with improvements in tasks regarding the intelligence coefficient (verbal and non-verbal), attention and numerical memory, in contrast with pre-training assessment. However, the authors do not detail the training methodology, or the cognitive activities assessed, which limits comparisons with future studies.

Adopting a structured and replicable methodology, ^{Ribeiro and Santos (2017}) evaluated the effect of non-instrumental musical training on the numerical cognition of children with DD. The study involved 46 children of 8 years old, with seven rhythmic sessions and seven auditory sessions, divided into four mixed groups. The pre-training cluster analysis divided participants into groups, one clinical and one control, coinciding with the numerical cognition battery scores specialized in the diagnosis of DD. However, in the post-training cluster analysis, some children in the clinical group, who increased their performance in numerical cognition, became identified among the controls.

^{Silva, Baldin, & Santos, 2017}) researched the effects of the numeracy musical training (NMT), a specific non-instrumental music training programme designed to stimulate numerical cognition. It was a prospective, randomized and controlled study in a sample of 57 preschoolers with typical development, aged 5 years. The Experimental group (n = 25) participated in eight NMT sessions while the control group (n = 32) did not take part in training but remained in school in regular pedagogical activities. All participants were assessed in abstract reasoning, verbal and visuospatial working memory and numerical cognition in two moments. The results of the post-training assessment of the experimental group suggested progress in learning skills, specifically in mathematics and working memory compared to pre-training, which were not observed in the control group.

The present research is the first study using the NMT for a sample of school children with low achievement in mathematics. In order to test this methodology in a more ecological way, NMT was performed in mixed but balanced groups, i.e., children with and without difficulties in mathematics. This approach has already been adopted in other case-control studies (^{Kucian et al., 2011}; ^{Michels, O'Gorman, & Kucian, 2017}) where controls also took part in the training. The collective training promotes inclusive behaviour and socialization during the intervention (^{Silva, Baldin, & Santos, 2017}).

In this way, the aim of this study was to investigate NMT effects on school performance, numerical cognition (number sense, number production and comprehension, calculation), working memory and behavioural aspects in school-age children with low or medium academic performance through neurocognitive assessments at two moments, Pre and Post NMT. Study hypothesis is that NMT will produce near transfer effects in arithmetic systems (explicitly stimulated) and less expressive far transfer effects in other non-stimulated cognitive abilities, such as visuospatial reasoning.

Method

Participants

Initially, 310 children were recruited for the study, all from two public schools in Bauru, São Paulo State, Brazil. The participating centres reached the national average in the Development Index of Basic Education (Ideb) of 2011, with a higher grade than 5.0 in the initial grades (first to fifth) corresponding to the OECD international level, based on evaluation of the Education Development Index of the State of São Paulo (IDESP) of 2013. The city has an HDI - Human Development Index of 0.80 classified as high (UNDP, 2010).

The case group, labelled as LA, met the following inclusion criteria: (a) mathematics difficulties reported by the child's teacher, b) cut-off grade in arithmetic test (^{Stein, 1994}) <9 (third grade children) or <14 fourth grade children), provided in the School Performance Test handbook, as a result lower than expected for the grade taken, however, with expected abstract reasoning for their chronological age. Twenty-one children fulfilled the criteria, corresponding to 6.7% of the sample. Then a control group was matched to them.

For the control group, labelled as AA, besides general criteria, inclusion criteria were not repeating a school grade, no learning disorders, nor complaints of learning difficulties. For both groups, exclusion criteria were the presence of previously diagnosed psychiatric or neurological comorbidities and the use of psychoactive substances.

Participants were 42 children, 18 boys and 24 girls, aged between 8 and 10 years, were selected for the study. Based on screening results, i.e. the measure of school performance in arithmetic, b) children were assigned in two groups: low arithmetic achievement (LA = 21) and average arithmetic achievement (AA = 21). Portuguese was the mother tongue of all children and they did not receive independent music classes for music training, nor did the schools have a music teacher. Data is presented in Table 1.

Table 1.  Sociodemographic aspects of the sample. 

Results

In clinical records, no significant differences were found between the LA and AA groups in birth and weight variables. As regards neuropsychomotor development, LA children took longer to acquire cervical control (t = 3.08, p = .04, d = .98) and trunk (t = 2.04, p = .04, d = 2.90). Learning to walk and language were similar in comparison. There were similarities between groups regarding the mother's personal history; presented for each group: 2 cases of hypertension, 1 threat of miscarriage and 1 haemorrhage, representing 19.05% of incidence for each group. As for socioeconomic status, families receive the equivalent of about two monthly minimum wages in Brazil (salary: M = 1680.00, SD = 134.40, or ⸗ 394 €).

In Abstract Reasoning analysis the Percentile showed time effects, reporting an improvement for both AA and LA groups in the second evaluation (ps = .01), neither Group effect nor interaction effect were found.

Regarding Numerical Cognition, the Zareki-R subtest was analysed to research the implications of exposure to numerical cognition training in children and scores (Mean (standard deviation)) are presented in Table 2. The only tasks where no differences were found between pre and post-NMT evaluations nor between LA x AA groups nor interactions between groups and evaluations were the counting dots and perceptive estimation.

Table 2.  Scores (Mean (Standard Deviation)) obtained by groups LA and AA for abstract reasoning, School Achievement Test and ZAREKI-R. 

In subtests, counting backward, dictation of numbers and reading numbers, the LA group had lower scores compared to the AA group and differences were significant. The pre- and post-NMT comparisons showed improvements given by the superior results in the second evaluation. The results of the interactions between groups and evaluations indicate that although both groups achieved better scores in these analysis areas in the second evaluations, differences remained between the groups, the AA children presented higher results than LA children (ps = .01).

There were differences between the LA and AA groups for the oral comparison subtest: for the other tasks, written comparison and contextual estimation, results were similar for LA and AA. On the other hand, there was a time effect in comparison between NMT pre - post evaluations for the three subtests. However, no significant interactions were found for either three subtests, the groups maintained similar results among themselves, obtaining better scores in post-training evaluation (ps <.01); written comparison showed better scores in the post-NMT evaluation where AA children presented higher results than their LA peers (ps = .49). For contextual estimation, significant differences were found between the pre- and post-NMT evaluations. In the first, the LA children showed lower results to their peers, which were compensated in the second evaluation where both groups of children obtained similar results. (ps <.01)

For the subtest related to Calculus, there was a group effect where the performance of children with AA was higher than children with LA. Analyses also reported expressive improvements in post-NMT evaluations: for mental calculation and problem solving, accompanied by group interactions and evaluations for both. Even when significant improvements in the post-NMT evaluation were perceived, the results of LA children were much lower than those of their AA peers, AA children remained with similar scores in both calculation evaluations (ps <.01) and problem solving (ps <.001).

The result of the total Zareki-R score, which is a parameter for diagnosis of Developmental Dyscalculia, in pre-NMT evaluation it was seen that 100% of participants with LA had lower than expected results with scores equivalent to the delay of one school grade. In addition, 13 cases (61.90%) of the 21 LA children (N = 5 of third grade and N = 8 of fourth grade), did not achieve expected results for two previous school grades. There were more severe deficits for 3 of the 16 children in the fourth grade whose results did not reach those expected for children with typical development from the first grade of primary education.

In post-NMT evaluation, 6 of the 21 (28.47%) LA children achieved expected results for their age, 15 of the 21 LA children (71.43%) continued without reaching the expected average for their school grade, 12 of whom (57.14%) were below average for a previous grade, only 5 children showed moderate to severe losses (23.81%), three children from the third grade and two from the fourth grade, these latter two (N = 5) had a performance that from a clinical viewpoint corresponds to DD (^{Kaufmann et al., 2013}). From the AA group 100% was initially the average expected for their age and 100% remained in the mean (^{Santos et al., 2012}).

Regarding School Achievement, children from both groups showed better in the re-evaluation of the Writing, Arithmetic and Reading subtest, as well as in the total result, compared to performance in the first evaluation (Table 2).

Detailed analysis of the subtest evaluating school achievement reported lower results from the LA group in relation to the AA group; however, the most representative advances in Post-NMT evaluation were obtained precisely by the LA children, Writing Subtest (ps <.001), Arithmetic Subtest (ps <.001), Reading Subtest (ps = .001). Accordingly, comparisons of the total row score also showed significant differences, where the LA group showed inferior performance compared to the AA group, the LA children always achieved lower results than their AA peers, but the most representative gains in Post-NMT evaluation were obtained precisely by that group of children (ps <.001).

Analysis of the composites of Working Memory showed that children with LA obtained higher gains than their peers with AA after NMT for verbal short-term memory, however in the Pre and Post-NMT assessments the LA group obtained lower results than the AA group (ps <.001). For visuospatial short-term memory, the AA group maintained similar results in the pre- and post-NMT evaluations, but were always better compared to children with LA, despite the fact that they had greater gains in the Post-NMT evaluation (ps <.001). For verbal working memory there was a group effect, the LA group obtained inferior results to the MR group in the two evaluations, the perceived gains in the post-NMT evaluation were proportionally the same for both groups (ps <.001). For the visuospatial working memory, despite improvements in the scores of the LA children in post-NMT evaluation, they still presented lower results than those obtained by their AA peers who maintained equivalent results in the two evaluations (ps <.001). The mean scores and standard deviation are presented in Table 3.

Table 3.  Scores (Average (Standard deviation)) obtained by LA and AA groups in AWMA battery. 

In Behavioral Scales, specifically in the Child Stress Symptom Inventory, children with LA showed more evidence of physical stress reactions compared to their MR peers, although these differences between groups were not significant, there was a decrease in physical reactions after the NMT for both groups (ps = .14); the same for psychological reactions (ps = .60); as for psychological reactions with depressive components (ps = .98) and psychophysiological stress reactions (ps = .48), there were significant differences between groups, the LA children showing more pronounced stress reactions than AAs, which was reduced in the post-NMT evaluation. However, some children showed signs of stress distributed as follows: 5 children from the AA group and 11 from the LA group, representing 23.81% and 52.38%, respectively.

Regarding the Anxiety Scale for Children, a decrease in math anxiety was observed in both groups of children in post-NMT assessment, the LA children were slightly more anxious than the AA children in the NMT - pre and post assessments (ps = .02).

The results of the Self-Efficacy Scale for Children indicated that those from the LA and AA groups self-evaluated in the same way in the academic, social and emotional areas, improvements were seen in post-NMT evaluation for social self-efficacy (ps = .58). (Data from the three behavioural scales presented in Table 4).

Table 4.  Scores (Average (Standard deviation)) obtained by LA and AA groups in Behavioural Scales. 

Discussion

NMT (numeracy musical training) was originally employed to stimulate the numerical cognition of preschool children (^{Silva, Baldin, & Santos, 2017}). In the present study this programme has been tested as an intervention to ameliorate numerical cognition deficits in children with low performance in mathematics. The groups were paired in socioeconomic level, abstract visual reasoning (intellectually upper middle) and age. For this purpose, we carried out a case-control, double-blind, and prospective study contrasting school age children with low (LA) and average arithmetic achievement (AA) in cognitive abilities, evaluated in two moments: pre-NMT and post-NMT. Findings suggest that 8 NMT sessions produce near transfer (counting backward, dictation and number reading, mental calculation and problem solving) and partially a far transfer (verbal working memory). However, the findings of far transfers should be met with caution (^{Melby-Lervåg & Hulme, 2013}; ^{Sala & Gobet, 2017}) considering circumstances of the research discussed further.

Based on Brazilian normative data for numerical cognition (^{Santos et al., 2012}), the LA children found difficulty in performing addition and subtraction tasks in pre-NMT, in understanding numerical symbols and their reproduction; as well as severe deficits in calculation which are substantial for diagnosis of DD (^{Kaufmann & von Aster, 2012}; ^{Kaufmann et al., 2013}; ^{Kucian et al., 2011}). Consequently, the present sample is expected to present low response to training since deficits of numerical cognition are usually persistent in children with DD (^{Mazzocco, & Räsänen, 2013}). It should be noted that the persistence of deficits in the systems of numerical cognition following intervention is a confirmatory element in the diagnosis of specific learning disabilitity (^{Kaufmann et al., 2013}).

Overall numerical cognition in the first evaluation showed that LA children obtained worse scores compared to AA children for nine Zareki-R subtests. But impairment reduced for seven subtests in the second evaluation. In fact, 28.47% of children obtained expected results for their age in the second evaluation, according to the normative study (^{Santos et al., 2012}). Their School Achievement score after NMT was congruent to this, some children on LA group still showed results below those expected for the grade. However, children from both groups generally achieved better scores in school performance, even in the arithmetic subtest. Usually business as usual do not produce significant changes in cognitive performance in such small interval of time, suggesting a possible effect of the intervention (^{Katsh & Merle-Fishman, 1985}).

As for counting backwards, reading and dictation of numbers, as well as mental calculation tests and problem solving, the performance of the LA group even after intervention, was lower in relation to normative data (^{Santos et al., 2012}). On the other hand, according to interactions, NMT promoted enhancement in children as regards these tasks. Results are consistent with the findings of ^{Spelke (2008}) and ^{Ribeiro and Santos (2017}) suggesting that NMT helps develop logical-mathematical reasoning (^{Rauscher et al., 1997}) and the manipulation of symbols (^{Smedt, Noël, Gilmore, & Ansari, 2013}; ^{von Aster & Shalev, 2007}). These findings are promising, since previously advance has been seen after only 8 NMT sessions. Although it must be noted that in the study by ^{Silva, Baldin, & Santos, 2017}) children performed NMT as a way of stimulating skills, without there being deficits; in the present study children with LA were initially below expectations.

According to medical records, the observed gains do not appear related to the ages of the parents in years, level of schooling and work situation, neither to complications during pregnancy, childbirth (birth weight and prematurity) as there were no differences between groups in this regard. In addition, socioeconomic level seems not to have influenced the results of the neuropsychological evaluations as groups were equivalent (^{Arias-Rodríguez, Nascimento, & Santos, 2017}; Santos et al., 2012).

Despite the musical intervention offered, some aspects of numerical cognition in children with DD were resistant to treatment (Ribeiro, 2013; Ribeiro & Santos, 2017), suggesting that perhaps more intervention time is required, e.g. Ribeiro and Santos (2017) obtained improvements in children with DD after 14 sessions of music training. For instance, we did not observe group effect, time or interactions in the counting dots and perceptive estimation tasks, achieving the expected score for age. Although there were differences between groups in the written comparison and contextual estimation tasks, there were no improvements in the post-test of these specific skills.

Regarding far transfers, two cognitive measures of our protocol must be accounted. Pre-and post-NMT comparison showed progress regarding abstract visual reasoning similar to previous studies (^{Moreno, et al., 2011}; ^{Rauscher & Zupan, 2000}; ^{Schellenberg & Mankarious, 2012}), however there was no interaction, consequently no direct relationship with music training can be inferred. As for working memory, the findings of the present study show that the LA group was inferior to the AA group in the two evaluations. Deficits in working memory are actually common in children with numerical cognition dysfunction (^{Alloway & Passolunghi, 2011}; ^{Geary, 1993}), and often observed in children with secondary DD (^{Kaufmann, et al., 2013}). Despite better scores in post-training, no interactions were seen for the visuospatial short-term and working memory tasks, because the LA group maintained a worse performance in post-test compared to the AA group (^{Ribeiro & Santos, 2012}). The benefits of NMT were limited to the verbal component of the working memory, corroborating the meta-analysis of ^{Melby-Lervåg and Hulme (2013}), and partially the study by ^{Ribeiro and Santos (2012}), where better scores were observed for storing information and processing control following musical stimulation.

As for results of behavioural scales, groups did not differ statistically nor present clinical symptoms of depression, stress, or low self-efficacy in the post-NMT evaluation; except some children from both groups who showed signs of stress (^{Lipp & Lucarelli, 1998}), perhaps because the school year final exams coincided with the post-NMT evaluations. An important finding was the decrease mathematical anxiety in children from both groups after NMT evaluation suggesting that music has an impact on emotional aspects, bringing about improved self-esteem and self-confidence, providing self-discipline and promoting motivation for learning in general (^{Gan, Lim, & Haw, 2016}). However, it can be inferred that extending NMT exposure time could provide greater progress for children with DD (Ribeiro, 2013).

Conclusions

After the 8 group sessions of NMT we observed a decrease in mathematics anxiety and improvements in counting backwards, reading and dictation of numbers, arithmetic as well as word problem solving, and to a lesser extent in verbal working memory. Despite evolving in relation to deficits presented in the first evaluation, children still showed problems in some areas such as calculation, and in working memory tasks (non-numerical cognitive dysfunctions). Certain difficulties (e.g. oral and written comparisons, contextual estimation) were resistant to treatment, at least with a short duration. This profile suggests the diagnosis of secondary DD. In conclusion, NMT provided beneficial effects in the performance of numerical cognition tasks for children with DD.