Precursors of mathematics learning: identification and intervention

2013/2014The studies presented in this thesis contribute to the growing body of knowledge regarding the relationships between domain general precursors, domain specific precursors and mathematical learning. Given the previous inconsistent results about the link between ANS acuity and mathematical abilities (i.e. Kolkman et al., 2013) we were interested in the investigation of the relation between the ANS and early mathematics abilities in preschool years. This relationship was explored by using two different approaches: in Chapter 2 the relationship between ANS and different mathematical measures, measured concurrently, were investigated; Chapter 3 examined the possibility to improve the ANS abilities in preschool children by using intensive adaptive training over a relatively short period. Moreover, the transfer effects of the training on mathematical abilities were also examined. The second aim of this dissertation concerned the potential role played by STM and WM skills in supporting domain-specific precursors of mathematics. The few studies that considered the role of WM in simple quantity comparison abilities did not provide a strong basis for any firm conclusions on such a relationship (Mussolin, Nys, Leybaert, & Content, 2012; Soltész, Szűcs, & Szűcs, 2010) since they didn’t assess all of the WM components. Chapter 2 focuses on uncovering which specific WM component is involved in non-symbolic approximate quantity comparison processing in the preschool age. Moreover, in ordered to further explore the relationship between WM abilities and ANS, Chapter 3 investigated whether training focusing on the improvement of ANS abilities produced a far-transfer effect on WM abilities. Several studies found that WM abilities are related to overall mathematical skills (Gersten et al., 2005; Jordan et al., 2006; Passolunghi & Lanfranchi, 2012), therefore Chapter 4 investigated a possible causal relationship between domain-general working memory abilities and domain-specific numerical competence through a training study during the preschool years. The third aim of this dissertation was to explore the malleability of cognitive precursors of mathematical learning. In Chapter 3, the possibility to improve ANS abilities was investigated, whereas Chapter 4 aimed to verify and compare the effects on early numerical competence of two types of training. One type of training focused on the enhancement of domain-general precursors, working memory abilities; while the other focused on the enhancement of domain-specific precursors, early numeracy abilities. In the field of intellectual disabilities, some studies suggested that WM skills of children with neurodevelopmental disorders (like Down’s syndrome) tend to be impaired and very poor compared to typically developing children of a similar mental age (Gathercole & Alloway, 2006). In Chapter 5, the efficacy of a school-based visuo-spatial WM training on STM and WM skills for two individuals with DS was examined. The main findings emerging from overall studies and limitations, future directions and implications of the research are finally discussed in Chapter 6.  Gli studi presentati in questa tesi forniscono nuovi risultati riguardo la relazione tra precursori dominio-generali e precursori dominio-specifici dell’apprendimento matematico. Le ricerche che si sono occupate di indagare la relazione tra Approximate Numer System (ANS) e abilità matematiche finora hanno prodotto risultati contrastanti (Kolkman et al., 2013). Ci siamo quindi interessati ad approfondire lo studio della relazione tra ANS e matematica in età prescolare utilizzando due diversi approcci. Nel capitolo 2 è stato indagata la relazione tra ANS e diverse abilità matematiche, misurati contemporaneamente. Nel Capitolo 3 è stata indagata la possibilità di potenziare le abilità di ANS in bambini in età prescolare, utilizzando un training intensivo per un periodo relativamente breve. Inoltre, sono stati esaminati gli effetti del training di ANS sulle abilità matematiche. Il secondo obiettivo di questa tesi è stato quello di indagare il ruolo svolto della memoria a breve termine e della memoria di lavoro nello sviluppo dei precursori dominio-specifici dell’apprendimento matematico. I pochi studi che hanno considerato la relazione tra abilità di memoria e abilità di confronto di quantità non sono giunti a conclusioni certe su tale relazione in quanto non hanno analizzato in maniera completa il ruolo svolto da ogni componente della memoria di lavoro (Mussolin, Nys, Leybaert, & Content, 2012; Soltész, Szűcs, & Szűcs, 2010). Con lo studio descritto nel Capitolo 2 abbiamo voluto indagare quale specifica componete della memoria di lavoro sia coinvolta nel processo di comparazione di quantità in età prescolare. Inoltre, per analizzare ulteriormente tale relazione nel Capitolo 3 abbiamo indagato l’effetto, sulle abilita di memoria di lavoro, di un training focalizzato sul potenziamento delle abilità di ANS. Dato che diversi studi confermano la relazione tra memoria di lavoro e abilità matematiche (Gersten et al., 2005; Jordan et al., 2006; Passolunghi & Lanfranchi, 2012), nel capitolo 4 abbiamo esaminato la possibile relazione causale tra le capacità dominio-generali di memoria di lavoro e precursori dominio-specifici (abilità matematiche precoci) tramite uno studio di training in età prescolare. Il terzo obiettivo di questa tesi è stato quello di esplorale la malleabilità dei precursori cognitivi dell’apprendimento matematico. Nel Capitolo 3 è stata indagata la possibilità di potenziare le abilità di ANS mentre nel capitolo 4 abbiamo voluto verificare l’efficacia di due diversi tipi di training sul potenziamento della abilità matematiche in età prescolare. Il primo training era focalizzato sul potenziamento della memoria di lavoro (precursore dominio-generale) mentre il secondo training era focalizzato sul potenziamento delle abilità matematiche precordi (precursore dominio-specifico). Gli studi nel campo delle disabilità intellettive, suggeriscono che le competenze di memoria di lavoro di bambini con disturbi dello sviluppo neurologico (come la sindrome di Down) tendono ad essere compromesse e molto scarsa in confronto alle abilità di memoria di bambini con sviluppo tipico ma stessa età mentale (Gathercole e Alloway, 2006). Nel capitolo 5, è stata indagata l'efficacia di un training di memoria di lavoro visuo-spaziale sulle abilità di memoria a breve termine e di memoria di lavoro di due persone con sindrome di Down. I principali risultati della tesi, le direzioni future e le implicazioni delle ricerche sono state discusse nel Capitolo 6.XXVII Ciclo198

Working Memory in Individuals With Down Syndrome: A Treatment Case Study

The great majority of people with Down Syndrome (DS) tend to be more limited in their short term memory (STM) and working memory (WM) abilities than other people. Moreover, research has found that individuals with DS experience relatively more difficulty storing and retrieving verbal information than they do with storing and retrieving visual information. Previous studies focusing on the enhancement of memory skills in DS generally used a rehearsal training strategy to investigate the possibility of increasing verbal memory span durably (Comblain, 1999; Conners, 2008). Therefore, these studies focused on the enhancement of shortterm memory skills. The aim of our study was to evaluate the impact of a school-based visuo-spatial working memory training on the short-term memory and working memory skills a teenage child (A.S.) with Down’s syndrome. The results showed that A.S. improved his WM and STM performance in the trained and non-trained working memory tasks proposed. These results suggest that school-based visuospatial memory training in a school setting could be both feasible and effective for individuals with DS

Improving approximate number sense abilities in preschoolers : PLUS games

Previous studies in both typically and atypically developing children have shown that approximate number system (ANS) abilities predict formal mathematical knowledge later on in life. The current study investigated whether playing specially designed training games that targets the ANS system using nonsymbolic stimuli only would improve preschool children\u2019s ANS abilities. Thirty-eight preschool children were randomly allocated to either the training or control group. For 5 weeks, 20 preschoolers (9 girls) in the training group played daily games for 10 min that included guessing and comparing numerosities, whereas 18 control children (6 girls) were involved in interactive picture book reading sessions. Children\u2019s ANS abilities were assessed using a computerized task before and after the training program. An analysis of covariance with posttraining ANS scores as dependent variable and pretraining scores as a covariate showed that the children in the training group had higher ANS abilities after the training, in contrast to children in the control group. This study provides evidence that ANS abilities can be improved in preschool children through a daily training program that targets the ANS specifically. These findings provide support for further training programs for preschool children who show mathematical difficulties early on in life

The contributions of working memory domains and processes to early mathematical knowledge between preschool and first grade

Working Memory (WM) plays a crucial role in supporting children\u2019s mathematical learning. However, there is no consensus on the relative contributions of different WM domains (i.e., verbal, visuo-spatial, and numerical\u2013verbal) and processes (i.e., low-control and high-control) to mathematical performance, specifically before and after the onset of formal education. This cross-sectional study examined the relations between WM domains and processes and early mathematical knowledge, comparing a group of children in the second year of preschool (N = 66) to a group of first graders (N = 110). Results of multigroup path analysis showed that whereas visuo-spatial low-control WM significantly predicted early mathematical knowledge only among preschoolers, verbal low-control WM was a significant predictor only among first graders. Instead, the contribution of visuo-spatial high-control WM emerged as significant for both age groups, as well as that of numerical\u2013 verbal WM, although the latter to a greater extent among preschoolers. These findings provide new insights into the WM domains and processes most involved in early mathematical knowledge at different developmental stages, with potential implications for the implementation of age-appropriate training interventions targeting specific WM skills before and after the onset of formal education

Cognitive abilities as precursors of the early acquisition of mathematical skills

Specific role of the domain-general and domain-specific precursors in the development of early mathematics is still under debate. In addition, there is no general agreement about the detection of the main components of domain-specific precursors. The aim of the present work is to get more insight in the role played by a fundamental domain-general precursor, the working memory, in the development of mathematical skills. Other goal for this paper is to propose a general framework for the classification of domain-specific precursors of mathematical learning

Approximate Number System (ANS) acuity, mathematical ability and inhibitory control: A longitudinal perspective from First to Second grades

Recent research has turned increasing attention to the cognitive underpinnings of early mathematical competence. The Approximate Number System (ANS) is a non-verbal primitive cognitive system that allows to represent and estimate numerical quantity in an imprecise and intuitive way. It is present at birth, is amodal and it supports basic numerical computations like adding, subtracting and comparing quantities without using counting or numerical symbols. ANS acuity is the degree of precision of the internal quantity representation and there are considerable individual differences in ANS precision. Recent findings demonstrating a relationship between ANS acuity and mathematical ability have suggested that the ANS plays a foundational role on later-learned math skills. However, not all studies have found links between the ANS and mathematical ability in children and the evidence for a relationship in adults is mixed. It is still unclear if the relationship between the ANS and mathematical ability found in some studies could be mediated by other general cognitive abilities. Indeed, recent research findings suggest that this relationship may be the result of inhibition skills, rather than the precision of nonsymbolic representations. The primary purpose of this 2-year longitudinal study was to investigate the relationship between the ANS and mathematical ability, analysing the data separately for congruent and incongruent dot comparison task trials. We measured ANS acuity, verbal intelligence and different aspects of numerical and mathematical competence in 110 children twice, in first and second grade (Time 1 and Time 2), a year apart. ANS acuity was assessed with a computerized non-symbolic numerical comparison task. At Time 1, correlational analyses indicated that ANS acuity was associated with mathematical ability. This association was significant for both congruent and incongruent trials of the dot comparison task. In particular, it was found that the subgroup of children with higher ANS performance showed better mathematical ability than the subgroup of children with lower ANS performance. Also at Time 2, correlational analyses suggested an association between ANS acuity and mathematical ability and this positive correlation was significant for both congruent and incongruent trials again. These results are in contrast with the hypothesis that the relationship between the ANS and mathematical ability may be an artefact of the inhibitory control demands of incongruent trials in the dot comparison task. This study thereby suggests a tight link between the ANS and mathematical ability

¿Qué necesitamos para aprender a multiplicar?: El rol de las habilidades numéricas básicas y la ansiedad

The mastering of single-digit multiplication is a challenge for many children. Recent research has suggested that some basic numerical skills, such as those measured in Arabic numeral comparison tasks (i.e. access to numerical representations from symbols), are good predictors of children’s performance on single-digit multiplication. On the other hand, emotional variables, such as math anxiety, play a role in learning arithmetic facts. The aim of the current study was to study the role of several basic numerical cognitive skills and emotional factors in single-digit multiplication solving in a longitudinal study including school-age children. Math anxiety, trait anxiety, non-symbolic numerical comparison, Arabic numeral comparison, and subtraction fluency were assessed in first-, second-, and third-grade Spanish children, (N = 141; boys = 66, girls = 75; age range: 6 – 9 years). We assessed multiplication fluency 3 years later. The results showed that performance on Arabic numeral comparison and subtraction tasks significantly predicted performance on multiplication tasks. This finding supports previously published results. Regarding anxiety factors, it was also found that the participant’s score in trait anxiety, but not math anxiety, was a marginally significant predictor. The results highlight the relevance of access to numerical semantic codes from symbolic representations and fluency in combining these representations in subtraction in predicting children’s performance in single-digit multiplication. Anxiety factors seem to be of less relevance.Llegar a dominar las multiplicaciones de un solo dígito es un desafío para una gran cantidad de niños y niñas durante la educación primaria. Investigaciones recientes han sugerido que ciertas habilidades numéricas básicas, como la medida en tareas de comparación de arábigos, son buenas predictoras del rendimiento de los niños y niñas en multiplicaciones de un solo dígito. Por otro lado, otros estudios encuentran que ciertas variables emocionales, como la ansiedad ante las matemáticas, afectan al aprendizaje de los hechos aritméticos. El objetivo de esta investigación es estudiar el papel de varias habilidades numéricas básicas junto con factores emocionales en la resolución de multiplicaciones de un solo dígito en un estudio longitudinal con niños y niñas en edad escolar. La ansiedad ante las matemáticas, la ansiedad-rasgo, la comparación numérica no-simbólica, la comparación de arábigos y la fluidez al resolver sumas y restas se evaluaron en niños y niñas españoles de primer, segundo y tercer curso (N = 139, hombres = 66, mujeres = 73, edades 6 a 9). Tres años después se evaluó la fluidez al resolver multiplicaciones. Los resultados mostraron que la fluidez en multiplicaciones fue predicha significativamente por el rendimiento en las tareas de comparación de arábigos y fluidez en restas, lo que respalda hallazgos anteriores. Entre las variables de ansiedad, la puntuación de los participantes en la ansiedad-rasgo resultó ser un predictor marginalmente significativo, no así la ansiedad ante las matemáticas. Los resultados señalan la importancia del acceso a los códigos semántico-numéricos a partir de las representaciones simbólicas y las habilidades para manejar estas representaciones de forma fluida en restas, en la predicción del rendimiento en las multiplicaciones. Las variables de ansiedad, sin embargo, parecen jugar un papel menor