Mathematics Experience and Format-specific Effects in Numerical Cognition

A persistent issue in numerical cognition research is how the format of numerical information influences numerical processing. The format-independent view postulates that information from various formats (e.g. ‘3’ or ‘three’) is represented in a uniform numerical code and that format should thus have no influence on number manipulation. The format-specific view assumes separate representational pathways for arabic digits and number words, which come into play during number processing as well as manipulation. Five experiments were undertaken with methods ranging from behavioural measures of reaction time to more refined measures of cognitive processes such as eye-tracking and eventrelated potentials (ERPs). In each experiment, effects of format were investigated at different levels of mathematics experience, in order to examine how the processing of numbers might differ in this regard. The first three experiments focused on basic number processing and processing differences that can occur for arabic digits, number words and quantifier words. In Experiment 1, a modified counting Stroop task was employed to investigate cognitive interference of arabic digits and number words. Participants took longer to respond on incongruent trials (e.g. 4 4 4; how many numbers are present? Correct response: ‘3’) relative to neutral (e.g. * * *; Correct response: ‘3’) and congruent (e.g. 3 3 3; Correct response: ‘3’) trials. Individuals with high mathematics experience showed greater interference on digit trials, whereas no effect of mathematics experience was found for word trials (e.g. three three; respond ‘2’). This suggests that the influence of format on number processing can be regulated by mathematics experience. Experiment 2 investigated this effect further by considering numerical (e.g. 5 2; which number is higher?) and physical size (e.g. 5 2; which number is physically bigger?) comparisons of digit and word stimuli. For both formats, participants responded faster on trials with a large numerical distance (e.g. 2 7) compared to trials with a small numerical distance (e.g. 2 3) suggesting that specific number meanings are accessed spontaneously from digits and number words, however the size congruity effect only occurred for digit stimuli. Individuals with greater mathematics experience showed an overall advantage for numerical comparison, regardless of format. Based on the findings from Experiments 1 and 2, Experiment 3 modified the counting Stroop task (Experiment 1) to investigate if mathematics experience can be related to the processing of quantifier words (e.g. many, few, each). Stimuli were presented as either specific (e.g. both both; correct response ‘2’) or general (e.g. some some) quantifier words and participants were required to count the items on-screen. While the effects were minimal in comparison with Experiment 1, any effects related to the congruity of the stimuli only emerged for the highly mathematics experienced participants, suggesting the involvement of number experience in quantifier word processing, and in turn for extracting number meaning from language in general. As the first three experiments demonstrated format-specific effects in basic number processing, the second part of the thesis investigated these effects for more advanced numerical processing such as arithmetic. The second part of the thesis also employed more refined measures of cognitive processing (eyetracking and event-related potential [ERP] technology) to investigate effects that might not be evident from behavioural data alone. Experiment 4 employed eye tracking technology to compare effects of problem size, operation and format at different levels of mathematics experience. Fixation patterns supported the format-specific view of number processing by suggesting that in comparison with digit-format, word-format impeded the use of direct memory retrieval in arithmetic, an effect that seemed to be more pronounced for individuals with low mathematics experience. Eye-tracking data also supported behavioural data as well as self-report data that have been noted in reports on strategy use in arithmetic. From this, inferences were made regarding the degree to which surface format influences subsequent calculation processes and how this might be moderated by mathematics experience. Experiment 5 investigated the interaction between the encoding and answer-retrieval stages in digit- and word-format arithmetic by separating the presentation of the first operand and the rest of the equation in a true–false verification task (e.g. ‘3’ and ‘x 4 = 12’; correct response ‘true’). Before each test block, participants were told which operation was to follow (addition or multiplication). ERP findings suggested that operands presented in the same format were encoded in the same way, with effects of operation only emerging during the second part of the equation, after participants had seen the operation sign (‘+’ or ‘x’). Regardless of format, the High Maths group showed greater left anterior potentials for multiplication than addition, suggesting an advantage for arithmetic fact retrieval. In the final chapter of the thesis the findings are discussed in relation to existing theoretical accounts on the influence of format in numerical cognition, with specific focus on the benefit of considering mathematics experience in this regard

Perceptions, difficulties and working memory capacity related to mathematics performance

There is a general view that students of not have a positive attitude towards mathematics. In general, mathematics is considered a ‘difficult’ subject and sometimes there is a lack of enjoyment. Mathematics is often portrayed as being abstract and unrelated to life. In the light of the key role mathematics has in the curriculum, the aim of this study is to explore the difficulties and self-perceptions of students aged about 10-12 in Pakistan as they undertake their studies in mathematics. The study uses a survey of student perceptions, working with samples of students drawn from both Urdu and English medium schools (N = 813). In addition, working memory capacity of those in grade 5 (age about 10) was measured and information was gained about their performance in mathematics examinations. The data is analysed to consider how their self-perceptions related to their experiences in learning mathematics which varies with age, language background and gender. Any relationships between these self perceptions, mathematics marks and measured working memory capacity are explored as well. The observed outcomes can be used to inform the agenda for action or further study. It was found that the vast majority (English medium and Urdu medium) appreciate the role and the importance of studies in mathematics although topics like geometry, fractions, topics with life applications, statistics are causing problems. It is almost certain that these topics place demands on working memory which make understanding very difficult. In the Urdu medium schools, the curriculum in grade 6 is clearly causing major problems while, in both systems, pressures for success based on examination performance have generated a complete industry of private tutors. Many of the gender differences can be interpreted in terms of the social roles in Pakistani society. However, girls do seem more positive and more committed in relation to their studies in mathematics. The study has revealed two major issues which need careful consideration. One is the whole issue of memorisation and understanding. The goal of meaningful learning must be stressed more if positive attitudes are to be retained. The whole issue of making the mathematics studied become related in some way to the lifestyle of the learner seems very important but this is not easy without overloading working memory. In considering both of these issues, the critical role of assessment has to be addressed: if assessment offers rewards almost entirely for the recall and correct execution of mathematical procedures, then this will be reflected in textbooks and teaching approaches. Along with curriculum design and teaching approaches which are consistent with the known limitations of working memory, assessment is perhaps the single most important issue to be considered. Very significant correlations were found for grade 5 students when their measured working memory capacity was related to their mathematics examination performance. Indeed, the correlation value for Urdu medium students is the highest such correlation which has been found in any discipline. This suggests major curriculum design problems in the national syllabus for Urdu medium schools as well as assessment problems. The study has pinpointed many areas of success along with specific areas where there are serious problems. In this way, an agenda for future research and action has been described

The morphology of the intraparietal sulcus in children prenatally exposed to alcohol in a sample of children from the Western Cape, South Africa and its potential relationship with number processing

The intraparietal sulcus (IPS) is a prominent feature in the parietal lobe and extends posteriorly from the postcentral sulcus through the parietal lobe to end in the occipital. It is involved in visuospatial functions and is known to play a critical role in number processing. Fetal alcohol spectrum disorders (FASD) result from prenatal exposure to alcohol and are particularly prevalent in the Western Cape region of South Africa. Arithmetic is a domain of cognitive function that is particularly sensitive to prenatal alcohol exposure, and effects on arithmetic remain significant after controlling for lower IQ. Magnetic resonance imaging (MRI) was used to investigate the morphology of the IPS and whether this morphology had a relation to the number processing abilities of children prenatally exposed to alcohol in a Western Cape community. Participants were 9 to 14-year-old children from the same community in Cape Town, South Africa, who formed part of a study aimed at investigating the effects of prenatal alcohol exposure (PAE) on brain structure and function particularly during number processing. Mothers were interviewed regarding alcohol consumption during pregnancy using a timeline follow-back approach. The first analysis included designing a protocol for manually parcellating the IPS into two regions of interest (ROI): the medial wall (MIPS) and the lateral wall (LIPS) respectively. The neuroimaging program MultiTracer was used for the manual tracing and to calculate the volume of the cortex of both the MIPS and LIPS. The purpose of this first analysis was to examine the effects of PAE on IPS volume and asymmetry using manual tracing, the relation between IPS volume and number processing performance, and potential moderation by PAE of the relation between IPS volume and number processing performance. Results indicated that when comparing the FAS/PFAS (Fetal Alcohol Syndrome/Partial FAS) children to the controls, PAE had an effect on the left LIPS and higher arithmetic scores were associated with larger bilateral MIPS volumes suggesting that the effect of PAE on math may not be moderated by IPS volume. The left LIPS was significantly smaller in FAS/PFAS individuals when compared by FASD diagnosis, and this remained a trend after controlling for potential confounders. In the second analysis, the automated neuroimaging software program FreeSurfer was used to parcellate the IPS. These volumes were then compared with our previously manually traced volumes. Intra-rater reliability testing was statistically significant for consistency and absolute agreement indicating good retraceability of the designed protocol for manual tracing. Both left and right IPS volumes were significantly larger with the manually traced method compared to automated tracing. The manually traced left IPS yielded stronger results when comparing volumes by diagnostic groups, conversely the automated volumes showed stronger associations with alcohol measures. A possible explanation is that FreeSurfer parcellated the IPS differently to our protocol and does not take into account the extensive variability of the morphology of the sulcus. BrainVoyager QX, another neuroimaging software program was used in the third analysis when looking at the BOLD fMRI data of the participants. For this analysis, the manually traced MIPS and LIPS were subdivided into five ROI's for the left and right hemispheres respectively: (1) the superior MIPS, (2) the medial branch of the MIPS, (3) the inferior MIPS, (4) the superior LIPS, and (5) the inferior LIPS. The percent signal change were calculated for each participant for the proximity judgement (PJ) tasks they performed inside the scanner. Associations of the percent signal change of the ROI's of the PAE children with absolute alcohol per occasion (oz) were all significant even after controlling for IQ except the left inferior LIPS, supporting what is found in the literature. The current findings, in agreement with previous studies, demonstrate that PAE is associated with both structural and functional changes in the brain. While the morphology of the IPS may not moderate the effects of PAE on arithmetic function, some cortical volumes within the IPS were sensitive to PAE. Moreover, altered activation of the IPS in the performance of magnitude comparison tasks was strongly associated with PAE. The IPS is an extremely variable structure whose anatomy is often misunderstood, which emphasises the importance of anatomical knowledge for imaging studies. Future research will refine the protocol for manual tracing of the IPS, which may lead to greater understanding of the functions of the different areas. It is to be hoped that these findings will give more insight into understanding the functioning of children and adults with FASDs and contribute to more effective therapeutic interventions for these individuals

Third graders’ verbal reports of multiplication strategy use: How valid are they?

This study investigates whether children’s verbal reports accurately represent their thinking processes when solving simple multiplication problems. A total of 106 third graders in Dutch mainstream primary schools solved simple multiplication problems and retrospectively reported how they had done this. The degree to which verbal reports predict children’s problem-solving performance in ways that correspond to known patterns of response latency, accuracy, errors and strategy choice was assessed. The analyses took account of relevant problem characteristics and child cognitive characteristics (i.e., math ability, verbal ability, phonological decoding speed) known to affect the relation between strategy use and multiplication performance. The verbal reports were largely consistent with known patterns, supporting the use of verbal reports in assessing multiplication strategy use. Moreover, verbal reports provide valuable information that can alert teachers and educational researchers to specific issues that students face when solving simple multiplication problems. Considerations for soliciting reliable verbal reports are suggested

Développement et déterminants précoces de la connaissance des nombres et des habiletés mathématiques à l'enfance

Tableau d'honneur de la Faculté des études supérieures et postdoctorales, 2016-2017La connaissance des nombres, un précurseur des habiletés mathématiques, est essentielle à la maitrise de concepts fondamentaux en début de scolarisation. On sait toutefois peu de choses des mécanismes qui sous-tendent la connaissance des nombres, des facteurs qui influencent son développement et de sa contribution à long terme au développement des mathématiques. Dans le cadre de cette thèse doctorale, des trajectoires de développement de la connaissance des nombres ont été établies entre l’âge de 4 et 7 ans auprès de 1597 enfants. Quatre trajectoires ont été identifiées, dont l’une est constituée d’enfants (10%) qui se caractérisent par une connaissance des nombres constamment inférieure aux autres. Ces enfants ont été comparés aux autres sur leurs compétences en mathématiques à 8 et 10 ans, et ont également été évalués sur différents aspects de leur environnement familial et sur leurs habiletés cognitives à 41 mois. Les résultats montrent que les enfants avec une faible connaissance des nombres à l’âge préscolaire demeurent avec un rendement en mathématiques inférieur à celui des autres enfants et ce, jusqu’à la fin de l’école primaire. Ces enfants se caractérisent d’ailleurs par un revenu familial moindre, une faible scolarité du père, et des habiletés visuospatiales, une capacité de rétention et un développement cognitif général inférieurs à ceux des autres enfants. De plus, des modélisations génétiques effectuées à 5, 7, et 10-12 ans montrent que l’environnement commun aux jumeaux (p.ex., l’éducation familiale) explique principalement les variations individuelles de la connaissance des nombres à l’enfance alors qu’en vieillissant, ces variations s’expliquent davantage par les facteurs génétiques et spécifiques à l’environnement de chaque individu. Ces résultats sont similaires pour les garçons et les filles. Les résultats montrent également que la variance génétique est associée à la stabilité de la connaissance des nombres et à son association prédictive au rendement en mathématiques. Ces facteurs génétiques expliquent aussi des changements qui sont spécifiques au rendement en mathématiques, ce qui suggère l’apport de nouveaux gènes au rendement en mathématiques à la fin de l’école primaire. Les facteurs de l’environnement, commun et unique à chacun des jumeaux, contribuent tous deux à la stabilité de l’association entre la connaissance des nombres et le rendement en mathématiques, sans apport additionnel significatif de ces facteurs après la période préscolaire. Ensemble, les résultats de cette thèse révèlent que la période préscolaire s’avère la plus propice pour intervenir auprès d’enfants afin de prévenir les difficultés en mathématiques.Number knowledge and skills (NKS), the conceptual and procedural understanding of whole numbers, predicts later scholastic achievement. However, little is known about the mechanisms underlying the NKS, its antecedents in early childhood, and its predictive validity to later math achievement. Children’s NKS was assessed four times at regular intervals between the ages 4 and 7 years in a large, representative population-based sample. Developmental trajectories of NKS were established for 1597 children. Four different groups of preschoolers were identified. About 10% of the children belonged to a trajectory of constantly and significantly lower performance compared with the other trajectories. These children were compared with others on their mathematics achievement at ages 8 and 10, and were also evaluated with respect to several features of their family environment at 5, 18 and 30 months, as well as their cognitive skills at age 41 months. The results showed significant differences between the trajectories of NKS with respect to later math achievement in elementary school, with the low trajectory-group remaining low throughout these years. The onset and developmental course of low NKS were associated with low household income and father educational background, low children’s early cognitive development, and more specifically, weak visual-spatial skills and memory span. Children with low cognitive abilities and poor living condition are at risk of low NKS profile from late preschool to school entry, and therefore, deserve special attention to alleviate later mathematic difficulties. Moreover, genetic multivariate analysis at ages 5, 7, and 10-12 years showed that shared environmental factors between twins of the same family (e.g. sharing the same home environment) mainly explained individual variations in preschool NKS, with increased heritability with time – genetic factors play the dominant role in later math achievement, suggesting different mechanisms in math-related tasks over the years. However, these mechanisms were similar for boys and girls. Genetic factors accounted for continuity from preschool NKS to late primary math achievement, but also explained specific variations in mathematics achievement, which suggest activation of new genes relevant to mathematics in late primary school years. The shared and non-shared environmental factors involved in preschool NKS were carried over to mathematics achievement, with no additional age-specific effect after the preschool period. Altogether, the results of this thesis highlight the preschool age as an optimum window for prevention and intervention of math difficulties. Given this, screening for early NKS and math difficulties should be afforded before school entry in order to provide additional support as soon as difficulties emerge in this area