This thesis concerns the formulation of novel developmental robotics models of embodied phenomena in number learning. Learning to count is believed to be of paramount importance for the acquisition of the remarkable fluency with which humans are able to manipulate numbers and other abstract concepts derived from them later in life. The ever-increasing amount of evidence for the embodied nature of human mathematical thinking suggests that the investigation of numerical cognition with the use of robotic cognitive models has a high potential of contributing toward the better understanding of the involved mechanisms. This thesis focuses on two particular groups of embodied effects tightly linked with learning to count. The first considered phenomenon is the contribution of the counting gestures to the counting accuracy of young children during the period of their acquisition of the skill. The second phenomenon, which arises over a longer time scale, is the human tendency to internally associate numbers with space that results, among others, in the widely-studied SNARC effect. The PhD research contributes to the knowledge in the subject by formulating novel neuro-robotic cognitive models of these phenomena, and by employing these in two series of simulation experiments. In the context of the counting gestures the simulations provide evidence for the importance of learning the number words prior to learning to count, for the usefulness of the proprioceptive information connected with gestures to improving counting accuracy, and for the significance of the spatial correspondence between the indicative acts and the objects being enumerated. In the context of the model of spatial-numerical associations the simulations demonstrate for the first time that these may arise as a consequence of the consistent spatial biases present when children are learning to count. Finally, based on the experience gathered throughout both modelling experiments, specific guidelines concerning future efforts in the application of robotic modelling in mathematical cognition are formulated.

The purpose of this grounded theory study was to explore the interplay of prospective secondary mathematics teachers’ affect, metacognition, and mathematical cognition in a problem-solving context. From a social constructivist epistemological paradigm and using a constructivist grounded theory approach, the main research question guiding the study was: What is the characterization of the interplay among prospective teachers’ mathematical beliefs, mathematical behavior, and mathematical knowledge in the context of solving mathematics problems? I conducted four interviews with four prospective secondary mathematics teachers enrolled in an undergraduate mathematics course. Participant artifacts, observations, and researcher reflections were regularly recorded and included as part of the data collection. The theory that emerged from the study is grounded in the participants’ mathematics problem-solving experiences and it depicts the interplay among affect, metacognition, and mathematical cognition as meta-affect, persistence and autonomy, and meta-strategic knowledge. For the participants, “Knowing How and Knowing Why” mathematics procedures work and having the ability to justify their reasoning and problem solutions represented mathematics knowledge and understanding that could empower them to become productive problem-solvers and effective secondary mathematics teachers. The results of the study also indicated that the participants interpreted their experiences with difficult, challenging problem-solving situations as opportunities to learn and understand mathematics deeply. Although they experienced fear, frustration, and disappointment in difficult problem-solving and mathematics-learning situations, they viewed such difficulty with the expectation that feelings of satisfaction, joy, pride, and confidence would occur because of their mathematical understanding. In problem-solving situations, affect, metacognition, and mathematics cognition interacted in a way that resulted in mathematics understanding that was productive and empowering for these prospective teachers. The theory resulting from this study has implications for prospective teachers, teacher education, curriculum development, and mathematics education research.

<p>The uniquely human mathematical mind sets us apart from all other animals. Although humans typically think about number symbolically, we also possess nonverbal representations of quantity that are present at birth and shared with many other animal species. These primitive numerical representations are thought to arise from an evolutionarily ancient system termed the Approximate Number System (ANS). The present dissertation aims to determine how these preverbal representations of quantity may serve as the foundation for more complex quantitative reasoning abilities. To this end, the five studies contained herein investigate the relations between representations of number, representations of other magnitude dimensions, and symbolic math proficiency in infants, children, and adults. The first empirical study, described in Chapter 2, investigated whether infants engage the ANS to represent the full range of natural numbers. The study presented in Chapter 3 compared infants' acuity for detecting changes in contour length to their acuity for detecting changes in number to assess whether representations of continuous quantities are primary to representations of number in infancy. The study presented in Chapter 4 compared individual differences in acuity for number, line length, and brightness in children and adults to determine how the relations between these magnitudes may change over development. Chapter 5 contains a longitudinal study investigating the relation between preverbal number sense in infancy and symbolic math abilities in preschool-aged children. Finally, the study presented in Chapter 6 investigated the mechanisms underlying the maturation of the number sense and determined which features of the number sense are predictive of symbolic math skill. Taken together, these findings confirm that number is a salient feature of the environment for infants and young children and suggest that approximate number representations are fundamental for the acquisition of symbolic math.</p> / Dissertation

The purpose of the present thesis was to test and contrast hypotheses about the cognitive conditions that support the development of mathematical learning disability (MLD). Following hypotheses were tested in the thesis: a) domain general deficit, the deficit is primarily located in the domain general systems such as the working memory, b) number sense deficit, the deficit is located in the innate approximate number system (ANS), c) numerosity coding deficit, the deficit is located to a exact number representation system, d) access deficit, the deficit is in the mapping between symbols and the innate number representational system (e.g., ANS), e) multiple deficit hypothesis states that MLD could be related to more than one deficit. Three studies examined the connection between cognitive abilities and arithmetic. Study one and three compared different groups of children with or without MLD (or risk of MLD). Study two investigated the connection between early number knowledge, verbal working memory and the development of arithmetic ability. The results favoring the multiple deficit hypothesis, more specifically the result indicate that number sense deficit together with working memory functions constitutes risk-factors to the development of MLD in children. A simple developmental model that is based on von Asters and Shalev´s (2007) model and the present results is suggested, in order to understand the development of MLD in children. / Avhandlingens syfte var att testa och kontrastera hypoteser om vilka kognitiva förutsättningar som är centrala för utvecklandet av matematiska inlärningssvårigheter (MLD) hos barn. De hypoteser som prövas i avhandlingen är följande: a) den domängenerella hypotesen, detta innebär att den förmodade störningen/nedsättningen finns primärt i barnets generella förmågor, främst då i arbetsminnes funktioner. b) en nedsättning i den medfödda approximativa antalsuppfattningen. c) nedsättning i den exakta antalskodningen. d) nedsättning gällande kopplingen mellan den kulturellt betingande symboliska nivå (räkneord och siffror) samt den medfödda antalsuppfattningen (eller antalskodningen). e) slutligen prövas även hypotesen att MLD kan härröras från flera nedsättningar i dessa förmågor. I tre studier undersöktes kopplingen mellan kognitiva förmågor och aritmetik. i studie1 och 3 jämfördes grupper av barn med MLD (eller risk för MLD) med grupper av barn som inte hade MLD i studie 2 undersöktes kopplingen mellan förmågorna verbalt arbetsminne och tidig sifferkunskap samt tidig aritmetiskförmåga. Sammantaget indikerar resultaten från denna avhandling att det kan vara både multipla och enstaka kognitiva förmågor, primärt i den approximativa antalsuppfattningen samt i arbetsminnesfunktioner, som kan fungera som riskförutsättningar för utvecklande av MLD hos barn. Dock måste dessa förmågor samspela med andra faktorer som kan fungera kompensatoriskt eller riskhöjande för utvecklandet av MLD. En förenklad utvecklingsmodell med utgångspunkten i resultaten från studierna samt von Asters och Shalevs (2007) modell föreslås. Syftet med modellen är att den ska kunna användas som teoretiskt ramverk för att förstå utvecklingen av MLD hos barn.

Traditionally, mathematics and metaphor have been thought of as disparate: the former rigorous, objective, universal, eternal, and fundamental; the latter imprecise, derivative, nearly - if not patently - false, and therefore of merely aesthetic value, at best. A growing amount of contemporary scholarship argues that both of these characterizations are flawed. This dissertation shows that there are important connexions between mathematics and metaphor that benefit our understanding of both. A historically structured overview of traditional theories of metaphor reveals it to be a notion that is complicated, controversial, and inadequately understood; this motivates a non-traditional approach. Paradigmatically shifting the locus of metaphor from the linguistic to the conceptual - as George Lakoff, Mark Johnson, and many other contemporary metaphor scholars do - overcomes problems plaguing traditional theories and promisingly advances our understanding of both metaphor and of concepts. It is argued that conceptual metaphor plays a key role in explaining how mathematics is grounded, and simultaneously provides a mechanism for reconciling and integrating the strengths of traditional theories of mathematics usually understood as mutually incompatible. Conversely, it is shown that metaphor can be usefully and consistently understood in terms of mathematics. However, instead of developing a rigorous mathematical model of metaphor, the unorthodox approach of applying mathematical concepts metaphorically is defended.

Comment le cerveau humain parvient-il à conceptualiser des idées abstraites ? Quelle est en particulier l’origine de l’activité mathématique lorsqu'elle est associée à un haut niveau d’abstraction ? La question de savoir si la pensée mathématique peut exister sans langage intéresse depuis longtemps les philosophes, les mathématiciens et les enseignants. Elle commence aujourd’hui à être abordée par les neurosciences cognitives. Alors que les études précédentes se sont principalement focalisées sur l’arithmétique élémentaire, mon travail de thèse privilégie l’étude de la manipulation d’idées mathématiques plus avancées et des processus cérébraux impliqués dans leur apprentissage. Les travaux présentés dans cette thèse révèlent que : (1) la réflexion sur des concepts mathématiques de haut niveau appris depuis de nombreuses années n’implique pas les aires du langage ; (2) l’activité mathématique, quels qu’en soient la difficulté et le domaine, implique systématiquement des régions classiquement associées à la manipulation des nombres et de l’espace, y compris chez des personnes non-voyantes; (3) l’apprentissage non-verbal de règles géométriques repose sur un langage de la pensée indépendant du langage parlé naturel. Ces résultats ouvrent la voie à de nouvelles questions en neurosciences. Par exemple, l’apprentissage de concepts mathématiques enseignés à l’école par le truchement des mots se passe-t-il également du langage ? Ou enfin, que signifie réellement "faire des mathématiques" pour le cerveau humain ? / How does the human brain conceptualize abstract ideas? In particular, what is the origin of mathematical activity, especially when it is associated with high-level of abstraction? Is mathematical thought independent of language? Cognitive science has now started to investigate this question that has been of great interest to philosophers, mathematicians and educators for a long time. While studies have so far focused on basic arithmetic processing, my PhD thesis aims at further investigating the cerebral processes involved in the manipulation and learning of more advanced mathematical ideas. I have shown that (1) advanced mathematical reflection on concepts mastered for many years does not recruit the brain circuits for language; (2) mathematical activity systematically involves number- and space-related brain regions, regardless of mathematical domain, problem difficulty, and participants' visual experience; (3) non-verbal acquisition of geometrical rules relies on a language of thought that is independent of natural spoken language. Finally, altogether these results raise new questions and pave the way to further investigations in neuroscience: - is the human ability for language also irrelevant to advanced mathematical acquisition in schools where knowledge is taught verbally? - What is the operational definition of the fields of “mathematics” and “language” at the brain level?

Poor mathematics performance in South African schools is of national concern. An attempt to gain insight into the problem prompted a study into the possibility of a relationship between visual perceptual skill and mathematic ability. A theoretical review revealed that inherent limitations of traditional psychological theories hinder an adequate explanation for the possible existence of such a relationship. The theory of situated cognition seems to be better suited as an explanatory model, and simultaneously clarifies the nature of both visual perception and mathematics. A small exploratory study, with a sample of 70 Grade 6 learners, provided empirical evidence towards the plausibility of the relationship. Specifically, it proved the hypothesis that visual perceptual skill positively correlates with scholastic mathematics achievement. The results of the study, interpreted within the situated cognitive framework, suggest that a conceptual emphasis in mathematics education – as opposed to a factual emphasis – might improve mathematic ability, which may credibly reflect in scholastic performance. / Psychology / M. Sc. (Psychology)

Poor mathematics performance in South African schools is of national concern. An attempt to gain insight into the problem prompted a study into the possibility of a relationship between visual perceptual skill and mathematic ability. A theoretical review revealed that inherent limitations of traditional psychological theories hinder an adequate explanation for the possible existence of such a relationship. The theory of situated cognition seems to be better suited as an explanatory model, and simultaneously clarifies the nature of both visual perception and mathematics. A small exploratory study, with a sample of 70 Grade 6 learners, provided empirical evidence towards the plausibility of the relationship. Specifically, it proved the hypothesis that visual perceptual skill positively correlates with scholastic mathematics achievement. The results of the study, interpreted within the situated cognitive framework, suggest that a conceptual emphasis in mathematics education – as opposed to a factual emphasis – might improve mathematic ability, which may credibly reflect in scholastic performance. / Psychology / M. Sc. (Psychology)