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Research in communication disorders suggests that training linguistically complex forms will generalize to untrained, simpler forms with similar structural properties (see Thompson, 2007 for review). The present study investigated generalization patterns from transitive verbs to two classes of intransitive verbs following administration of Verb Network Strengthening Treatment (VNeST; Edmonds, Nadeau & Kiran, 2009). Based on the Argument Structure Complexity Hypothesis (ASCH; Thompson, 2003), it was predicted that greater generalization would occur to unergatives because unergatives bear a structural relationship to transitive verbs and unaccusatives do not. Results at post-treatment supported the hypothesis with both intransitive verb types showing generalization and slightly higher effect sizes observed for unergative than for unaccusative verbs. At maintenance, this pattern was not maintained due to improvements in production of unaccusative verbs. Results support the findings of Edmonds et al. (2009) that administration of VNeST results in gains on measures of untrained, semantically related verbs as well as standardized measures of lexical retrieval and connected speech. These results also suggest that training transitive verbs results in slight generalization to untrained intransitive verbs; however, it is inconclusive whether unergative and unaccusatives intransitives show differential improvement. / Communication Sciences

Ce mémoire se veut une synthèse, destinée à la communauté combinatoricienne, de quelques outils développés pour aborder le problème d'Hurwitz ainsi qu'une présentation des résultats récoltés. Le problème d'Hurwitz consiste à évaluer, dans un groupe symétrique, le nombre (dit d'Hurwitz) de factorisations transitives de la permutation identité dont on a imposé le type cyclique des facteurs. Nous décrivons tout d'abord les origines topologiques de ce problème à travers le dénombrement des revêtements ramifiés de la sphère. Nous présentons également un cadre algébrique naturel, le monoïde des permutations scindées, qui permet d'exprimer les nombres d'Hurwitz comme coefficients de structure de l'algèbre de ce monoïde, plus précisément de la sous-algèbre engendrée par les classes de conjugaison, dont une base naturelle est indexée par les multipartitions (ou partitions scindées). La théorie des représentations de cette algèbre fournit un algorithme pour calculer les nombres d'Hurwitz à une partition dont la complexité (minimale, uniforme et exponentielle) est bien meilleure que celle d'une approche naïve. Ce cadre algébrique donne par ailleurs une formule décrivant les séries d'Hurwitz à plusieurs partitions comme polynômes en les séries d'Hurwitz à une seule partition. Nous présentons secondement le cadre géométrique dans lequel s'expriment d'une part la formule ELSV, laquelle décrit les nombres d'Hurwitz à une partition comme fonctions de certaines intégrales, d'autre part un théorème de M. Kazarian exprimant les séries de Hurwitz à une partition comme polynômes en certaines séries formelles dont l'étude asymptotique est achevée. Une fois décrit le fonctionnement de ce cadre intégral, nous récoltons l'asymptotique de tous les nombres d'Hurwitz / This thesis is meant to be a digest, adressed to the combinatorician community, of some tools developped to tackle the problem of Hurwitz, as well as an exhibition of the thus-harvested results. The problem of Hurwitz consists of computing, in a symmetric group, the (so-called Hurwitz) number of transitive factorisations of the identity permutation whose factors have prescribed cyclic types. We first describe the topological layout of this problem through the enumeration of the ramified coverings of the sphere. We also present a natural algebraic frame, the monoid of split permutations, which allows to describe Hurwitz numbers as structure coeffcients of the algebra of this monoid, more precisely of the subalgebra spanned by the conjugacy classes, whose natural basis is indexed by multipartitions (or split partitions). The representation theory of this algebra yields an algoithm to compute one-partition Hurwitz numbers whose complexity (minimal, uniform and exponential) is far better than that of a naive edging about. This algebraic frame yields a formula describing several-partition Hurwitz series as polynomials in one-partition Hurwitz series. We secondly present the geometric frame in which are been expressed on the one hand the ELSV formula, which describes one-partition Hurwitz numbers as functions of some integrals, one the other hand a theorem of M. Kazarian expressing one-partition Hurwitz series as polynomials in some formal power series whose asymptotics is completly understood. Once the using of this integration frame has been described, we derive the asymptotics of all Hurwitz numbers

Chordal graphs are an important class of perfect graphs. The beautiful theory surrounding their study varies from natural applications to elegant characterizations in terms of forbidden subgraphs, subtree representations, vertex orderings, and to linear time recognition algorithms. Haskins and Rose introduced the class of chordal digraphs as a digraph analogue of chordal graphs. Chordal digraphs can be defined in terms of vertex orderings and several results about chordal graphs can be extended to chordal digraphs. However, a forbidden subdigraph characterization of chordal digraphs is not known and finding such a characterization seems to be a difficult problem. Meister and Telle studied semi-complete chordal digraphs and gave a forbidden subdigraph characterization of this class of digraphs. In this thesis, we study chordal digraphs within the classes of quasi-transitive, extended semi-complete, and locally semi-complete digraphs. For each of these classes we obtain a forbidden subdigraph characterization of digraphs which are chordal. We also introduce in this thesis a new variant of chordal digraphs called semi-strict chordal digraphs. We obtain a forbidden subdigraph characterization of semi-strict chordal digraphs for each of the classes of semi-complete, quasi-transitive, extended semi-complete, and locally semi-complete digraphs. / Graduate

O estudo do responder controlado por relações de seleção ou rejeição insere-se no âmbito de investigações sobre as condições necessárias e suficientes para a formação de classes de estímulos equivalentes. O presente projeto pretendeu investigar aspectos relacionados ao tema em experimentos conduzidos com humanos e pombos. O primeiro experimento teve por objetivo avaliar se o registro do comportamento de olhar, em termos do tempo de observação dos estímulos e padrões de rastreamento exibidos ao longo do treino de discriminações condicionais, poderia fornecer medidas auxiliares sobre o processo de aquisição dos controles por seleção e rejeição durante o treino de discriminações condicionais em participantes humanos. Para esta finalidade era importante preparar uma situação experimental que, previsivelmente, gerasse diferenças nos resultados da aprendizagem discriminativa. Desse modo, seria possível verificar se o olhar poderia fornecer dicas adicionais sobre o estabelecimento do responder controlado por relações de seleção ou rejeição. Estudos anteriores mostraram que, em procedimentos de MTS, utilizar o teclado ou o mouse produz diferentes resultados nos testes de formação de classes. Objetivou-se assim verificar se diferentes topografias de resposta gerariam resultados diferentes na aquisição das discriminações condicionais e nos testes de formação de classes e, adicionalmente, se gerariam padrões de rastreamento diferentes em termos do tempo de observação dos estímulos. Estudantes universitários foram submetidos aos treinos de discriminações condicionais envolvendo seis conjuntos de estímulos (treinos EF, DE, CD, BC e AB) enquanto utilizam um equipamento que registrava o comportamento de olhar. Os resultados sugerem que diferentes topografias de respostas dão origem a diferentes padrões de rastreamento dos estímulos, entretanto não foram encontradas evidências de que tais padrões estariam relacionados ao estabelecimento de relações controladas por seleção ou rejeição. O segundo experimento teve por objetivo verificar a existência de relações de transitividade a partir de um treino condicional envolvendo estímulos temporais. Foram utilizados 12 pombos (Columba lívia) experimentalmente ingênuos e mantidos a 80% do seu peso ad lib. O equipamento utilizado consistia em uma caixa experimental padrão contendo três discos de respostas dispostos horizontalmente. Uma tentativa de treino ocorria da seguinte forma: a luz ambiente e o disco central de respostas eram acesos ao mesmo tempo. Se, por exemplo, o estímulo temporal programado fosse de 4 s, uma luz branca no disco central permanecia ligada por esse período. Após 4 s, a luz do disco central era apagada e cores eram apresentadas nos discos laterais. Uma bicada em qualquer um dos discos laterais encerrava a tentativa. Respostas corretas eram seguidas por um curto período de acesso ao comedouro e um IET de 20 s. Respostas incorretas eram seguidas pelo IET e pela reapresentação da mesma tentativa (procedimento de correção). O procedimento era composto por três fases de treino: nas tentativas do Treino A, os sujeitos aprenderam a bicar o disco iluminado pela cor vermelha diante do modelo 1 s e a bicar o disco iluminado pela cor verde diante de 4 s. Nas tentativas do Treino B, eles aprenderam a bicar o disco iluminado pela cor azul diante de 4 s e a bicar o disco iluminado pela cor amarela diante de 16 s. Durante o Treino C, os sujeitos aprenderam a bicar o disco iluminado pela cor azul diante do modelo verde e a bicar o disco iluminado pela cor amarela diante do modelo vermelho. Considerando os treinos A e B, as cores azul e verde foram relacionadas ao mesmo estímulo temporal 4 s, enquanto que as cores vermelho e amarelo não partilharam qualquer estímulo comum durante o treino anterior. Os resultados sugerem que as relações entre azul e verde foram adquiridas mais rapidamente, fato de parece comprovar a formação de relações de transitividade em fases ix anteriores do treino / The study of responding controlled by selection or rejection relations is within the scope of research on the necessary and sufficient conditions for the formation of equivalent stimuli classes. This project sought to investigate aspects related to this subject by performing experiments with humans and pigeons. The first experiment evaluated whether the record of eye movement - in terms of stimuli observation time and tracking patterns exhibited throughout conditional discrimination training - could provide ancillary measures on the acquisition process of control by selection rejection during the conditional discrimination training on human participants. For this purpose it was important to prepare an experimental situation that predictably generate differences in the discriminative learning outcomes. Thus, it would be possible to verify if the eye movement could provide additional clues about the establishment of responding controlled by relations of selection or rejection. Previous studies have shown that using the keyboard or mouse, in MTS tasks, produces different results in class formation testing. The objective was, therefore, to check if different response topographies would generate different results in conditional discrimination acquisition and class formation tests and, furthermore, if it would generate different tracking patterns, in terms of stimuli observation time. College students were subjected to conditional discriminations training involving six sets of stimuli (training EF, DE, CD, BC and AB) while using a device that recorded eye movement. Results suggest that different response topographies originate different patterns of stimulus tracking, however there was no evidence that such patterns referred to the establishment of relations controlled by selection or rejection. The second experiment sought verifying the existence of transitivity relations from conditional training involving temporal stimuli. Subjects were 12 pigeons (Columba livia) experimentally naive and kept at 80% of their ad lib weight. The equipment used consisted of a box containing three standard experimental response keys horizontally arranged. A training attempt occurred as follows: house light and the central key were lit at the same time. If, for example, the programmed interval was 4 s, a white light in the central key remained on for that period. After 4 s, the light from the central key was turned off and colors were presented on the side keys. A peck on either of the side keys ended the attempt. Correct answers were followed by a short period of access to the feeder and an ITI of 20 s. Incorrect answers were followed by the ITI and the repetition of the same trial (correction procedure). The procedure consisted of three phases of training: in Training A attempts, the subjects learned to choose the red colored key when the 1 s model was presented, and to choose the green colored key when the 4 s model was presented. In Training B attempts, they learned to choose the blue colored key when the 4 s model was presented and to choose the yellow colored key when the 16 s model was presented. During Training C, subjects learned to choose the blue colored key when green color was presented and to choose yellow colored key when red color model was presented. Regarding A and B trainings, blue and green stimuli were related to the same temporal stimulus 4 s, while red and yellow ones did not share any common temporal stimulus during earlier training. Results suggest that relations between blue and green were acquired more quickly, a fact that seems to demonstrate formation of transitivity relations in earlier stages of training

O estudo do responder controlado por relações de seleção ou rejeição insere-se no âmbito de investigações sobre as condições necessárias e suficientes para a formação de classes de estímulos equivalentes. O presente projeto pretendeu investigar aspectos relacionados ao tema em experimentos conduzidos com humanos e pombos. O primeiro experimento teve por objetivo avaliar se o registro do comportamento de olhar, em termos do tempo de observação dos estímulos e padrões de rastreamento exibidos ao longo do treino de discriminações condicionais, poderia fornecer medidas auxiliares sobre o processo de aquisição dos controles por seleção e rejeição durante o treino de discriminações condicionais em participantes humanos. Para esta finalidade era importante preparar uma situação experimental que, previsivelmente, gerasse diferenças nos resultados da aprendizagem discriminativa. Desse modo, seria possível verificar se o olhar poderia fornecer dicas adicionais sobre o estabelecimento do responder controlado por relações de seleção ou rejeição. Estudos anteriores mostraram que, em procedimentos de MTS, utilizar o teclado ou o mouse produz diferentes resultados nos testes de formação de classes. Objetivou-se assim verificar se diferentes topografias de resposta gerariam resultados diferentes na aquisição das discriminações condicionais e nos testes de formação de classes e, adicionalmente, se gerariam padrões de rastreamento diferentes em termos do tempo de observação dos estímulos. Estudantes universitários foram submetidos aos treinos de discriminações condicionais envolvendo seis conjuntos de estímulos (treinos EF, DE, CD, BC e AB) enquanto utilizam um equipamento que registrava o comportamento de olhar. Os resultados sugerem que diferentes topografias de respostas dão origem a diferentes padrões de rastreamento dos estímulos, entretanto não foram encontradas evidências de que tais padrões estariam relacionados ao estabelecimento de relações controladas por seleção ou rejeição. O segundo experimento teve por objetivo verificar a existência de relações de transitividade a partir de um treino condicional envolvendo estímulos temporais. Foram utilizados 12 pombos (Columba lívia) experimentalmente ingênuos e mantidos a 80% do seu peso ad lib. O equipamento utilizado consistia em uma caixa experimental padrão contendo três discos de respostas dispostos horizontalmente. Uma tentativa de treino ocorria da seguinte forma: a luz ambiente e o disco central de respostas eram acesos ao mesmo tempo. Se, por exemplo, o estímulo temporal programado fosse de 4 s, uma luz branca no disco central permanecia ligada por esse período. Após 4 s, a luz do disco central era apagada e cores eram apresentadas nos discos laterais. Uma bicada em qualquer um dos discos laterais encerrava a tentativa. Respostas corretas eram seguidas por um curto período de acesso ao comedouro e um IET de 20 s. Respostas incorretas eram seguidas pelo IET e pela reapresentação da mesma tentativa (procedimento de correção). O procedimento era composto por três fases de treino: nas tentativas do Treino A, os sujeitos aprenderam a bicar o disco iluminado pela cor vermelha diante do modelo 1 s e a bicar o disco iluminado pela cor verde diante de 4 s. Nas tentativas do Treino B, eles aprenderam a bicar o disco iluminado pela cor azul diante de 4 s e a bicar o disco iluminado pela cor amarela diante de 16 s. Durante o Treino C, os sujeitos aprenderam a bicar o disco iluminado pela cor azul diante do modelo verde e a bicar o disco iluminado pela cor amarela diante do modelo vermelho. Considerando os treinos A e B, as cores azul e verde foram relacionadas ao mesmo estímulo temporal 4 s, enquanto que as cores vermelho e amarelo não partilharam qualquer estímulo comum durante o treino anterior. Os resultados sugerem que as relações entre azul e verde foram adquiridas mais rapidamente, fato de parece comprovar a formação de relações de transitividade em fases ix anteriores do treino / The study of responding controlled by selection or rejection relations is within the scope of research on the necessary and sufficient conditions for the formation of equivalent stimuli classes. This project sought to investigate aspects related to this subject by performing experiments with humans and pigeons. The first experiment evaluated whether the record of eye movement - in terms of stimuli observation time and tracking patterns exhibited throughout conditional discrimination training - could provide ancillary measures on the acquisition process of control by selection rejection during the conditional discrimination training on human participants. For this purpose it was important to prepare an experimental situation that predictably generate differences in the discriminative learning outcomes. Thus, it would be possible to verify if the eye movement could provide additional clues about the establishment of responding controlled by relations of selection or rejection. Previous studies have shown that using the keyboard or mouse, in MTS tasks, produces different results in class formation testing. The objective was, therefore, to check if different response topographies would generate different results in conditional discrimination acquisition and class formation tests and, furthermore, if it would generate different tracking patterns, in terms of stimuli observation time. College students were subjected to conditional discriminations training involving six sets of stimuli (training EF, DE, CD, BC and AB) while using a device that recorded eye movement. Results suggest that different response topographies originate different patterns of stimulus tracking, however there was no evidence that such patterns referred to the establishment of relations controlled by selection or rejection. The second experiment sought verifying the existence of transitivity relations from conditional training involving temporal stimuli. Subjects were 12 pigeons (Columba livia) experimentally naive and kept at 80% of their ad lib weight. The equipment used consisted of a box containing three standard experimental response keys horizontally arranged. A training attempt occurred as follows: house light and the central key were lit at the same time. If, for example, the programmed interval was 4 s, a white light in the central key remained on for that period. After 4 s, the light from the central key was turned off and colors were presented on the side keys. A peck on either of the side keys ended the attempt. Correct answers were followed by a short period of access to the feeder and an ITI of 20 s. Incorrect answers were followed by the ITI and the repetition of the same trial (correction procedure). The procedure consisted of three phases of training: in Training A attempts, the subjects learned to choose the red colored key when the 1 s model was presented, and to choose the green colored key when the 4 s model was presented. In Training B attempts, they learned to choose the blue colored key when the 4 s model was presented and to choose the yellow colored key when the 16 s model was presented. During Training C, subjects learned to choose the blue colored key when green color was presented and to choose yellow colored key when red color model was presented. Regarding A and B trainings, blue and green stimuli were related to the same temporal stimulus 4 s, while red and yellow ones did not share any common temporal stimulus during earlier training. Results suggest that relations between blue and green were acquired more quickly, a fact that seems to demonstrate formation of transitivity relations in earlier stages of training

Les travaux présentés dans cette thèse sont lies aux problèmes de vérification de l'atteignabilité et de la terminaison de programmes qui manipulent des données entières non-bornées. On décrit une nouvelle méthode de vérification basée sur une technique d'accélération de boucle, qui calcule, de manière exacte, la clôture transitive d'une relation arithmétique. D'abord, on introduit un algorithme d'accélération de boucle qui peut calculer, en quelques secondes, des clôtures transitives pour des relations de l'ordre d'une centaine de variables. Ensuite, on présente une méthode d'analyse de l'atteignabilité, qui manipule des relations entre les variables entières d'un programme, et applique l'accélération pour le calcul des relations entrée-sortie des procédures, de façon modulaire. Une approche alternative pour l'analyse de l'atteignabilité, présentée également dans cette thèse, intègre l'accélération avec l'abstraction par prédicats, afin de traiter le problème de divergence de cette dernière. Ces deux méthodes ont été évaluées de manière pratique, sur un nombre important d'exemples, qui étaient, jusqu'a présent, hors de la portée des outils d'analyse existants. Dernièrement, on a étudié le problème de la terminaison pour certaines classes de boucles de programme, et on a montré la décidabilité pour les relations étudiées. Pour ces classes de relations arithmétiques, on présente un algorithme qui s'exécute en temps au plus polynomial, et qui calcule l'ensemble d'états qui peuvent générer une exécution infinie. Ensuite on a intégré cet algorithme dans une méthode d'analyse de la terminaison pour des programmes qui manipulent des données entières.

On considère R une relation d’équivalence borélienne standard de type I I1 sur un espace de probabilités (X, µ). On étudie une certaine propriété d’homogénéité pour un graphage fixé de la relation R : on suppose que les feuilles du graphage sont toutes isomorphes à un certain graphe transitif (connexe, infini, localement fini) Γ. Que peut-on dire sur la relation ? Dans ce cas, en considérant une action "à la Mackey", on montre qu’il existe (Z ,η) un revêtement standard probabilisé de (X, µ), une action libre (qui préserve η) sur Z du groupe G (localement compact, à base dénombrable d’ouverts) des automorphismes du graphe et un isomorphisme stable des groupoïdes mesurés associés. On fait le lien entre les propriétés du groupe G et celles de la relation de départ ; en particulier la propriété (T), (H) et la moyennabilité "passent" du graphe à la relation et réciproquement. On déduit aussi de la construction quelques couplages d’équivalence mesurée (ou plus généralement des "randembeddings") entre certains sous-groupes des automorphismes de Γ et tout groupe qui contient orbitalement la relation R. Dans un deuxième chapitre, on aborde le cas particulier de la propriété (T) relative pour les paires de groupes (ΓxZ^2, Z^2), où Γ est un sous-groupe non moyennable de SL(2,Z). Cette propriété a d’abord été prouvée par Marc Burger, puis "re-démontrée" plus "visuellement" quelques années plus tard dans le cas de SL(2,Z)xZ^2 par Y. Shalom, en utilisant des découpages du plan. On reprend cette technique dans le cas général du théorème de Burger afin d’obtenir par un algorithme des constantes de Kazhdan explicites pour toute paire (ΓxZ^2, Z^2). / We consider a measure preserving standard borel equivalence relation R on a standard probability space (X,µ). We study a particular property of homogeneity for a fixed graphing of the relation R : We assume that the leaves of the graphing are all isomorphic to a given transitive graph Γ (connected, infinite, locally finite). What can be known about the relation ?In this case, considering a « Mackey action », we show that there exists a standard covering of (X,µ) i.e. a standard space Z; a probability measure η; a free, measure-preserving action on Z of G the (locally compact, second countable) group of all graph automorphisms of Γ and a stable isomorphism of the associated measured groupoid with R. We investigate some links between properties of G (resp. of the graph Γ) and those of R. In particular, Kazhdan property (T), Haagerup property (H) and amenability are preserved from the graph to the relation and conversely. We also deduce from the construction some couplings of measured equivalence (more generally some randembeddings) between subgroups of G and any group orbitally containing R. In a second chapter, we deal with the relative property (T) for the pairs (ΓxZ^2,Z^2), where Γ is a non-amenable subgroup of SL(2,Z). This property was first proved by M. Burger. Later on, Y. Shalom gave a more geometrical proof in the case of SL(2,Z)xZ^2, by using partitions of the plane. Following the same techniques in the general case of Burger's theorem, we develop an algorithm producing explicit constants for all pairs (ΓxZ^2,Z^2).

Les dyscalculie se caractérise par d’importantes difficultés d’apprentissage des maths, malgré une scolarisation adéquat et des capacités intellectuelles dans la norme. Même si ce trouble affecte 3 à 7% des enfants d’âge scolaire, ses causes restent encore peu connues. Il a été proposé que la dyscalculie consiste en un déficit spécifique de représentation des quantités numériques, causé par des anomalies cérébrales au niveau du sillon intra-pariétal (IPS). Cependant, de plus en plus d’études suggèrent que la dyscalculie serait dûe à des atteintes cognitives générales. Cette thèse s’est intéressée au lien entre apprentissage des maths et une forme de raisonnement déductif, à savoir le raisonnement transitif (A>B, B>C donc A>C). Dans une première étude en Imagerie par Résonance Magnétique fonctionnelle (IRMf), nous avons comparé l’activité cérébrale d’enfants dyscalculiques de 9 à 12 ans à celle d’enfants neurotypiques de même âge pendant l’écoute d’histoires comprenant des problèmes transitifs. Dans une seconde étude, nous avons évalué l’amélioration en maths des enfants neurotypiques 1,5 an après la session IRMf. Nous avons déterminé si les mesures cérébrales associées au raisonnement transitif pouvaient prédire l’amélioration en maths. Nous avons mis en évidence l’implication de l’IPS gauche dans le raisonnement transitif chez les enfants neurotypiques mais pas chez les dyscalculiques, qui étaient moins précis pour évaluer les problèmes transitifs. Par ailleurs, les capacités de mémoire de travail étaient corrélées à la performance et à l’activité de l’IPS lors du raisonnement transitif. Enfin, l’activité cérébrale de l’IPS pendant le traitement des problèmes transitifs prédisait l’amélioration en maths chez les enfants neurotypiques. Ces résultats soulignent l’importance du raisonnement transitif pour l’apprentissage des maths et suggèrent qu’un déficit de traitement des problèmes transitifs dans l’IPS pourrait contribuer à la dyscalculie / Children who struggle to learn math despite normal intelligence and adequate schooling may suffer from dyscalculia. Although this learning disability may affect 3-7% of children worldwide, its causes remain poorly understood. Previous research has suggested that dyscalculia was due to a specific deficit in the processing of numerical magnitude that results from neural anomalies to the Intraparietal Sulcus (IPS). However, a growing body of studies has highlighted the domain-general deficits exhibited by children with dyscalculia. The goal of the present thesis was to investigate the neural relationship between math learning and a type of deductive reasoning, namely transitive reasoning (A>B, B>C, therefore A>C). In a first study, we used functional Magnetic Reasonance Imaging (fMRI) to measure brain activity in both typically developing (TD) children and chidren with dyscalculia while they listened to stories that included transitive relations. In a second study, we followed TD children longitudinally and to test whether brain measures of transitive reasoning could predict math improvement. First, the processing of transitive relations was associated with enhanced activity in the left IPS in TD children, but not in children with dyscalculia. Second, children with dyscalculia made more errors when processing transitive relations than TD peers. Third, IPS activity and behavioral performance was correlated with working-memory skill across all participants, suggesting that working memory impairments contribute to impaired transitive reasoning skills. Fourth, math gain in TD children was predicted by brain activity around the IPS during the processing of transitive relations. Therefore, the present findings show that measures of transitive reasoning are associated with math achievement. Our results further suggest that impaired neural processing of transitive relations in the IPS may contribute to math difficulties in dyscalculia