This study investigates the potential of using a newly developed sub-game in the Magical Garden research platform as a method of teaching preschool children subitizing. Thirty preschool children played the game, identifying formation with four, five and six objects. Some formations with five objects were scaffolded by changing the sprites of some of the objects. Results showed that children were significantly faster and accurate at formations with four objects than five and six. No significant difference was found between five and six objects, which were also considerably similar in both time and accuracy. Analysis suggests that perceptual subitizing was used to a greater degree when presented with four objects and that counting was used for higher numerosities. The study showed that there were some problems with the study design, with formations being more difficult at formations with five objects. Suggestions are made for how the game should be altered, including a dynamic difficulty changing component to account for the large individual differences. Additionally, it is suggested to lower the amount of time the formations are shown in order to elicit the use of conceptual subitizing and focus on lowering the difficulty in terms of the number of items and formation pattern rather than altering time. There was no significant difference for performance when comparing scaffolded formations with non-scaffolded formations. The reason could be that the formations were not subitized. Future studies should include the suggested changes and conduct longitudinal studies looking at improvements over time and whether children are improving.

Subitizing, a quick apprehension of the numerosity of a small set of items, is inconsistently utilized by preschool educators to support early number understandings (Sarama & Clements, 2009). The purpose of this qualitative study is to investigate the relationship between children’s number understanding and subitizing activity. Sarama and Clements (2009) consider students’ subitizing activity as shifting from reliance upon perceptual processes to conceptual processes. Hypothesized mental actions carried into subitizing activity by children have not yet been empirically investigated (Sarama & Clements, 2009). Drawing upon Piaget’s (1968/1970) three mother structures of mathematical thinking, the theoretical implications of this study consider expanding the scope of Piaget’s (1968/1970) definition of topological thinking structures to include patterned orientations. Increasing the scope of this definition would allow for the investigation of the development of topological thinking structures and subitizing activity. An 11-week teaching experiment was conducted with six preschool aged children in order to analyze student engagement with subitizing tasks (Steffe & Ulrich, in press). To infer what perceptual and conceptual processes students relied upon when subitizing, tasks were designed to either assess or provoke cognitive changes. Analysis of interactions between students and the teacher-researcher informed this teacher-researcher of cognitive changes relative to each student’s thinking structure. Results indicated that students rely upon the space between items, symmetrical aspects of items, and color of items when perceptually subitizing. Seven different types of subitizing activity were documented and used to more explicitly describe student reliance upon perceptual or conceptual processes. Conceptual subitizing activity was redefined in this study, as depending upon mental reversibility and sophisticated number schemes. Students capable of conceptual subitizing were also able to conserve number. Students capable of conserving number were not always capable of conceptual subitizing. The symmetrical aspects of an item’s arrangement elicited students’ attention towards subgroups and transitioning students’ perceptual subitizing to conceptual subitizing. Combinations of counting and subitizing activity explained students’ reliance upon serial and classification thinking structures when transitioning from perceptual subitizing to conceptual subitizing. Implications of this study suggest effectively designed subitizing activity can both assess students’ number understandings, and appropriately differentiate preschool curriculum. / Ph. D.

Visual saliency computation is about detecting and understanding salient regions and elements in a visual scene. Algorithms for visual saliency computation can give clues to where people will look in images, what objects are visually prominent in a scene, etc. Such algorithms could be useful in a wide range of applications in computer vision and graphics. In this thesis, we study the following visual saliency computation problems. 1) Eye Fixation Prediction. Eye fixation prediction aims to predict where people look in a visual scene. For this problem, we propose a Boolean Map Saliency (BMS) model which leverages the global surroundedness cue using a Boolean map representation. We draw a theoretic connection between BMS and the Minimum Barrier Distance (MBD) transform to provide insight into our algorithm. Experiment results show that BMS compares favorably with state-of-the-art methods on seven benchmark datasets. 2) Salient Region Detection. Salient region detection entails computing a saliency map that highlights the regions of dominant objects in a scene. We propose a salient region detection method based on the Minimum Barrier Distance (MBD) transform. We present a fast approximate MBD transform algorithm with an error bound analysis. Powered by this fast MBD transform algorithm, our method can run at about 80 FPS and achieve state-of-the-art performance on four benchmark datasets. 3) Salient Object Detection. Salient object detection targets at localizing each salient object instance in an image. We propose a method using a Convolutional Neural Network (CNN) model for proposal generation and a novel subset optimization formulation for bounding box filtering. In experiments, our subset optimization formulation consistently outperforms heuristic bounding box filtering baselines, such as Non-maximum Suppression, and our method substantially outperforms previous methods on three challenging datasets. 4) Salient Object Subitizing. We propose a new visual saliency computation task, called Salient Object Subitizing, which is to predict the existence and the number of salient objects in an image using holistic cues. To this end, we present an image dataset of about 14K everyday images which are annotated using an online crowdsourcing marketplace. We show that an end-to-end trained CNN subitizing model can achieve promising performance without requiring any localization process. A method is proposed to further improve the training of the CNN subitizing model by leveraging synthetic images. 5) Top-down Saliency Detection. Unlike the aforementioned tasks, top-down saliency detection entails generating task-specific saliency maps. We propose a weakly supervised top-down saliency detection approach by modeling the top-down attention of a CNN image classifier. We propose Excitation Backprop and the concept of contrastive attention to generate highly discriminative top-down saliency maps. Our top-down saliency detection method achieves superior performance in weakly supervised localization tasks on challenging datasets. The usefulness of our method is further validated in the text-to-region association task, where our method provides state-of-the-art performance using only weakly labeled web images for training.

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Universally across modern cultures children acquire the meaning of the words one, two, and three in order. While much research has focused on how children acquire this knowledge and what this knowledge represents, the question of why children learn numbers in order has been comparatively neglected. To address this question, a non-verbal anticipatory looking task was implemented. In this task, 35 14- to 23-month-old infants were assessed on their ability to form implicit category structures for the numbers one, two, and three. We hypothesized that children would be able to form the implicit category structure for the number one but not for two or three because sets of two and three objects would exceed the working memory capacities of infants. We found results consistent with this hypothesis; infants (regardless of age) were able form a category for sets with one object, as evidenced by their looking behavior while the looking behavior for the numbers two and three did not demonstrate a statistically significant pattern. We interpret our results as consistent with our hypothesis and discuss implications for parallel individuation, number acquisition theories, and the development of working memory resources.

Le sens du nombre est un des piliers des apprentissages en arithmétique. Son acquisition permet, entre autres, de comprendre notre système de numération. Dans les années 1980, des chercheures se sont intéressées aux difficultés associées à l’apprentissage de la numération et ont énoncé une série de recommandations pour favoriser la compréhension de ce concept. Ces recherches sont encore aujourd’hui des recherches phares et plusieurs études s’en sont inspirées. Des études plus récentes montrent cependant que les difficultés liées à l’apprentissage de la numération demeurent les mêmes pour les élèves d’aujourd’hui. L’objectif général de la présente recherche est de mieux comprendre comment se développe le sens du nombre et de la numération, de la petite enfance jusqu’à l’âge de 7-8 ans et de faire ressortir les conditions qui favorisent ce développement. Des recherches montrent que le développement du sens du nombre s’appuie sur la construction de représentations mentales dynamiques et imagées. Pour favoriser cette construction, les élèves doivent avoir accès à des représentations concrètes et imagées aussi variées que fécondes. Des tâches de résolution de problèmes dans lesquelles les élèves s’engagent doivent aussi être prévues pour favoriser les apprentissages. Des recherches montrent enfin que le sens du nombre peut être décrit sous forme de continuum qui se développe du préscolaire à l’âge adulte. Or, aucune étude connue ne s’est intéressée à ce type de progression et n’a tenté d’identifier les conditions qui favorisent ce développement en tenant compte des éléments ci-haut mentionnés. Dans la présente recherche, une proposition de continuum du sens du nombre et de la numération de la petite enfance à 8 ans, s’appuyant sur ces recherches, a été établie. Ce continuum identifie les éléments clés du développement de la compréhension des élèves. Un outil d’évaluation a été construit. Il permet de situer l’élève sur ce continuum. Une séquence didactique a été mise en place. Elle donne l’occasion à l’élève de développer sa compréhension selon ce continuum. Ces instruments s’adressent aux élèves de la fin de la 2e année et du début de la 3e année, soit des enfants de 7-8 ans. La construction de ces instruments constitue un des objectifs spécifiques de cette recherche. Un deuxième objectif est de vérifier la viabilité en contexte de ces instruments auprès de professionnels de l’éducation. Les objectifs de l’étude ont été atteints : les instruments ont été créés, puis leur viabilité en contexte a été évaluée par des professionnels du milieu de l’éducation. Selon l’analyse qualitative des commentaires des participants, l’outil d’évaluation permettrait d’évaluer le niveau de développement du sens du nombre des élèves et de dépister ceux qui ont des difficultés à apprendre ces concepts. Il donnerait aussi l’occasion aux élèves de développer leur sens du nombre, selon leur niveau de compréhension, à travers une séquence d’activités. Une analyse fine des commentaires fait clairement ressortir que le sens du nombre, de même que les conditions à mettre en place pour favoriser son développement, n’occupent pas une assez grande place dans l’enseignement actuel de l’arithmétique au primaire ni dans le Programme de formation de l’école québécoise. Il demeure cependant un prédicteur important de réussite scolaire. C’est pour cette raison que d’autres travaux doivent porter sur les concepts ciblés dans la présente étude afin de mieux accompagner les élèves et leurs enseignants vers la réussite. / Number sense is one of the pillars of learning arithmetic. Its acquisition allows, among other things, to understand our decimal and positional numeral system. In the 1980s, researchers became interested in the difficulties associated with learning numeration by elementary school children. They proposed a framework and a series of recommendation to promote the understanding of this concept. Their research is still a reference today and several studies have been inspired by it. More recent studies, however, show that the difficulties associated with learning numeration remain the same for today’s students. The general objective of this research is to better understand how number sense develops from infancy up to the age of 7-8 years and to highlight the conditions that increase this development. Research shows that the development of number sense and the understanding of mathematical concepts is based on the construction of dynamic and image-based mental representations. To promote this construction, students must be given access to concrete and image-based representations that are as varied as they are fruitful. Problem-solving tasks in which students are called upon to engage must also be included in the planning of mathematical activities to promote student learning. Finally, research shows that number sense can be described as a continuum that develops from preschool to adult life. However, no known study has been focusing on in this type of progression leading to the understanding of numeration and has attempted to identify the conditions to promote the development of number sense by taking into account the above-mentioned elements. In the present study, a proposal for a number sense continuum, from infancy to age 8, has been established. This continuum identifies key elements in the development of student understanding. An evaluation tool has been built. It helps situate the student on this continuum. A didactic sequence has also been built. It gives students the opportunity to develop their understanding along this continuum. These tools were intended for students at the end of Grade 2 and the beginning of Grade 3, i.e. children aged 7-8. The construction of these tools was one of the specific objectives of this research. A second objective was to verify viability in context of these tools with education professionals. The objectives of the study were achieved: the device was created and then the viability in context was evaluated by professionals in the education community. According to the qualitative analysis of participants' comments, the device could make it possible to assess the level of development of the number sense of the students and to identify those who have difficulty in learning this concept. It also could give students the opportunity to develop their number sense, to their level of understanding, through a sequence of activities. A detailed analysis of the comments clearly shows that number sense, as well as the conditions that need to be put in place to promote its development, do not occupy a sufficiently large place in the current teaching of arithmetic at the elementary school level or in the Quebec Education Program. However, it remains an important predictor of academic success. For this reason, further work must be done on the concepts targeted in this study in order to better guide students and their teachers towards success.