Thesis advisor: Sara Cordes / Human infants show relatively sophisticated abilities to track and use proportional information. However, by the age of 6, children tend to make predictable errors in their proportional reasoning and later encounter significant challenges in many aspects of formal fraction learning. Thus, one of the central questions motivating this research is to identify the factors leading to these difficulties, in light of evidence of early intuitions about these concepts. In the current dissertation, I address this question by investigating the tradeoff between attending to proportional magnitude information and discrete numerical information about the components (termed “numerical interference”) across both spatial (i.e., area models, number lines) and symbolic (fractions, decimals) representations of proportion information. These explorations focus on young children (5-7 year olds) who have yet to receive formal fraction instruction, older children (9-12 year olds) who are in the process of learning these concepts, and adults who have already learned formal fractions. In Project 1, I investigated how older children and adults map between symbolic and spatial representations, particularly focusing on their strategies in highlighting componential information versus magnitude information when solving these mapping tasks. In Projects 2 and 3, I explore the malleability of individual differences in this numerical interference in 4- to 7-year-old children. Across the three projects, I suggest that although numerical interference does impact proportional reasoning, this over-attention to number can be reduced through modifying early experiences with proportional information. These findings have implications for education and the way we conceptualize numerical interference more generally. / Thesis (PhD) — Boston College, 2017. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Psychology.

The current study investigated variability in student performance on a concurrent verbalization measure based on a grade-level sample math word problem and sought to determine to what extent the variability in verbalization scores is related to scores on a reliable measure of reading (DIBELS Next) and math (easyCBM) and to student factors (e.g. sex, grade, economic status). In light of the 2014 implementation of the Common Core State Standards and related measures of student performance, both of which contain components of language in mathematics curriculum and assessment, it was the intent of this study to identify factors associated with verbalization on sample math word problems that could be correlated with student performance on reliable, commonly used assessments of reading and math. The sample for analysis included 105 intermediate-grade students from one elementary school in the Pacific Northwest. Results support a relation between students' verbalizations about math word problems and benchmark assessments in reading and math. Limitations of the study, considerations for future research, and implications for practice are discussed.

Games as a means of education have been starting to become more of an everyday reality. Not only are games used in classrooms, but they are used in industry to train soldiers, medical staff, and even surgeons. This thesis focuses on physically active (i.e. embodied) multiplayer games as a means of education; not only by having students play, but also by having students create games. The embodied multiplayer aspect allows for a more interactive experience between players and their environment, making the game immersive and collaborative. In order to create these games, students must exercise their computational thinking abilities. The Wearable Learning Cloud Platform has been developed to enable students to design, create, and play multiplayer games for STEM. This platform allows users to create, edit and manage the behavior of mobile technologies that act as support to players of these games, specified as finite-state machines. The platform provides a means of testing created games, as well as executing (running) these games wirelessly by serving them to smartphones (or smart watches) so that students can play them with other students, in teams, or as individual players. The platform features a full drag and drop game editor with a sophisticated validation engine that prevents users from making syntax errors. Visually programmed games transpile to JavaScript for execution on the game server and provide two separate levels of programming abstraction. This platform has been successfully tested with 18 participants and they have shown significant improvements in their understanding of Finite State Machines and have shown an increase in their Computational Thinking abilities.

abstract: This mixed methods action research dissertation examines the effects of implementing growth mindset teaching practices in third grade math as a means to improve student math self-efficacy, math mindset and student achievement. Since the transition to the Pennsylvania Core Standards, students across the state including those in this district have been experiencing a decrease in math achievement in grades three through eight according to the Pennsylvania System of School Assessment (PSSA) the standardized achievement test all public school students take. Locally, traditional interventions such as worksheets, boxed programs, computer-based programs and extra practice have not yielded gains so this intervention focused on developing growth mindset teaching practices in math to answer four research questions. Framed in Dweck’s Implicit Theories of Personal Attributes (1995), Bandura’s description of self-efficacy (1997) and Hall and Hords’ work with teachers in bridging research into practice (2011), this study used Jo Boaler’s, Mathematical Mindset (2015) in a book study with the third-grade teachers. The dissertation study analyzed pre and post survey data from the third-grade class (n=57) on both mindset and self-efficacy. The study also analyzed pre and post survey data from the teachers (n=2) on mindset along with pre and post intervention interviews with the teachers. Qualitative and quantitative data analysis revealed the intervention had a positive effect on teacher mindsets and practices, a positive effect on student mindsets and a positive effect on student math self-efficacy. While the study did not reveal the intervention to have a positive impact on student achievement at this time, previous research included in the literature review cites improvement in student achievement through developing growth mindset thinking. This gives reason to predict that with more time, these students will experience improved achievement in math. Implications from this study include that we should train all math teachers in incorporating growth mindset practices, and that administrators should build the bridge between research and practice for teachers as they implement new teaching practices in effort to positively affect student performance. / Dissertation/Thesis / Doctoral Dissertation Leadership and Innovation 2019

Over the course of a college mathematics degree, students are inevitably exposed to elementary physics. The derivation of the equations of motion are the classic examples of applications of derivatives and integrals. These equations of motion are easy to understand, however they can be expressed in other ways that students aren't often exposed to. Using the Lagrangian and the Hamiltonian, we can capture the same governing dynamics of Newtonian mechanics with equations that emphasize physical quantities other than position, velocity, and acceleration like Newton's equations do. Building o of these alternate interpretations of mechanics and understanding gauge transformations, we begin to understand some of the mathematical physics relating to gauge theories. In general, gauge theories are eld theories that can have gauge transformations applied to them in such a way that the meaningful physical quantities remain invariant. This paper covers the buildup to gauge theories, some of their applications, and some computational approaches to understanding them.

The local problem that drove this study is that a high school in an upper middle class suburban city in Pennsylvania wants to improve its student scores on its end-of-course Algebra 1 Keystone Exam. The purpose of this study was to conduct a quantitative, quasi-experimental assessment of an online high school mathematics remediation program to determine if the remediation program was successful in its endeavor to remediate students. This research study, informed by the self-efficacy and the behaviorist learning theories, attempted to determine whether students who (a) scored below proficient on the May algebra exam and were placed in the Math Lab course improved statistically significantly compared with the students who (b) scored below proficient on the May algebra exam and who retook the exam in January but were not placed in the Math Lab course. Using a convenience sample, an independent samples t test was performed on the difference scores (original Keystone Exam and retest) of 408 students. The study determined that the online remediation program did not increase student scores for the students at the Pennsylvania high school compared with students who were not in the remediation program. The second literature review and white paper provide six research-based recommendations for the SEPSD to improve the Math Lab course. The recommendations include eliminating the course, purchasing a different remediation program, or modify elements of the current program. The students in the SEPHS would benefit from the research with a better remediation program. The research based suggestions, once implemented, should lead to the improvement of mathematics achievement.

Dans cette thèse, nous allons considérer un système de contrôle dont l'état est donné par la solution de l'équation de Korteweg-de Vries (KdV) posée sur un intervalle borné. On imposera des conditions Dirichlet homogènes aux bords et le contrôle portera sur la condition Neumann à droite de l'intervalle. Nous allons considérer deux types de problèmes qui sont étroitement liés : la contrôlabilité et la stabilisation. Les chapitres 2 et 3 sont consacrés a étudier la contrôlabilité sur quelques domaines pour lesquels le système linéaire n'est pas contrôlable. Notre but est de démontrer que malgré cette perte de contrôlabilité du système linéaire, la non-linéarité nous permet d'obtenir la contrôlabilité pour le système non linéaire. Pour faire ceci nous allons utiliser la méthode de développement en séries entières, introduite dans le cadre de la dimension infinie par J.-M. Coron et E. Crépeau dans [J. Eur. Math. Soc. (JEMS) 6, no. 3, pp. 367-398, 2004]. La méthode consiste à bouger le système le long des directions manquantes pour le système linéaire par des développements d'ordre supérieur à un, et puis à appliquer un théorème de point fixe. Dans le chapitre 4, on étudiera la stabilisation pour notre système. Le but de cette partie est de construire des lois de feedback tel que le système en boucle fermée ait une décroissance exponentielle vers zéro avec un taux de décroissance arbitraire. La méthode utilisée est due a J. M. Urquiza qui l'a introduite dans [SIAM J. Control Optim., V. 43, no. 6, pp 2233-2244, 2005]. Pour être en mesure d'appliquer cette méthode, une analyse spectrale de l'opérateur de KdV stationnaire est nécessaire.

La thèse contient un premier chapitre avec des préliminaires sur les langages bidimensionnels, sur les résultats principaux et sur les différentes caractérisations des langages reconnaissables par systèmes de pavages qui jouent un rôle central dans la thèse. Ensuite, nous décrivons la structure algébrique des familles des langages locaux. Nous prouvons que cette structure est un treillis par rapport à l'inclusion et nous étudions les propriétés de ce treillis. Par ailleurs, nous traitons des problèmes informatiques de décidabilité et nous donnons la position, dans la hiérarchie arithmétique, des problèmes classiques sur des langages de mots appliquées aux langages bidimensionnelles. Dans la thèse, après quelques définitions de base sur les polyominos, nous traitons la reconnaissabilité de plusieurs classes des polyominos par des langages reconnaissables par systèmes de pavages. En particulier, nous donnons les systèmes de pavages pour des langages représentant les classes des polyominos convexes, h-convexes ou parallélogrammes. Ensuite, nous étudions la reconnaissabilité des polyominos L-convexes. En conclusion, la dernière partie de la thèse est consacrée à l'application des langages reconnaissables par systèmes de pavages au calcul d'ADN. Nous donnons l'idée de la construction avec de l'ADN de quelques classes des polyominos (par exemple la classe des polyominos parallélogrammes) obtenues à travers la famille des langages reconnaissables par systèmes de pavages.

Cette thèse a pour objectif d'étudier les applications cryptographiques des courbes elliptiques sur l'anneau Fp["], où Fp représente un corps fini d'ordre premier p et où " vérifie"2 = 0. Après avoir décrit ces courbes définies sur un anneau, nous en étudions l'aspect algorithmique en proposant des solutions concrètes d'implémentations des éléments et de la loi de groupe. Enfin, nous illustrons leur intérêt cryptographique, en proposant : une attaque du problème du logarithme discret elliptique (sur un corps fini) utilisant ces courbes ; un cryptosystème de type ElGamal sur ces courbes, dont nous étudions les propiétés de sécurité.

Il s'agit d'un document qui survole de manière informelle les travaux quej'ai présenté pour obtenir l'habilitation à diriger des recherches.<br />Ces travaux concernent les sous-variétés legendriennes, étudiées du point de vue de la théorie de Morse et de la combinatoire des fronts d'ondes.