Abstract The vocational math teachers at a vocational secondary school lack the adequate skills needed to be effective in math classrooms. The vocational teachers who teach math at the local vocational school have been identified as noncertified teachers based on their certification with the state department of education. The purpose of this project study was to determine the best professional development sessions needed to improve the instructional practice of untrained math teachers at the school. Their work is critical to the academic and career development of students. The social development theory of Vygotsky, which states that social interaction has an effect on individuals' growth and development, was the conceptual framework for this study. The research question focused on teachers' perception of the best approach to improve instructional strategies of vocational teachers in math. A qualitative case-study design was used to collect data from 8 teachers using interviews at the local vocational school. Data were analyzed and coded based on common themes. Member checking and triangulation of the data were used to ensure accuracy and credibility. The findings of this study showed that teachers at the study site need professional development, specifically a district policy ensuring that vocational math teachers receive professional development training, common planning time, participation in professional learning communities, and support in understanding the math objectives they are expected to teach to ensure their continuous improvement. This project may influence social change by spurring administrators to provide additional training to vocational educators, which may improve the quality of teachers' level of instruction and subsequent student outcomes.

This study identified variations in the learning trajectories of Tier II students when learning equivalent fraction concepts using physical and virtual manipulatives. The study compared three interventions: physical manipulatives, virtual manipulatives, and a combination of physical and virtual manipulatives. The research used a sequential explanatory mixed-method approach to collect and analyze data and used two types of learning trajectories to compare and synthesize the results. For this study, 43 Tier II fifthgrade students participated in 10 sessions of equivalent fraction intervention. Pre- to postdata analysis indicated significant gains for all three interventions. Cohen d effect size scores were used to compare the effect of the three types of manipulatives—at the total, cluster, and questions levels of the assessments. Daily assessment data were used to develop trajectories comparing mastery and achievement changes over the duration of the intervention. Data were also synthesized into an iceberg learning trajectory containing five clusters and three subcluster concepts of equivalent fraction understanding and variations among interventions were identified. The syntheses favored the use of physical manipulatives for instruction in two clusters, the use of virtual manipulatives for one cluster, and the use of combined manipulatives for two clusters. The qualitative analysis identified variations in students’ resolution of misconceptions and variations in their use of strategies and representations. Variations favored virtual manipulatives for the development of symbolic only representations and physical manipulatives for the development of set model representations. Results also suggested that there is a link between the simultaneous linking of the virtual manipulatives and the development of multiplicative thinking as seen in the tendency of the students using virtual manipulative intervention to have higher gains on questions asking students to develop groups of three or more equivalent fractions. These results demonstrated that the instructional affordances of physical and virtual manipulatives are specific to different equivalent fraction subconcepts and that an understanding of the variations is needed to determine when and how each manipulative should be used in the sequence of instruction.

This study was designed to explore the relationship between maternal attitudes about math, mothers' attitudes about their prekindergarten child's math abilities, mother-child math interactions, and prekindergarten children's math achievement. Using a Vygotskian framework, an intervention was developed. An intervention group of 18 mother-child dyads from Hyrum, Utah, area and a comparison group of 17 mother-child dyads from the Weston, Idaho, area were invited to participate in this research project. Measures included the Early Mathematics Concepts (EMC) assessment, the School Readiness Composite (SRC) of the Bracken Basic Concepts Scale-Revised (BBCS-R), My Attitudes Scale (MAS), and My Attitudes About My Child's Abilities Scale (MAAMCAS). Research hypotheses predicted that the intervention group would score higher on the EMC and each of the EMC's four focus areas at the posttest. Results did not support these hypotheses. Research hypotheses also predicted that the intervention group would have a larger difference in MAS and MAAMCAS scores from pretest to posttest and .that MAS and MAAMCAS scores would predict EMC scores. Neither of these hypotheses was supported. The final research hypothesis predicted a relationship between EMC focus area scores and mathrelated subscales of the BBCS- R. The strongest relationship appeared to be between the EMC number focus and the BBCS-R number subscale. The results were interpreted in accordance with current research and possible limitations of the present study.

Logical statements are prevalent in mathematics, the sciences, law, and many areas of everyday life. The most common logical statements are conditionals, which have the form “If H..., then C...,” where “H” is a hypothesis (or condition) to be satisfied and “C” is a conclusion to follow. Reasoning about conditionals is a skill that is only superficially understood by most individuals and depends on four main conditional contexts (e.g., intuitive, abstract, symbolic, or counterintuitive). The purpose of this study was to test a theory about the effects of context ordering on reasoning about conditionals. To test the theory, the researcher developed, tested, and revised a virtual manipulative educational mathematics application, called the Learning Logic App. This study employed a convergent parallel mixed methods design to answer an overarching research question and two subquestions. The overarching research question was “How does the order of teaching four conditional contexts influence reasoning about conditionals?” The two subquestions examined this influence on reasoning in terms of performance and perceptions. This study involved two phases. During Phase I, 10 participants interacted with the Learning Logic App in a clinical setting. The researcher used information gathered in Phase I to revise the Learning Logic App for Phase II. During Phase II, 154 participants interacted with the Learning Logic App in a randomly assigned context ordering in an online setting. In both phases, the researcher collected quantitative and qualitative data. After independent analyses, the researcher made meta- inferences from the two data strands. The results of this study suggest that context ordering does influence learners’ reasoning. The most beneficial context ordering for learners’ performance was symbolic-intuitive-abstract-counterintuitive. The most beneficial context ordering for learners’ perceptions was intuitive-abstract-counterintuitive-symbolic. Based on these results, the researcher proposed a new context ordering: symbolic-intuitive-abstract-counterintuitive-symbolic. This progression incorporates a catalyst at the beginning (symbolic context) which aids the learner in reassessing their prior knowledge. Then, the difficulty of the contexts progresses from easiest to hardest (intuitive-abstract-counterintuitive-symbolic). These findings are important because they provide an instructional sequence for teaching and learning to reason about conditionals that is beneficial to both learners’ performance and their perceptions.

Many universities and colleges have a considerable number of students enroll whose entrance examination scores indicate deficiencies in high school and pre-high school mathematics. The purpose of this study was to investigate the effectiveness of programed texts, as compared with conventional procedures, for teaching basic mathematics to remedial mathematics students. The effects of mental ability, study skills and attitudes on achievement through the use of programed texts also carne under investigation. For one quarter an experimental group of 73 subjects used a set of three linear programed texts for their sole mode of instruction in mathematics. A control group of 58 subjects were taught the same content material by traditional lecture-discussion procedures. At the beginning of the 1965 Fall quarter, the subjects were given a mental ability test, a study skills and attitude survey and a standardized mathematics pretest. The subjects' scores on these three measures served as covariates for a covariance analysis of the mathematics posttest scores. Analysis of variance showed no significant differences (NSD) between the means of the experimental and control groups scores on the mathematics posttest. However, analysis of covariance showed a significant mean score difference in favor of the experimental group for the questions pertaining to mathematics fundamentals (computation) and NSD for the questions pertaining to reasoning (problem solving) . The experimental group went from a mathematics pretest mean score grade placement of about 8.5 to a mathematics posttest mean score grade placement of about 10.5. The control group went from a mathematics pretest mean score grade placement of about 9.0 to a mathematics posttest mean score grade placement of about 10.5. The two different teaching methods did not bring about significant differences in the variability of the subjects' mathematics test scores. The correlation between mental ability scores and mathematics test scores was moderate (about .50). As would be expected, the correlation between mathematics pretest and posttest scores was high (about .80). The correlation between study skills scores and mathematics test scores was low (about .26 for the experimental group and about .04 for the control group). Individual rates of progress, made possible by programed texts, enabled a considerable number of students in the experimental group to complete the equivalent of a quarter's study in basic mathematics in less than a quarter's time. A survey questionnaire concerning interest and attitude of the subjects toward mathematics, programed instruction and the remedial mathematics course was given at the end of the quarter in which the study was conducted. Chi-square analysis of the responses to the questions generally showed the subjects' responses were independent of the type of instruction they had received. The subjects were also asked to comment on what they thought were the most favorable characteristics of the course and what they thought were the least favorable characteristics of the course. The favorable comment listed most frequently by control subjects pertained to the slow group pace. They explained that it was the slow pace, coupled with a very understanding instructor, which enabled them to learn mathematics which they had missed in high school. However, it was also the slow group pace which drew the most number of control subjects' responses as to what they liked least about the course. The favorable comment listed most frequently by the subjects who learned from programed texts pertained to the opportunity the programed texts had provided for each student to progress at his own rate. The unfavorable comment listed most frequently by the programed learning group pertained to no teacher-student interaction and no class discussion when programed texts were used.

Thesis (M.A.)--Wheaton College Graduate School, 2004. Action Research Paper (M.A.)--Wheaton College Graduate School, 2004. / Abstract. Includes bibliographical references (leaves 64-70).

Ce travail met en jeu plusieurs systèmes dynamiques sur des tores en dimension finie, pour lesquels on sait établir des théorèmes limites, qui permettent de préciser leur comportement stochastique. On généralise d'abord le théorème limite local usuel sur un sous-shift de type fini, en ajoutant un terme de perturbation, en reprenant la preuve classique, par des techniques d'opérateurs. On en déduit un théorème limite local pour les sommes de « Riesz-Raïkov unitaires étendues », et des observables höldériennes. Pour cela, on reprend une méthode employée par Bernard Petit, en utilisant des codages symboliques, et le théorème limite local avec perturbation. Puis, on présente plusieurs situations de composées d'automorphismes hyperboliques du tore en dimension deux pour lesquelles on sait établir un théorème limite central quelque soit le choix de la composée. En particulier, on aborde le cas des matrices à coefficients entiers positifs.

Cette thèse a pour objet d'affiner les critères pour l'indépendance algébrique et les mesures d'indépendance algébrique démontrés par P. Philippon et par E.M. Jabbouri d'une part, et l'indépendance linéaire et les mesures d'indépendance linéaire établis par Y.V. Nesterenko et par P. Bundschuh et T. Töpfer d'autre part, réalisant ainsi une jonction entre les problèmes d'indépendance linéaire et algébrique sur un corps de nombres. Ils diffèrent des critères démontrés par P. Philippon et par E.M. Jabbouri par l'utilisation de nouvelles hauteurs et de nouvelles distances qui coïncident, dans le cas linéaire, à celles utilisées par Y.V. Nesterenko et par P. Bundschuh et T. Töpfer. Ils énoncent de façon générale, que dans un espace projectif, lorsqu'on a un système de formes définies sur un corps de nombres, prenant des valeurs petites en un point, mais n'ayant pas de zéro commun trop proche de ce point, alors on peut minorer la distance de ce point à toute variété définie sur le corps de nombres, de dimension, degré et hauteur bornés en fonction des formes

Nous désignons par D l'anneau des germes à à l'origine d'opérateurs différentiels linéaires à coefficients analytiques. Nous étudions les résolutions libres minimales de D-modules, introduites par M. Granger, T. Oaku et N. Takayama. Plus précisément nous considérons des modules admettant une V - filtration le long d'une hypersurface lisse, et les résolutions minimales sont adaptées à cette filtration. Nous nous intéressons particulièrement aux rangs d'une telle résolution minimale, appelés nombres de Betti, ce sont des invariants du module. En premier lieu, nous donnons des résultats généraux : nous ramenons le calcul des nombres de Betti à une situation d'algèbre commutative et nous définissons les résolutions minimales génériques. Ensuite, nous considérons une singularité d'hypersurface complexe f = 0 et le module N = D x , t Fs introduit par B. Malgrange, dont la restriction le long de t=0 fournit la cohomologie locale algébrique du faisceau des fonctions analytiques à support dans f = 0. Le module N est naturellement muni de la V -filtration le long de t = 0, nous étudions les nombres de Betti correspondants. Ces nombres sont des invariants analytiques pour l'hypersurface f = 0. Nous les calculons pour f une singularité isolée quasi homogène ou un monôme. Lorsque f est à singularité isolée, nous caractérisons la quasi-homogénéité en termes des nombres de Betti.

Un espace de Banach spectral p-adique est un espace de~Banach p-adique muni d'une algèbre de fonctions analytiques à valeurs dans un corps complet et algébriquement clos C. Un espace de Banach-Colmez est un espace de Banach spectral qui s'obtient par extensions et quotients à partir de C et Qp. Ces espaces forment une catégorie abélienne, qui est naturellement munie de fonctions additives « dimension » et « hauteur » ; on retrouve ainsi une démonstration du théorème « faiblement admissible implique admissible » (Colmez-Fontaine, 2000). De plus, il existe une sous-catégorie pleine qui admet une filtration canonique par les pentes de l'action du Frobenius, décroissante et indexée par les rationnels positifs.