This diploma thesis deals with the topic negative numbers, especially with strategies and difficulties of 7th and 8th grade primary school pupils when solving selected tasks with negative numbers. This thesis is divided into theoretical and experimental parts. The theoretical part deals with negative numbers, their development, didactic difficulty and work with them. The thesis deals with selected international comparative and foreign researches with the focus on negative numbers. The most extensive part of the thesis contains an analysis of mathematics textbooks used in schools in the Czech Republic with the focus on negative numbers. The experimental part, which aim is the identification of pupils' difficulties and the analysis of mistakes they make when working with negative numbers, is the key part of the thesis. 185 pupils of 7th and 8th grade from three different schools took part in the research. The methodology consisted of the preparation of a didactic test on negative numbers and assigning it to the pupils. After completing the test, selected pupils were asked to explain their written solutions. The didactic test was prepared on the basis of mentioned research results and the analysis of mathematics textbooks. The data were analysed in terms of pupils' strategies and mistakes. The thesis...

L'instabilité de Rayleigh-Taylor (IRT) est notamment rencontrée lors des expériences de Fusion par Confinement Inertiel, et son développement est un obstacle à la réussite de ces expériences. L'objet de cette thèse est d'étudier la croissance de l'IRT pour différents régimes de compressibilité, au moyen de simulations numériques directes réalisées à l'aide d'un code pseudo-spectral multidomaine de type Chebyshev-Fourier-Fourier.La méthode du développement asymptotique permet d'établir des modèles à bas nombre de Mach pour lesquels la contribution acoustique est éliminée. L'implantation dans le code de simulation du modèle anélastique, qui met en jeu des fluides stratifiés et capture les effets thermiques, est améliorée. Le modèle de Boussinesq est ajouté au code. La précision de la méthode numérique est étudiée pour différents découpages en sous-domaines. Plusieurs éléments de validation sont présentés, dont la comparaison avec une expérience.La première simulation présentée, réalisée avec le modèle de Boussinesq, s'intéresse à la croissance auto-semblable de l'IRT. Les lois d'échelle de la vorticité et de la dissipation sont dégagées. La structure de la turbulence et du mélange entre les deux fluides est discutée. Certaines propriétés de la turbulence homogène et isotrope sont retrouvées, mais on note la persistance d'anisotropie aux petites échelles. Les premières simulations 3D de l'IRT avec le modèle anélastique sont présentées. L'influence des effets de compressibilité sur les premières phases de la croissance est étudiée. En outre, une couche de mélange anélastique en faible stratification est analysée et présente des effets de compressibilité non négligeables. / The Rayleigh-Taylor instability (RTI) is especially observed in inertial confinement fusion experiments, and its development prevents the success of these experiments. The purpose of this work is to study the growth of the RTI for different compressibility regimes by using a multidomain pseudospectral Chebyshev-Fourier-Fourier simulation code. The asymptotic expansion method allows to establish several low Mach number models which do not contains acoustics. The implantation of the anelastic model, which deals with stratified fluids and captures thermal effects, has been improved. Moreover, the Boussinesq model is added to the simulation code. The accuracy of the entire numerical method is studied, as a function of the subdomain separation, and several validation elements are shown, including a comparison with an experimental study. The first simulation to be analyzed is achieved with the Boussinesq model. We focus on the self-similarity of the RTI growth. The temporal scalings of vorticity and dissipation are displayed, and the structures of turbulence and mixing are discussed. Some properties of isotropic and homogeneous turbulence are observed, however some anisotropy remains at small scales. The first three-dimensional anelastic simulations are presented. The influence of compressibility effects on the first stages of the growth is studied. Finally, a developed anelastic mixing layer involving weakly stratified fluids is described and was found to display non-negligible compressibility effects.

Understanding base-ten numbers is one of the most important mathematics topics taught in the primary school, and yet also one of the most difficult to teach and to learn. Research shows that many children have inaccurate or faulty number conceptions, and use rote-learned procedures with little regard for quantities represented by mathematical symbols. Base-ten blocks are widely used to teach place-value concepts, but children often do not perceive the links between numbers, symbols, and models. Software has also been suggested as a means of improving children's development of these links but there is little research on its efficacy. Sixteen Queensland Year 3 students worked cooperatively with the researcher for 10 daily sessions, in 4 groups of 4 students of either high or low mathematical achievement level, on tasks introducing the hundreds place. Two groups used physical base-ten blocks and two used place-value software incorporating electronic base-ten blocks. Individual interviews assessed participants' place-value understanding before and after teaching sessions. Data sources were videotapes of interviews and teaching sessions, field notes, workbooks, and software audit trails, analysed using a grounded theory method. There was little difference evident in learning by students using either physical or electronic blocks. Many errors related to the "face-value" construct, counting and handling errors, and a lack of knowledge of base-ten rules were evident. Several students trusted the counting of blocks to reveal number relationships. The study failed to confirm several reported schemes describing children's conceptual structures for multidigit numbers. Many participants demonstrated a preference for grouping or counting approaches, but not stable mental models characterising their thinking about numbers generally. The independent-place construct is proposed to explain evidence in both the study and the literature that shows students making single-dimensional associations between a place, a set of number words, and a digit, rather than taking account of groups of 10. Feedback received in the two conditions differed greatly. Electronic feedback was more positive and accurate than feedback from blocks, and reduced the need for human-based feedback. Primary teachers are urged to monitor students' use of base-ten blocks closely, and to challenge faulty number conceptions by asking appropriate questions.