Real algebraic morphisms and Del Pezzo surfaces of degree 2

Let X and Y be affine nonsingular real algebraic varieties. A general problem in Real Algebraic Geometry is to try to decide when a smooth map f : X -> Y can be approximated by regular maps in the space of smooth mappings from X to Y, equipped with the compact-open topology. In this paper we give a complete solution to this problem when the target space is the usual 2-dimensional sphere and the source space is a geometrically rational real algebraic surface. The approximation result for real algebraic surfaces rational over R is due to J. Bochnak and W. Kucharz. Here we give a detailed description of the more interesting case, namely a real Del Pezzo surfaces of degree 2

Una propuesta colaborativa para enriquecer la formación matemática inicial y continua de maestros de infantil

En este trabajo presentamos los resultados de una experiencia realizada conjuntamente por maestros en activo, estudiantes para maestro y formadores investigadores en el área de didáctica de las matemáticas en Educación Infantil. El primer objetivo de esta experiencia es enriquecer la formación de los maestros implicados, conectando la formación con la práctica profesional. Impulsados por el interés compartido de promover la flexibilidad matemática de los alumnos a través del uso y conversiones entre distintos modos de representación, el trabajo se desarrolla sobre una actividad en la que se tratan aspectos del número y de la geometría en tres momentos: diseño de la tarea, puesta en práctica en el aula por maestras en ejercicio, reflexión conjunta sobre la puesta en práctica del diseño, mediante el visionado de los vídeos obtenidos. El interés de los formadores-investigadores es identificar los conocimientos matemáticos movilizados por las maestras en formación en el diseño de una tarea y por las maestras en ejercicio al analizar y llevar al aula esa tarea. Se seguirá para tal fin el modelo de Conocimiento Especializado del Profesor de Matemáticas (MTSK) sobre las transcripciones de las sesiones de trabajo conjunto audiograbadas. Esta experiencia permitirá elaborar materiales para mejorar la formación

Which One Is the “Best”: a Cross-national Comparative Study of Students’ Strategy Evaluation in Equation Solving

This cross-national study examined students’ evaluation of strategies for solving linear equations, as well as the extent to which their evaluation criteria were related to their use of strategies and/or aligned with experts’ views about which strategy is the best. A total of 792 middle school and high school students from Sweden, Finland, and Spain participated in the study. Students were asked to solve twelve equations, provide multiple solving strategies for each equation, and select the best strategy among those they produced for each equation. Our results indicate that students’ evaluation of strategies was not strongly related to their initial preferences for using strategies. Instead, many students’ criteria were aligned with the flexibility goals, in that a strategy that takes advantages of task context was more highly valued than a standard algorithm. However, cross-national differences in strategy evaluation indicated that Swedish and Finnish students were more aligned with flexibility goals in terms of their strategy evaluation criteria, while Spanish students tended to consider standard algorithms better than other strategies. We also found that high school students showed more flexibility concerns than middle school students. Different emphases in educational practice and prior knowledge might explain these cross-national differences as well as the findings of developmental changes in students’ evaluation criteria

Exploring students’ procedural flexibility in three countries

BackgroundIn this cross-national study, Spanish, Finnish, and Swedish middle and high school students’ procedural flexibility was examined, with the specific intent of determining whether and how students’ equation-solving accuracy and flexibility varied by country, age, and/or academic track. The 791 student participants were asked to solve twelve linear equations, provide multiple strategies for each equation, and select the best strategy from among their own strategies.ResultsOur results indicate that knowledge and use of the standard algorithm for solving linear equations is quite widespread across students in all three countries, but that there exists substantial within-country variation as well as between-country variation in students’ reliance on standard vs. situationally appropriate strategies. In addition, we found correlations between equation-solving accuracy and students’ flexibility in all three countries but to different degrees.ConclusionsAlthough it is increasingly recognized as an important construct of interest, there are many aspects of mathematical flexibility that are not well-understood. Particularly lacking in the literature on flexibility are studies that explore similarities and differences in students’ repertoire of strategies for solving algebra problems across countries with different educational systems and curricula. This study yielded important insights about flexibility and can push the field to explore the extent that within- and between-country differences in flexibility can be linked to differences in countries’ educational systems, teaching practices, and/or cultural norms around mathematics teaching and learning

Exploring students’ procedural flexibility in three countries

BackgroundIn this cross-national study, Spanish, Finnish, and Swedish middle and high school students’ procedural flexibility was examined, with the specific intent of determining whether and how students’ equation-solving accuracy and flexibility varied by country, age, and/or academic track. The 791 student participants were asked to solve twelve linear equations, provide multiple strategies for each equation, and select the best strategy from among their own strategies.ResultsOur results indicate that knowledge and use of the standard algorithm for solving linear equations is quite widespread across students in all three countries, but that there exists substantial within-country variation as well as between-country variation in students’ reliance on standard vs. situationally appropriate strategies. In addition, we found correlations between equation-solving accuracy and students’ flexibility in all three countries but to different degrees.ConclusionsAlthough it is increasingly recognized as an important construct of interest, there are many aspects of mathematical flexibility that are not well-understood. Particularly lacking in the literature on flexibility are studies that explore similarities and differences in students’ repertoire of strategies for solving algebra problems across countries with different educational systems and curricula. This study yielded important insights about flexibility and can push the field to explore the extent that within- and between-country differences in flexibility can be linked to differences in countries’ educational systems, teaching practices, and/or cultural norms around mathematics teaching and learning.</div

Pauta de observación de la enseñanza de las matemáticas en Educación Secundaria en España (POEMat.ES)

POEMat.ES es una pauta de observación de las prácticas de enseñanza de profesores de matemáticas de Educación Secundaria grabadas en vídeo. Este instrumento permite recoger información sobre las acciones de los profesores de matemáticas en el aula desde 3 dimensiones diferentes, organizadas a su vez en 17 subdimensiones. Cada subdimensión se puede valorar en cuatro niveles de desempeño que describen características concretas de las acciones realizadas por el profesor observadas en un fragmento de vídeo. POEMat.ES ha sido desarrollada con el objetivo inicial de ofrecer una visión equilibrada y multidimensional de la enseñanza de las matemáticas en Educación Secundaria en el contexto español