Análisis de la interacción profesor-alumno al resolver problemas no rutinarios en aulas de primaria

[ES] La investigación educativa ha mostrado que cuando maestro y alumnos resuelven de forma conjunta problemas en el aula, apenas existe razonamiento y el grado de participación de los alumnos es prácticamente inexistente. Es por ello que la presente Tesis Doctoral pretende analizar si la tarea que se realiza en las aulas determina el comportamiento de los docentes, para que éstos promuevan mayor razonamiento en las aulas y los alumnos sean más participativos en la construcción de su propio aprendizaje. Para ello se lleva a cabo dos estudios empíricos con diez maestros y sus alumnos en aulas de Primaria. En el primero se resuelve de forma conjunta un problema con tres apartados diferentes de distintos dominios cognitivos. En el segundo se resuelve un problema no rutinario en el mismo contexto que el del primer estudio. Se analiza las interacciones que surgen en dichas resoluciones según los procesos que se promueven y el grado de participación que tienen los alumnos, así como los perfiles docentes según diferentes dimensiones que se tendrán en cuenta. Los resultados obtenidos muestran que a medida que la complejidad cognitiva de la tarea es superior los procesos cognitivos avanzados, como el razonamiento, y el grado de participación de los alumnos también aumenta. Del mismo modo, en cuanto a los perfiles de los maestros se refiere, los docentes se vuelven menos directos y promueven el razonamiento en mayor medida

Dynamics of a multipoint variant of Chebyshev-Halley's family

In this paper, a complex dynamical study of a parametric Chebyshev-Halley type family of iterative methods on quadratic polynomial is presented. The stability of the fixed points is analyzed in terms of the parameter of the family. We also calculate the critical points building their corresponding parameter planes which allow us to analyze the qualitative behavior of this family. Moreover, we locate some dynamical planes showing different pathological aspects of this family. (C) 2016 Elsevier Inc. All rights reserved.The authors thank to the anonymous referees for their suggestions to improve the readability of the paper. This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P.Campos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel, P. (2016). Dynamics of a multipoint variant of Chebyshev-Halley's family. Applied Mathematics and Computation. 284:195-208. doi:10.1016/j.amc.2016.03.009S19520828

Stability of King's family of iterative methods with memory

[EN] In the literature exist many iterative methods with memory for solving nonlinear equations, the most of them designed in the last years. As they use the information of (at least) the two previous iterates to generate the new one, usual techniques of complex dynamics are not useful in this case. In this paper, we present some real multidimensional dynamical tools to undertake this task, applied on a very well-known family of iterative schemes; King's class. It is showed that the most of elements of this class present a very stable behavior, visualized in different dynamical planes. However, pathological cases as attracting strange fixed points or periodic orbits can also be found. (C) 2016 Elsevier B.V. All rights reserved.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P.Campos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel Cañas, P. (2017). Stability of King's family of iterative methods with memory. Journal of Computational and Applied Mathematics. 318:504-514. https://doi.org/10.1016/j.cam.2016.01.035S50451431

Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family

[EN] The study of the dynamical behaviour of the operators defined by iterative methods help us to know more deeply the regions where these methods have a good performance. In this paper, we follow the dynamical study of a multipoint variant of the known Chebyshev-Halley's family, showing the existence of attractive periodic orbits of period 2 for some values of the parameter.This research was partially supported by Ministerio de Econom´ı a y Competitividad MTM2014-52016-C02-2-PCampos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel Cañas, P. (2016). Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family. Numerical Algorithms. 73(1):141-156. https://doi.org/10.1007/s11075-015-0089-0141156731Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11(1), 85–141 (1984)Beardon, A.F.: Iteration of Rational Functions, Graduate Texts in Mathematics. Springer-Verlag, New York (1991)Behl, R., Kanwar, V.: Variants of Chebyshev’s method with optimal order of convergence. Tamsui Oxf. J. Inf. Math. Sci. 29(1), 39–53 (2013)Campos, B., Cordero, A., Magreñan, A., Torregrosa, J.R., Vindel, P.: Study of a bi-parametric family of iterative methods. Abstr. Appl. Anal. 2014. Art. ID 141643, 12 ppCampos, B., Cordero, A., Torregrosa, J.R., Vindel, P.: Bifurcations in the dynamics of a variant of Chebyshev method. In: Proceedings of the 15th International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE 2015, ISBN 978-84-617-2230-3, pp. 291–299 (2015)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. The Scientific World Journal Volume 2013 Article ID 780153Varona, J.L.: Graphic and numerical comparison between iterative methods. Math. Intelligencer 24, 37–46 (2002)Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A: Math. Sci. 10, 3–35 (2004)Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)Cordero, A., Torregrosa, J.P., Vindel, P.: Dynamics of a family of Chebyshev-Halley type method. Appl. Math. Comput. 219, 8568–8583 (2013)Gutiérrez, J.M., Hernández, M.A., Romero, N.: Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233, 2688–2695 (2010)Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equation. App. Math. Comput. 227, 567–592 (2014)Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007

Recursos didácticos para el aula de matemáticas

La utilización en el aula de diferentes recursos puede ayudar a mejorar la actitud de los estudiantes hacia las matemáticas, y conseguir así un aprendizaje más significativo. uno de los recursos que mayor número de adeptos tiene en las aulas es el libro de texto. En este capítulo se pretende mostrar cómo el uso exclusivo del libro de texto presenta algunas limitaciones que se pueden solventar con un correcto acompañamiento de otros recursos didácticos. Se mostrarán algunos recursos y materiales manipulativos con los que se pueden trabajar matemáticas, exponiendo las ventajas e inconvenientes de su utilización. También se propondrán pautas para su uso en las aulas de matemáticas, con el fin de animar a los lectores a introducirlos en el aula y ayudar así al aprendizaje de sus estudiantes

Validation of a Scale to Assess the Reversibility of Thought in Verbal Arithmetic Problems

Background: Reversibility is a key concept for the understanding and development of mathematical thinking. There is an agreement regarding problem-solving as a fundamental part of mathematical competence, and some authors regard reversible thinking as a requirement for it. Objectives: We want to validate an instrument that assesses the reversibility of thought when solving verbal arithmetic problems (word problems) involving various operations, semantic-mathematical structures and proximity of situational information. Design: A qualitative study was carried out from the data obtained by experts, and a quantitative study was carried out to determine the validity and reliability of the instrument. Setting and Participants: 318 students from different Spanish schools attending primary education (6 to 12 years) participated. Data collection and analysis: Participants performed 180 mathematical tasks distributed over three theoretical scales, two operations, and four semantic configurations. Results: To determine the consistency of the data, a reliability analysis was performed globally and on each of the scales, all values being greater than 0.90. Exploratory factor analysis resulted in three factors that explained more than 70%. To analyse the validity of the instrument, confirmatory factor analysis was performed, and its indices showed an adjustment of the models. Conclusions: We consider that the designed instrument is sufficiently robust to assess the reversibility of the basic addition and subtraction operations and, in addition, to analyse the discrimination of word problems according to the semantic-mathematical structure and their situational context

A multidimensional dynamical approach to iterative methods with memory

[EN] A dynamical approach on the dynamics of iterative methods with memory for solving nonlinear equations is made. We have designed new methods with memory from Steffensen’ or Traub’s schemes, as well as from a parametric family of iterative procedures of third- and fourth-order of convergence. We study the local order of convergence of the new iterative methods with memory. We define each iterative method with memory as a discrete dynamical system and we analyze the stability of the fixed points of its rational operator associated on quadratic polynomials. As far as we know, there is no dynamical study on iterative methods with memory and the techniques of complex dynamics used in schemes without memory are not useful in this context. So, we adapt real multidimensional dynamical tools to afford this task. The dynamical behavior of Secant method and the versions of Steffensen’ and Traub’s schemes with memory, applied on quadratic polynomials, are analyzed. Different kinds of behavior occur, being in general very stable but pathologic cases as attracting strange fixed points are also found. Finally, a modified parametric family of order four, applied on quadratic polynomials, is also studied, showing the bifurcations diagrams and the appearance of chaos.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P.Campos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel, P. (2015). A multidimensional dynamical approach to iterative methods with memory. Applied Mathematics and Computation. 271:701-715. https://doi.org/10.1016/j.amc.2015.09.05670171527

Problemas realistas para la formación de docentes de Educación Infantil, Primaria y Secundaria en aulas de Matemáticas

Memoria ID-033. Ayudas de la Universidad de Salamanca para la innovación docente, curso 2018-2019

Una herramienta para analizar el grado de participación en la interacción de maestro y estudiantes cuando resuelven conjuntamente tareas matemáticas

Analizar la práctica real del aula permite conocer aspectos de cómo se desarrolla la enseñanza y el aprendizaje. En este sentido, la forma de participación de los estudiantes cuando se desarrollan interacciones en el aula de forma conjunta con el profesor influye en el aprendizaje de los mismos. Valorar esa participación permite conocer particularidades que favorecen una interacción más rica. En este trabajo se pretende desarrollar una herramienta que analice la participación de los estudiantes en las interacciones que se producen entre docente y estudiantes cuando resuelven tareas matemáticas en el aula, basada en trabajos previos. Considera cuatro categorías discretas de análisis en función del grado de participación de los alumnos, desde una nula participación del estudiante hasta una completa participación del mismo. Esta herramienta, que se depurará a partir de su aplicación en contextos de interacción reales, puede tener implicaciones educativas al permitir clarificar aspectos concretos de la interacción referido al grado de responsabilidad de los estudiantes