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

Analysis of teacher-student interaction in the joint solving of non-routine problems in primary education classrooms

[EN] The analysis of teacher–student interaction when jointly solving routine problems in the primary education mathematics classroom has revealed that there is scarce reasoning and little participation on students’ part. To analyze whether this fact is due to the routine nature of the problems, a sample of teachers who solved, together with their students, a routine problem involving three questions with di erent cognitive di culty levels (task 1) was analyzed, describing on which part of the problem-solving process (selection of information or reasoning) they focused their interaction. Results showed that they barely focused the interaction on reasoning, and participation of students was scarce, regardless of the cognitive di culty of the question to be answered. To check whether these results could be due to the routine nature of the problem, a nonroutine problem (task 2) was solved by the same sample of teachers and students. The results revealed an increase in both reasoning and participation of students in processes that required complex reasoning. This being so, the main conclusion of the present study is that including nonroutine problem solving in the primary education classroom as a challenging task is a reasonable way to increase students’ ability to use their own reasoning to solve problems, and to promote greater teacher–student collaboration. These two aspects are relevant for students to become creative, critical, and reflective citizens.The authors are participants in the following projects: Project Science, Innovation, and University Ministry, Spain [PGC2018–100758-B-100]; Project University of Salamanca, Spain [MODELGEO-CEI 18.K133)]; and Project Junta of Castilla-León, Spain [SA050G19

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

Field trips and other teaching resources in natural and social sciences: educational implications from past experiences in Spanish primary schools

[EN] This paper describes the use of Natural and Social Sciences teaching resources in Spanish primary schools during the second half of the 20th century, based on quantitative and qualitative analysis of 250 recollections of retired teachers. Differences were identified between groups of teachers determined by gender, previous teaching context (urban vs rural) and higher education qualifications. Our results show that the academic background had a direct impact on the variety and type of teaching resources used and was a determinant factor to promote experiential learning. The study also highlights the comprehensive use of school field trips in the past as teaching resources to motivate students and develop different inquiry skills. Finally, the importance of mastering different educational resources in the teaching profession is discussed.This study is part of the Project Estudio y análisis de la cultura escolar a través de los testimonios de docentes (REF K117), funded by the University of Salamanca

Analysis of the interaction of teachers when they solve realistic problems along with their students in primary classrooms, taking into account their teaching experience

[ES] La formación de docentes de matemáticas de Primaria necesita conocer y analizar la práctica real del aula. Este trabajo se centra en la resolución conjunta de problemas en las aulas. La investigación educativa ha mostrado que existe escaso razonamiento y participación de los alumnos, cuando maestro y alumnos resuelven de forma conjunta problemas rutinarios en aulas de matemáticas de Primaria. Algunos trabajos muestran que ese razonamiento y participación aumenta cuando resuelven problemas no rutinarios, algo que no siempre ocurre cuando los maestros son noveles. Por ello se pretende analizar qué sucede cuando se utiliza un tipo especial de problemas no rutinarios como los realistas, que la literatura ha considerado con interés, teniendo en cuenta la experiencia del docente. De forma concreta, el objetivo de este trabajo es analizar la interacción entre dos maestras, una experta y una novel, cuando resuelven conjuntamente problemas realistas con sus estudiantes en su aula habitual, atendiendo a los procesos que se promueven, el grado de participación y la experiencia docente de las maestras. Los resultados mostraron que aumentó tanto el razonamiento como la participación de los alumnos, y que la maestra experta promovió más el razonamiento y realizó una mejor interpretación realista de los problemas en la resolución que la novel. Aunque se trata de una muestra escasa, estos resultados permiten dar un paso para abrir futuras líneas de investigación que podrían tener implicaciones educativas, tanto para los docentes en formación como para la formación continua de los que están ejerciendo.[EN] Training of Primary mathematics teachers needs to know and analyze the real practice of the classroom. This work focuses on a joint resolution of problems in the classroom. Educational research has shown that there is little reasoning and participation of students when teachers and students jointly solve routine problems in Primary math classrooms. Some studies showed that this reasoning and participation increases when they solve non-routine problems, but it does not always happen when teachers are novel. Therefore, it is intended to analyze what happens when a special type of non-routine problems such as realistic problems is used, which the literature has considered with interest, taking into account the experience of the teacher. Specifically, the objective of this work is to analyze the interaction between two teachers, an expert and a novel, when they solve realistic problems along with their students in their usual classroom, attending to the processes, degree of participation, and teaching experience of the teachers. The results showed that both reasoning and participation of the students increased, and the expert teacher promoted the reasoning more than the novel and also, she made a better realistic interpretation of the problems in the resolution. Although it is a small sample, these results allow us to take a step to open future lines of research that could have educational implications for both preservice teachers and in-service teachers

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