The aim of this article is to carry out a work of networking theories which combines two perspectives on the mathematical activity involved in a modelling process, in order to answer the following question: To what extent does the application of the onto-semiotic tools complement the analysis from a cognitive perspective of a mathematical modelling process? To this end, we considered two theoretical frameworks: on the one hand, the onto-semiotic approach, which provides tools for the analysis of any mathematical activity and which here we applied to the activity of modelling; on the other hand, the modelling cycle from a cognitive perspective, which is a reflection on the specific mathematical activity of modelling. Then, we took a modelling problem that we applied to prospective mathematics teachers (at undergraduate and postgraduate level), and we analysed it from the perspective of both frameworks, in order to identify concordances and complementarities between these two ways of analysing the mathematical activity involved in the modelling process. The main conclusion is that both frameworks complement each other for a more detailed analysis of the mathematical activity that underlies the modelling process. Specifically, the analysis with the tools provided by the onto-semiotic approach reveals the phases or transitions of the modelling cycle as a conglomerate of mathematical practices, processes, and the primary objects activated in these practices.
Green et al. (2019) provide methodological recommendations for improving research on the effects of behavioural interventions aimed at cognitive enhancement. After addressing some points made by Green et al. with which we (partially) disagree, we discuss additional recommendations for making progress in cognitive training research, focusing on the use of computer modelling and the analysis of detailed data to understand participants’ knowledge and strategies. We then discuss the importance of using meta-analyses to understand the effects of cognitive training and show that meta-analyses have resolved the apparent lack of consensus in the field. Finally, taking a broader perspective, we conclude that the available empirical evidence on cognitive training and other fields of research suggests that the likelihood of finding reliable and robust far-transfer effects is low. Thus, research efforts should be redirected to near transfer or to other methods for improving cognition.
Few studies on federalism analyze the role of courts as safeguards of the federal arrangement, and those that do tend to be too optimistic about what courts can do. This article analyzes the effect of judicial review on the interaction between the central and a regional government in a federation in order to understand the conditions under which courts may or may not enforce compliance with federalism. It argues that politicians of either level of government anticipate the likelihood of a judicial challenge and an eventual veto, and it finds distinct equilibria in the interaction between central and regional governments (imposition, auto-limitation, negotiation and litigation). Only under auto-limitation do courts effectively prevent transgressions to the federal arrangement. In all other scenarios, defection may take place despite the presence of courts. These findings show that as the court's jurisprudence becomes more solid and defined, the chances for governments to successfully exceed their powers increase. Not only do transgressions take place despite the presence of the court, but because of it.
El objetivo de este estudio es analizar la reflexión realizada por un futuro profesor (estudiante de un máster de formación inicial de profesores de matemáticas de secundaria) sobre el papel de la modelización matemática en la mejora de un proceso de instrucción para enseñar trigonometría. En particular, después de explicar el contexto institucional de formación y de cómo se desarrolla la reflexión sobre la propia práctica en este máster, nos centraremos en un estudio de caso de un profesor analizando la reflexión que hace en su Trabajo Final de Máster, donde valora la implementación de una secuencia de tareas en su periodo de prácticas para hacer una propuesta de rediseño orientada a su mejora. El interés se centra en estudiar el uso que hace el futuro docente de los criterios de idoneidad didáctica (propuestos por el Enfoque Onto-Semiótico) en la memoria escrita de su trabajo final de máster. La metodología consiste en realizar un análisis de contenido de la reflexión realizada en esta memoria escrita y, desde el punto de vista de la extensión, se trata de un estudio de caso. Un primer resultado es que, de la reflexión y de las tareas que propone, se infiere que el futuro profesor no diferencia claramente entre los procesos ‘resolución de problemas’, ‘contextualización’ y ‘modelización’. Un segundo resultado es que este profesor asocia la incorporación de tareas para desarrollar el proceso de modelización con la mejora sobre todo de las idoneidades epistémica y ecológica del proceso de instrucción implementado. O objetivo deste estudo é analisar a reflexão feita por um futuro professor (aluno de um curso de mestrado de formação inicial de professores de matemática do ensino secundário na Espanha) sobre o papel da modelagem matemática no aprimoramento de um processo de instrução para o ensino de trigonometria. Em particular, depois de explicar o contexto institucional de formação e o modo como se desenvolve a reflexão sobre a própria prática nesse mestrado, focaremos no estudo de caso de um professor, analisando sua reflexão desenvolvida na sua Dissertação de Mestrado, onde avalia uma sequência de tarefas, implementadas no seu período de estágio, para fazer uma proposta de redesenho visando o aprimoramento dessas atividades. O interesse central deste estudo é verificar o uso que o futuro professor faz dos critérios de idoneidade didática (propostos pelo Enfoque Onto-Semiótico) evidenciado no seu trabalho de conclusão de curso. A metodologia consiste na realização de uma análise de conteúdo da reflexão feita na dissertação de mestrado e, do ponto de vista da extensão, é um estudo de caso. Um primeiro resultado é que, da reflexão e das tarefas que propõe, infere-se que o futuro professor não faz distinção clara entre os processos ‘resolução de problemas’, ‘contextualização’ e ‘modelagem matemática’. Um segundo resultado é que esse professor associa a incorporação de tarefas para desenvolver o processo de modelagem com o aperfeiçoamento, sobretudo, das idoneidades epistêmica e ecológica do processo de instrução implementado. The aim of this study is to analyse the reflection made by a future teacher (a student of a master’s degree in initial secondary school mathematics teachers’ training) on the role of mathematical modelling for improving an instructional process for the teaching of trigonometry. Particularly, after explaining the institutional context of training and how the reflection on the own practice is made in this master program, we will focus on a case study of a teacher analysing the reflection that he makes in his Master’s Thesis, where he values the implementation of a sequence of tasks in his internship period to make a redesign proposal aimed at its improvement. The interest of this work focuses on how the future teacher uses the didactic suitability criteria (proposed by the Onto-Semiotic Approach) in the written report of his master’s thesis. The methodology consists of carrying out a content analysis made in his written report and, from the point of view of its extension, it is a case study. The first result is that, based on the reflection and the tasks that he proposes, it is interred that the future teacher does not clearly differentiate between the processes ‘problem solving’, ‘contextualisation’, and ‘modelling’. The second result is that this teacher relates the addition of tasks to develop the modelling process with the improvement of the epistemic and ecological suitability of the implemented instructional process.
Durante las últimas décadas el número de trabajos sobre el papel de la modelización matemática en los procesos de enseñanza ha aumentado de modo significativo. La Teoría Antropológica de lo Didáctico (TAD) concibe la modelización matemática como procesos de construcción de praxeologías de complejidad creciente y sitúa el análisis de las condiciones y restricciones que afectan su viabilidad en el centro de la actividad investigadora. Presentamos tres procesos de modelización en Educación Infantil, Secundaria y Universidad, así como su análisis ecológico. Mostramos las complejidades existentes para hacer vivir tales procesos en las instituciones docentes y sus similitudes y diferencias.
Aunque durante este año se ha hablado, reflexionado e investigado ampliamente sobre el efecto de la pandemia de la COVID19 en la Educación Matemática, en cierta manera es inevitable, en esta editorial del último número de Bolema del año 2020, continuar hablando de esta problemática, dado que parece que la pandemia nos va a acompañar más tiempo del que, en algún momento, se pensó. Al escribir esta editorial ya parece claro que el 2021 será un año lleno de incertidumbres como consecuencia de la pandemia, en particular lo será para la Educación Matemática. En la editorial del número anterior de Bolema (escrito durante lo que se llamó, seguramente de manera no muy precisa, la primera oleada de la pandemia) Miarka y Maltempi (2020) se preguntaban ¿qué será de la Educación Matemática después del coronavirus? En estos momentos parece que la respuesta es que la investigación en Educación Matemática será diferente en muchos aspectos que ya, más o menos, sabemos cuáles son y que lo será, también, en otros que aún no conocemos. Un primer aspecto, previsiblemente, será un aumento de la investigación sobre las acciones que tomaron los profesores en general (y los de matemáticas en particular) en este periodo de pandemia. Al inicio de la pandemia, una de las medidas adoptadas para afrontarla
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