The development of algebraic thinking in children of primary school

Este artículo presenta los resultados de investigaciones focalizadas en el desarrollo del pensamiento algebraico en niños de escolaridad primaria. Se centra, específicamente, en el desarrollo del pensamiento funcional y en la práctica representacional, atendiendo a la apropiación de las tablas y las letras para representar cantidades indeterminadas por parte de niños de primer grado. Estos estudios muestran los recursos que manifiestan los niños para desarrollar su pensamiento funcional y para usar tablas y letras en contextos algebraicos. Estos resultados se oponen a los de estudios clásicos que habían enfatizado las grandes dificultades de estudiantes adolescentes a la hora de aprender álgebra. El artículo finaliza con reflexiones a nivel metodológico y teórico en cuanto a estos estudios de investigación.This article presents an overview of the results of studies that have focused on the development of algebraic thinking among young elementary school children. Specifically, it centers on the development of functional thinking and the algebraic practice of representation, with a focus on the appropriation of tables and notation for variables among first grade children. These studies show the resources that children have to develop functional thinking and to use tables and notation for variables in algebraic contexts. These results are contrasted with earlier studies that had emphasized the great difficulties that adolescent students have in learning algebra. In closing, the article provides methodological and theoretical reflections related to these research studies.Facultad de Psicologí

The development of algebraic thinking in children of primary school

Este artículo presenta los resultados de investigaciones focalizadas en el desarrollo del pensamiento algebraico en niños de escolaridad primaria. Se centra, específicamente, en el desarrollo del pensamiento funcional y en la práctica representacional, atendiendo a la apropiación de las tablas y las letras para representar cantidades indeterminadas por parte de niños de primer grado. Estos estudios muestran los recursos que manifiestan los niños para desarrollar su pensamiento funcional y para usar tablas y letras en contextos algebraicos. Estos resultados se oponen a los de estudios clásicos que habían enfatizado las grandes dificultades de estudiantes adolescentes a la hora de aprender álgebra. El artículo finaliza con reflexiones a nivel metodológico y teórico en cuanto a estos estudios de investigación.This article presents an overview of the results of studies that have focused on the development of algebraic thinking among young elementary school children. Specifically, it centers on the development of functional thinking and the algebraic practice of representation, with a focus on the appropriation of tables and notation for variables among first grade children. These studies show the resources that children have to develop functional thinking and to use tables and notation for variables in algebraic contexts. These results are contrasted with earlier studies that had emphasized the great difficulties that adolescent students have in learning algebra. In closing, the article provides methodological and theoretical reflections related to these research studies.Facultad de Psicologí

The development of algebraic thinking in children of primary school

Este artículo presenta los resultados de investigaciones focalizadas en el desarrollo del pensamiento algebraico en niños de escolaridad primaria. Se centra, específicamente, en el desarrollo del pensamiento funcional y en la práctica representacional, atendiendo a la apropiación de las tablas y las letras para representar cantidades indeterminadas por parte de niños de primer grado. Estos estudios muestran los recursos que manifiestan los niños para desarrollar su pensamiento funcional y para usar tablas y letras en contextos algebraicos. Estos resultados se oponen a los de estudios clásicos que habían enfatizado las grandes dificultades de estudiantes adolescentes a la hora de aprender álgebra. El artículo finaliza con reflexiones a nivel metodológico y teórico en cuanto a estos estudios de investigación.This article presents an overview of the results of studies that have focused on the development of algebraic thinking among young elementary school children. Specifically, it centers on the development of functional thinking and the algebraic practice of representation, with a focus on the appropriation of tables and notation for variables among first grade children. These studies show the resources that children have to develop functional thinking and to use tables and notation for variables in algebraic contexts. These results are contrasted with earlier studies that had emphasized the great difficulties that adolescent students have in learning algebra. In closing, the article provides methodological and theoretical reflections related to these research studies.Facultad de Psicologí

Examining how teachers use graphs to teach mathematics during a professional development program

There are urgent calls for more studies examining the impact of Professional Development (PD) programs on teachers’ instructional practices. In this study, we analyzed how grades 5-9 mathematics teachers used graphs to teach mathematics at the start and end of a PD program. This topic is relevant because while many studies have investigated students’ difficulties with graphs, there is limited research on how teachers use graphs in their classrooms and no research on how PD impacts the way teachers use graphs in class to teach mathematics. Participant teachers took three graduate level semester-long courses focused on mathematics and student mathematical thinking. The program provided teachers with multiple opportunities for exploration and discussion, systematic feedback, contexts for collaboration and collegial sharing, and extended follow-up support. We analyzed all lessons where teachers used graphs in class at the start and end of the program, finding that teachers’ use of graphs was qualitatively more sophisticated in the end lessons. Results suggest that the features of the PD program had a positive effect on teachers’ classroom practices regarding the use of graphs

Mathematics teachers’ ideas about mathematical models: a diverse landscape

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds

Ideas de profesores de matemáticas sobre modelos matemáticos: un panorama diverso

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds.Este artículo describe las ideas que tienen profesores de matemáticas (grados 5-9) acerca de los modelos matemáticos de fenómenos del mundo real y explora cómo esas ideas difieren dependiendo de la formación académica de los profesores. Analizamos las respuestas de 56 profesores en ejercicio estadounidenses a tres preguntas abiertas, mediante un análisis de contenido. Identificamos un panorama variado de ideas sobre las entidades que constituyen el modelo matemático, sobre si los datos pertenecen o no al modelo, y sobre si el modelo es más o menos informativo que los datos. Encontramos diferencias interesantes entre profesores con y sin formación matemática.This study was funded by the National Science Foundation (NSF), Grant # DUE- 0962863, “The Poincaré Institute: A Partnership for Mathematics Education.” The ideas expressed herein are those of the authors and do not necessarily reflect the ideas of the funding agency

Mathematics teachers’ ideas about mathematical models: A diverse landscape

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds.Ideas de profesores de matemáticas sobre modelos matemáticos: un panorama diversoEste artículo describe las ideas que tienen profesores de matemáticas (grados 5-9) acerca de los modelos matemáticos de fenómenos del mundo real y explora cómo esas ideas difieren dependiendo de la formación académica de los profesores. Analizamos las respuestas de 56 profesores en ejercicio estadounidenses a tres preguntas abiertas, mediante un análisis de contenido. Identificamos un panorama variado de ideas sobre las entidades que constituyen el modelo matemático, sobre si los datos pertenecen o no al modelo, y sobre si el modelo es más o menos informativo que los datos. Encontramos diferencias interesantes entre profesores con y sin formación matemática.Handle:  http://hdl.handle.net/10481/3323

Mathematics teachers’ ideas about mathematical models: A diverse landscape

This paper describes the ideas that mathematics teachers (grades 5-9) have regarding mathematical models of real-world phenomena, and explores how teachers’ ideas differ depending on their educational background. Participants were 56 United States in-service mathematics teachers. We analyzed teachers’ written responses to three open-ended questions through content analysis. A varied landscape of ideas was identified. Teachers referred to different entities as constituting models, expressed different ideas about whether data points can be part of models, and whether models convey more information than data. Interesting differences according to educational background were identified, especially between teachers with and without mathematics backgrounds.Ideas de profesores de matemáticas sobre modelos matemáticos: un panorama diversoEste artículo describe las ideas que tienen profesores de matemáticas (grados 5-9) acerca de los modelos matemáticos de fenómenos del mundo real y explora cómo esas ideas difieren dependiendo de la formación académica de los profesores. Analizamos las respuestas de 56 profesores en ejercicio estadounidenses a tres preguntas abiertas, mediante un análisis de contenido. Identificamos un panorama variado de ideas sobre las entidades que constituyen el modelo matemático, sobre si los datos pertenecen o no al modelo, y sobre si el modelo es más o menos informativo que los datos. Encontramos diferencias interesantes entre profesores con y sin formación matemática.Handle:  http://hdl.handle.net/10481/3323

Concepciones de estudiantes de piano sobre el aprendizaje, la enseñanza y la evaluación

The Version of Record of this manuscript has been published and is available in Estudios de Psicología 2012 http://www.tandfonline.com/10.1174/021093912799803872Artistic domains of knowledge, such as music performance, have not raised the interest of researchers focused on conceptual change. By adopting the framework of implicit theories, this article addresses the conceptions of learning and instruction held by students at professional music conservatories. More specifically, our aims were to study the conceptions of learning, teaching, and assessment/evaluation held by piano students at three developmental/instructional levels, and analyze whether their conceptions constitute theoretically consistent profiles. The participants, 215 students of Intermediate and Tertiary levels, were selected according to three levels of the combined variable “Age/Level of Instruction”. Data were collected through a multiple-choice questionnaire, and analyzed with descriptive and non-parametric methods. The findings suggested that: a) students’ conceptions tend to be more sophisticated as their age and education level increase; b) each developmental/instructional group is typically associated to different conceptions; c) three increasingly sophisticated profiles of conceptions can be identified among these students. Implications for conceptual change research and limitations of the study are discussed. Further lines of research are suggestedEl campo del conocimiento artístico, como la interpretación musical, no ha despertado el interés de los investigadores que estudian el cambio conceptual. Adoptando el marco de teorías implícitas, este estudio aborda las concepciones sobre el aprendizaje y la instrucción de estudiantes de conservatorios profesionales de música. Nuestros objetivos eran estudiar las concepciones sobre aprendizaje, enseñanza, evaluación/acreditación mantenidas por estudiantes de piano de tres niveles evolutivo/educativos, y analizar si dichas concepciones constituían perfiles teóricamente consistentes. Los participantes, 215 estudiantes de Grado Medio y Superior, fueron seleccionados de acuerdo a tres niveles de la variable combinada “Edad/Nivel de Instrucción”. Los datos fueron recogidos mediante un cuestionario de opción múltiple,y analizados mediante métodos descriptivos y no-paramétricos. Los resultados sugieren que: a) las concepciones de estos estudiantes tienden a ser más sofisticadas a medida que avanza su edad y nivel educativo; b) cada grupo evolutivo/educativo está típicamente asociado a diferentes concepciones; c) entre los estudiantes, pueden identificarse tres perfiles de concepciones de sofisticación creciente. Se discuten las implicaciones para la investigación sobre cambio conceptual, las limitaciones del estudio,y se sugieren futuras líneas de investigació