The Didactical Problem of Elementary Algebra: A Macro- Ecological Analysis Based on the Anthropological Theory of Didactics (original) (raw)
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Journal of Research in Mathematics Education, 2015
We present a global, synthetic and updated vision of the contributions made by the anthropological theory of the didactic to the problem of teaching elementary algebra. We start by summarising the first results obtained by Yves Chevallard in the 80s that have marked the trend of research on elementary algebra in the anthropological approach. A reference epistemological model of elementary algebra is then proposed. This model provides a new interpretation of the role of elementary algebra in current school mathematics and guides the study of its ecology, that is, the constraints which partially explain the current situation and the conditions that could modify it in a given direction.
2020
El artículo se enmarca dentro del proceso de modelización matemática como un tema de investigación emergente en didáctica de las matemáticas, a partir de la enseñanza del Álgebra Lineal, en el marco de la Teoría Antropológica de lo Didáctico (TAD). Las Actividades de Estudio y de Investigación (AEI), como dispositivos didácticos dentro de la TAD, propone estudiar las matemáticas a partir de un proceso que se organiza para dar respuestas a las preguntas que surgen de los problemas y así construir los Objetos Matemáticos (OM). Para lograrlo se contempla un conjunto de praxeologías, teorías, tecnologías y tareas. En este orden, la modelización matemática en el marco de la TAD, es un proceso, que va a surgir, emerger, naturalmente cuando se implementan estos dispositivos didácticos, como una nueva alternativa de enseñanza en el modelo de la pedagogía de la investigación y del cuestionamiento del mundo.The article is framed within the mathematical modeling process as an emerging research...
La Teoría Antropológica de lo Didáctico en el Aula de Matemática
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2018
En este articulo, centramos el interes en la asignatura Matematica General que cursan los estudiantes de las Tecnicaturas Universitarias en Industrias Lacteas e Inspeccion de Alimentos de la Universidad Nacional de Lujan (UNLu). Nos planteamos indagar sobre el nivel de desarrollo que los estudiantes de esta materia tienen en torno a cuestiones vinculadas con la comunicacion de los saberes en Matematica y, a su vez, con los errores que circundan a este aspecto del saber. Atendiendo a este proposito, nos hemos planteado disenar e implementar una evaluacion diagnostica que involucre dos temas prioritarios de la asignatura, funciones lineales y cuadraticas, que se vinculan con todos los otros contenidos de la disciplina. En este trabajo, nos proponemos compartir algunos de los criterios seguidos en el diseno y organizacion del instrumento de evaluacion, asi como el analisis de los resultados de algunas actividades incluidas en el. Palabras clave: actividad matematica; evaluacion diagnos...
Didactic engineering for developing algebraic sense in prospective elementary school teachers
Avances de Investigación en Educación Matemática, 2014
In this paper we analyze a formative experience directed to prospective primary school teachers, which was aimed at developing their competence to discriminate algebraic objects and the different algebraization levels of school mathematical activity. The experience was performed in a Teaching and Learning primary school mathematics course, where elementary algebraic reasoning was a transversal topic for the remaining mathematical themes. The methodology was based on didactic engineering, which was understood in a generalized sense and was based on the onto semiotic approach to mathematical knowledge and instruction. The activities designed to develop elementary algebraic reasoning were carried out by 56 students. Our preliminary analysis suggest the relevance of this content for teacher education, although the recognition of algebraic object and the assignment of algebraization levels were difficult to achieve with the resources allocated in the implemented training process.
El problema del álgebra elemental en la teoría antropológicade lo didáctico
2017
Los primeros trabajos sobre la transposicion didactica del algebra, llevados a cabo por Yves Chevallard en la decada de los 80 del siglo pasado, pusieron de manifiesto que, en primera instancia, el algebra es un instrumento —el instrumento algebraico— que culmina en la modelizacion algebraica y transforma completamente las condiciones del trabajo matematico. En coherencia con esta posicion epistemologica, las investigaciones desarrolladas a lo largo de los ultimos 25 anos sobre el algebra elemental en el ambito de la teoria antropologica de lo didactico proponen considerar este dominio como un proceso de algebrizacion, ampliando asi el ambito de la actividad matematica escolar en el que se suele situar el problema didactico correspondiente. Hemos estudiado, como componentes de dicho problema, la relacion entre «lo numerico» y «lo algebraico», la algebrizacion de la proporcionalidad, la construccion de los numeros enteros como objetos algebraicos y el desarrollo de la modelizacion al...
Didactic transposition of the biodiversity concept. A study of Spanish high-school textbooks
2014
This article examines the existence and complexity of the treatment of the biodiversity concept in middle and high school Spanish textbooks. There is a consensus that science teaching should provide students with a theoretical foundation and a practical guide to support decision-making skills related to biodiversity conservation. However, many textbooks treat the concept of biological diversity as being decontextualized from the social sphere, even though the ethical and political dimensions are substantial. Initially studied in the framework of mathematics education, Chevallard stated that the route of reference knowledge (“scholarly” or “original” knowledge) up to the moment it is implemented in the classroom (“knowledge to be taught” and that effectively “taught”) requires a great deal of transpositive work (or “didactic transposition”), whereas the construction of new knowledge arises with epistemological and functional statuses that differ from the reference knowledge. The themes to be taught then take on a new shape, accompanied by a process of depersonalizing, desyncretizing, decontextualizing, publicity and programmability to be recontextualized into a new discourse in the textbooks and teaching materials. In this context, the main objectives of this research were three-fold: (a) to develop a conceptual analysis grid of the biodiversity concept, which is feasible to apply to textbooks of secondary education, (b) to analyze the didactic transposition of the concept of biodiversity that is promoted in high (“bachillerato”) and middle school (“escuela secundaria obligatoria, ESO” / “compulsory secondary education”) textbooks, and (c) to establish the levels of complexity of the biodiversity treatment in textbooks of secondary education by defining a set of variables. With this purpose, scientific papers from the "expert community" were reviewed in order to define the biodiversity conceptual framework, and secondly, the national curriculum proposed by the Spanish Ministry of Education was analyzed in order to determine the presence of the biodiversity concept. Finally, a third phase consisted of an analysis of the presence/absence of the concept in textbooks. A total of 60 textbooks from middle (n=27) and high schools (n=33) were selected (older than 1995), with the response variable being the presence or absence of treatment in each category’s specified content analysis. After carrying out an analysis of “expert” definitions of biodiversity occurring in scientific papers and encyclopedias, it was found that the conceptualization has become wider over the recent decades. After syllabi analysis, the biodiversity concept was observed to be present in the first and the fourth courses of middle school in the “Natural Sciences” and “Biology and Geology” subjects, respectively, as well as in the first and second high school courses in the “Biology I” and “Earth and Environmental Sciences” subjects, respectively. The conceptualization of biodiversity occurred in most of the analyzed textbooks, with “species” being the most prominent component, while other components such as genetic/population, community/ecosystem, landscape and functional groups were absent in the vast majority of textbooks. These findings reveal that the transpositive process for conceptualizing biodiversity is outdated with respect to the definitions given in the scientific community. This phenomenon, denotated by Chevallard as "biological aging", occurs when the gap between the scientific community knowledge and the "knowledge to be taught" increases. Regarding biodiversity attributes, “richness” was the one most frequently occurring in the manuals of both educational levels, although “species richness” and “relative abundance” attributes were more complex in the second year of high school, with estimates of species diversity indices being proposed. To investigate the levels of complexity, a Principal Component Analysis was performed, and it was found that most of the second-course high school textbooks represented the highest level in the conceptual treatment of biodiversity, while the lowest level was largely composed of first-course middle school books. Minor changes from the first to the fourth middle-school courses were also observed. Taking into account all the above findings, we recommend the use of the ‘earth and environmental sciences’ textbooks to transpose the biodiversity concept for all secondary education courses. Additionally, the intricate process of transposition revealed that the official national curriculum does not guarantee a conceptualization of biodiversity more in line with that of the “scientific community".
The Nature of Elementary Algebraic Reasoning
BOLEMA: Mathematics Education Bulletin, 2012
The introduction of algebraic reasoning in primary education is a subject of interest for research and curricular innovation in mathematics education, which supposes an extended vision of the nature of school algebra. In this paper we propose a way to conceive of algebraic reasoning based on the types of mathematical objects and processes introduced in the onto-semiotic approach to mathematical knowledge. In particular, considering a mathematical practice as algebraic is based on the intervention of generalization and symbolization processes, along with other objects usually considered as algebraic, such as binary relations, operations, functions and structures. This way to conceive of elementary algebra is based on and compared to the characterizations given by other authors. We also propose a typology of algebraic configurations that allows defining degrees of algebrization of mathematical activity.