RESUMEN • El desarrollo del razonamiento algebraico elemental desde los primeros niveles educativos es un objetivo propuesto en diversas investigaciones y orientaciones curriculares. En consecuencia, es importante que el profesor de educación primaria conozca las características del razonamiento algebraico y sea capaz de seleccionar y elaborar tareas matemáticas adecuadas que permitan la progresiva introducción del razonamiento algebraico en la escuela primaria. En este trabajo, presentamos un modelo en el que se diferencian tres niveles de razonamiento algebraico elemental que puede utilizarse para reconocer características algebraicas en la resolución de tareas matemáticas. Presentamos el modelo junto con ejemplos de actividades matemáticas, clasificadas según los distintos niveles de algebrización. Estas actividades pueden ser usadas en la formación de profesores a fin de capacitarlos para el desarrollo del sentido algebraico en sus alumnos. PALABRAS CLAVE: álgebra elemental; niveles de algebrización; tareas matemáticas; formación de profesores; sentido algebraico. ABSTRACT • Developing elementary algebraic thinking since the earliest levels of education is a goal proposed in different research works and curricular guidelines. Consequently, primary school teachers should know the characteristics of algebraic reasoning and be able to select and develop appropriate mathematical tasks that serve to gradually introduce algebraic reasoning in primary school. In this paper we present a model that distinguish three levels of elementary algebraic thinking and is useful in analyzing the algebraic features in solving mathematical tasks. We describe this model with examples of mathematical activities, classified according to the different levels of algebraization. These activities can be used in the education of teachers to prepare them to develop their students' algebraic sense.

En el marco del enfoque ontosemiótico del conocimiento y la instrucción matemáticos se ha propuesto una caracterización del razonamiento algebraico en Educación Primaria basada en la distinción de tres niveles de algebrización. Tales niveles se definen teniendo en cuenta los tipos de representaciones usadas, los procesos de generalización implicados y el cálculo analítico que se pone en juego en la actividad matemática correspondiente. En este trabajo ampliamos el modelo anterior mediante la inclusión de otros tres niveles más avanzados de razonamiento algebraico que permiten analizar la actividad matemática en los niveles de Educación Secundaria. Estos niveles están basados en la consideración de 1) el uso y tratamiento de parámetros para representar familias de ecuaciones y funciones; 2) estudio de las estructuras algebraicas en sí mismas, sus definiciones y propiedades. Asimismo, se analizan las concordancias y complementariedades de este modelo con las tres etapas del proceso de algebrización propuestas en el marco de la teoría antropológica de lo didáctico.

The semiotic approach to mathematics education introduces the notion of "semiotic system" as a tool to describe mathematical activity. The semiotic system is formed by the set of signs, the production rules of signs and the underlying meaning structures. In this paper, we present the notions of system of practices and configuration of objects and processes that complement the notion of semiotic system and help to understand the complex nature of mathematical objects. We also show in what sense these notions facilitate the description and comprehension of building and communicating mathematical knowledge, by applying them to analyze semiotic systems involved in the teaching and learning of some elementary arithmetic concepts. Natural numbers as equivalence classes of setsThe understanding of the nature of mathematical concepts is a complex question as is revealed in the following class episode. This extract of the interaction between a lecturer and a group of future primary education teachers shows the predominance in the lecturer of a formalist understanding of natural numbers which contrasts with the informal use of these numbers. As we shall see in the following section, from an educative point of view, it is necessary to assume a wider perspective of the nature of the numbers to that shown by this lecturer with this group of students.Does anyone know what a number is? 1The lecturer begins the class on "natural numbers" by saying: First we will work on the concept of number, the idea and then we will think about the language in which we are going to write it. What are numbers?; for example, What is number five? We are posed with a problem, we have been using numbers from a very early age. However, when we are asked what a number is, we have difficulty in answering. He asks the students: Does anyone know what a number is? One student replies, A sign that refers to a quantity. The lecturer asks again: What is number four? The students do not reply. The lecturer writes the symbol 4 on the board and says: This is no more than a sign. What would the idea behind this be, how could we define it? The lecturer answers the question himself: If I want to communicate what number four means we put examples of groups of four, like for example: four pieces of chalk, four fingers, four people, four chairs, etc. What these sets have in common is what we call the idea of being four. How do we work in Preschool and Primary Education?The numbers are first shown as tools; however as future teachers we are going to take them as an object of study.The lecturer continues the class by explaining the Logicist construction of the natural numbers as the elements of the quotient set determined on the set of finite sets by the relationship of equivalence or coordinability between sets.Two aspects are reported from this episode:-The lecturer's teaching strategy: The lecturer's questions are rhetorical since he is assuming the discourse weight. The initial reply given by the student ("a sign that refers to a quantity") is neither considered n...

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The main objective of this paper is to apply the onto-semiotic approach to analyze the mathematical concept of different coordinate systems, as well as some situations and university students' actions related to these coordinate systems. The identification of objects that emerge from the mathematical activity and a first intent to describe an epistemic network that relates to this activity were carried out. Multivariate calculus students' responses to questions involving single and multivariate functions in polar, cylindrical, and spherical coordinates were used to classify semiotic functions that relate the different mathematical objects.Keywords Double and triple integration . Multivariate functions . Spherical and cylindrical coordinates . Semiotic registers . Onto-semiotic approach . Personal-institutional duality 1 Different coordinate systemsThe mathematical notion of different coordinate systems is introduced formally at a precalculus level, with the polar system as the first topological and algebraic example. The emphasis is placed on the geometrical (topological) representation and transformations between systems are introduced as formulas, under the notion of equality (x ¼ r cos q; r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y 2 p , etc.). The polar system is usually revisited as part of the calculus sequence; in single variable calculus, the formula for integration in the polar context is covered, as a means to calculate area. In multivariate calculus, work with polar coordinates, and transformations in general, is performed in the context of multivariable functions. It is in calculus applications that the different systems become more than geometrical representations of curves, some familiar (the circle) and some exotic (the rose of 'so many' petals, the Educ Stud Math (2009) 72:139-160