object may have different representations and that there are representations that are well formed syntactically but which do not represent a mathematical object. Moreover, a mater ial mathematical sign can be considered to represent, depending on what is most appropriate, a particular, general or ideal object. In mathematical discourse it is considered, whether explicitly or implicitly, that mathematical objects exist in a special way (non-ostensive, virtual, ideal, mental, abstract, general, etc., depending on the theoretical perspective) that is different from the way in which physical objects exist, and which particularly differs from the material symbols that represent them. In line with that stated by various authors (Presmeg, 1997; Sfard, 2000; Lakoff & Nunez, 2000; Acevedo, 2008) we assume that speaking about the existence of mathematical objects, as objects that exist in a form that is different from that of their material symbols, is essentially a metaphorical question. Somehow one of the properties of objects, such as chairs, trees, stones, etc., is translated into the world of mathematical objects. Mathematical discourse moves flexibly between repre sentations and the mathematical objects they represent. In mathematical practice it is not always considered necessary to distinguish explicitly the representation from the object represented, as this distinction is taken for granted. How ever, at times it is worth making a clear distinction between the two of them; for example, when a new representation is introduced, when speaking of representations that are well formed syntactically but which do not represent a mathematical object, or when it is necessary to distinguish between the particular or general nature of the object repre sented. In the case of the professional practice of mathematicians, the use of the object metaphor (explained later), as well as that of the synecdoche, which treats a par ticular case as if it was a generality, does not seem to create any conflict. A basic aim of mathematics education is that students learn to move flexibly between representations and the mathematical objects they represent. This is not easy for students to learn, and it also poses a challenge for teachers because they are not always aware of the complexity of this language game (Wittgenstein, 1953). This article presents classroom vignettes that illustrate how students and teach ers make use of representations of mathematical objects and how they refer to them in terms of existence. This involves a metaphorical discourse that, under certain circumstances, poses problems for students' understanding, and which may hinder, among other things, the processes of idealization and generalization. The aim of this paper is to illustrate how teachers and stu dents speak in class about mathematical objects and their representations. Some conflictive uses of a particular object to refer to a general object (synecdoche) are also discussed.

This article presents an analysis of a phenomenon that was observed within the dynamic processes of teaching and learning to read and elaborate Cartesian graphs for functions at high-school level. Two questions were considered during this investigation: What types of metaphors does the teacher use to explain the graphic representation of functions at high-school level? Is the teacher aware of the use he/she has made of metaphors in his/her speech, and to what extent does he/she monitor them? The theoretical framework was based on embodied cognition theory. Our findings include teachers’ expressions that suggest, among other ideas: (1) orientation metaphors, such as “the abscissa axis is horizontal”; (2) fictive motion, such as “the graph of a function can be considered as the trace of a point that moves over the graph”; (3) ontological metaphors; and (4) interaction of metaphors. We also show that teachers were not aware of using metaphors.

In this paper we are interested in the understanding of how the classroom discourse helps to develop the students' comprehension of the non ostensive mathematical objects as objects that have "existence". First, we examine the role of the objectual metaphor in the understanding of the mathematical entities as "objects with existence", as well as in some of the conflicts that the use of this type of metaphor can provoke in the students' interpretations. Second, we examine the mathematics discourse from the perspective of the ostensives representing non ostensives that do not exist.

O objetivo deste artigo é oferecer uma reflexão teórica sobre um fenômeno que é observado na dinâmica dos processos de ensino e de aprendizagem sobre gráficos de funções na aula de cálculo. A teoria da Cognição Corporificada (Embodiment Cognition) nos parece frutífera para tal: O professor de matemática no intuito de facilitar ou simplificar o conteúdo sobre gráfico de funções, para os estudantes, utiliza em seu discurso, às vezes sem se dar conta, expressões que sugerem, entre outras, (1) metáforas orientacionais, p.e., “o eixo das abscissas é horizontal”, (2) movimento fictivo, “o gráfico da função é o rastro de um ponto que se move sobre o gráfico”, (3) metáforas ontológicas e (4) montagens conceituais. O impacto de tais metáforas, entretanto, pode não facilitar o aprendizado dos estudantes. Apresentamos para ilustração uma análise do ocorrido em um curso de bachillerato na Espanha.

En este trabajo aplicamos herramientas de la teoria de Lakoff y Nunez (2000) y de la teoria de las funciones semioticas (Godino, Contreras y Font, 2004) al analisis de una sesion de clase de bachillerato en la que se estudia la representacion grafica de funciones. Como unidad primaria de analisis didactico se propone la configuracion didactica, constituida por las interacciones profesor-alumno a proposito de una tarea matematica y usando unos recursos materiales especificos. Dentro de cada configuracion didactica enfocamos nuestro analisis a los fenomenos relacionados con el uso de metaforas en el discurso del profesor y en el de los alumnos. Terminamos con algunas consideraciones sobre las posibles causas de estos fenomenos.

To investigate students’ ways of working with concrete materials in mathematics, a three-dimensional static artefact was constructed and made available to upper secondary students, with pre-knowledge only in two-dimensional coordinate geometry, for solving problems about planes and straight lines in space. Artefact interactivity was generally high, even students also disregarded the model to work only numerically with the coordinates, building on knowledge about lines in two dimensions. The model was used when trying to convince other students in the group.