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Metaphors in mathematics classrooms: analyzing the dynamic process of teaching and learning of graph functions

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Abstract

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.

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Notes

  1. Bachillerato in Spain (17-18 years old)

  2. See Oakley (2007) for a presentation of its terminological history, the review of a range of studies illustrating the application of image schemas, as well as the studies' review that establish the psychological and neuropsychological reality of image schemas.

  3. For us, to learn mathematics is to become able to carry out a practice and, above all, to perform a discursive reflection about it that would be recognized as mathematical by expert interlocutors. From this perspective, we see a teacher’s speech as a component of his professional practice. The objective of this practice is to generate not only a type of practice within the student, but also a discursive reflection about what can be considered as mathematics.

  4. The term existence can be understood from an absolute or a relative perspective, as Carnap discussed in Empiricism, semantics and ontology (Carnap, 1950). In relation to the existence of abstract entities, he considered “internal” and “external” questions. “If someone wishes to speak in his language about a new kind of entity, he has to introduce a system of new ways of speaking, subject to new rules; we shall call this procedure the construction of a linguistic framework for the new entities in question. Now we must distinguish two kinds of questions of existence: first, questions of the existence of certain entities of the new kind within the framework, which we can call internal questions; and second, questions concerning the existence or reality of the system of entities as a whole, called external questions. Internal questions and possible answers to them are formulated with the help of the new forms of expressions. The answers may be found either by purely logical methods or by empirical methods, depending upon whether the framework is a logical or a factual one. An external question has a problematic nature which is in need of closer examination” (Carnap, 1950, p. 20). As Carnap said, the internal existence within a particular linguistic framework is not problematic, in comparison to the external existence of the system of entities as a whole.

  5. At this school level, in our country, students are not taught about complex numbers.

  6. Other investigators also consider that metaphoric processes play a key role in the existence of mathematical objects. For example, Sfard (2000, p. 322) states the following: “To begin with, let me make clear that the statement on the existence of some special beings (that we call mathematical objects) implicit in all these questions is essentially metaphorical.”

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Acknowledgment

The research work reported in this article was carried out as part of the following projects: EDU 2009-08120/EDUC.

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Authors and Affiliations

  1. Departament de Didàctica de les CCEE i la Matemàtica, Facultat de Formació del Professorat, Universitat de Barcelona, Passeig de la Vall d’Hebron, 171, 08035, Barcelona, Catalonia, Spain

    Vicenç Font & Jorge Acevedo

  2. Programa de Pós-Graduação em Educação Matemática, Universidade Bandeirante de São Paulo, Av. Braz Leme, 3029 - São Paulo - SP - CEP: 02022-011, Brazil

    Janete Bolite

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  1. Vicenç Font
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Correspondence to Vicenç Font.

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Font, V., Bolite, J. & Acevedo, J. Metaphors in mathematics classrooms: analyzing the dynamic process of teaching and learning of graph functions. Educ Stud Math 75, 131–152 (2010). https://doi.org/10.1007/s10649-010-9247-4

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