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Connecting theories in mathematics education: challenges and possibilities

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Abstract

This paper is a commentary on the problem of networking theories. My commentary draws on the papers contained in this ZDM issue and is divided into three parts. In the first part, following semiotician Yuri Lotman, I suggest that a network of theories can be conceived of as a semiosphere, i.e., a space of encounter of various languages and intellectual traditions. I argue that such a networking space revolves around two different and complementary “themes”—integration and differentiation. In the second part, I advocate conceptualizing theories in mathematics education as triplets formed by a system of theoretical principles, a methodology, and templates of research questions, and attempt to show that this tripartite view of theories provides us with a morphology of theories for investigating differences and potential connections. In the third part of the article, I discuss some examples of networking theories. The investigation of limits of connectivity leads me to talk about the boundary of a theory, which I suggest defining as the “limit” of what a theory can legitimately predicate about its objects of discourse; beyond such an edge, the theory conflicts with its own principles. I conclude with some implications of networking theories for the advancement of mathematics education.

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Notes

  1. It is this flexible characteristic of communication that the Russian literary critic Mikhail Bakhtin seems to have been referring to when he asserted that any language can always in principle be deciphered, that is, translated into other languages (Bakhtin, 1986).

  2. Data collection is not necessarily intended here in the positivist empirical sense of the natural sciences; data collection can also refer to hermeneutical, phenomenological, epistemological and other processes of producing and endowing data with relevance and meaning.

  3. For the sake of clarity, let me add that theoretical principles can be of various sorts. Among others, they include (interrelated) principles of psychological, epistemological and ontological natures. Among the psychological principles, we find ideas about the “cognizing subject,” the role of others in knowledge acquisition, etc. Epistemological principles include ideas about what the theory understands by learning, the role of cultural institutions and society, ways of understanding and interpreting the teaching and learning of mathematics, etc. Ontological principles have to do with the status that the theory attributes to mathematical knowledge and the realities the theory deals with.

  4. We could easily imagine the difficulties that would have arisen had Prediger asked this question in terms of, say, Arzarello, Bosch, Gascón, and Sabena’s embodied perspective.

  5. “It is entirely to be expected” says Lotman, giving an example of the dynamics of internal organization and external disorganization, “that the rational positivistic society of nineteenth-century Europe should create images of the ‘pre-logical savage,’ or the irrational subconscious as anti-spheres lying beyond the rational space of culture” (Lotman, 1990, p. 142).

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Acknowledgments

I wish to thank the referees for their insightful comments on a previous version of this paper.

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  1. École des sciences de l’éducation, Université Laurentienne, Sudbury, ON, Canada, P3E 2C6

    Luis Radford

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  1. Luis Radford
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Correspondence to Luis Radford.

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This article is a result of a research program funded by The Social Sciences and Humanities Research Council of Canada/Le Conseil de recherches en sciences humaines du Canada.

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Radford, L. Connecting theories in mathematics education: challenges and possibilities. ZDM Mathematics Education 40, 317–327 (2008). https://doi.org/10.1007/s11858-008-0090-3

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