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Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts

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Abstract

The aim of this paper is to investigate the progressive manner in which students gain fluency with cultural algebraic modes of reflection and action in pattern generalizing tasks. The first section contains a short discussion of some epistemological aspects of generalization. Drawing on this section, a definition of algebraic generalization of patterns is suggested. This definition is used in the subsequent sections to distinguish between algebraic and arithmetic generalizations and some elementary naïve forms of induction to which students often resort to solve pattern problems. The rest of the paper discusses the implementation of a teaching sequence in a Grade 7 class and focuses on the social, sign-mediated processes of objectification through which the students reached stable forms of algebraic reflection. The semiotic analysis puts into evidence two central processes of objectification—iconicity and contraction.

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Notes

  1. These calculations were carried out in the context of word-problem solving (concerning Babylonian algebra, see e.g. Høyrup 2002) and pattern investigations (concerning Greek mathematics, see e.g. Manitius 1888, Ver Eecke 1926, and a summary in Radford 2001).

  2. Although strictly speaking, there is nothing that guarantees that C applies to the other terms, there is no reason to believe the opposite either, i.e. that at a certain point the terms will start behaving differently. The way the question is asked tilts the scale towards the idea that, all things being equal, the commonality should apply to the terms that follow.

  3. The concept of induction has been the object of a vast number of investigations in epistemology and in education; (see e.g. Peirce in Hoopes, 1991, pp. 59–61; Polya, 1945, pp. 114–121; Poincaré, 1968 p. 32 ff.).

  4. The team was comprised of two teachers (Tammy Cantin and Rita Venne-Beaudry), three researchers (Serge Demers, Monique Grenier and Luis Radford) and four research assistants (Isaias Miranda, Emilie Fielding, Mia Coutu and Julie Deault).

  5. In the previous problems, there was essentially just one form of expressing the general term of the sequence. Thus, as discussed previously, in the first problem given to the students, the general term was 12 + 3n. In the second and following problems, the nth term of the sequence can be expressed by an infinite number of polynomials. Generally speaking, a sequence p 1 , p 2 , p 3,…,p k of k terms (k ≥ 1) can be interpolated by polynomials of degree k − 1, k, k + 1, etc. One of the reviewers suggested that we should have used sequences having one solution only. I do not think so. As we shall see, the fact that there were an infinite number of solutions to express the general term in the second and following problems never occurred to the students, nor did it become a problem during their first contacts with algebra. Sensitivity to this problem can only arise if one already knows a great deal about polynomials, which of course was not the case with our Grade 7 students. In using patterns as a route to algebraic generalizations, our focus was on linear patterns.

  6. Otherness is in fact a crucial aspect of knowledge objectification, as Hegel remarked in another context (Hegel, 1977).

  7. Prosody refers to all those features to which speakers resort in order to mark the ideas conveyed in conversation in a distinctive way. Typical prosodic elements include intonation, prominence (as indicated by the duration of words) and perceived pitch. Some works on prosody include Bolinger (1983), Goodwin, Goodwin & Yaeger-Dror (2002), Roth (2007) and Selting (1994).

  8. One of the reviewers found my reference to different theories (epistemology, semiotics, psychology, and phenomenology) unnecessary, if not unpleasant, in my account of the students' processes of generalization. As all teaching and learning phenomena, the students’ processes of generalization are very complex. A single viewpoint (be it epistemological, psychological, or semiotic) is certainly insufficient for offering a satisfactory scientific explanation. The traditional psychologism from which Mathematics Education emerged in the 1970s is no longer a possibility. Cognition is much too complicated a phenomenon; understanding it requires that we make recourse to several viewpoints. My paper is an attempt at going beyond self-contained disciplines. It is about a coherent multidisciplinary approach. That which my reviewer sees as a move into the darkness of theorizing, I see as the emergence of light.

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  1. École des Sciences de l’éducation, Laurentian University, Sudbury, ON, P3E 2C6, Canada

    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 (SSHRC/CRSH).

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Radford, L. Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM Mathematics Education 40, 83–96 (2008). https://doi.org/10.1007/s11858-007-0061-0

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