Algebraic thinking from a cultural semiotic perspective

Abstract

In this article, I introduce a typology of forms of algebraic thinking. In the first part, I argue that the form and generality of algebraic thinking are characterised by the mathematical problem at hand and the embodied and other semiotic resources that are mobilised to tackle the problem in analytic ways. My claim is based not only on semiotic considerations but also on new theories of cognition that stress the fundamental role of the context, the body and the senses in the way in which we come to know. In the second part, I present some concrete examples from a longitudinal classroom research study through which the typology of forms of algebraic thinking is illustrated.

Keywords:

• algebraic thinking
• semiotics
• embodiment

Acknowledgements

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 (SSHRC/CRSH). A previous version of this paper was presented at the Sixth Congress of the European Society for Research in Mathematics Education (CERME 6), in Lyon, France, in January 2009. I wish to thank Heinz Steinbring for his thoughtful and stimulating reaction to my CERME presentation.

Notes

1. This point was well made by Nemirovsky in an interesting article published in 1994. Nemirovsky complained about the emphasis put on symbolic systems and the students’ understanding of symbolic systems’ rules: “Countless studies,” he said, referring to previous research, “describe how students’ mistakes related to specific ‘alternative’ rules.” (1994, 391).

2. See e.g. the emphasis on notations in contemporary research on early algebra.

3. My use of the term objectification differs hence from other current uses where objectification is conceived of as referring to something external and objective (regardless of the culture) or as a process transformed into object. The former has been developed in epistemological research informed by Realism; the latter by the linguistic tradition. My use of objectification comes from Hegel, Vygotsky, and Husserl's phenomenological work.

4. The examples have been chosen because they are representative of the ideas discussed in the article. They are also strongly representative of what happened in the classroom – without meaning that they are representations of a kind of “average” of the students’ behaviour in the sense of quantitative studies.

5. To avoid confusions, figures in the article will be numbered using Roman numerals to distinguish them from numbers that refer to figures in the sequences investigated by the students.

6. The adjective factual stresses the idea that this generalisation occurs within an elementary layer of generality – one in which the universe of discourse does not go beyond particular figures, like Figure 1000, Figure 3245, and so on.

7. In our current research with Grade 2 students these mechanisms of rhythmic coordination are also present, but they do not reach the subtle sensorial synchrony that we observe in older students, as reported here.

8. It still supposes a spatially situated relationship between the individual and the object of knowledge that gives sense to expressions like ‘top’ and ‘bottom’.

9. As epistemological analyses show, algebra has never been about guessing. Algebra since Babylonian and Greek times has always been about direct procedures to answer questions and solve problems characterised by the analytic manner in which indeterminate quantities (e.g., unknowns, variables, parameters; see Radford 2001) are dealt with. The advent of algebraic symbolism in the Renaissance and a concomitant interest in the devising of general methods to solve problems resulted in a focus on structures, although the ‘structural turn’ was not specific to algebra. In the case that we are discussing here, the meaning of the formula n×2 + 4 does not include this analytic structural dimension. The formula was obtained by simple guessing. It includes indeterminate quantities (symbolised by ‘n’), but lacks the analytic component. Its justification results from a numerical match between a guessed formula and a few observed cases, a match that is hoped to hold for all numbers. It is a form of naïve arithmetic generalisation. Bills and Rowland (1999) make a similar point in their interesting distinction between structural versus empirical generalisations without being concerned, however, by the question of analyticity. It might be the case that a generalisation can be structural without being algebraic, as there are also arithmetic and geometric structures (e.g., arithmetic false position methods exhibit a sort of structural component, without including the analytical component proper to algebraic thinking). I do not have the space here to go into further details, nor do I have space to say more about the delicate distinction between algebraic and arithmetic formulas. For a further discussion of the latter point, see (Radford 2006).

10. Rhythmic gestures, in this passage, were very important. As in factual algebraic thinking, they allowed the teacher to link various visual, linguistic, and symbolic elements together. However, rhythm here is not as prominent as it usually is in factual algebraic thinking. The cognitive difference in rhythmicity in both types of algebraic thinking is a matter of further investigation. At this point, I cannot say more. I am indebted to one of the reviewers for bringing this interesting point to discussion.

11. It might not be useless to remind here that these were the reasons that led Vygotsky (1981) to argue that education is the artificial development of the individuals.

12. Our current research on equations suggests that the typology presented here applies also to other domains of algebra (Radford, Demers and Miranda 2009).