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Reactor: Heinz Steinbring (Duisburg-Essen University)
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SUMMARY

In this presentation I will deal with the ontogenesis of algebraic thinking.
Drawing on a cultural semiotic perspective, informed by current anthropological and embodied theories of knowing and learning, in the first part of my talk I will comment on the shortcomings of traditional mental approaches to cognition. In tune with contemporary research in neuroscience, cultural psychology, and semiotics, I will contend that we are better off conceiving of thinking as a sensuous and signmediated activity embodied in the corporeality of actions, gestures, and artifacts. In the second part of my talk, I will argue that algebraic thinking can be characterized in accordance with the semiotic means to which the students resort in order to express and deal with algebraic generality. I will draw upon results obtained in the course of a 10-year longitudinal classroom research project to offer examples of students’ forms of algebraic thinking. Two of the most elementary forms of algebraic thinking identified in our research are characterized by their contextual and embodied nature; they rely extensively upon rhythm and perceptual and deictic (linguistic and gestural) mechanisms of meaning production. Furthermore, keeping in line with the situated nature of the students’ mathematical experience, signs here usually designate their objects in an indexical manner. These elementary forms of algebraic thinking differ from the traditional one—based on the standard alphanumeric symbolism—in that the latter relies on sign distinctions of a morphological kind. Here signs cease to designate objects in the usual indexical sense to give rise to symbolic processes of recognition and manipulation governed by sign shape.

The aforementioned conception of thinking in general and the ensuing distinction of forms of algebraic thinking shed some light on the kind of abstraction that is entailed by the use of standard algebraic symbolism. They intimate some of the conceptual shifts that the students have to make in order to gain fluency in a cultural sophisticated form of mathematical thinking. Voice, gesture, and rhythm fade away. Embodied and contextual ways of signifying are then replaced with a perceptual activity where differences and similarities are a matter of morphology, and where meaning becomes relational.