The functional use of a mathematical sign

ISSN： 0013-1954

Source： Educational Studies in Mathematics, Vol.55, Iss.1-3, 2004-01, pp. : 81-102

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Abstract

The question of how a mathematics studentat university-level makes sense of a newmathematicalsign, presented to her or him in the formof a definition, is a fundamental problemin mathematicseducation. Using an analogy with Vygotsky’stheory (1986, 1994) of how a child learns anew word, Iargue that a learner uses a newmathematical sign both as an object withwhich to communicate (likea word is used) and as an object on whichto focus and to organise her or hismathematical ideas(again as a word is used) even before sheor he fully comprehends the meaning of thissign. Throughthis sign usage, I claim that themathematical concept evolves for thatlearner so that it eventually haspersonal meaning, like the meaning of a newword does for a child; furthermore, becausethe usageis socially regulated, I claim that theconcept evolves for the learner so that itsusage concurs with itsusage in the mathematical community. Inline with Vygotsky, I call this usage ofthe mathematical signbefore mature understanding, ‘functionaluse’. I demonstrate ‘functional use’ ofsigns (manipulations,imitations, template-matching andassociations) through an analysis of aninterview in which amathematics university student engages witha ‘new’ mathematical sign, the improperintegral, usingpedagogically designed tasks and a standardCalculus textbook as resources.