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The past decade has been a time for much discussion about the influence of social interaction on the development of mathematical understanding. The roots of this discussion can be traced back to the ideas of Lev Vygotsky, a Russian psychologist who conducted research during the 1920s and 1930s. He was interested in how children conceptualize the meanings of words. He concluded that social interaction and communication are essential components in this conceptualization process. To show how children learn mathematical language, this article examines a classroom vignette and demonstrates how Vygotsky's ideas can be put in action in the mathematics classroom.
The NCTM's Standards documents (1989, 1991) emphasize the importance of social interaction and communication in learning mathematics. Mathematics as communication is a common thread woven throughout all levels of these documents. "Communication plays an important role in helping children construct links between their informal, intuitive notions and the abstract language and symbolism of mathematics; it also plays a key role in helping children make important connections among physical, pictorial, graphic, symbolic, verbal, and mental representations of mathematical ideas" (NCTM 1989, 26). What do communication and interaction in the classroom have to do with Vygotsky's ideas about learning mathematical language? In what ways must new words be learned to enrich a child's understanding of mathematics?
Vygotsky (1994) believed that as children talk, they internalize the meanings of words that they say. Only through communicating ideas can language be internalized. Children learn new words by reflecting on, and picturing the meanings of, the words in their minds as they interact. Through verbal expression of thoughts, children begin to reason for themselves. As children begin to use new words in the presence of a knowledgeable other person, they often find themselves in what Vygotsky called the zone of proximal development (ZPD), a place for learning that is located somewhere between the child's current understanding and potential understanding (Vygotsky 1978). A knowledgeable person can add meaning to what is familiar to the child when he or she enters the child's ZPD. In this zone, the child's rich but sometimes disorganized informal concepts meet with the systematic, formal reasoning of a knowledgeable person (Vygotsky 1994). This conception of the ZPD suggests that a teacher can assist a child by...