Communicating with Young Children in Mathematics: A Unique Challenge
Schwartz, Sydney L., Brown, Anna Beth, Teaching Children Mathematics
For prekindergarten and kindergarten children, whose communication skills fall far behind their mathematical understandings, meeting the NCTM's Standard 2: Mathematics as Communication poses unique challenges (NCTM 1989, 26-28). Young children's meanings and understandings of mathematical ideas take place in an action-based learning environment as they use concrete materials as tools with which to think and talk. They construct these mathematical understandings as they manipulate the objects; they test their mathematical understandings through what adults view as endless repetition; they use their understandings of mathematical relationships to build models of their ideas; and they use mathematical skills and understandings to solve problems in all aspects of their lives. They construct their understanding and build power in using the skills, both in school and out of school, with others and alone. Finally, through communication, their mathematical ideas and understandings resolutely move from the intuitive to the conscious level and increasingly become organized for retrieval, use, and development (Kamii 1982; Katz and Chard 1989).
Whereas the core of their mathematical learning revolves around using physical materials and observing events, prekindergarten and kindergarten children are at that stage of language development in which their thinking far outpaces their ability to verbalize. Attempts to elicit explanations about their mathematical thinking often meet with simplistic statements that grossly understate the complexity of the ideas involved. Even less revealing are the shrugs that mean "I don't know" or "I just knew."
The essential ingredients, then, for enhancing children's mathematical thinking are (1) presentation of a large selection of classroom materials to use in making sense of their experiences and refining their mathematical understandings, (2) allocation of blocks of time for children to interact with their peers and adults as they use the materials, and (3) participation in stimulating curriculum experiences to feed the process.
Teaching Strategies That Foster Communication
The major thrust of the current mathematics education literature for early childhood is to "catch them thinking mathematics" (Greenberg 1993). Teachers face a dilemma when fulfilling the purposes of communication in mathematics with four- and five-year-old children. How do we enter the child's thinking world and make a contribution without overriding the thinking process in mathematics? The teaching strategies that can be used are to validate, to review, and to challenge. Each can support the goals of mathematics as communication in a specific way.
The validating strategy
Validating supports the child's growing sense of mathematical relationships. By agreeing with the child's expressed ideas, we are supplying further stimulus for the child to continue to grow in understanding. If, in addition, we explain the basis for our agreement, we further feed the process by modeling the process of checking.
The reviewing strategy
Reviewing serves to strengthen skills and clarify understandings. As the mathematics education community shifts the focus of review from skill to mathematical understanding, the related teaching strategies have moved from emphasis on isolated skill practice to talking about how the child thinks about mathematical relationships.
The challenge strategy
A challenge furthers the child's consideration of mathematical relationships by extending the child's thinking without additional manipulation of the materials or stimulating further action with the materials to provoke more complex thinking.
It is important to note that to meet the goal of promoting mathematics as communication, the primary focus of these strategies needs to be on the child's mathematical thinking rather than on the child as a person. Such teaching responses as, "Good," "Great," "How smart you are," and "That's wonderful" are examples of value judgments that are externally imposed. …