Interaction and Children's Mathematics

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In this article we propose the beginnings of a constructivist model of mathematical learning that supersedes Piaget's and Vygotsky's views on learning. First, we analyze aspects of Piaget's and Vygotsky's grand theories of learning and development. Then, we formulate our superseding model, which is based on the interrelations between two types of interaction in constructivism—the basic sequence of action and perturbation, and the interaction of constructs in the course of re-presentation or other previously constructed items. When teaching children, we base our interactions with them on the schemes we infer by observing the children's interactions in a medium. This emphasis makes contact with both Piaget's and Vygotsky's ideas of spontaneous development. In our model, learning is understood as being spontaneous rather than provoked. To maintain our emphasis on spontaneity, we separate the unintentionality of the learner from the intentionality of the teacher. To ground our model, we describe how two ten-year-old children, Arthur and Nathan, used their partitioning and iterating operations while they worked with a teacher in a computer microworld. We conclude by stressing the teacher's responsibilty to induce perturbations in children that are emotionally bearable and in the child's zone of potential construction.