UNDERSTANDING ZERO AND INFINITY IN THE EARLY SCHOOL YEARS

For most children mathematical knowledge begins with counting. This dissertation addresses the issue of how early counting knowledge is extended to form new mathematical concepts. Two extensions of counting knowledge were considered: (1) the child\u27s knowledge of zero and its role as the identity element and (2) the child\u27s understanding of infinity in relation to closure. A brief history of mathematics is reported demonstrating the psychological barriers these concepts pose. However, historically, once these concepts became a part of the number system, mathematical thought was greatly enhanced. The close relationship of these two concepts to the counting numbers makes it likely that they will be among the first movements beyond counting that children will make. Their inclusion in the child\u27s number system may similarly enhance mathematical thought. One group of children (kindergartners and first graders) were asked to solve zero and non-zero problems and then interviewed about their knowledge of zero. While the younger children often made errors, by first grade, children were virtually perfect at solving zero problems. Still, these children often said that zero was not a number. Error patterns as well as the children\u27s comments suggested that zero knowledge grows out of knowledge of nothing , exists somewhat separately from counting knowledge for some time and only later is coordinated with existing number knowledge. Another group of children (kindergartners, first, second and third graders) were interviewed to determine their knowledge of infinity. The results suggested that recognition that there is no greatest number is related to knowledge of numbers greater than one hundred, the ability to add systematically and the ability to induce that one may always add one to make a larger number. These conditions, however, were not sufficient for a child to recognize that there is no largest number. Such a conclusion may require that the child is able to recognize and to resolve contradictions in his thought. That many of the first and second graders and most of the third graders did recognize the infinite nature of the number system was quite impressive. The children in the two studies demonstrated rather powerful mathematical skills. They show that children in the early school years are able to make inductions, resolve contradictions and use systematic procedures without the aid of manipulable objects. With such skills, children may begin the task of creating a mathematical system