students construct meaning for formal symbols and procedures associated with rational numbers ( Hiebert, 1988; Hiebert & Carpenter, 1992; Kaput, 1987; Kieren, 1988). Recently, researchers such as Behr et al. and Kieren (chapters 2 & 3 respectively, in this volume) have constructed conceptual analyses of the domain of rational numbers to identify elements that are essential for the development of students' understanding and to suggest possible sequences in which the development may proceed. Although these conceptual analyses are not yet finished, there is an emerging consensus that a complete understanding of rational number depends on first developing a broad conception of rational number.
However, it is possible for students to construct meaning for formal symbols and procedures by building on a conception of rational number that is initially limited in perspective ( Leinhardt, 1988; Mack, 1990). Therefore, it may not be necessary that initial instruction concentrate on developing a broad understanding of rational number. A viable alternative may be to develop a strand of rational number based on partitioning and then to expand that conception to other strands once students can relate mathematical symbols and procedures to their informal knowledge and can reflect on these relations. There is some evidence that suggests this is an effective alternative for developing students' understanding of fractions because it redresses the initial limitations of students' informal knowledge ( Mack, 1990). It is not yet clear how students' understanding of other rational number strands can be developed through partitioning or how students can move from one rational number strand to another as their informal conception broadens. However, by building on what students already know, their informal conceptions can be extended in meaningful ways.
Although the evidence is tentative at this point, the potential of students' informal knowledge in the development of their understanding of rational numbers should not be dismissed. One of the most critical questions currently facing us as we seek to gain deeper insights into the ways that students can construct meaning for rational number concepts and procedures is whether students can develop a broad understanding of rational number by building on their informal conception of partitioning.
I would like to thank The Graduate School of Northern Illinois University (Summer 1990 Research Grant) for partial support while writing this chapter. The opinions expressed in this chapter are those of the author and not necessarily those of The Graduate School of Northern Illinois University.