LEARNING RATIONAL NUMBERS WITH UNDERSTANDING: THE CASE OF INFORMAL KNOWLEDGE
Nancy K. Mack
University of Pittsburgh
Students come to instruction with a rich store of informal knowledge related to rational number concepts and procedures. Initially this informal knowledge is limited in three ways: (a) Students' informal strategies treat rational number problems as whole number partitioning problems, (b) students' informal conception of rational number influences their ability to reconceptualize the unit, and (c) students' informal knowledge initially is disconnected from their knowledge of formal symbols and procedures associated with rational numbers. However, appropriate instruction can extend students' informal knowledge so that these limitations are redressed and the informal knowledge provides a base for developing an understanding of formal symbols and procedures.
For a number of years, researchers have been concerned with issues related to students' understanding of mathematics and the nature of its development ( Hiebert & Carpenter, 1992; Hiebert & LeFevre, 1986; Romberg & Carpenter, 1986). Although many of the intricacies of understanding still elude us, researchers concur that understanding depends on relationships the individual forms between new and existing knowledge ( Brownell & Sims, 1946; Carpenter, 1986; Greeno, 1978; Hiebert & LeFevre, 1986; Nickerson, 1982; Resnick & Ford, 1981; Riley, Greeno, & Heller, 1983). Recent theories concerning the development of students' understanding have attempted to characterize the ways that students form relationships between new and existing knowledge by focusing on the ways that students construct meaning for mathematical symbols. Although all of the theories are not yet fully developed and some differences exist among them, most of the theoretical discussions suggest that students construct meaning for