There is a growing theoretical consensus that the concepts of ratio and proportion do not develop in isolation. Rather, they are part of the individual's multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers. The current study attempted to clarify the beginning of this developmental process. One fifth-grade student, Bruce, was encouraged to schematize his trial-and-error-based method, which was effective in solving so-called missing-value tasks. This study describes several advancements Bruce made during the teaching experiment and analyzes the challenges Bruce faced in attempting to schematize his method. Finally, the mathematical knowledge Bruce needed to further develop his ratio and proportion concepts is identified. The findings provide additional support for the view that the development of ratio and proportion concepts is embedded within the development of the multiplicative conceptual field.
Jane-Jane Lo, Visiting Scholar, Department of Mathematics, Cornell University, Ithaca, New York, 14850-7901; e-mail: firstname.lastname@example.org
Tad Watanabe, Assistant Professor, Department of Mathematics, Towson State University, Towson, Maryland, 21252-7097; e-mail: email@example.com