RATIO AND PROPORTION: CHILDREN'S COGNITIVE AND METACOGNITIVE PROCESSES
Susan J. Lamon Marquette University
In this chapter, children's preinstructional cognitive and metacognitive functions while solving ratio and proportion problems are discussed. Their thinking is analyzed in terms of a theoretical framework for building increasingly complex quantity structures. The formation of composite units is viewed as a mechanism by which more sophisticated reasoning develops and a ratio is viewed as a complex unit resulting from several compositions. Student protocols are used to illustrate that under certain conditions, children naturally view a ratio as a unit, and a strong metacognitive component in the children's thinking suggests that internal monitoring and regulation propels the unitizing process.
In well-defined mathematical domains such as whole number addition and subtraction, rich descriptions of children's thinking processes are linked to specific content ( Carpenter & Moser, 1983). Knowledge of the interplay between content and children's thinking enhances the classroom teacher's pedagogical content knowledge and better informs instructional decision- making. In more complex domains such as ratio and proportion, it is difficult to specify content and many more variables affect student performance. Contexts, task variables, semantic nuances, and psychological and mathematical development are all known influences on a child's ability to operate within the domain.
Although the exact content of ratio and proportion cannot be mapped out in a comprehensive way, we have some knowledge upon which instruction might build. We have detailed accounts of children's strategies and errors, including their preinstructional, preproportional thinking. We can distinguish various developmental states and order them according to