Although students are actually doing algebra less formally in the earlier grades, they are not forced to generalize their knowledge to a more formal level, nor to operate at a more formal level, before they have had sufficient experience with the underlying concepts. Thus, students may move back and forth among levels of formality depending on the problem situation or on the mathematics involved.
Central to curriculum frameworks such as “progressive formalization” are questions about what is developmentally appropriate to teach at various ages. Such questions represent another example of overlap between learnercentered and knowledge-centered perspectives. Older views that young children are incapable of complex reasoning have been replaced by evidence that children are capable of sophisticated levels of thinking and reasoning when they have the knowledge necessary to support these activities (see Chapter 4). An impressive body of research shows the potential benefit of early access by students to important conceptual ideas. In classrooms using a form of “cognitively guided” instruction in geometry, second-grade children’s skills for representing and visualizing three-dimensional forms exceeded those of comparison groups of undergraduate students at a leading university (Lehrer and Chazan, 1998). Young children have also demonstrated powerful forms of early algebraic generalization (Lehrer and Chazan, 1998). Forms of generalization in science, such as experimentation, can be introduced before the secondary school years through a developmental approach to important mathematical and scientific ideas (Schauble et al., 1995; Warren and Rosebery, 1996). Such an approach entails becoming cognizant of the early origins of students’ thinking and then identifying how those ideas can be fostered and elaborated (Brown and Campione, 1994).
Attempts to create environments that are knowledge centered also raise important questions about how to foster an integrated understanding of a discipline. Many models of curriculum design seem to produce knowledge and skills that are disconnected rather than organized into coherent wholes. The National Research Council (1990:4) notes that “To the Romans, a curriculum was a rutted course that guided the path of two-wheeled chariots.” This rutted path metaphor is an appropriate description of the curriculum for many school subjects:
Vast numbers of learning objectives, each associated with pedagogical strategies, serve as mile posts along the trail mapped by texts from kindergarten to twelfth grade…. Problems are solved not by observing and responding to the natural landscape through which the mathematics curriculum passes, but by mastering time tested routines, conveniently placed along the path (National Research Council, 1990:4).
An alternative to a “rutted path” curriculum is one of “learning the landscape” (Greeno, 1991). In this metaphor, learning is analogous to learning