plore and test big ideas (American Association for the Advancement of Science, 1989; National Research Council, 1996). As noted in Chapter 2, the Third International Mathematics and Science Study (Schmidt et al., 1997) characterized American curricula in mathematics and science as being “a mile wide and an inch deep.” (Examples of teaching for depth rather than breadth are illustrated in Chapter 7.)

As discussed in the first part of this book, knowledge-centered environments also include an emphasis on sense-making—on helping students become metacognitive by expecting new information to make sense and asking for clarification when it doesn’t (e.g., Palincsar and Brown, 1984; Schoenfeld, 1983, 1985, 1991). A concern with sense-making raises questions about many existing curricula. For example, it has been argued that many mathematics curricula emphasize

…not so much a form of thinking as a substitute for thinking. The process of calculation or computation only involves the deployment of a set routine with no room for ingenuity or flair, no place for guess work or surprise, no chance for discovery, no need for the human being, in fact (Scheffler, 1975:184).

The argument here is not that students should never learn to compute, but that they should also learn other things about mathematics, especially the fact that it is possible for them to make sense of mathematics and to think mathematically (e.g., Cobb et al., 1992).

There are interesting new approaches to the development of curricula that support learning with understanding and encourage sense making. One is “progressive formalization,” which begins with the informal ideas that students bring to school and gradually helps them see how these ideas can be transformed and formalized. Instructional units encourage students to build on their informal ideas in a gradual but structured manner so that they acquire the concepts and procedures of a discipline.

The idea of progressive formalization is exemplified by the algebra strand for middle school students using Mathematics in Context (National Center for Research in Mathematical Sciences Education and Freudenthal Institute, 1997). It begins by having students use their own words, pictures, or diagrams to describe mathematical situations to organize their own knowledge and work and to explain their strategies. In later units, students gradually begin to use symbols to describe situations, organize their mathematical work, or express their strategies. At this level, students devise their own symbols or learn some nonconventional notation. Their representations of problem situations and explanations of their work are a mixture of words and symbols. Later, students learn and use standard conventional algebraic notation for writing expressions and equations, for manipulating algebraic expressions and solving equations, and for graphing equations. Movement along this continuum is not necessarily smooth, nor all in one direction.



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