Kelly, 1999; Griffin and Case, 1997). A summary of their cognitive theory for the development of whole-number sense is presented in Box 5–1. Drawing from their extensive research on how children develop mathematical understanding as well as the work of other cognitive development researchers— such as Gelman, Siegler, Fuson, and Piaget—Case, Griffin and colleagues have constructed a detailed theory of how children develop number sense. This theory describes the understandings that children typically exhibit at various stages of development, the ways they approach problems, and the processes they use to solve them. The theory also describes how children typically progress from the novice state of understanding to expertise.

Case, Griffin, and colleagues have used their model of cognition and learning to design mathematics readiness programs for economically disadvantaged young children. The model has enabled them to (1) specify what knowledge is most crucial for early success in mathematics, (2) assess where any given population stands with regard to this knowledge, and (3) provide children who do not have all this knowledge with the experience they need to construct it (Case et al., 1999). These researchers have implemented their Rightstart program in different communities in Canada and the United States and have consistently found that children in the experimental program perform significantly better on a variety of measures of number sense than those in control groups (Griffin and Case, 1997; Griffin, Case, and Sandieson, 1992; Griffin, Case, and Siegler, 1994). Later in this chapter we present an assessment they have developed to assess student understanding relative to this theory.

Features of the Model of Cognition and Learning

The model of learning that informs assessment design should have several key features. First, it should be based on empirical studies of learners in the domain. Developing a model of learning such as the example in Box 5–1 requires an intensive analysis of the targeted performances, using the types of scientific methods described in Chapter 3. The amount of work requirfsed should not be underestimated. Research on cognition and learning has produced a rich set of descriptions of domain-specific performance that can serve as the basis for assessment design, particularly for certain areas of mathematics and science (e.g., National Research Council [NRC], 2001; American Association for the Advancement of Science, 2001) (see also the discussion of domain-based models of learning and performance in Chapter 3). Yet much more research is needed. The literature contains analyses of children’s thinking conducted by various types of professionals, including teachers, curriculum developers, and research psychologists, for a variety of purposes. Existing descriptions of thinking differ on a number of dimensions: some are highly detailed, whereas others are coarser-grained; some

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