The references that follow are organized to match chapters and sections of Benchmarks, which in turn mostly match those of Science for All Americans. The list is very selective and includes only those references that met two criteria. One was relevance—some excellent papers were not included because they did not bear on one of the Benchmarks topics. The other criterion was quality—papers, however relevant, were bypassed if they were seen to have design flaws or their evidence or argument was weak. Even then, however, not all relevant and good papers are included. In many cases, a single paper has been used as representative of a number of similar reports.
It will immediately be clear that mathematics and the physical sciences have had the benefit of many more studies than have other fields. Perhaps that is because the subject matter lends itself to research more easily; in the next few years, though, perhaps the attention to cognitive research will increase in all fields.
Research related to students' beliefs about the nature of mathematics has started to receive increasing attention. For literature reviews of the available research see McLeod (1992) and Schoenfeld (1992). Studies of the National Assessment of Educational Progress have recently included items related to student beliefs about mathematics as a discipline (Brown et al., 1988; Carpenter et al., 1983; Dossey et al., 1988). In addition, research on mathematical problem solving has recently included investigations of the beliefs students hold about the nature of mathematics (Schoenfeld, 1985, 1989a, 1989b, 1992). These studies have examined students' perceptions of mathematics as rule-oriented versus process-oriented or as a static versus a dynamic discipline, students' beliefs about the nature of mathematical problem solving, and students' perceptions about the role of memorization in learning mathematics. Little emphasis has been given to students' understanding of mathematics as the study of patterns and relationships, or to the relationships between mathematics, science, and technology, or to the nature of mathematical inquiry as a modeling process.
2a. Patterns and Relationships
Preliminary research hints that students have difficulty making connections between mathematical expressions, sentences, and sequences that share common structural patterns. They focus instead upon incidental similarities or differences (Ericksen, 1991).
2b. Mathematics, Science, and Technology
Middle-school and high-school students think that mathematics has practical, everyday uses and tend to think mathematics is more important for society than for them personally (Brown et al., 1988).
2c. Mathematical Inquiry
Typical student beliefs about mathematical inquiry include the following: There is only one correct way to solve any mathematics problem; mathematics problems have only one correct answer; mathematics is done by individuals in isolation; mathematical problems can be solved quickly or not at all; mathematical problems and their solutions do not have to make sense; and that formal proof is irrelevant to processes of discovery and invention (Schoenfeld, 1985, 1989a, 1989b). These beliefs limit students' mathematical behavior (Schoenfeld, 1985). Further research is needed to assess when and how students can understand that mathematical inquiry is a cycle in which ideas are represented abstractly, the abstractions are manipulated, and the results are tested against the original ideas. We must also learn at what age students can begin to represent something by a symbol or expression, and what standards students use to judge when solutions to mathematical problems are useful or adequate.
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