Advances in Computer-Based
JOSEPH I. LIPSON, JOSEPH FALETTI, and MICHAEL E. MARTINEZ
Mathematician/philosopher Alfred North Whitehead has been credited with saying that civilization advances by extending the number of important operations which we can perform without thinking of them. In this century, an insufficiently appreciated task of education has become to extend automated skill in key operations to all citizens. As a consequence, it becomes the task of educational assessment to determine the presence of these automated skills. Often, students will demonstrate these automated operations in familiar educational settings with traditional assessments, but are unable to apply them to noneducational situations (for example, see Erlwanger, 1973; Ros nick and Clement, 1980; diSessa, 1982; McCloskey et al., 1980; and a general discussion in Davis, 1984).
Many educational reformers believe that only through radical restructuring can our schools show the needed improvement. However, two narrow external pressures can be applied to improve education significantly without the complete restructuring requested by some reformers ( Holden, 1989).
The first is curriculum change. By changing the courses offered and deciding which of these courses are required, we change the concepts and skills that students should learn. Regardless of course quality, students required to take several years of intensive mathematics will be different from students for whom advanced mathematics courses are either elective or not available. A significant tool for such change is the creation of new standards such as those of the N ational Council of Teachers of Mathematics ( NCTM, 1989).
A second prod for reform in education is a change in tests ( Fredericksen, 1984). Tests can influence course goals; a new consensus on tests often crystallizes an educational problem in the public mind and leads to curriculum change. Recent examples include the National Geographic Society's test of geography knowledge, Hirsch's ( 1987) specification of cultural literacy (although not strictly a test), and the National Assessment of Educational Progress (NAEP), all of which have fueled curriculum debate. In the rhetoric of education, the evidence from tests is often persuasive.
In this chapter, we explore ways of improving mathematics assessment so that it better supports the goals of education and the needs of students. Many of these improvements require operations that can only be done efficiently with a computer, so the design and architecture of a computer-based system for improved assessment is also discussed. In view of the increasing availability of low-cost, powerful computer workstations, and developments in cognitive science research, we will explore the possibility of large-scale mathematics assessment/instruction involving "intelligent" computer-based systems. Given