Students' Theories About
Mathematics and Their
JOHN G. NICHOLLS, PAUL COBB, ERNA YACKEL, TERRY WOOD, and GRAYSON WHEATLEY
The use of conventional standardized academic achievement tests to evaluate educational practices and the progress of individual students has long been accompanied by a chorus of dissenting academics -- who often stand far off stage -- and by muffled grumbling in the ranks of teachers who are constrained to administer these instruments to their charges. But the psychometric juggernaut rolls on, endowed by the arcane language of validity and reliability with an aura of scientific infallibility. The tests are called objective. The scores appear more substantial than any alternative. Opposing these hard and shining instruments, the carping critics offer their own, students', or teachers' subjective impressions.
In one sense, standardized achievement tests do help us avoid some of the problems that can spring from subjective self-interest. Binet's intelligence test, from which current academic achievement tests derive, was constructed to deal with the fact that teachers would sometimes attempt to remove able but troublesome students from their classes by declaring them to be in need of remedial help. An independently administered ability or achievement test can avoid this form of bias. But technical innovations such as tests cannot save us from bias. Teaching to the test, for example, can create a blatantly false impression of students' mathematical competence. Whatever test is used to evaluate teaching, it almost inexorably biases teaching toward that test ( Fredericksen, 1984; Johnston, 1989). The result can be proficiency with the forms of tasks used on the tests but little deep understanding or appreciation of the knowledge that the test constructors hoped to assess. Tests are never objective in the sense of being untarnished by human concerns. Their impact, their usefulness, and their validity depend on how they are construed by humans. Human purposes are, in turn, shaped by the tests and the ways they are used.
For a teacher who seeks to foster students' higher order mathematical thinking. no conventional achievement test is likely to be of immediate help. What teachers can use for this enterprise is more-or-loss constant feedback on how students interpret the