ABSTRACT
The current “top down” approach to mathematics curriculum in the USA is exemplified in such notions as intended and implemented curricula and in ideal, implemented, and achieved curricula, where each is viewed as an approximate subset of the one above it. The intended or ideal curriculum is of primary concern in some of the available documents proposing curriculum changes in the USA in that redistributions of mathematical content, with the inclusions of some currently missing parts and omissions of some currently used parts, have been recommended. We also have the recommendations that “standardized tests of achievement… should be administered at major transition points from one level of schooling to another” (A Nation at Risk, 1983) and “clear standards for achievement must be established at each grade level in order to create an institutionalized climate of expectations to which students will respond” (McKnight 1987). The top down principle of curriculum design is caught nicely in a statement made by Griffiths (1983) at ICME-4.
In countries like the U.S. A., the problems of a mobile population and the low esteem of teaching as a job have induced the dream of a superficially simple description of a curriculum by means of an algorithm: the pupils are to pass through stages SI, S2,… , Sj, and when a pupil is judged to be in stage Sn at T; the algorithm moves him into stage Sn + 1 by time t + h where h depends on n. (p. 358)
Although the two factors isolated by Griffiths unquestionably contribute to the top-down principle, the principle that the mathematics curriculum exists independently of the teachers and students who use it is a more pervasive factor. This principle has been undermined by the results of the Second International Mathematics Study (SIMS, 1986). McKnight (1987) isolated teaching and learning strategies called rote.This use… suggests a view that learning for most teachers should be passive—teachers transmit knowledge to students who receive it and remember it mostly in the form in which it was transmitted. This might be considered as an approach of “rote teaching” and “rote learning.” (p. 81)
Teachers and students clearly do not use ideal mathematics curriculum as intended. It is understandable that teachers who have studied mathematics extensively might view it as being somehow disembodied from human experience because of the dominance of structuralism and formalism (Byers, 1983). At the beginning of this century, Brouwer (1913), in his inaugural address at the University of Amsterdam, noted that in the case of formalism there is a “presupposition of a world of mathematical objects, a world independent of the thinking individual.” This presupposition permeates mathematics education at all levels and is a basis for the principle that mathematics curriculum exists independently of the teachers and students who use it. Brouwer (1913) provided an alternative view, “the question of where mathematical exactness does exist, is answered differently by the two sides; the intuitionist says: in the human intellect, the formalist says: on paper.”