Learning to See the Invisible: What Skills and Knowledge are Needed to Engage with Students' Mathematical Ideas?
Growing concern that our country's students are not learning to reason mathematically-concern validated by a series of national assessments (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981; Kouba et al., 1988), has converged with several decades of research into children's natural abilities for mathematical thought (cf. Grouws, 1992) to encourage the development of a pedagogy "aim[ing] to take seriously both mathematics as a discipline and children's mathe matical ideas" (Ball, 1998, p. 7; see also Cobb, Wood, & Yackel, 1990; Fennema et al., 1996; Hiebert et al., 1996; Lampert, 1988). Outlined in the National Council of Teachers of Mathematics (NCTM) Standards (1989, 1991, 2000), embodied in new curricular materials (cf. Education Development Center, 1998, 2000; The University of Chicago School Mathematics Project, 1998; TERC, 1998), Bus pedagogy is being put into play in school systems around the United States.