dren's learning, researchers and teachers must have optimal educational environments in which teachers (and researchers) can simultaneously observe and facilitate children's mathematical thinking. In addition, this approach brings values and goals of mathematics education to the forefront. In too many curricula and studies, these values and goals are only implicit.
A critic might argue that this is an egocentric approach to research-- that it limits implications of the research to a specific curriculum. Certainly, there is some truth in this position. We would respond, however, that in every study, children respond within the context of the environment, even if it is so rudimentary as to be nearly invisible. This caveat, therefore, applies to all studies. Further, many of the environments (e.g., those in clinical studies) have been limited in educing that mathematical thinking of which children are known to be capable.
We argue, then, the benefits of multiple foci: researched-based curriculum development, curriculum-guided research on children's thinking, and acknowledgment of the inextricable issues of computer and noncomputer environments, and teacher beliefs and knowledge. Such perspectives are, palpably, demanding. Requirements of curriculum development (e.g., deadlines from publishers and funding agencies) and of the classroom (e.g., teachers' need to "cover" other material; often tight, inflexible schedules) must be balanced with those of research. Controlling all the variables is absurd. A flexible, multifaceted approach can achieve a balance that synergistically contributes to curriculum development, classroom teaching, and mathematics educational research.
The work described in this report was funded in part by the National Science Foundation (grant no. ESI-8954664 and grant no. ESI-9050210). Opinions expressed are those of the authors and not necessarily those of the foundation.
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