Past investments in R&D have produced curricular interventions to address each of the two problems raised above. With respect to the first, several curricula have been developed that introduce children to whole-number mathematics, with particular attention to the needs of young children who have had little preparation outside school. The most extensively researched of these is the Number Worlds curriculum, which has been tested in more than 20 matched, controlled trials. The results suggest that well-planned activities designed to put each step required in mastering the concept of quantity securely in place can allow disadvantaged students to catch up to their more advantaged peers right at the start of formal schooling (see Box 3.3). The curriculum has a companion assessment tool (the Number Knowledge Test) to help the teacher monitor and guide instruction. If results in controlled trials could be attained in schools across the country that serve disadvantaged populations, this would represent a major success with respect to narrowing the achievement gap, a long-standing national goal that has proven difficult to realize. Number Worlds is not the only curriculum designed to achieve this end. Others include Big Math for Little Kids (Ginsburg and Greenes, 2003) and Children’s Math Worlds (Fuson, 2003). While research to compare these curricula on a variety of dimensions is in order, it is clear that the tools to better prepare disadvantaged children for mathematics are now available.
With respect to the second concern—building children’s mathematical reasoning ability—controversy persists. While there is evidence that procedural knowledge without conceptual understanding leads to poor mathematical reasoning, it is also well documented that procedural knowledge is a critical element of mathematical competence (National Reasearch Council, 2001a; Haverty, 1999). Without adequate procedural knowledge, not only are children unable to engage in more challenging problem solving, but also, they are unable to engage in basic everyday transactions, like making change. The goal, then, must be one of strengthening mathematical reasoning without sacrificing procedural knowledge.
Research done in the 1990s investigated the effects on student achievement of instruction that builds on informal understandings and emphasizes mathematical concepts and reasoning. Cobb et al.’s problem-centered mathematics project (Wood