the psychology of learning and motivation - Percepts and Concepts ...

172 Robert S. Siegler et al.other numerical activities. Detailed analyses of learning patterns indicatedthat this theoretically motivated, game-based intervention exercises mostof its effects by enhancing and refining children’s representations of numericalmagnitudes.1. INTRODUCTIONWhen children enter kindergarten, they already differ greatly inmathematical knowledge. These early differences have large and lastingconsequences. Initial knowledge predicts subsequent success in manydomains (Bransford, Brown, & Cocking, 1999), but the relation betweenearly math knowledge and subsequent math achievement is particularlystrong and persistent (e.g., roughly twice as strong as the relation betweenearly and later reading achievement; Duncan et. al, 2007). Limited mathematicalknowledge has large, deleterious economic and occupationalconsequences in adulthood (McCloskey, 2007; Rivera-Batiz, 1992).Children from low-income families are particularly at risk for difficultieslearning mathematics. They tend to begin school with less numericalknowledge than their classmates from middle- or high-income households,and over the course of schooling, they fall steadily further behind(e.g., N. C. Jordan, Kaplan, Olah, & Locuniak, 2006). Again, similarpatterns are present in other areas such as reading, but the differences inmathematical knowledge tend to be greater (Case, Griffin, & Kelly, 1999).Recognition of these problems has led a number of researchers todevise programs intended to improve the mathematical knowledge ofpreschoolers from low-income families and thus to reduce the gapbetween them and children from more affluent backgrounds. However,these large-scale interventions are costly, both financially and in the timeneeded to train teachers to implement them. Because these interventionsare so multifaceted, they also yield limited information regarding how toimprove future interventions and future theories of numerical cognition.In this chapter, we briefly describe the early development of numericalknowledge, differences between the numerical knowledge of preschoolersfrom low-income and middle-income backgrounds, and three broad-basedcurricular interventions that have been found to improve that knowledge.We then present a theoretical analysis that suggests that underdevelopedknowledge of numerical magnitudes is a key source of the mathematicaldifficulties of children from low-income backgrounds. Next, we describean intervention, based on the theoretical analysis, designed to improveyoung children’s knowledge of numerical magnitudes. Finally, we presentevidence regarding the effectiveness of the intervention and discuss generalimplications of the findings.