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Early mathematics knowledge is a strong predictor of later academic achievement, but children from low-income families enter school with weak mathematics knowledge. An early math trajectories model is proposed and evaluated within a longitudinal study of 517 low-income American children from ages 4 to 11. This model includes a broad range of math topics, as well as potential pathways from preschool to middle grades mathematics achievement. In preschool, nonsymbolic quantity, counting, and patterning knowledge predicted fifth-grade mathematics achievement. By the end of first grade, symbolic mapping, calculation, and patterning knowledge were the important predictors. Furthermore, the first-grade predictors mediated the relation between preschool math knowledge and fifth-grade mathematics achievement. Findings support the early math trajectories model among low-income children.

This study examined whether practice with arithmetic problems presented in a nontraditional problem format improves understanding of mathematical equivalence. Children (M age = 8;0; N = 90) were randomly assigned to practice addition in one of three conditions: (a) traditional, in which problems were presented in the traditional "operations on left side" format (e.g., 9 + 8 = 17); (b) nontraditional, in which problems were presented in a nontraditional format (e.g., 17 = 9 + 8); or (c) no extra practice. Children developed a better understanding of mathematical equivalence after receiving nontraditional practice than after receiving traditional practice or no extra practice. Results suggest that minor differences in early input can yield substantial differences in children's understanding of fundamental concepts.

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Providing exploratory activities prior to explicit instruction can facilitate learning. However, the level of guidance provided during the exploration has largely gone unstudied. In this study, we examined the effects of 1 form of guidance, feedback, during exploratory mathematics problem solving for children with varying levels of prior domain knowledge. In 2 experiments, 2nd- and 3rd-grade children solved 12 novel mathematical equivalence problems and then received brief conceptual instruction. After solving each problem, they received (a) no feedback, (b) outcome feedback, or (c) strategy feedback. In both experiments, prior knowledge moderated the impact of feedback on children's learning. Children with little prior knowledge of correct solution strategies benefited from feedback during exploration, but children with some prior knowledge of a correct solution strategy benefited more from exploring without feedback. These results suggest that theories of learning need to incorporate the role of prior knowledge and that providing feedback may not always be optimal.

Feedback can be a powerful learning tool, but its effects vary widely. Research has suggested that learners' prior knowledge may moderate the effects of feedback; however, no causal link has been established. In Experiment 1, we randomly assigned elementary schoolchildren (N ϭ 108) to a condition based on a crossing of 2 factors: induced strategy knowledge (yes vs. no) and immediate, verification feedback (present vs. absent). Feedback had positive effects for children who were not taught a correct strategy, but negative effects for children with induced knowledge of a correct strategy. In Experiment 2, we induced strategy knowledge in all children (N ϭ 101) and randomly assigned them to 1 of 3 conditions: no feedback, immediate correct-answer feedback, or summative correct-answer feedback. Again, feedback had negative effects relative to no feedback. Results provide evidence for a causal role of prior knowledge and indicate that minimal feedback can both help and hinder learning.