Children's failure to reason often leads to their mathematical performance being shaped by spurious associations from problem input and overgeneralization of inapplicable procedures rather than by whether answers and procedures make sense. In particular, imbalanced distributions of problems, particularly in textbooks, lead children to create spurious associations between arithmetic operations and the numbers they combine; when conceptual knowledge is absent, these spurious associations contribute to the implausible answers, flawed strategies, and violations of principles characteristic of children's mathematics in many areas. To illustrate mechanisms that create flawed strategies in some areas but not others, we contrast computer simulations of fraction and whole number arithmetic. Most of their mechanisms are similar, but the model of fraction arithmetic lacks conceptual knowledge that precludes strategies that violate basic mathematical principles. Presenting balanced problem distributions and inculcating conceptual knowledge for distinguishing flawed from legitimate strategies are promising means for improving children's learning. Expected final online publication date for the Annual Review of Developmental Psychology, Volume 2 is December 15, 2020. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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The purpose of this replication study was to provide replication evidence not currently available of the effects of a research-based mathematics program, schema-based instruction, on the mathematical problem-solving performance of 7th-grade students. The replication was implemented in 36 schools in 5 districts; 59 mathematics teachers and their students (N ϭ 1,492) participated in the study. Multilevel hierarchical linear analyses revealed statistically significant differences between conditions on proximal and distal measures of mathematics problem solving, with effects sizes similar to those reported in Jitendra et al. (2015). Educational Impact and Implications StatementThe present study provides support for classroom practices linked to an educational program (schema-based instruction [SBI]) that is designed to improve middle school students' understanding of solving problems involving proportional relationships, which is critical for mastering the higherlevel mathematics students encounter later. The results of a large study of SBI's impact showed that students generally benefitted from SBI although there were differences favoring White students versus Black, Hispanic, and multiracial students, whereas Asian students outperformed White students. The latter results highlight the need to identify ways to eliminate differences among racial subgroups to ensure all students have the opportunity to benefit from SBI.

Although almost everyone agrees that the environment shapes children's learning, surprisingly few studies assess in detail the specific environments that shape children's learning of specific content. The present article briefly reviews examples of how such environmental assessments have improved understanding of child development in diverse areas, and examines in depth the contributions of analyses of one type of environment to one type of learning: how biased distributions of problems in mathematics textbooks influence children's learning of fraction arithmetic. We find extensive parallels between types of problems that are rarely presented in US textbooks and problems where children in the US encounter greater difficulty than might be expected from the apparent difficulty of the procedures involved. We also consider how some children master fraction arithmetic despite also learning the textbook distributions. Finally, we present findings from a recent intervention that indicates how children's fraction learning can be improved.