Variations in the effective use of analogies in math instruction across countries may contribute to performance differences in the TIMSS studies.

Testing previously studied information enhances long-term memory, particularly when the information is successfully retrieved from memory. The authors examined the effect of unsuccessful retrieval attempts on learning. Participants in 5 experiments read an essay about vision. In the test condition, they were asked about embedded concepts before reading the passage; in the extended study condition, they were given a longer time to read the passage. To distinguish the effects of testing from attention direction, the authors emphasized the tested concepts in both conditions, using italics or bolded keywords or, in Experiment 5, by presenting the questions but not asking participants to answer them before reading the passage. Posttest performance was better in the test condition than in the extended study condition in all experiments--a pretesting effect--even though only items that were not successfully retrieved on the pretest were analyzed. The testing effect appears to be attributable, in part, to the role unsuccessful tests play in enhancing future learning.

Many students graduate from K–12 mathematics programs without flexible, conceptual mathematics knowledge. This article reviews psychological and educational research to propose that refining K–12 classroom instruction such that students draw connections through relational comparisons may enhance their long-term ability to transfer and engage with mathematics as a meaningful system. We begin by examining the mathematical knowledge of students in one community college, reviewing results that show even after completing a K–12 required mathematics sequence, these students were unlikely to flexibly reason about mathematics. Rather than drawing relationships between presented problems or inferences about the representations, students preferred to attempt previously memorized (often incorrect) procedures (Givvin, Stigler, & Thompson, 2011 Givvin, K. B. , Stigler, J. W. and Thompson, B. J. 2011. What community college developmental mathematics students understand about mathematics, Part II: The interviews. The MathAMATYC Educator,, 2(3): 4–18. [Google Scholar] ; Stigler, Givvin, & Thompson, 2010 Stigler, J. W. , Givvin, K. B. and Thompson, B. 2010. What community college developmental mathematics students understand about mathematics. The MathAMATYC Educator,, 10: 4–16. [Google Scholar] ). We next describe the relations between the cognition of flexible, comparative reasoning and experimentally derived strategies for supporting students’ ability to make these connections. A cross-cultural study found that U.S. teachers currently use these strategies much less frequently than their international counterparts (Hiebert et al., 2003 Hiebert, J. , Gallimore, R. , Garnier, H. , Givvin, K. B. , Hollingsworth, H. , Jacobs, J. and … Stigler, J. W. 2003. Teaching mathematics in seven countries: Results from the TIMSS 1999 video study, Washington, D.C. : National Center for Education Statistics, United States Department of Education. (NCES 2003-013) [Google Scholar] ; Richland, Zur, & Holyoak, 2007 Richland, L. E. , Zur, O. and Holyoak, K. J. 2007. Cognitive supports for analogy in the mathematics classroom. Science,, 316: 1128–1129. [Crossref], [PubMed], [Web of Science ®] [Google Scholar] ), suggesting that these practices may be correlated with high student performance. Finally, we articulate a research agenda for improving and studying pedagogical practices for fostering students’ relational thinking about mathematics.

Analogical reasoning is a core cognitive skill that distinguishes humans from all other species and contributes to general fluid intelligence, creativity, and adaptive learning capacities. Yet its origins are not well understood. In the study reported here, we analyzed large-scale longitudinal data from the Study of Early Child Care and Youth Development to test predictors of growth in analogical-reasoning skill from third grade to adolescence. Our results suggest an integrative resolution to the theoretical debate regarding contributory factors arising from smaller-scale, cross-sectional experiments on analogy development. Children with greater executive-function skills (both composite and inhibitory control) and vocabulary knowledge in early elementary school displayed higher scores on a verbal analogies task at age 15 years, even after adjusting for key covariates. We posit that knowledge is a prerequisite to analogy performance, but strong executive-functioning resources during early childhood are related to long-term gains in fundamental reasoning skills.