Teaching the Conceptual Structure of Mathematics

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.