Mathematical Understanding: An Introduction

Karen C. Fuson, Mindy Kalchman, and John D. Bransford

For many people, free association with the word “mathematics” would produce strong, negative images. Gary Larson published a cartoon entitled “Hell’s Library” that consisted of nothing but book after book of math word problems. Many students—and teachers—resonate strongly with this cartoon’s message. It is not just funny to them; it is true.

Why are associations with mathematics so negative for so many people? If we look through the lens of How People Learn, we see a subject that is rarely taught in a way that makes use of the three principles that are the focus of this volume. Instead of connecting with, building on, and refining the mathematical understandings, intuitions, and resourcefulness that students bring to the classroom (Principle 1), mathematics instruction often overrides students’ reasoning processes, replacing them with a set of rules and procedures that disconnects problem solving from meaning making. Instead of organizing the skills and competences required to do mathematics fluently around a set of core mathematical concepts (Principle 2), those skills and competencies are often themselves the center, and sometimes the whole, of instruction. And precisely because the acquisition of procedural knowledge is often divorced from meaning making, students do not use metacognitive strategies (Principle 3) when they engage in solving mathematics problems. Box 5-1 provides a vignette involving a student who gives an answer to a problem that is quite obviously impossible. When quizzed, he can see that his answer does not make sense, but he does not consider it wrong because he believes he followed the rule. Not only did he neglect to use metacognitive strategies to monitor whether his answer made sense, but he believes that sense making is irrelevant.

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