disposition toward mathematics, as do, of course, their experiences in learning mathematics.

The chapters that follow on whole number, rational number, and functions look at the principles of How People Learn as they apply to those specific domains. In this introduction, we explore how those principles apply to the subject of mathematics more generally. We draw on examples from the Children’s Math World project, a decade-long research project in urban and suburban English-speaking and Spanish-speaking classrooms.3

PRINCIPLE #1: TEACHERS MUST ENGAGE STUDENTS’ PRECONCEPTIONS

At a very early age, children begin to demonstrate an awareness of number.4 As with language, that awareness appears to be universal in normally developing children, though the rate of development varies at least in part because of environmental influences.5

But it is not only the awareness of quantity that develops without formal training. Both children and adults engage in mathematical problem solving, developing untrained strategies to do so successfully when formal experiences are not provided. For example, it was found that Brazilian street children could perform mathematics when making sales in the street, but were unable to answer similar problems presented in a school context.6 Likewise, a study of housewives in California uncovered an ability to solve mathematical problems when comparison shopping, even though the women could not solve problems presented abstractly in a classroom that required the same mathematics.7 A similar result was found in a study of a group of Weight Watchers, who used strategies for solving mathematical measurement problems related to dieting that they could not solve when the problems were presented more abstractly.8 And men who successfully handicapped horse races could not apply the same skill to securities in the stock market.9

These examples suggest that people possess resources in the form of informal strategy development and mathematical reasoning that can serve as a foundation for learning more abstract mathematics. But they also suggest that the link is not automatic. If there is no bridge between informal and formal mathematics, the two often remain disconnected.

The first principle of How People Learn emphasizes both the need to build on existing knowledge and the need to engage students’ preconceptions—particularly when they interfere with learning. In mathematics, certain preconceptions that are often fostered early on in school settings are in fact counterproductive. Students who believe them can easily conclude that the study of mathematics is “not for them” and should be avoided if at all possible. We discuss these preconceptions below.



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