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Learning mathematics requires learning to use culturally specific mathematical language, formats, and methods (math tools). To use these math tools effectively in a problem situation, one must learn to identify the mathematical elements of that problem situation; i.e., one must learn to mathematize. In traditional approaches to mathematics learning, these aspects are often separated, with problem solving following learning about mathematical tools. We present in this chapter a model for learning mathematics with understanding that highlights the kinds of connections that can facilitate sense-making by the learner. We exemplify this model with a new approach to the learning of ratio and proportion. This approach addresses two major learning difficulties in this domain (e.g., Behr, Harel, Post, & Lesh, 1993; Harel & Confrey, 1994; Kaput & West, 1994; Lamon, 1999). First, students typically use additive rather than multiplicative solution methods (e.g., to solve 6:14 =?:35, they find the difference between 6 and 14 and subtract it from 35 to find 27:35 rather than seek multiplicative relationships). Second, they have difficulty moving from easy problems that use the basic ratio (e.g., 3:7 =?:14) to middle-difficulty nondivisible problems in which neither ratio is a multiple of the other (e.g., 6:14 =?:35).
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