ABSTRACT
The vision we present of classrooms where students learn with understanding requires that teachers understand the mathematics they are teaching and that they understand their students’ thinking about that mathematics. For teachers in the middle grades, this means that they themselves must have a deep understanding of a rich, interrelated set of concepts and the ways in which children develop understanding of these concepts. First, students must extend whole-number concepts and reasoning to rational number concepts and reasoning. This extension in turn depends on a deepening understanding of the role of the unit (whole number) and mathematical consequences of partitioning the unit, of the many ways we represent rational numbers, and of the ways these representations are connected and can be used to perform operations on rational numbers. Second, recognition of situations that are multiplicative rather than additive in nature and, therefore, demand a different type of reasoning plays a central role in mathematics at this level. Third, another central but closely related idea is understanding the role of proportionality in many mathematical situations. When teachers understand this complex of concepts and related forms of reasoning, and have had opportunities to form appropriate expectations of students’ growth of understanding and reasoning about these ideas, they are able to deal with students’ insights and misconceptions, to recognize and seize opportunities for fruitful digressions, and to choose appropriate tasks, tools, and representations for promoting understanding.
