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Students enter grade 3 with an interest in learning mathematics. Nearly three-quarters of U.S. fourth graders report liking mathematics, seeing it as practical and important. If mathematics continues to be seen as interesting and understandable, students will remain engaged. If learning becomes simply a process of mimicking and memorizing, students' interest is likely to diminish.

Interwoven through the Content Standards for grades 3–5 are three crucial mathematical themes--multiplicative thinking, equivalence, and computational fluency. The focus on multiplicative reasoning develops knowledge that students build on as they move into the middle grades, where the emphasis is on proportional reasoning. As a part of multiplicative reasoning, students in grades 3–5 should build their understanding of fractions as a part of a whole and as division.

The concept of equivalence helps students learn different mathematical representations and offers a way to explore algebraic ideas. Students should develop computational fluency-- efficient and accurate methods for computing that are based on well-understood properties and number relationships. For example, 298 42 can be thought of as (300 42) – (2 42), or 41 16 can be computed by multiplying 41 8 to get 328 and then doubling 328 to get 656. When these three themes are emphasized, the expectations for grades 3–5 reinforce two major objectives of mathematics learning: making sense of mathematical ideas and acquiring the skills and understandings needed to solve problems.

In grades 3–5, algebraic ideas emerge and are investigated by children. For example, students in these grades are able to make a general statement about how one variable is related to another variable. If a sandwich costs \$3, you can figure out how many dollars any number of sandwiches cost by multiplying that number by 3. In this case, students have developed a model of a proportional relationship: the value of one variable is always 3 times the value of the other, or C = 3 n.

Given their central role in shaping the mathematics learning of students in these grades, teachers must recognize the need to develop mathematical expertise. Some elementary schools identify a "mathematics teacher-leader," who can support other teachers in their instruction and professional development. Other schools use "mathematics specialists" at the upper elementary grade levels, who assume primary responsibility for teaching mathematics to larger groups of students. Each of these models needs to be explored to enhance the mathematics education of students in grades 3–5.

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