Most students enter grade 3 with enthusiasm for, and interest in, learning mathematics. In fact, nearly three-quarters of U.S. fourth graders report liking mathematics (Silver, Strutchens, and Zawojewski 1997). They find it practical and believe that what they are learning is important. If the mathematics studied in grades 3–5 is interesting and understandable, the increasingly sophisticated mathematical ideas at this level can maintain students' engagement and enthusiasm. But if their learning becomes a process of simply mimicking and memorizing, they can soon begin to lose interest. Instruction at this level must be active and intellectually stimulating and must help students make sense of mathematics.

This chapter presents a challenging set of mathematical content and processes that students in grades 3–5 can and should learn. It also emphasizes teaching that fosters and builds on students' mathematical understanding and thinking. The Content and Process Standards described here form the basis for a significant and interconnected mathematics curriculum. Interwoven through these Standards are three central mathematical themes—multiplicative reasoning, equivalence, and computational fluency. They are briefly discussed here and elaborated on throughout the chapter.

Students entering grade 3 should have a good grasp of, and much experience with, additive reasoning. Their understanding of whole numbers is often based on an additive model—a sequence of numbers used to count in different ways—and their computing strategies usually involve counting on or counting back. In grades 3–5, multiplicative reasoning should become a focus. Multiplicative reasoning is more than just doing multiplication or division. It is about understanding situations in which multiplication or division » is an appropriate operation. It involves a way of viewing situations and thinking about them (Thompson forthcoming). For example, to estimate the height of an adult, students might use their own heights as a benchmark and then think of the situation from an additive perspective (the adult is about 50 centimeters taller than the student) or a multiplicative perspective (the adult is one quarter again as tall as the student).

In grades 3–5, multiplicative reasoning emerges and should be discussed and developed through the study of many different mathematical topics. Students' understanding of the base-ten number system is deepened as they come to understand its multiplicative structure. That is, 484 is 4100 plus 810 plus 41 as well as a collection of 484 individual objects. Multiplicative reasoning is further developed as students use a geometric model of multiplication, such as a rectangular array, and adapt this model for computing the area of shapes and the volume of solids. They also begin to reason algebraically with multiplication, looking for general patterns. For example, they explore problems such as, What is the effect of doubling one factor and halving the other in a multiplication problem? The focus on multiplicative reasoning in grades 3–5 provides foundational knowledge that can be built on as students move to an emphasis on proportional reasoning in the middle grades.

Equivalence should be another central idea in grades 3–5. Students' ability to recognize, create, and use equivalent representations of numbers and geometric objects should expand. For example, 3/4 can be thought of as a half and a fourth, as 6/8, or as 0.75; a parallelogram can be transformed into a rectangle with equal area by cutting and pasting; 825 can be thought of as 855 or as 450; and three feet is the same as thirty-six inches, or one yard. Students should extend their use of equivalent forms of numbers as they develop new strategies for computing and should recognize that different representations of numbers are helpful for different purposes. Likewise, they should explore when and how shapes can be decomposed and reassembled and what features of the shapes remain unchanged. Equivalence also takes center stage as students study fractions and as they relate fractions, decimals, and percents. Examining equivalences provides a way to explore algebraic ideas, including properties such as commutativity and associativity.

A major goal in grades 3–5 is the development of computational fluency with whole numbers. Fluency refers to having efficient, accurate, and generalizable methods (algorithms) for computing that are based on well-understood properties and number relationships. Some of these methods are performed mentally, and others are carried out using paper and pencil to facilitate the recording of thinking. Students should come to view algorithms as tools for solving problems rather than as the goal of mathematics study. As students develop computational algorithms, teachers should evaluate their work, help them recognize efficient algorithms, and provide sufficient and appropriate practice so that they become fluent and flexible in computing. Students in these grades should also develop computational-estimation strategies for situations that call for an estimate and as a tool for judging the reasonableness of solutions.

Teachers in grades 3–5 make decisions every day that influence their students' opportunities to learn and the quality of that learning. The classroom environment they create, the attention to various topics of mathematics, and the tools they and their students use to explore mathematical ideas are all important in helping students in grades 3–5 gain increased mathematical maturity. In these grades teachers should help students learn to work together as part of building a mathematical community of learners. In such a community, students' ideas are valued and serve as a source of learning, mistakes are seen not as dead ends but rather as potential avenues for learning, and ideas are valued because they are mathematically sound rather than because they are argued strongly or proposed by a particular individual (Hiebert et al. 1997). A classroom environment that would support the learning of mathematics with meaning should have several characteristics: students feel comfortable making and correcting mistakes; rewards are given for sustained effort and progress, not the number of problems completed; and students think through and explain their solutions instead of seeking or trying to recollect the "right" answer or method (Cobb et al. 1988). Creating a » classroom environment that fosters mathematics as sense making requires the careful attention of the teacher. The teacher establishes the model for classroom discussion, making explicit what counts as a convincing mathematical argument. The teacher also lays the groundwork for students to be respectful listeners, valuing and learning from one another's ideas even when they disagree with them.

Because of the increasing mathematical sophistication of the curriculum in grades 3–5, the development of teachers' expertise is particularly important. Teachers need to understand both the mathematical content for teaching and students' mathematical thinking. However, teachers at this level are usually called on to teach a variety of disciplines in addition to mathematics. Many elementary teacher preparation programs require minimal attention to mathematics content knowledge. Given their primary role in shaping the mathematics learning of their students, teachers in grades 3–5 often must seek ways to advance their own understanding.

Many different professional development models emphasize the enhancement of teachers' mathematical knowledge. Likewise, schools and districts have developed strategies for strengthening the mathematical expertise in their instructional programs. For example, some elementary schools identify a mathematics teacher-leader (someone who has particular interest and expertise in mathematics) and then support that teacher's continuing development and create a role for him or her to organize professional development events for colleagues. Such activities can include grade-level mathematics study groups, seminars and workshops, and coaching and modeling in the classroom. Other schools use mathematics specialists in the upper elementary grades. These are elementary school teachers with particular interest and expertise in mathematics who assume primary responsibility for teaching mathematics to a group of students—for example, all the fourth graders in a school. This strategy allows some teachers to focus on a particular content area rather than to attempt being an expert in all areas.

Ensuring that the mathematics outlined in this chapter is learned by all students in grades 3–5 requires a commitment of effort by teachers to continue to be mathematical learners. It also implies that districts, schools, and teacher preparation programs will develop strategies to identify current and prospective elementary school teachers for specialized mathematics preparation and assignment. Each of the models outlined here—mathematics teacher-leaders and mathematics specialists—should be explored as ways to develop and enhance students' mathematics education experience. For successful implementation of these Standards, it is essential that the mathematical expertise of teachers be developed, whatever model is used.