Middle-grades students should see mathematics as an exciting, useful, and creative field of study. As they enter adolescence, students experience physical, emotional, and intellectual changes that mark the middle grades as a significant transition point in their lives. During this time, many students will solidify conceptions about themselves as learners of mathematics—about their competence, their attitude, and their interest and motivation. These conceptions will influence how they approach the study of mathematics in later years, which will in turn influence their life opportunities. Middle-grades students are drawn toward mathematics if they find both challenge and support in the mathematics classroom. Students acquire an appreciation for, and develop an understanding of, mathematical ideas if they have frequent encounters with interesting, challenging problems.

In the middle-grades mathematics classroom, young adolescents should regularly engage in thoughtful activity tied to their emerging capabilities of finding and imposing structure, conjecturing and verifying, thinking hypothetically, comprehending cause and effect, and abstracting and generalizing. In these grades, each student follows his or her own developmental timetable. Some mature early, and others late. Some progress rapidly, others more slowly. Thus, every middle-grades teacher faces the challenge of dealing with many aspects of diversity. Yet students also display some commonalities. For example, young adolescents are almost universally sensitive to the influence of their peers. The differences in intellectual development and emotional maturity and the sensitivity of individuals to peer-group perceptions make it especially important for teachers to create classroom environments in which clearly established norms support the learning of mathematics by everyone.

An ambitious, focused mathematics program for all students in the middle grades is proposed in these Standards. Ambitious expectations are identified in algebra and geometry that would stretch the middle-grades program beyond a preoccupation with number. In recent years, the possibility and necessity of students' gaining facility in algebraic thinking have been widely » recognized. Accordingly, these Standards propose a significant amount of algebra for the middle grades. In addition, there is a need for increased attention to geometry in these grades. Facility in geometric thinking is essential to success in the later study of mathematics and also in many situations that arise outside the mathematics classroom. Moreover, geometry is typically the area in which U.S. students perform most poorly on domestic and international assessments of mathematics proficiency. Therefore, significantly more geometry is recommended in these Standards for the middle grades than has been the norm. The recommendations are ambitious—they call for students to learn many topics in algebra and geometry and also in other content areas. To guard against fragmentation of the curriculum, therefore, middle-grades mathematics curriculum and instruction must also be focused and integrated.

Specific foci are identified in several content areas. For example, in number and operations, these Standards propose that students develop a deep understanding of rational-number concepts, become proficient in rational-number computation and estimation, and learn to think flexibly about relationships among fractions, decimals, and percents. This facility with rational numbers should be developed through experience with many problems involving a range of topics, such as area, volume, relative frequency, and probability. In algebra, the focus is on proficiency in recognizing and working effectively with linear relationships and their corresponding representations in tables, graphs, and equations; such proficiency includes competence in solving linear equations. Students can develop the desired algebraic facility through problems and contexts that involve linear and nonlinear relationships. Appropriate problem contexts can be found in many areas of the curriculum, such as using scatterplots and approximate lines of fit to give meaning to the concept of slope or noting that the relationship between the side lengths and the perimeters of similar figures is linear, whereas the relationship between the side lengths and the areas of similar figures is nonlinear.

Curricular focus and integration are also evident in the proposed emphasis on proportionality as an integrative theme in the middle-grades mathematics program. Facility with proportionality develops through work in many areas of the curriculum, including ratio and proportion, percent, similarity, scaling, linear equations, slope, relative-frequency histograms, and probability. The understanding of proportionality should also emerge through problem solving and reasoning, and it is important in connecting mathematical topics and in connecting mathematics and other domains such as science and art.

In the recommendations for middle-grades mathematics outlined here, students will learn significant amounts of algebra and geometry throughout grades 6, 7, and 8. Moreover, they will see algebra and geometry as interconnected with each other and with other content areas in the curriculum. They will have experience with both the geometric representation of algebraic ideas, such as visual models of algebraic identities, and the algebraic representation of geometric ideas, such as equations for lines represented on coordinate grids. They will see the value of interpreting both algebraically and geometrically such important mathematical ideas as the slope of a line and the Pythagorean relationship. They also will relate algebraic and geometric ideas to other topics—for example, when they reason about percents using visual models or equations or when they represent an approximate line of fit for a scatterplot both geometrically and » algebraically. Students can gain a deeper understanding of proportionality if it develops along with foundational algebraic ideas such as linear relationships and geometric ideas such as similarity.

Students' understanding of foundational algebraic and geometric ideas should be developed through extended experience over all three years in the middle grades and across a broad range of mathematics content, including statistics, number, and measurement. How these ideas are packaged into courses and what names are given to the resulting arrangement are far less important than ensuring that students have opportunities to see and understand the connections among related ideas. This approach is a challenging alternative to the practice of offering a select group of middle-grades students a one-year course that focuses narrowly on algebra or geometry. All middle-grades students will benefit from a rich and integrated treatment of mathematics content. Instruction that segregates the content of algebra or geometry from that of other areas is educationally unwise and mathematically counterproductive.

Principles and Standards for School Mathematics proposes an ambitious and rich experience for middle-grades students that both prepares them to use mathematics effectively to deal with quantitative situations in their lives outside school and lays a solid foundation for their study of mathematics in high school. Students are expected to learn serious, substantive mathematics in classrooms in which the emphasis is on thoughtful engagement and meaningful learning.

For those who make decisions about the design and organization of middle-grades mathematics education, it would be insufficient simply to announce new and more-ambitious goals like those suggested here. School system leaders need to commit to and support steady, long-term improvement and capacity building to accomplish such goals. The capacity of schools and middle-grades teachers to provide the kind of mathematics education envisioned needs to be built. Special attention must be given to the preparation and ongoing professional support of teachers in the middle grades. Teachers need to develop a sound knowledge of mathematical ideas and excellent pedagogical practices and become aware of current research on students' mathematics learning. Professional development is especially important in the middle grades because so little attention has been given in most states and provinces to the special preparation that may be required for mathematics teachers at these grade levels. Many such teachers hold elementary school generalist certification, which typically involves little specific preparation in mathematics. Yet teachers in the middle grades need to know much more mathematics than is required in most elementary school teacher-certification programs. Some middle-grades mathematics teachers hold secondary school mathematics-specialist certification. But middle-grades teachers need to know much more about adolescent development, pedagogical alternatives, and interdisciplinary approaches to teaching than most secondary school teacher-certification programs require. In order to accomplish the ambitious goals for the middle grades that are presented here, special teacher-preparation programs must be developed.