Algebra has its historical roots in the study of general methods for solving equations. The Algebra Standard emphasizes relationships among quantities, including functions, ways of representing mathematical relationships, and the analysis of change. Functional relationships can be expressed by using symbolic notation, which allows complex mathematical ideas to be expressed succinctly and change to be analyzed efficiently. Today, the methods and ideas of algebra support mathematical work in many areas. For example, distribution and communication networks, laws of physics, population models, and statistical results can all be represented in the symbolic language of algebra. In addition, algebra is about abstract structures and about using the principles of those structures in solving problems expressed with symbols.

Much of the symbolic and structural emphasis in algebra can build on students' extensive experiences with number. Algebra is also closely linked to geometry and to data analysis. The ideas included in the Algebra Standard constitute a major component of the school mathematics curriculum and help to unify it. Algebraic competence is important in adult life, both on the job and as preparation for postsecondary education. All students should learn algebra.

By viewing algebra as a strand in the curriculum from prekindergarten on, teachers can help students build a solid foundation of understanding and experience as a preparation for more-sophisticated work in algebra in the middle grades and high school. For example, systematic experience with patterns can build up to an understanding of the idea of function (Erick Smith forthcoming), and experience with numbers and their properties lays a foundation for later work with symbols and algebraic expressions. By learning that situations often can be described using mathematics, students can begin to form elementary notions of mathematical modeling.

Often, algebra has not been treated explicitly in the school curriculum until the traditional algebra course offered in middle school or high school. By promoting algebra as a strand that is begun in the early grades, Principles and Standards supports other possibilities for configuring programs in the middle grades and secondary schools. The Standards for grades 6–8 include a significant emphasis on algebra, along with much more geometry than has normally been offered in the middle grades, and call for the integration of these two areas. The » Standards for grades 9–12, assuming that this strong foundation in algebra will be in place by the end of the eighth grade, describe an ambitious program in algebra, geometry, and data analysis and statistics and also call for integration and connections among ideas.

Early experiences with classifying and ordering objects are natural and interesting for young children. Teachers might help children notice that red-blue-blue-red-blue-blue can be extended with another red-blue-blue sequence or help them predict that the twelfth term is blue, assuming that the red-blue-blue pattern repeats indefinitely. Initially, students may describe the regularity in patterns verbally rather than with mathematical symbols (English and Warren 1998). In grades 3–5, they can begin to use variables and algebraic expressions as they describe and extend patterns. By the end of secondary school, they should be comfortable using the notation of functions to describe relationships.

In the lower grades, students can describe patterns like 2, 4, 6, 8, ... by focusing on how a term is obtained from the previous number—in this example, by adding 2. This is the beginning of recursive thinking. Later, students can study sequences that can best be defined and computed using recursion, such as the Fibonacci sequence, 1, 1, 2, 3, 5, 8, ..., in which each term is the sum of the previous two terms. Recursive sequences appear naturally in many contexts and can be studied using technology.

As they progress from preschool through high school, students should develop a repertoire of many types of functions. In the middle grades, students should focus on understanding linear relationships. In high school, they should enlarge their repertoire of functions and learn about the characteristics of classes of functions.

Many college students understand the notion of function only as a rule or formula such as "given n, find 2ⁿ for n = 0, 1, 2, and 3" (Vinner and Dreyfus 1989). By the middle grades, students should be able to understand the relationships among tables, graphs, and symbols and to judge the advantages and disadvantages of each way of representing relationships for particular purposes. As they work with multiple representations of functions—including numeric, graphic, and symbolic—they will develop a more comprehensive understanding of functions (see Leinhardt, Zaslavsky, and Stein 1990; Moschkovich, Schoenfeld, and Arcavi 1993; NRC 1998).

Students' understanding of properties of numbers develops gradually from preschool through high school. While young children are skip-counting by twos, they may notice that the numbers they are using end in 0, 2, 4, 6, and 8; they could then use this algebraic observation to extend the pattern. In grades 3–5, as students investigate properties of whole-number operations, they may find that they can multiply 18 by 14 mentally by computing 1810 and adding it to 184; they are » using the distributive property of multiplication over addition. Sometimes geometric arguments can be understood long before students can reasonably be expected to perform sophisticated manipulations of algebraic symbols. For example, the diagram in figure 3.2 might help lead upper elementary school students to the conjecture that the sum of the first n odd numbers is n². Middle school students should be able to understand how the diagram relates to the equation. Students in high school should be able to represent the relationship in general, with symbols, as 1 + 3 + ... + (2n – 1) =  n², and they should be able to prove the validity of their generalization.

Fig. 3.2. Demonstration that 1 + 3 + 5 + 7 = 42

Research indicates a variety of student difficulties with the concept of variable (Küchemann 1978; Kieran 1983; Wagner and Parker 1993), so developing understanding of variable over the grades is important. In the elementary grades, students typically develop a notion of variable as a placeholder for a specific number, as in __ + 2 = 11. Later, they should learn that the variable x in the equation 3x + 2 = 11 has a very different use from the variable x in the identity 0 x = 0 and that both uses are quite different from the use of r in the formula A = r². A thorough understanding of variable develops over a long time, and it needs to be grounded in extensive experience (Sfard 1991).

The notion of equality also should be developed throughout the curriculum. As a consequence of the instruction they have received, young students typically perceive the equals sign operationally, that is, as a signal to "do something" (Behr, Erlwanger, and Nichols 1976; Kieran 1981). They should come to view the equals sign as a symbol of equivalence and balance.

Students should begin to develop their skill in producing equivalent expressions and solving linear equations in the middle grades, both mentally and with paper and pencil. They should develop fluency in operating with symbols in their high school years, with by-hand or mental computation in simple cases and with computer algebra technology in all cases. In general, if students engage extensively in symbolic manipulation before they develop a solid conceptual foundation for their work, they will be unable to do more than mechanical manipulations (NRC 1998). The foundation for meaningful work with symbolic notation should be laid over a long time.

In grades 3–5 students should use their models to make predictions, draw conclusions, or better understand quantitative situations. These uses of models will grow more sophisticated. For instance, in solving a problem about making punch, middle-grades students might describe the relationships in the problem with the formula P = (8/3)J, where P is » the number of cups of punch and J is the number of cups of juice. This mathematical model can be used to decide how much punch will be made from fifty cups of juice.

High school students should be able to develop models by drawing on their knowledge of many classes of functions—to decide, for instance, whether a situation would best be modeled with a linear function or a quadratic function—and be able to draw conclusions about the situation by analyzing the model. Using computer-based laboratories (devices that gather data, such as the speed or distance of an object, and transmit them directly to a computer so that graphs, tables, and equations can be generated), students can get reliable numerical data quickly from physical experiments. This technology allows them to build models in a wide range of interesting situations.