Representing and analyzing mathematical situations and structures is a major component of algebraic thinking. To build concepts for later work with properties such as commutativity (a + b = b + a), associativity ([a + b] + c = a + [b + c]), and equivalent forms (a + b = c), children need many experiences with mathematical situations and structures through representations and analyses of equality. As noted in the math standards for prekindergarten through second grade, “Equality is an important algebraic concept that students must encounter and begin to understand in the lower grades” (NCTM 2000, 94). If children experience and begin to understand equality in the early grades, they can readily solve for unknown variables.

With this in mind, we need to offer young children many experiences with recognizing, defining, creating, and maintaining equality. Pan balance scales are one way to demonstrate equality. Young children need to use scales and make scales. They need to talk about equal/not equal, same/different, more/less, and balanced/unbalanced. It is through authentic dialogue that children construct meaning related to the concept of algebraic equality. Children can manipulate the scales with everyday objects to show equality through the idea of balance. Prekindergartner Joey explains, “You know it’s balanced when it’s really straight." "Yeah," adds Robert, "it’s not going to one side—and that’s what balance is all about" (see Figure 6). In this learning situation the children recognize and define equality through the idea of balance. They are ready for experiences with creating and maintaining equality (see "First-Graders Discover Concept of Balance in a Simple Experiment," pp. 18–19).

Many other mathematical situations emphasize equality. We know that children often announce, "She has more than me!" or "I don’t have enough—it’s not fair." Instead of concentrating only on the social implications of these statements, we can focus on the mathematics. By asking how many are needed to make the quantities the same or the situation fair, we incorporate the concept of equality, making algebraic thinking part of everyday life.

Additionally, children can engage in a variety of tasks involving mathematics situations and structures with an assortment of manipulatives. In a classroom rich with algebraic thinking in action, children link cubes to make towers that are equal in quantity and thus equal in height. Others string beads and discuss why one string is longer than the other. One kindergartner says to a classmate, "Ours are not the same. Mine is longer because I have two more beads than you." The children also use the arrangement of ten-frames to show equality by means of different arrangements—one shows six red counters and four yellow counters while another shows five red counters and five yellow counters. The children explain that both frames of 10 show the same value. "See," says Justin, “they are both 10. It doesn't matter that mine has more red than his.” Such opportunities with mathematical situations and structures offer the experiences young children need to build strong foundations in algebraic thinking.

See also:
    ALGEBRA in the Early Years? Yes! [PDF, 407KB]