Children's Understanding of Equality: A Foundation for Algebra
Falkner, Karen P., Levi, Linda, Carpenter, Thomas P., Teaching Children Mathematics
Many states and school districts, as well as Principles and Standards for School Mathematics: Discussion Draft (NCTM 1998), recommend that algebra be taught in the early childhood years. Although young children often understand much more than traditionally thought, adults can have difficulty conceptualizing what would constitute appropriate algebra for the early childhood years. Fifteen teachers and three university researchers are currently involved in a project to define what algebra instruction can and should be for young children. In this article, we discuss the concept of equality, which is a crucial idea for developing algebraic reasoning in young children.
Misconceptions about the Equals Sign
Even though teachers frequently use the equals sign with their students, it is interesting to explore what children understand about equality and the equals sign. At the start of this project, many teachers asked their students to solve the following problem:
8 + 4 =  + 5
At first, this problem looked trivial to many teachers. One sixth-grade teacher, for example, said, "Sure, I will help you out and give this problem to my students, but I have no idea why this will be of interest to you." This teacher found that all twenty-four of her students thought that 12 was the answer that should go in the box. She found this result so interesting that before we had a chance to check back with her, she had the other sixth-grade teachers at her school give this problem to their students. As shown in table 1, all 145 sixth-grade students given this problem thought that either 12 or 17 should go in the box.
Why did so many children have trouble with this problem? Clearly, children have a limited understanding of equality and the equals sign if they think that 12 or 17 is the answer that goes in the box. Many young children do, however, understand how to model a situation that involves making things equal. For example, Mary Jo Yttri, a kindergarten teacher, gave her students the problem 4 + 5 =  + 6. All the children thought that 9 should go in the box. Yttri then modeled this situation with the children. Together, they made a stack of four cubes, then a stack of five cubes. In another space, they made stacks of nine and six cubes. Yttri asked the children if each arrangement had the same number of cubes. The children knew that the groupings did not have the same number of cubes and were able to tell her which one had more. Several children were able to tell the teacher how they could make both groupings have the same number of cubes. Even after doing this activity, however, the children still thought that 9 should go in the box in the equation.
This incident surprised Yttri and the researchers. We had assumed that kindergarten children would have little experience with the equals sign and would not yet have formed the misconceptions about equality demonstrated by older children. Even kindergarten children, however, appear to have enduring misconceptions about the meaning of the equals sign that are not eliminated with one or two examples or a simple explanation. This incident also illustrates that children as young as kindergarten age may have an appropriate understanding of equality relations involving collections of objects but have difficulty relating this understanding to symbolic representations involving the equals sign. A concerted effort over an extended period of time is required to establish appropriate notions of equality. Teachers should also be concerned about children's conceptions of equality as soon as symbols for representing number operations are introduced. Otherwise, misconceptions about equality can become more firmly entrenched. (See "About the Mathematics" on p. 234.)
As Behr, Erlwanger, and Nichols (1975); Erlwanger and Berlanger (1983); and Anenz-Ludlow and Walgamuth (1998) have documented, children in the elementary grades generally think that the equals sign means that they should carry out the calculation that precedes it and that the number after the equals sign is the answer to the calculation. …