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Teaching Children Equations
Part of the job of teaching mathematics to children is uncovering and correcting misinformation they bring with them to the classroom. In research funded by the Office of Educational Research and Improvement, U.S. Department of Education, WCER researcher Tom Carpenter found that some children develop concepts about equations and the “equals” sign earlier than was supposed.
Kindergarten teacher Mary Jo Yttri gave her students the problem 4 + 5 = x + 6. To her surprise, every one of the children thought that the answer should be 9.
Intrigued, Yttri then used plastic cubes to model this equation with the children. To illustrate the problem, 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 they 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.
But even after doing this activity, the children still thought that the answer to the equation was 9. As other research has 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.” Elementary school children generally do not see the equals sign as a symbol that expresses the relationship “is the same as.”
Misconceptions about the meaning of the equals sign are not eliminated with one or two examples or a simple explanation, says Carpenter, who directs WCER’s OERI-funded National Center for Improving Student Learning and Achievement in Mathematics and Science. “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.”
Teachers must make a concentrated effort over an extended period of time to establish appropriate notions of equality, says Carpenter. “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.”
The Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) recommends that algebraic concepts be taught throughout the elementary years. The concept of equality is crucial to algebraic thinking, says Carpenter, and “it looks like we have to change the way we teach math in the early years if our students are to understand even such a foundational idea the way adults do.”
“Like a teeter-totter”
Karen Falkner teaches a first- and second-grade class in a school district in the Midwest. Children typically stay in her class for two years. Her students have progressed in their understanding of equality over the past year-and-a-half.
When Faulkner initially asked students to solve the number sentence 8 + 4 = x + 5, the students answered by putting 12 in the box. Some extended the sentence by adding = 17. Most said that 12 should go in the box because “eight plus four equals twelve.”
But some students objected. Lillie gave the most spirited explanation. “The equals sign means that it has to be even,” she explained. “The amount has to be the same on each side of the equals sign. [Gesturing with her hands]. It is just like a teeter-totter. It has to be level.” The class wrestled with this problem for some time.
“As we reflect on our introduction of the notion of equality and the equals sign to this class and others,” Carpenter says, “we continue to be amazed at the interest and excitement that the children bring to the discussions. Lillie uses her teeter-totter metaphor with the enthusiasm of a child ready to play on one.” Another student, Skip, is genuinely outraged that anyone should fill in a blank so that an equation reads 12 = 17.
These are not the bored comments of children looking forward to recess, but the excited contributions of children who are exploring a new world of thinking and communicating mathematically and who are enjoying the power of that new knowledge. These children are developing an understanding of equality as they learn about numbers and operations. This understanding will allow them to solve equations and will lay a firm foundation for later learning of algebra.
[This article originally appeared in different form in the journal Teaching Children Mathematics.]