A major reason many students don’t perform better in math and algebra is that they don’t understand the meaning of the equal sign, says a recent study in the *Journal of Educational Psychology*. Rather than interpret the equal sign to mean equivalence, many students seem to interpret it to mean “the answer comes next.”

The issue is not that students have difficulty grasping the concept of equivalence, researchers write, but that they have trouble working on equivalence problems expressed in symbols and numerals.

When 2nd graders are given equivalence problems with blocks, they not only are able to solve the problems with far greater accuracy than students who solve the same problems with numeric symbols, the researchers report. But they also are more likely to see the equal sign as a “relational” sysmbol rather than an “operator” symbol.

“Children’s superior performance on nonsymbolic versus symbolic problems suggests that children fail to map their understanding of equivalence onto problems presented with the symbols of arithmetic,” the researchers write.

### Transfer of equivalence problems

The good news for educators is that children’s exposure to equivalence problems with blocks or other manipulatives seems to transfer to solving equivalence problems in numerals and symbols and provides a simple and effective early intervention.

In a 2nd study, researchers report that children exposed first to equivalence problems in a nonsymbolic format performed better a week later on problems using numerals and symbols than children who solved the problems in the symbolic format first.

“Manipulatives may be an important tool for not only assessing children’s relational reasoning but perhaps for instructing students about equivalence and relational thought,” the researchers write. “These studies are the first to conclusively show that young children are capable, with little instruction or feedback, of solving complex equivalence problems when presented in a nonsymbolic context. Building upon such competencies may serve as an important educational tool for helping to remediate children’s failure on equivalence problems and misunderstanding of the equal sign.”

In the 1st study, 48 2nd-grade students were tested individually with 20 problems in 20-minute sessions. The two groups of students were presented with the same equivalence problems, but one group solved the problems in symbolic terms on flash cards and the intervention group solved the problems with blocks. A piece of blue cardboard, folded to look like a tent, separated the two sides of the equation. Wooden cylinder blocks were placed in opaque plastic bins to represent each term of the arithmetic expressions. On the right side was purple construction paper and on the left was orange construction paper to make the two sides distinct.

### Operator vs. relational interpretation

In the 1st study, children in the non-symbolic group not only performed better in solving the problems, but they also were more likely to report anĀ “relational” rather than an “operator” interpretation of the equal sign. In the 2nd study, 32 children from the same area as the children in the 1st study solved the same problems in both formats but in a different order. They were placed in two groups: symbolic/non-symbolic and non-symbolic/symbolic. They selected their answers from multiple choices. In the non-symbolic format, students selected one of four photographs of blocks placed in bins to signify both sides of an equivalence problem.

While students in the nonsymbolic/symbolic format performed better than students who solved the problems in the reverse order, they still had reported an “operator” interpretation rather than a “relational” interpretation of the equal sign. This outcome is surprising, the researchers write, but may point to children’s ability to change procedurally before they change conceptually.

Previous research has found that approximately 88% of 4th and 5th-grade students failed equivalence problems, the authors write. Other research has described how children have distinct microworlds in which they reason very differently about the same tasks in different concepts. Children’s experience with money, for example, makes the “money world” of addition more accessible than the same addition with paper and pencil.

“Children have competencies hidden by their misconceptions of mathematical symbols, particularly the equal sign, and both discovering and building upon these competencies are important for remediating children’s difficulty with equivalence and for preparing the way to higher algebraic and symbolic thought,” they write.

*“Equivalence in Symbolic and Nonsymbolic Contexts: Benefits of Solving Problems With Manipulatives,” by Jody Sherman and Jeffrey Bisanz, Journal of Educational Psychology, volume 101, Number 1, 2009, pp. 88-100.*