Is your math textbook misleading students about the meaning of the equal sign?

One major reason K-5 students commonly misinterpret the equal sign is that they are only exposed to equations that look like this: 2 + 3=5 or 8 – 3=5.
To better understand that the equal sign means “equivalent to” rather than “the answer is,” students need to be exposed to many atypical equations, says University of Virginia researcher Sarah R. Powell. They need to see flipped equations such as 8=5 + 3 or equations presented as two operations, such as 3 + 4=12 – 5.
Review your current or proposed math textbooks to make sure they frequently expose students to atypical equations. You also should should check teacher manuals for explicit instructions on the equal sign and for consistent definitions across all grade levels.
“Without formal instruction, students perform relatively well on verbal story problems yet poorly on corresponding symbolic equations. This indicates that symbolic representations and problem structure may hinder students’ ability to solve problems,” says the author.

In one study comparing student understanding of the equal sign in the US and China, only 28% of U.S. 6th-grade students could correctly solve equations with operations on both sides compared to 98% of Chinese 6th-grade students. One possible explanation for this difference is that teacher texts in China provide more instruction on teaching equivalency and the relational meaning of the equal sign than teacher texts in the United States, Powell reports.

For this study, Powell’s research team examined the frequency of use of atypical equations in 8 of the most popular K-5 math curricula and also reviewed instructions to teachers and definitions of equal signs at all grade levels. Coders classified the equations on every other page of the textbooks as standard or non-standard. What they found was that many curricula only expose students to the standard equation with the operation on the left side.

Everyday Mathematics stood out for exposing students to more nonstandard equations (8=5+3) where the operation was on the right side in addition, subtraction and multiplication. Math Expressions included many nonstandard equations for addition and Singapore Math had many nonstandard equations for multiplication.

But, in this review of the textbooks,  there were few equations with operations on both sides (3+4=12-5) , which is one of the more effective ways to teach children that the equal sign means “equivalent to.”

“Researchers hypothesize that as students see and work with typical teacher- or textbook-presented equations, where an answer always needs to be computed after the equal sign, students come to understand the equal sign as an operational indicator directing them to perform a calculation,” Powell says.

“In terms of viewing the equal sign as an operational symbol, most elementary students believe the equal sign signals them to “do something” or “find the total,” or that the “answer comes next.”

Explicit instruction for the equal sign

Across curricula, the equal sign was mentioned no more than 8 times in teacher manuals at each grade level, according to Powell. In her study, she details how each curriculum defines the equal sign and whether the definitions are consistent across the grades. She also notes whether the curriculum uses a balance scale to illustrate the meaning of the equal sign as equivalent to.

“No curriculum, however, provides the same definition at all grade levels, and some curricula provide different definitions across grade levels or within the same grade level,” the researcher writes. “This could prove confusing to students who learned one definition in first grade and are provided with another in second grade without understanding that the definitions may have similar meanings. “

Discussion of the equal sign occurred most often in grades K-2 and infrequently in grades 3-5. Powell says children should continue to receive instruction on the equal sign in later grades because they still misconstrue it.

“Therefore, explicit instruction at all grade levels is beneficial until researchers demonstrate that students do not continue to misinterpret the equal sign,” the researcher says. Classroom teachers may have to deviate from the curriculum to compensate for the shortcomings in use of nonstandard equations and in the instructions and definitions of the equal sign.

“Equations and the equal sign in elementary mathematics textbooks,” by Sarah R. Powell, The Elementary School Journal, 2012, Volume 112, Number 4, pps. 627-648.

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