Not all equal signs are created equal

03/21/2011  |  DR. HENRY BORENSON

Most young students readily understand the operational meaning of the equal sign — that is, when the sign is used to indicate the result of an operation such as 4 + 3 = 7. But they often do not grasp as easily the relational meaning of this sign. In fact, the National Mathematics Advisory Panel (NMAP) noted that many elementary and middle school students do not fully understand the multiple meanings of the equal sign, knowledge considered essential for success in algebra.

Understanding the relational meaning of the equal sign enables students to respond with 10, for example, in the problem 3 + 7 + 5 = ___ + 5, because they know that “both sides of the equal sign must have the same value.” Students who understand only the operational concept of equality will give 15 for the missing number because they add the 3, 7, and 5 which precede the equal sign, ignoring the + 5 that follows the blank.

In a 1999 study involving 752 students in first through sixth grade, fewer than five percent of the students were able to provide the correct response to 8 + 4 = __ + 5. In fact, none of the 145 sixth graders in the study could do so. Without direct instruction in the relational meaning of the equal sign, most students do not see the difference between this type of problem and calculations that require only the operational equal sign.

This conclusion should not be surprising. Mathematical symbols, like words, can have various meanings and uses. For example, “pet” can be both a noun and a verb, but at first glance “I want to pet your pet,” may be confusing to a young child. In mathematics, just because students understand the minus sign in 5 – 3 = 2 does not mean that they will immediately know how to make sense of 5 + (–3) = ___.  A similar kind of confusion surrounds the equal sign.

Experiencing the Relational Equal Sign

Students need to experience the relational meaning of the equal sign to understand this concept fully. An effective way to provide this experience is illustrated by the Hands-On Equations® program for introducing algebraic concepts to young students. The teacher creates this experiential learning by following a series of logical steps.

First the teacher verifies that students understand what a balance scale is, namely, that it has two sides. If one side is heavier, that side will go down. If both sides have the same weight, the scale will stay even, or in balance.

Using a steady demonstration scale, the teacher demonstrates the balance concept using pawns and number cubes. The teacher says, “On one side we have a cube with a weight of eight. On the other side we have two blue pawns. If both sides balance, what is each pawn worth?”

 Figure 1:  An introductory algebra problem is illustrated on the teacher's demonstration balance scale.

Students will answer that the pawn is worth 4. The teacher can then tell them that “x” is another name for the blue pawn. Therefore, the answer can be written as x = 4. The teacher also shows them that every example has a check. When students check this equation and find that both sides equal 8, they place a check mark over the equal sign in 8 = 8.

Taking this approach a step further, the teacher can show students a problem with the unknown on both sides of the balance scale:

Figure 2: This pictorial representation shows the unknown (the pawn) on both sides of the balance.


The teacher can illustrate solving this problem, at first by guessing and checking. For example, the teacher might suggest trying x = 4. This will yield a value of 10 on the left side but 12 on the right. So this response is written as x = 4, 10 12. Eventually students discover that x = 6, giving each side a value of 14. Thus through experience, students learn both that the correct value for the pawn makes the scale balance and that all of the pieces on each side need to be taken into account.

Relating Hands-On Experience and Abstraction

The next step is to help students connect hands-on learning to abstract algebraic representation. For example, the teacher presents an equation such as 4x + 2 = 3x + 9 and shows how to place this abstract equation on the students’ two-dimensional scale representation.

Figure 3: Student scale representation of 4x + 2 = 3x + 9.

Students soon realize that the guess-and-check approach is laborious. The teacher then helps them learn a basic algebraic procedure, namely that removing the same weight, or value, from each side of the scale will maintain balance while simplifying the equation. Physically taking a pawn from each side provides kinesthetic reinforcement of this principle. This action is termed a “legal move” and is done by the student using both hands to remove a pawn on each side simultaneously.

Figure 4: A third-grader uses a "legal move" on the teacher's scale to simplify 4x +2 = 3x +9.

Students repeat this legal move to reduce the pieces to a pawn and a 2 cube on the left and a 9 cube on the right.

Figure 5: The student has removed three blue pawns or three x's from each side of the scale.

After this simplification students easily calculate that x = 7. To perform the check, students reset to the original configuration and count 7 for each pawn then add the number cube value, producing a value of 30 on each side. Therefore the answer is represented as x = 7, 30 = 30.

Hands-On Experience Works

Research conducted between 2007 and 2009 with more than 2,500 third through ninth graders, has shown the effectiveness of using this experiential approach in teaching students to solve linear equations with the unknown on both sides. A portion of this research, for example, involved a class of fifth-graders in an inner-city school. Following only a few basic lessons using this hands-on approach, the students were presented with the problem 3 + 7 + 5 = __ + 5 without any prior comment by the teacher. Ninety-three percent of the students gave the correct response of 10, showing that they had gained a firm grasp of the relational meaning of equality.

An introductory algebra problem is illustrated on the teacher's demonstration scale.
Dr. Henry Borenson has been a mathematics educator for more than 40 years, teaching struggling students from the inner city as well as Math Olympiad students. In order to help students gain an understanding of abstract algebraic equations he developed Hands-On Equations and the Making Algebra Child’s Play workshop to provide training to teachers grades three to nine. For more information visit
Comments & Ratings

  5/17/2011 6:41:21 AM
Deborah Moran 

Mathematics Teacher, Eleventh Street Alternative School 
This article provides a clear and cohesive window into the power of Hands-On Equations for students of all ages. As a teacber in an alternative high school setting, I encounter many students who proclaim that they have never experienced success in math. These students lack understanding of many foundation concepts including the relational meaning of the equal sign. Hands-On Equations provides these students with a successful math experience, helps them understand relational concepts, and bridges the gap between concrete models and abstract algebraic concepts. Students who think they are not good at math suddenly become powerful mathematical thinkers who excel in the classroom. Hands-On is a wonderful tool for students of all abilities.
  5/3/2011 8:49:44 AM
Leslie Law 

Gifted Resource Teacher Virginia Beach City Public Schools 
On 5/3/2011 10:02 AM, Leslie Law wrote:
Dear Dr. Borenson and Team,

I think you captured the “aha” moment seen in all the faces of children who experience this system of learning algebra. For so long they thought of the equal sign as the end of something and now it is visualized as the center of a balanced system… a huge shift for kids in my experience!
Leslie Law
  5/2/2011 1:33:06 PM
Dr. Rima Binder 

Northeastern Illinois University, Adjunct Faculty 
As a teacher of gifted students, I taught the Hands-On Equations program for many years. Once, when I was teaching a sixth-grade class, a pair of fraternal twin boys were enrolled who had been in an algebra class in their previous school. Each boy had an IQ well above 140. One of the boys said that the Hands-On Equations pawns looked a little childish and so chose just to observe a few lessons while his brother became an active participant. Later, however, when everyone was working on a quiz, the observer boy came to me and quietly asked if he might use the “baggie of stuff” (the Hands-On Equations manipulatives). He did so and received a perfect score on the quiz. From then on he, too, was an active participant in each lesson. Many years later, this boy came back to thank me. By then, he was a Ph.D. mathematics student at MIT. He remembered “the stuff with chess pieces” and asked if I still used them. My answer was a resounding “Yes!” He smiled and said that Hands-On Equations gave him a “motion picture in his head” and so he was able to “see” mathematics. He said that this realization was the turning point for him. In his view a true mathematician had to be able to “see” mathematics. His story affirmed that Hands-On Equations really is effective for all students—even the very brightest—and especially for visual-spatial learners who can feel lost in the typical verbally intensive classroom.
  4/29/2011 6:22:09 PM
Millie Brezinski 

Mathematics Coach, Nine Mile Falls School District 
I think Dr. Borenson did a terrific job of articulating how mathematical operation signs, such as the equal sign, can have multiple meanings depending on the context. I agree with his statement that, "Students need to experience the relational meaning of the equal sign to understand this concept fully." Hands-On Equations most definitely teaches students this relational understanding.

  4/27/2011 11:29:25 AM
Merrie Skaggs, Ph.D. 

Chair, Undergraduate School of Education, Baker University 
The article is clearly written and should help others understand the valuable contribution of Hands-On Equations. I want to pass on a portion of a conversation we had at a family dinner. My first time teaching Hands-On Equations was in 1997 when I taught this to my youngest daughter’s sixth grade class. We realized that my great niece Lauren had Hands-On Equations in the third or fourth grade while she was a student in a Department of Defense School in Germany. My daughter Anna contributed that the concept of “variable” always made sense to her in later algebra classes because of Hands-On Equations. Both Lauren and Anna are strong students. My experience—and research—with Hands-On Equations confirms that these materials are powerful for all students, the very capable and the not-as-capable.
  4/22/2011 11:42:06 AM
Dr. John Helfen 

Math Consultant 
Hands-on Equations equals the best program for students of all ages to experience and understand the relational equal sign!