03/21/2011 | DR. HENRY BORENSON

Mathematics

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.

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Deborah MoranMathematics Teacher, Eleventh Street Alternative SchoolLeslie LawGifted Resource Teacher Virginia Beach City Public SchoolsDear 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

Dr. Rima BinderNortheastern Illinois University, Adjunct FacultyMillie BrezinskiMathematics Coach, Nine Mile Falls School DistrictMerrie Skaggs, Ph.D.Chair, Undergraduate School of Education, Baker UniversityDr. John HelfenMath Consultant