Algebraic Thinking: An Introduction to
Mathematical Equivalence in Grades K - 8

The Final Report of the National Mathematics Advisory Panel noted that many middle school students have a weak understanding of mathematical equality and that this lack of understanding impedes their learning of algebra.

This small little symbol, “=” is apparently not as simple for students as one might think.  For example, in one study when elementary school students are asked to solve a problem such 6 + 4 = ___ + 3, fewer than 10% of students in grades 2 - 6 could do so prior to teacher instruction. Instead, students typically provide either the response of 10 since that is the sum of 6 and 4, or otherwise they provide the response of 13 since that is the sum of all the numbers.

In light of how crucial a knowledge of equivalence is to student success in algebra and in light of the emphasis the NCTM has placed on integrating algebraic thinking into the curriculum from the earliest grades, we offer several suggestion which teachers can use to help students acquire a better understanding of the use of the equal sign when the intended meaning is that of equivalence.

To assist grade school students with examples such as 6 + 4 = __ + 3, the teacher is asked to consider posing this problem diagrammatically as shown below:

The teacher will find that this notation makes it much easier for students to understand what the problem is asking. The students obtain a visual image of equivalence between the contents of the two boxes. Since the total value inside the box at the left is 10, so must the total value inside the box on the right be 10. Since we are already given a 3, the missing number is 7.

This above examples illustrates an important principle of pedagogy: If students cannot do a problem, such as figuring out the missing number in the example 6 + 4 = __ + 3, it may be because they do not understand what they are being asked to do. Once the same problem is presented in a format that they students can understand, they may experience a high level of success.  

Figure 1.  1^st grader solving 4 = __ + 2.

With very young children who are still counting on their fingers, the teacher should select examples involving small numbers where the sum on either side is 5 or less. For example, with kindergarten students, the teacher can present the example shown below.

The students can use chips or counters to figure out the missing number.

Figure 2. 5-year old figuring out the missing number using chips.

The child can see visually that one chip needs to be added to the right side to make it have the same amount as the left side.

Hence, from the earliest days that a child is working with numbers, the child should also be working with the concept of equivalence. In China, where the students achieve at very high levels on international tests, the concept of equivalence is taught before there is any attempt to teach addition or subtraction.

The Final Report of the National Mathematics Advisory Panel made special mention of the concept of equivalence as a core arithmetic concept which students need to learn if they are not to be impeded in their learning of algebra. The above approach used throughout the elementary grades, and eventually without the boxes, is a helpful step in this direction.

Another approach, which enables teachers to present far more advanced equations to their students, and thereby teach the concept of equivalence at a much deeper level, is the approach used in Hands-On Equations. In only a few lessons the students, in grades 3 – 8, learn to solve equations with unknowns on both sides such as 4x + 3 = 3x + 9 and 2(2x+ 1) = x + 11. The kinesthetic component reinforces the idea that the balance must be maintained at each step of the solution process. More information on Hands-On Equations, including a listing of public workshops where teachers can receive training, can be found at www.borenson.com.