David Shilkitus on Why the Equal Sign Is One of the Most Misunderstood Symbols in Early Math

Students encounter the equal sign almost immediately when learning arithmetic, yet David Shilkitus explains that familiarity with the symbol does not always translate into a complete understanding of what it means. A student may correctly solve dozens of problems written as 7 + 5 = 12 while quietly developing the idea that the equal sign simply means “write the answer next.”

That interpretation may work for straightforward arithmetic exercises, but mathematics eventually demands something more sophisticated. The equal sign represents a relationship: the quantities or expressions on either side have the same value.

Helping students understand that distinction can strengthen number sense, algebraic reasoning, and their ability to interpret unfamiliar equations.

David Shilkitus on Moving Beyond “The Answer Comes Next”

Traditional arithmetic problems often follow a predictable format:

8 + 4 = __

The structure encourages students to perform an operation from left to right and place the result after the equal sign. With enough repetition, some students may begin associating “=” with an instruction rather than a relationship.

That misunderstanding becomes easier to see when the format changes.

Consider:

__ = 8 + 4

A student who understands equality should recognize that 12 belongs in the blank. A student who has learned that the equal sign means “calculate what comes next” may find the arrangement surprisingly confusing.

David Shilkitus emphasizes that changing the structure of familiar problems can reveal whether students understand the underlying mathematical idea or have simply learned a routine.

Equality Is About Balance

One useful way to develop a stronger understanding is to treat an equation as a statement of balance.

Consider:

6 + 3 = 5 + 4

There is no conventional “answer” sitting by itself on the right side. Instead, students must determine whether both expressions represent the same quantity.

Both sides equal nine, so the statement is true.

For David Shilkitus, examples like these can help shift attention away from completing a calculation and toward examining the relationship between two expressions.

That distinction becomes increasingly important as students advance. Mathematics is not always about producing a number at the end of a line. Often, it involves understanding relationships between quantities.

Unusual Equation Formats Can Expose Misconceptions

Teachers can learn a great deal by presenting equations in forms students do not expect.

For example:

7 + 6 = __ + 5

A student who approaches the problem relationally can reason that the left side equals 13, so the missing number must make the right side equal 13 as well. The answer is eight.

But a student who interprets the equal sign as an instruction to perform every visible calculation may add 7 + 6 + 5 or use another procedure that does not preserve equality.

David Shilkitus recognizes that mistakes on problems like these can be valuable. They may reveal not a weakness in addition but a misunderstanding of what an equation communicates.

That gives teachers something more useful than a simple wrong answer: insight into the student’s reasoning.

The Equal Sign Creates an Early Bridge to Algebra

A strong understanding of equality becomes especially valuable when students encounter variables.

An equation such as:

x + 4 = 11

requires students to think about two equivalent quantities. The objective is to determine which value of x makes the relationship true.

Later equations can become much more complicated, but the principle remains.

Operations performed while solving an equation must preserve equality.

David Shilkitus explains that students who already understand the equal sign as representing a relationship have a conceptual foundation for this transition. They are not suddenly being asked to assign an entirely new meaning to a familiar symbol when algebra begins.

The notation has remained consistent. The mathematics surrounding it has simply become more sophisticated.

True-or-False Equations Can Strengthen Understanding

Not every equality exercise needs to ask students to calculate a missing value.

Teachers can present mathematical statements and ask whether they are true or false:

10 = 10

4 + 6 = 7 + 3

12 – 5 = 8

15 = 9 + 6

These problems encourage students to examine both sides rather than automatically searching for an operation followed by an answer.

David Shilkitus notes that students can also be asked to explain how they know.

A learner might initially calculate both sides. With stronger number sense, that student may eventually recognize relationships without completing every calculation separately.

The exercise therefore develops more than knowledge of the equal sign. It can also encourage flexible mathematical reasoning.

Students Should Encounter Equations in Multiple Forms

If every classroom example uses the same format, students can become skilled at recognizing the format without fully understanding the concept.

Variety can make a difference.

Students can work with equations where:

• The result appears before the equal sign.
• Both sides contain operations.
• A missing value appears on either side.
• Several expressions represent the same quantity.
• An equation is deliberately false and must be corrected.
• Students create their own equivalent expressions.

For David Shilkitus, this kind of variation encourages students to pay attention to mathematical meaning rather than relying exclusively on visual patterns.

The goal is not to make basic arithmetic unnecessarily complicated. It is to ensure that students understand why the notation works.

Mathematical Language Matters

How adults describe equations can also influence student understanding.

Phrases such as “the answer goes after the equal sign” may be convenient when introducing arithmetic, but they can unintentionally reinforce a limited interpretation.

Instead, teachers can consistently use language involving ideas such as “the same value as,” “equivalent to,” or “balanced with.”

David Shilkitus emphasizes that precise language can support precise thinking.

When a student reads 5 + 7 = 12 as “five plus seven has the same value as twelve,” the mathematical relationship becomes explicit. The symbol is no longer simply a command separating a problem from its answer.

It communicates something specific about the quantities on both sides.

Small Misconceptions Can Become Larger Obstacles

The equal sign demonstrates why seemingly minor misunderstandings in early mathematics deserve attention.

A child may successfully complete years of conventional arithmetic exercises while maintaining an incomplete understanding of equality. The misconception may become noticeable only when equations begin appearing in unfamiliar formats.

By then, mathematics may appear to have suddenly changed its rules.

David Shilkitus explains that the better approach is to strengthen conceptual understanding while students are developing procedural skills.

Correct calculation remains important. Students need fluency with arithmetic. But knowing how to calculate and understanding what mathematical notation communicates are complementary skills rather than competing goals.

One Small Symbol Carries a Major Mathematical Idea

The equal sign occupies very little space on a page, but the concept behind it extends throughout mathematics.

It connects arithmetic expressions, supports equations, prepares students for algebra, and communicates relationships between quantities. Understanding it as more than an instruction to produce an answer can change how students interpret mathematical statements.

For David Shilkitus, that makes equality an important example of why early mathematics should go beyond memorizing procedures.

Students benefit when they are asked not only to calculate but also to interpret, compare, justify, and explain. Moving a blank, placing expressions on both sides of an equation, or asking whether an equality is true can expose misconceptions that ordinary worksheets may not reveal.

A student who understands that the equal sign means both sides have the same value has learned more than a symbol’s meaning. That student has begun developing a relational way of thinking that will continue to matter as mathematics becomes more complex.

David Shilkitus recognizes that some of the most important mathematical lessons can begin with the smallest details. In this case, two short parallel lines provide an early opportunity to teach students that mathematics is not simply about reaching an answer. It is about understanding the relationships that make the answer true.