What does “=” mean?

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They solve it but don’t understand it

In October, teachers of six classes in Bishkek, from grade 2 to grade 6, gave their students a sheet with five short tasks. The main one was:

8 + 4 = □ + 5. What number goes in the box?

The correct answer is 7: the left side is 12, so the right side must also be 12, and 7 + 5 = 12.

Of 73 students, 7 wrote 41. Another 17 wrote 12 or “12 and 7”: “8 plus 4 makes 12, so I write the answer.” Three wrote 17 or 18, adding up everything they saw.

For comparison: in 1999 the same problem was given to 752 children in grades 1–6 in Madison, USA. Between 0 and 10 % in each grade answered correctly, and all 145 sixth graders at one school wrote 12 or 17. Our students do noticeably better on this problem.

But in the second task we asked: what does the “=” sign mean? Students answered in their own words, and teachers copied the answers down word for word.

• 34 of the 73 students wrote that “=” is the answer, the sum or the result: “Well, it’s the answer to the sum, without an answer there is no sum,” “It’s the sign that ends the sum or gives the answer,” “The final number in the sum.”
• 7 students wrote that both sides are the same: “It’s the equality of two numbers,” “It means the same, or, for example, they weigh the same.”
• The rest wrote “it’s equals,” gave an example like “3 + 4 = 7,” or wrote nothing.

Now the key finding. In grades 5 and 6, 14 students solved the box problem correctly. All fourteen described “=” as the answer, the sum or the result. They solved the problem, yet they understand the sign as “this is where the answer goes.” Neither a mark nor a test will reveal this: after all, the answer is correct.

A year from now these students will be given an equation with x on both sides of the sign, and an expression to simplify. That is where “this is where the answer goes” stops working: misunderstanding the sign causes many errors in both equations and algebraic transformations.

One sign, six jobs

Let’s look at the jobs the “=” sign does in school exercise books from grade 1 to grade 9.

1. Calculate. 3 + 5 = 8. In the early years the sign almost always looks like this: an operation on the left, the result on the right.
2. I claim these are the same. 7 = 3 + 4; 3 + 4 = 5 + 2. A statement like this can be checked: it is either true or false.
3. Find when it becomes true. x + 5 = 12. This is an equation. Here the sign asks a question: for which x are the two sides equal?
4. Always true. a + b = b + a; 2(x + 3) = 2x + 6. This is an identity: it holds for any numbers.
5. Let’s call it this. S = a · b; “let a = 5.” This is a formula or a definition: the sign introduces notation.
6. Put it in the box. x = x + 1 in a Python program. This is a command: take x, add one, store the result back.

Look at these two statements:

2(x + 3) = 2x + 6

2(x + 3) = 14

They look almost the same. The first is true for every x: it is an identity, and it is proved. The second is true only for x = 4: it is an equation, and it is solved. The jobs are completely different, yet the sign is the same.

Mathematics itself tells these jobs apart. There is the sign ≡ for an identity and “:=” for a definition. Programmers write “=” for a command and “==” for the question “are these equal?” At school, all of this is the same pair of short lines.

Having several meanings is not a problem in itself. The problem is that nobody talks about it. A student sees the same sign and reasonably assumes it has one meaning. Which meaning is suggested by experience: thousands of examples of “operation, equals, answer.” Researchers counted the statements with an “=” sign in four series of American textbooks for grades 6–8: operations on both sides of the sign appeared on average in 5 % of cases. The other 95 % were “calculate.”

Not the only sign like this

The minus sign has a similar story. It has three jobs, and they too usually go unmentioned:

1. Subtract: 5 − 3 = 2.
2. A number less than zero: −3 is a single number, not an operation.
3. Take the opposite: −x. If x = 5, then −x = −5. But if x = −5, then −x = 5, and “minus x” turns out to be a positive number.

Hence −(−3) = 3, “minus times minus is plus,” and errors such as “−x is always negative.” We will devote a separate article in the series to the minus sign.

Why this happens: the Kyrgyz curriculum

We checked the KAO documents: the subject standards and curricula for mathematics and computer science.

• Grade 1 includes the signs >, <, = and “an equality with an unknown component”: □ + 4 = 7. The box is solved by trial and error.
• Grades 2–3 have simple equations, and grade 5 has linear equations. They are solved “through the unknown components”: the unknown addend equals the sum minus the known addend.
• The properties of equality (“you may add the same number to both sides”) and moving terms across the equals sign (transposition) are not named in any document.
• Identities appear in grade 6, and equivalent equations in grade 7. Nowhere is it explained separately how an identity differs from an equation when the sign is the same.
• The new standard for grades 1–6 does not even name the signs >, <, = themselves.
• In computer science, variables and programs start in grade 7, yet assignment is absent from every document, in Russian and in Kyrgyz alike.

So this is how it works out. The sign is introduced in grade 1 and never revisited, even though its job changes every two or three years. The “through the components” method in grade 5 also works for a student who reads “=” as “this is where the answer goes”: find the unknown by the rule and you’re done. That is why the misunderstanding goes unnoticed until grade 7.

Where it breaks down

An equation with x on both sides.

3x + 5 = x + 13

What is the “unknown addend” here? Where is the “answer”? A student for whom “=” means “this is where the answer goes” doesn’t know where to start: there is no answer on the right, just x again.

A student who reads “=” as a balance scale sees two pans in equilibrium. Take an x off both pans: 2x + 5 = 13. Take 5 off both: 2x = 8. Halve both sides: x = 4. Check by substitution: 3 · 4 + 5 = 17 and 4 + 13 = 17. The pans balance.

Transposition as a ritual. In grade 7 the same solution is written more briefly: 3x − x = 13 − 5. “We move the term to the other side and change its sign.” Why change it? A student who knows about the scale will answer: we take 5 off both pans, we just write it more briefly. A student without the scale will memorise the rule and get confused about which sign to change and in which direction.

A chain of equalities. A student calculates 25 + 15 + 7 and writes:

25 + 15 = 40 + 7 = 47

Every step is calculated correctly, but the statement is false: 25 + 15 does not equal 47. For this student “=” means “and then you get.” The teacher will correct it, but unless the reason is explained, the error will come back.

Transforming expressions. “Simplify: 2(x + 3) + 4x.” The student writes:

2(x + 3) + 4x = 2x + 6 + 4x = 6x + 6

Here every “=” is an identity: all three expressions are equal for any x. But the student reads the sign as “the result of the next step,” just as in the chain 25 + 15 = 40 + 7. As long as the steps are correct, the difference is invisible. It shows up in the errors:

• “Simplify: x/2 + x/3.” The student “gets rid of the denominators” as in an equation, multiplies by 6 and writes x/2 + x/3 = 3x + 2x = 5x. But while both sides of an equation may be multiplied by 6, multiplying an expression changes it: x/2 + x/3 = 5x/6, not 5x. For the student, “=” in an equation and “=” in a transformation are the same “and then you get,” so techniques carry over from one to the other.
• The reverse error is writing an equation as a chain, like a transformation: 2x + 5 = 13 = 2x = 8 = x = 4. Each line of the solution is correct, but the whole statement claims that 13 = 8 = 4.
• “Simplify 3x + 6,” and the student writes 3x + 6 = 0, x = −2. The sign the student added turned the expression into an equation and the task into a different one.

There is one rule: in a transformation, every “=” states that the new expression equals the previous one for all values of the letters. So an operation that changes the value of the expression, such as multiplying everything by 6 or dropping a denominator, is forbidden in a transformation. In an equation the same operation is allowed, provided it is done to both sides.

Checking by substitution. Solved an equation? Substitute the answer and see whether the two sides are equal. This advice only makes sense to someone who understands “=” as “both sides are equal.” For everyone else there is nothing to check: the answer is already written down.

Identity and equation. “Prove that 2(x + 3) = 2x + 6,” and the student solves it like an equation: 2x − 2x = 6 − 6, 0 = 0… and has no idea what that result means. Nobody told the student that here the same sign states “true for every x.”

How serious this is was tested in 2006 with 177 students in grades 6–8 in the USA. Those who understood “=” as “both sides are the same” made up 31 to 43 % of each grade. And they solved the equation 4m + 10 = 70 more often; the link held even after adjusting for overall mathematics achievement.

What to do

The main thing is to announce the multiple meanings. Tell students directly: this sign has several jobs, and each time you need to work out which one it is doing right now. And come back to this every time the sign’s job changes.

Grades 1–2: “Are the left and right the same?”

• The answer on the left: 7 = 3 + 4. True? True: seven and three plus four are the same amount.
• Operations on both sides: 3 + 4 = 5 + 2. Let’s work out each side: 7 and 7.
• An equality with no operations: 12 = 12.
• A box and operations on both sides: 8 + 4 = □ + 5; 9 + 6 = □ + 10; 20 − 5 = □ − 3.
• True or false: 6 + 2 = 4 + 4; 5 + 3 = 8 + 1; 10 = 10 − 0.

Each time, one question: are the left and right the same? And a pan balance. A 2008 experiment: second and third graders taught with statements like 28 = 28 later solved problems like 3 + 9 + 5 = 6 + □ better than those taught with the familiar 15 + 13 = 28. In Davydov’s curriculum, first graders first balance lengths and masses on a scale, without numbers, and only then write the sign.

Our own students suggested this picture: “they weigh the same,” wrote a fourth grader.

Grades 3–4: chains. Find the error in 25 + 15 = 40 + 7 = 47 and write it correctly: 25 + 15 = 40, 40 + 7 = 47. Or 25 + 15 + 7 = 40 + 7 = 47, where every “=” stands between equal numbers.

Grade 5: two methods. Alongside the “through the components” method, teach doing the same thing to both sides. First with numbers: 8 + 4 = 7 + 5; subtract 5 from both sides and get 7 = 7, the equality still holds. Then with equations: 8 + 4 = x + 5 → 12 = x + 5 → 7 = x. Then in grade 7 the equation 3x + 5 = x + 13 is solved with the same move.

Grades 6–7: which job? Before each new kind of statement, ask: what is the sign doing here: stating, asking, always true, introducing notation? Put 2(x + 3) = 2x + 6 and 2(x + 3) = 14 side by side and discuss them. In transformations, check each step by substitution: at x = 1 the expression x/2 + x/3 equals 5/6, while 5x equals 5, so the step is wrong.

Computer science: name assignment. As an equality, x = x + 1 is absurd: a number cannot be one more than itself. As a command, it is simple. Niklaus Wirth, the creator of Pascal, called the “=” for assignment, inherited from Fortran, a well-known example of a bad idea. That is why Algol and Pascal write assignment as “:=”, and C and Python introduced “==” for comparison. A student who knows that “=” has several jobs can easily tell them apart.

A little history

Until the 16th century, equality was written as a word: aequales, faciunt, gleich. In 1557 the Welsh physician and mathematician Robert Recorde proposed two parallel lines of equal length, because, as he put it, no two things can be more equal. Recorde drew the equality of two sides: two identical lines for two identical sides. The sign took on its other jobs later.

A question for tomorrow

Give your class the problem 8 + 4 = □ + 5 and ask them to write what the “=” sign means. Even if everyone solves it correctly, the answers to the second question will show which students are ready for equations with x on both sides.

What the thread rests on

On the YourAIZone platform:

• what “=” means, the problem 8 + 4 = □ + 5 and the balance scale: the lesson “Comparing Numerical Expressions”
• doing the same thing to both sides and moving terms across the equals sign: the lesson “Solving Linear Equations”

Check data: 73 students in grades 2–6, six classes in Bishkek, October 2026. This is a teachers’ observation, not a statistical study.

Sources

1. Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232–236. https://academic.sun.ac.za/mathed/174/AlgebraNCTM.pdf
2. Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312. https://doi.org/10.2307/30034852
3. McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., & Krill, D. E. (2006). Middle-school students' understanding of the equal sign: The books they read can't help. Cognition and Instruction, 24(3), 367–385. https://doi.org/10.1207/s1532690xci2403_3
4. McNeil, N. M. (2008). Limitations to teaching children 2 + 2 = 4: Typical arithmetic problems can hinder learning of mathematical equivalence. Child Development, 79(5), 1524–1537. https://doi.org/10.1111/j.1467-8624.2008.01203.x
5. Venenciano, L., & Dougherty, B. (2014). Addressing priorities for elementary school mathematics. For the Learning of Mathematics, 34(1), 18–24. https://flm-journal.org
6. Wirth, N. (2006). Good ideas, through the looking glass. Computer (IEEE), 39(1), 28–39. https://doi.org/10.1109/MC.2006.20
7. Recorde, R. (1557). The Whetstone of Witte. London.
8. Kyrgyz Republic Cabinet Academy of Education (KAO). Subject standard “Mathematics”, grades 1–6 (2025): kao.kg; Subject standard “Mathematics”, grades 1–4 (2022): kao.kg; Subject standard “Mathematics”, grades 5–11 (2022): kao.kg; Subject standard “Computer Science”, grades 5–9 (2023): kao.kg