Jimmy Discovers That the Equals Sign Does Not Mean “Here Comes the Answer”

I wrote a simple problem on the board:

8 + 4 = □ + 5

“What number belongs in the box?” I asked.

“Seven,” said several students in unison.

“Twelve,” said several others.

Jimmy said nothing. He was staring at the equals sign.

“Why did some of you choose twelve?” I asked.

“Because eight plus four equals twelve,” Nick replied.

“That calculation is correct. But let us put twelve into the box.”

I wrote:

8 + 4 = 12 + 5

Then I simplified both sides:

12 = 17

“That isn’t right,” I pointed out.

“But twelve is the answer to eight plus four,” Nick protested.

“Yes, but the box is not asking only for the answer to eight plus four. We need a number that makes both sides equal.”

I replaced twelve with seven:

8 + 4 = 7 + 5

12 = 12

Jimmy finally raised his hand.

“Sir, what if the students think the equals sign means ‘write the answer now’?”

That was a much better question than it initially appeared.

Does equals mean “the answer is”?

Many students first encounter the equals sign in calculations such as:

8 + 4 = 12

Everything to the left is the calculation. Everything to the right is the answer.

After hundreds of examples in that form, it is understandable that they begin to read the equals sign as an instruction:

Work out everything before me and write the answer after me.

“But that isn’t what it really means?” Jimmy asked.

“No. It states that the expressions on its two sides have the same value.”

“So could I write this?”

He went to the board and reversed the original equation:

12 = 8 + 4

“Yes,” I nodded.

Several students looked uncomfortable.

“That looks wrong,” Sarah said. “The answer is on the wrong side.”

Jimmy turned towards me.

“Is there an answer side?”

“No. Equality works in both directions. If one expression equals another, we may reverse them.”

Therefore:

8 + 4 = 12

and:

12 = 8 + 4

are equally correct.

Jimmy examined the two statements.

“Then why do we almost always put the calculation first?”

“Because it follows the way we usually think. We begin with a calculation and arrive at its value.”

“So it is a habit, not a rule?” Jimmy asked.

“Exactly.”

Jimmy had found the first crack in a symbol everyone thought they understood.

Which side contains the answer?

I wrote:

3 + 4 = 10 − 3

“Which side is the calculation,” Jimmy asked, “and which side is the answer?”

“Both sides are calculations.”

“And they both give seven?”

“Yes.”

“So the equals sign is not pointing from a question to an answer?”

“No. It is stating that both expressions represent the same value.”

I could have written:

7 = 3 + 4

or:

10 − 3 = 7

or:

3 + 4 = 10 − 3

Each statement expresses an equality.

Jimmy looked thoughtful.

“If the sign means that the two sides have the same value, why does a calculator use it to make the answer appear?”

“We will come to that.”

“I was afraid you might say that.”

The line of working that claims seven equals fourteen

I showed the class a common piece of mathematical working:

3 + 4 = 7 × 2 = 14 − 5 = 9

“Is this correct?” I asked.

Several students nodded.

Each individual calculation seemed reasonable:

• Three plus four is seven.

• Seven multiplied by two is fourteen.

• Fourteen minus five is nine.

Jimmy studied the complete line.

“Does it say that everything between the equals signs has the same value?”

“Yes.”

“Then doesn’t it say that seven equals fourteen?”

“It does.”

“And that fourteen equals nine?”

“Yes.”

“But the student got the correct answer at every step.”

“The arithmetic was correct. The written chain of equalities was not.”

I rewrote the calculations on separate lines:

3 + 4 = 7

7 × 2 = 14

14 − 5 = 9

Each line was now a true statement.

Jimmy frowned.

“How can someone do every calculation correctly and still write something mathematically false?”

“Because mathematical notation does more than record the final numbers. It also states the relationships between them.”

“Then the equals signs were lying, even though the numbers were right?”

“The signs were not lying. The student was asking them to say something untrue.”

That answer appeared to satisfy everyone except Jimmy.

When a chain of equals signs is correct

“Can we ever use several equals signs on the same line?” he asked.

“Certainly.”

I wrote:

7 = 14 ÷ 2 = 10 − 3 = 3 + 4

Every expression has the value, seven.

Therefore, the entire chain is true.

Jimmy pointed to the first and last expressions.

“Does that mean seven equals three plus four?”

“Yes.”

“And fourteen divided by two equals ten minus three?”

“Yes.”

“And each one equals every other one?”

“Yes.”

“So a chain works only if we could compare any two parts and still get a true statement?”

“That is an excellent test.”

He nodded slowly.

The equals sign was no longer marking the end of a calculation. It was joining several different expressions that all described the same number.

An equation is not a question

I wrote:

x + 3 = 7

“What does this tell us?” I asked.

“That we have to find

,” a student replied.

Jimmy raised his hand.

“Does the equals sign actually tell us to find it?”

“No. The instruction to solve the equation comes from the question surrounding it.”

“Then what does the equation itself say?”

“It says that x + 3 and 7 have the same value.”

“And that is true only when

?” Jimmy concluded.

“Yes.”

For other values of

, the statement is false.

I wrote another statement beneath the first equation:

x + 0 = x

“Do we solve this one as well?” Jimmy asked.

“Yes. What value of x makes it true?” I asked.

He looked at the equation.

“Every value.”

“Exactly. It is true for every value of

, so we call it an identity.”

“Then x + 3 = 7 is true only when

, but this one is always true?”

“Yes.”

“And they both use exactly the same equals sign?”

“Yes.”

“How are students supposed to know the difference?”

“By examining the statement.”

“That sounds suspiciously like ‘from the context’ again.”

It was becoming increasingly difficult to argue with him.

Can equals mean “is defined to be”?

I wrote:

f(x) = 2x + 1

Jimmy immediately raised his hand.

“Did we discover that those two things are equal, or did we decide that f(x) would mean

?”

“In this case, we are defining the function.”

“So does the equals sign now mean ‘is defined to be’?”

“In this context, yes.”

A more specialised notation sometimes used for definitions is:

f(x) := 2x + 1

The colon indicates that the expression on the left is being defined by the one on the right.

However, many textbooks simply use:

f(x) = 2x + 1

and expect the reader to recognise from the context that this is a definition.

Jimmy began counting on his fingers.

“Can the equals sign can appear in a calculation, an equation, an identity and a definition?”

“Yes, although its central purpose is still to connect expressions being treated as equal.”

“And we know exactly what it is doing by — ” I began.

“Context,” Jimmy interrupted.

“I knew you were going to say that,” I replied.

The calculator changes the meaning

Jimmy picked up his calculator and entered:

8 + 4

Then he pressed the equals key.

The number 12 appeared.

“What did the equals key do?” he asked.

“It told the calculator to evaluate the expression.”

“Then on a calculator, doesn’t equals mean ‘give me the answer now’?”

“As a button, it acts more like a command.”

“So the calculator uses the symbol in exactly the way teachers tell students not to?”

“It uses the same symbol for a related purpose,” I explained.

“But it doesn’t show two equal sides.”

“No.”

“It just performs the calculation.”

“Yes.”

He held up the calculator as though it were evidence in a court case.

“Then perhaps calculators helped create the misunderstanding.”

He may have been right.

A student repeatedly presses:

8 + 4 =

and sees:

12

The equals button appears to transform an unfinished calculation into a finished answer. That reinforces the belief that the symbol means:

Here comes the answer.

But in written mathematics:

8 + 4 = 12

is not an instruction followed by its result. It is a statement that the two sides have equal value.

Spreadsheets make matters worse

I opened a spreadsheet and entered:

= 8 + 4

The cell displayed 12.

Jimmy stared at the formula.

“Why is the equals sign at the beginning?”

“It tells the spreadsheet that what follows is a formula.”

“But there is nothing on its left.”

“No, there isn’t,” I replied.

“Then it cannot be saying that two sides are equal.”

“Not in the usual mathematical sense. Here, it signals that the program should calculate the expression.”

“In a spreadsheet, can equals mean ‘a formula begins here’?”

“Yes.”

“And on a calculator, it means ‘perform the calculation now’?”

“Yes.”

“But in mathematics, it means ‘the two sides have the same value’?”

“Yes.”

Jimmy looked around the room.

“And adults wonder why students become confused.”

Computers give the sign another job

The situation becomes stranger in computer programming.

In many programming languages, a single equals sign assigns a value:

x = 5

This may mean:

Store the value 5 in the variable called x.

To test whether x already equals 5, the language may use two equals signs:

x == 5

Jimmy compared the two expressions.

“In mathematics, x = 5 states that x and 5 are equal?”

“Yes.”

“But in a computer program, it can order x to take the value 5?”

“Yes.”

“And if the computer wants to ask whether they are equal, it may need two equals signs?”

“Correct.”

“So one equals sign makes it happen, while two equals signs ask whether it has happened?”

“In many programming languages, that is a reasonable way of describing it.”

“Does every programming language do that?”

“No.”

“Of course not,” Jimmy stated with frustration.

By now, Jimmy had learnt that whenever notation appeared straightforward, another convention was waiting around the corner.

What the symbol really says

We returned to the original problem:

8 + 4 = □ + 5

The correct number was 7:

8 + 4 = 7 + 5

12 = 12

The arithmetic was elementary. The meaning of the symbol was not.

In mathematics, the equals sign does not inherently mean:

• “Write the answer.”

• “Continue the calculation.”

• “Everything moves from left to right.”

• “Solve this equation.”

Its fundamental message is:

The expression on the left has the same value as the expression on the right.

That is why:

3 + 4 = 7

and:

7 = 3 + 4

say the same thing.

It is why:

3 + 4 = 10 − 3

is correct.

It is also why:

3 + 4 = 7 × 2

is false, even though both calculations might form consecutive steps in someone’s thinking.

Jimmy looked again at the chain:

7 = 14 ÷ 2 = 10 − 3 = 3 + 4

“Then the equals sign doesn’t finish a calculation?” he asked.

“No.”

“It connects different ways of describing the same value?”

“Yes.”

For once, Jimmy had reached the conclusion without creating another difficulty.

I should have known it would not last.

He returned to his seat, then immediately raised his hand.

“Sir, when you write ‘Class = quiet’ on the board, are you describing a fact, defining the class, or giving us a computer command?”

The room erupted in laughter.

I erased the equals sign and replaced it with an exclamation mark.

Some mathematical ambiguities are best resolved by changing the notation.