# How to convince a high school student that the $=$ symbol denotes identity?

In French language, arithmetic statements are often read, at the elementary school level, as , say, " deux et deux font quatre" , i.e. something like " two and two make four".

Out of this arises a belief according to which the $$\Large=$$ symbol expresses some sort of action , either an action performed by numbers themselves or by the person that operates the mental activity of computation which is supposed to be denoted by the $$\Large +$$ sign.

This first belief may, in the head of older students, be replaced by the idea that $$\Large =$$ means " has the same magnitude " or " has the same value as".

I tried to show to high school students that the supposedly active meaning of $$\Large =$$ does not work anymore when the equality is reversed : $$2+2$$ may ( arguably) " make" $$4$$ , but would one say that $$4$$ " makes " $$2+2$$ ?

But I did no manage to convince them that, at least in the case of arithmetic statements, the " has the same magnitude " interpretation is not correct.

The identity meaning seems simply unbelievable to students.

• As charming as it is, the French language is weird. One says "Jean has 16 years" to talk about age and "It is doing cold" to talk about weather. This can't be the first time your students have considered action verbs that are idiomatically used to describe the state of an object. Jan 6, 2022 at 22:12
• In your experience, do English speaking students easily admit that equality is identity? Jan 6, 2022 at 22:19
• That said, I appreciate your students' perspective that equality is a relation that describes when two arithmetic expressions have the same magnitude. $2\times2=3+1$ is a true mathematical statement that doesn't have an interpretation under your sense of identity (at least, if I understand your position correctly). Jan 6, 2022 at 22:21
• Is it really that your students aren't believing you when you say what the sign means, or is it that you can't get it to stick in their minds? Because in the latter case, I might try overdoing it with the language: change whatever you usually say in place of the sign (like "equals" in English) by some more verbose and explicit version, like "is the same number as". Jan 6, 2022 at 22:57
• I don't think there's anything peculiarly French about your situation. See matheducators.stackexchange.com/questions/7964/… . Slightly worse maybe. I wouldnt know...
– Rusi
Jan 7, 2022 at 9:23

Although some of the trouble may be language dependent, there is also the issue (here in the U.S., too, in English) that young students get the feeling, perhaps from teachers, even, that = is like saying "and here's what comes next".

I have seen research on this issue. I can't find exactly what I was looking for, but did find this:

"To put it simply: Some students ... use the equals sign (=) as a symbol for the word "then" or the phrase "the next step is." For instance, when asked to find the third derivative of x4+7x2–5, some students will write "x4+7x2–5 = 4x3+14x = 12x2+14 = 24x." Of course, those four expressions are not actually equal to one another.

"A slight variant of this error consists of connecting several different equations with equal signs, where the intermediate equals signs are intended to convey "equivalent to" --- for example, x = y – 3 = x+3 = y. This is very confusing and altogether wrong, because equality is transitive --- i.e., if a=b and b=c then a=c, but x certainly is not equal to x+3. It would be better to replace that middle equals sign with some other symbol. The most obvious symbol for this purpose is ≡, which means "is equivalent to," but that symbol has the disadvantage of looking too much like an equals sign, and thus possibly leading to the same confusion. Thus, a better choice would be ↔ or ⇔, both of which mean "if and only if." Thus, I would rewrite the example above as x = y – 3 ⇔ x+3 = y."

The author called this 'stream of consciousness' notation.

I have seen both of these issues, and I just keep reminding students that they are writing something incorrect.