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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.

Tommi
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Vince Vickler
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    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. – Matthew Daly Jan 06 '22 at 22:12
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    In your experience, do English speaking students easily admit that equality is identity? – Vince Vickler Jan 06 '22 at 22:19
  • I mean , is the problem I point out a well identfied problem in mathematcal teaching elsewhere than in France? – Vince Vickler Jan 06 '22 at 22:20
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    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). – Matthew Daly Jan 06 '22 at 22:21
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    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". – Vercassivelaunos Jan 06 '22 at 22:57
  • If American teachers do settle this with their students, it is before high school (and you'd need a primary teacher to tell you what that lesson is like). In the first year of algebra, I build on substitution of equivalent expressions when solving systems of equations. But they grasp it well with at most a little one-on-one time. – Matthew Daly Jan 06 '22 at 22:58
  • I have seen this plenty in college students. – Sue VanHattum Jan 07 '22 at 07:46
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    I don't think there's anything peculiarly French about your situation. See https://matheducators.stackexchange.com/questions/7964/issues-with-equals-where-does-this-come-from-and-how-do-i-combat-it . Slightly worse maybe. I wouldnt know... – Rusi Jan 07 '22 at 09:23
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    Ontology ontologically precedes epistemology. But epistemology psychologically precedes ontology. IOW for young 'uns there is at most the vaguest "is" but a strong sense of "do" backed on an even stronger base of feelings, sensations -- «Why is maman leaving me in the dark??» «I'm hungry!» etc The ultimate business of the math-teacher is not math edu per se but to convey a feel for an outlook independent of changes in space and time... Aka the Platonic world. And the cognition of identity is key to that world. Dunno that there's any short road. It's nevertheless a primary task – Rusi Jan 07 '22 at 09:40
  • I draw ur attention particularly to this comment – Rusi Jan 07 '22 at 10:00
  • In many programming languages, for example those of the C family, the symbol $=$ denotes asignment. The expression $a = b$ is nonsymmetric, as it assigns to the variable $a$ the value of the variable $b$, and this value is the value of the expression. Such languages use $==$ to check equality (so $a == b$ returns a $0$ or $1$). This is just to say that what students assume about $=$ is so natural that it's built into some programming languages. Mathematicians use $=$ in the same way when they use $A = B$ to mean $A$ is defined to equal $B$. – Dan Fox Jan 13 '22 at 17:29
  • "How to convince a high school student that the = symbol denotes identity?"

    • $\quad$ The word ‘identity’ has multiple meanings in mathematics and formal logic, and while your usage is not wrong, it is probably better to call the = in $x+2=7$ ‘equality’ and the =/≡ in $x^2-y^2=(x+y)(x-y)$ and $x^2-y^2≡(x+y)(x-y)$ ‘identity’.

    – ryang Aug 05 '22 at 18:33
  • "a belief according to which the $ \Large= $ symbol expresses some sort of action performed by the person that operates the mental computation $\Large +$ on the numbers"

    • $\quad$ The “action”/operation interpretation of = actually has an alternative sense, which turns out to be more correct: instead of the input being its LHS and the output being its RHS, here the inputs are both its LHS and RHS and the output is True/False. This interpretation, aptly, circles back to the ‘equality’ meaning of = (above).

    – ryang Aug 05 '22 at 18:34