# Equality as “makes” vs equality as “equals”

A problem I often encounter while introducing students to equations is that of changing the conceptual image of the equation symbol $$=$$ from "results to" to "is equal to". To be more precise:

1. In the expression $$2+3=5$$, equality is often conceptualized by students as a predicate that denotes that the quantity on the LHS results to the quantity on the RHS. So, under this view, the reflexive property of equality as an equivalence relation is ignored.
2. In the expression $$2x+4=5-3$$, equality is/should be concpetualized as "the LHS is equal to the LHS and vice versa" - more or less in a sense similar to that of "balancing" on the two sides of a libra.

The first notion of equality is often and strongly interferring with the second one, something which could be realized e.g. due to students having dfficulties in recognizing $$x=3$$ and $$3=x$$ as equaly valid mathematical expressions of the same thing.

I do realize that this problem comes alongside with the introduction of mathematic notation that is structurally different from the way natural language is structured. Namely, a typical sentece has the form: $$\text{Subject - Verb - Object}$$ where the object usually is the receiver of the subject's action(s) or their result(s). So, $$x=3$$ is naturally translated to "(The subject) $$x$$ equals (the object) $$3$$". However, it would seem odd to translate $$3=x$$ to "(The subject) $$3$$ equals (the object) $$x$$".

So, do you have any suggestions/activities that would help students nurture themselves to equality as an equivalence relation against maintaining the false image of equality as a "results" process?

• Instead of having them try to decipher these equations, why not provide the context in plain English? – Peter Saveliev Dec 23 '19 at 23:04
• I meant that one would tell the students either: "Simplify the expression 2+3" or: "Solve the equation 2x+4=5-3". – Peter Saveliev Dec 23 '19 at 23:21
• In appropriate circs, I read $=$ as something like “results in” and see no problem with the students doing it, too. The problem is that the students the OP describes have ingrained in their minds that it is the only reading of $=$. This causes trouble in other contexts. Extending the meaning of a word to other ones in different contexts is a common cognitive problem. The OP seeks advice on how to do it for $=$. The previous comments do not seem to read the OP’s question in the same way. – user1027 Dec 24 '19 at 0:09
• I'm just nitpicking here, but technically I am not so sure about the way you are identifying subject and object in those sentences. I am not a grammar expert (nor a mother-tongue English speaker), but I think that "equals" is a copula and thus "3" in your first example is a subject complement, not an object. – Federico Poloni Dec 24 '19 at 18:18
• @XanderHenderson Subject/Object implies an asymmetry, which is a lot less pronounced with copulas. I am not so sure that "equals" in this construction can be considered a transitive verb, but you are right that it would make a good question for Linguistics or English Language & Usage. – Federico Poloni Dec 24 '19 at 19:46

Grammatically, I suspect that part of the problem is that the verbs "to make" and "to equal" are transitive verbs (they take a direct object). Thus there is real distinction in the grammatical structure between the left-hand side and the right-hand side of an equation when read aloud. To see the effect of this, consider what happens when you elide the direct object (the right-hand side of an equation):

Two plus three makes.

Makes what? What does two plus three make? Where is the end of the sentence? If you replace "makes" with "equals", you have the same problem—the sentence requires a direct object in order make sense.

Hence I would suggest using a entirely different grammatical structure.For example, use "is" as a synonym for "equals". Note that the following is (in fact) a complete sentence:

Two plus three is.

Here, "two plus three" is an object which exists, on its own. The sentence makes sense without a direct object—indeed, the verb "to be" does not take a direct object at all. The sentence simply declares that this thing, two plus three, exists. The notation $$2+3 = 5$$ may be expanded to "two plus three is five", which provides additional information about the nature of two plus three. Grammatically, the word "five" here is a predicate noun, which means that it falls into a somewhat distinct grammatical category from a direct object (the predicate noun is not being acted upon, but rather completes the verb phrase beginning with "is"). You can add verbosity for clarity:

Two plus three is exactly the same as five.

• Well, that was a nice view on the topic. Would you consider meaningful extending the former trick by also using expressions much like "five is two plus three", so as to highlight the symmetry included in the $=$ symbol? – Βασίλης Μάρκος Dec 24 '19 at 16:16
• @ΒασίληςΜάρκος Absolutely. If two plus three is five, then five is two plus three. They are one and the same. – Xander Henderson Dec 24 '19 at 16:34
• See my comment to the question about the grammar here. – Federico Poloni Dec 24 '19 at 18:18
• @FedericoPoloni I fail to see how this is relevant to the discussion of mathematical pedagogy here, but I have added clarification. – Xander Henderson Dec 24 '19 at 19:42
• Closely related: this question on math stackexchange, math.stackexchange.com/questions/2738360/… and my answer math.stackexchange.com/questions/2738360/… – Ethan Bolker Dec 25 '19 at 21:53