**The Overloaded Equals Sign: Equivalence**

In this series about mathematical notation and its impact on the learning of mathematics, the previous post started to look at the overloading of the equals sign – so many different meanings and uses cramped in this single notation “=”. Sometimes, like the “=” key on a calculator, the meaning is “do this now”, perform the calculation and get me a single number.

Other times, the “=” sign is used to indicate equivalence between two different ways of describing what is (in some sense) the same thing. If you compare with , you may agree that they look different, but that they have something very important in common. That is, by some measures they are different, by another measure they are the same. This phenomenon is known as *equivalence*. Equivalence is a very powerful and general idea. It is different from equality, in that two equivalent things need not be the same. They just need to match in the particular attribute that you care about. In textbooks, as well as in actual practice, the thing you care about is most often left implicit. When we say that and amount to the same thing, what we are focusing on is the values you get when the expression is evaluated for a particular choice of x, and then noticing that these values will be the same for both expressions regardless of what number we choose for x. By focusing on the values, and on nothing else, we say that and are the same. And yet they are not the same in other aspects. These other aspects may be considered trivial, but that doesn’t make them less real. For example, many teachers would not accept as a correct answer to a problem, saying that it should be simplified to . This is typical for the realm of equivalence: some things are more equal than others.

The particular kind of equivalence where we focus on the values when choosing a particular value for x, this kind of equivalence is often called *identity*. The standard mathematical notation for both equivalence and identity is “≡”, though ‘standard’ doesn’t mean universal. Most people and textbooks write . The idea is that both sides are equal, regardless of the value of x. The part of “regardless of the value of x” is not made explicit.

The use of “=” for identity is often confused by students with the use of “=” for equations. When we write , we are not saying or suggesting that both sides are equal regardless of the value of x. We’re saying almost exactly the opposite: we’re saying that by telling you that , we’ve given you a powerful hint or clue as to what value x should have. Students often rely on verbal clues like “simplify” or “expand” or “evaluate” or “prove” or “solve” to figure out what they are supposed to do. Teachers’ consistent use of “≡” for identity, even if the student is never asked to do the same, and even if the textbook doesn’t play along, is still a powerful tool for students to become aware of the different meanings for “=” they see in their textbooks.

Hence **I propose** that teachers use “≡” instead of “=” whenever the intended meaning is that the two sides are identical, that is, they evaluate to the same value whatever is chosen as the value for x. In my experience, both students and teachers find this helpful.

Hi Bert,

I have to agree with all your observations regarding the equals sign. I’ve often attributed the confusion you point out to students writing in a sort of “stream of consciousness” style – chaining together expressions using the equals sign as a sort of conjunction. I think you are right that many people think in terms of the calculator’s =, which allows you to just keep going – whatever happened on the left hand side of the = is just left behind as we push forward with our calculation.

Is the answer to introduce new notation, or is it better, just as in natural language, to continue to allow the overloading, but make both students and teachers more aware of how the meaning of the = changes in context? I think it is valid to try to include cues so that students can recognize context better, but I don’t think we’ll be completely successful in eliminating confusion through additional notation. I suppose there is a balance to be sought between recognizing context and eliminating ambiguity.

Mathematical notation, like all conventions, does change over time, and we may see some ideas like this become more standard, particularly as more people are exposed to programming languages (and the use of equals as assignment and identity) in school.

— Dan

Dan,

In this series I’m looking at things an individual teacher can do – without any support from textbooks or other teachers – to deal with ambiguity and confusion among students. Notational issues alone will never solve or sidestep all issues students have. Yet notation does have power.

From my experience, carefully considered notation helps the teacher as much as the student. It is often the teacher who misses the possible confusions and doesn’t see the ambiguity that the students are coping with. So yes, it’s precisely so that both students and teachers are more aware of how the meaning of things changes in context that I’m writing this series. It is addressed to teachers, more than to students.

In half a century of rethinking of mathematics in the field of computers and computer languages, almost nothing of that huge effort has penetrated or informed math education. It will take more than my little series to impact that!

Bert

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You write: ‘The part of “regardless of the value of x” is not made explicit.’ Why not, then, make it explicit by writing “for any x, x+1+1 = x+2”?