The equals sign is a sophisticated, subtle tool. You may not think of it as such, but it is deep. It is hard to learn and use and understand.

I first understood the challenge while teaching a calculus class. I had two expressions, something like (x – 2)(x + 3)(x – 7) and (x – 2)^3 – x + 50, and our goal was to show that they were equal. I took one, and I simplified it to x^3 – 6x^2 – 13x + 42, a chain of equals signs spread across the blackboard. I took the other, and simplified it to the same thing. So I proudly declared that they were equal, and thanks to the magic of my boardwork, the two identical expressions had ended up next to each other. I drew an equals sign between them, creating a giant chain of equals signs connecting the two original expressions. I thought this was a very clever presentation.

They didn’t get it.

I’d heard before that this might be a challenge. They’d all been drilled in elementary school on problems like 5 + 2 = ?. They thought of equals as meaning the outcome of an operation, which is why they felt comfortable using it sloppily. (You know what this looks like: asked to compute “five squared plus two”, they write 5^2 = 25 + 2 = 27.)

I thought, then, that it was time to correct this misunderstanding. So I speechified at length about how they had been tragically mis-taught about the equals sign. It actually means that two things are the same thing! I pointed to (x – 2)(x + 3)(x – 7) = (x – 2)(x^2 – 4x – 21) and said “these two things are the same thing!” Then I pointed to (x – 2)(x^2 – 4x – 21) = x(x^2 – 4x – 21) – 2(x^2 – 4x – 21) and proclaimed, “these two things are the same thing too!” Since the first expression was the same as the second, and the second was the same as the third, the first and third must be the same, and a pointed and said it. I went on, through all the equals signs, until I had discovered that the two things on opposite sides of equals signs were the same thing. I discussed why I didn’t like the way they did a problem like “five squares plus two”. I was proud of my improvisation, but somehow, it didn’t seem like the students had the great moment of realization (and repentance for their mathematical sins!) that I wanted.

I say it again: the equals sign is a sophisticated, subtle tool. Despite my oratorical skill, I had failed to bring together a deeper understanding of all the possible uses of equals. There are at least four:

- As the outcome of an operation (used in school, but mathematically the wrong way to look at it).
- To declare that two things are equal all the time, as in 4(x + 2) = 4x + 8. This is an
**equvalence**. There is a hidden universal quantifier. - To declare that two things will be equal for the right value of x, as in 4x + 2 = 3x + 9. (This is disambiguated from the previous case by explicitly saying, “Find x so that…” To a mathematically sophisticated reader, it is clear that this is the same = sign, just with an existential quantifier first. This is not clear to someone just learning mathematics.)
- As a definition, such as f(x) = 3x^2 – 4x.

Without any explicit guidance, we expect students to recognize these different situations. To recognize quantifiers and definitions without any discussion of what “equals” really means. All this while learning how to “solve” these problems. Also without discussions of the other subtleties of equals: symmetry, transitivity, reflexivity.

Embedded here is also the understanding that two things can look completely different and can be the same thing, can be equal. 30/3 and 10 are different representations of the same number. 7 + 3 and 5 + 5 are all different representations of that same number. All of these things are equal! They are also equal to 2x when x = 5. The mathematical symbols alone are not enough; the words nearby must be used to interpret them.

Worse, uses of the equals sign get conflated in different settings. For example, suppose that you are trying to find an x so that 4x + 2 = (x + 2)^2. You might write that (x + 2)^2 = x^2 + 4x + 4. One of your equals signs is only true for a few values of x; the other equals sign is true for all values of x, all in the same math problem. Imagine how confusing it would be if you wrote 4x + 2 = (x + 2)^2 = x^2 + 4x + 4!

The same thing can happen when you are defining a function. You might say “Let f(x) = (x + 2)^2 = x^2 + 4x + 4”. In one line, two different uses of the equals sign.

To me personally, these uses of = are all the same thing, stemming from the same definition. To a student just learning algebra, however, there are layers upon layers of subtlety, and they must be addressed explicitly or students will not truly understand the mathematics they are doing.

Dan,

This is a fantastic post. I think many math educators (myself included) use the equals sign flexibly without always explicitly discussing with students the subtle differences you point out.

You’ve very clearly explicated the problem; in terms of the solution, I wonder how we can draw students in so they can be more active in complicating their understanding of notation. For example I could imagine an activity presenting students with a variety of math statements and asking them to color-code the equals signs based on your bullet points above (perhaps consolidating the first two).

Best,

Tom

Color coding the different meanings makes sense! I also think you can introduce the different meanings one at a time, and explicitly discuss each use when it comes in and discuss how the surrounding language (“find x so that…”) causes the language to make sense. You might also give students a dialogue between two characters who are debating if two things are equal, have them read it aloud as characters, and then continue the discussion themselves and come to a decision as a class. Finally, I would return to simple examples. For example, maybe you might give students a bunch of numbers and ask them to break them into groups that are all equal, which subtly emphasizes the transitive property of equality.

Part of the problem is that we never get students as a blank slate (unless perhaps we are early elementary school teachers). My students always have a preconceived idea of what “equals” means (even if it is a fuzzy and confused one) coming from years of prior schooling, so I don’t see how I could introduce the different meanings one at a time as if “equals” were a new concept they hadn’t seen before.

Yes, fair enough. I agree that my comment was aimed largely at teachers who are first introducing algebra, in order to make the transition from “outcome of an operation” to “is the same as”.

That said, even in my college classes, I’ve started off discussions by asking, “what does equals mean?” and following it from there.

In some parts of mathematics, notably formal logic and type theory and the computerization of mathematics, these different meanings of equals are in fact notated differently as a matter of course.

The middle two (“equal for all x” and “equal for some x”) are, as you say, just caused by an omitted quantifier. (I think the omission of quantifiers is the cause of a lot more confusion than this in learning mathematics.)

The first one (“the outcome of an operation”) is, roughly, what is called “reduction” and sometimes denoted with an arrow or squiggly arrow, perhaps subscripted to denote the sort of reduction.

The fourth (“as a definition”) is called “definitional equality”, and is sometimes denoted by a colon-equals or a triple-equals.

Note that the first and fourth meanings are directional, while the second and third are not (one reason to prefer colon-equals over triple-equals). Since the expansion of definitions is part of evaluating the outcome of an operation, and sometimes even all of it (particularly if you include the defining equations of recursive functions as “definitions”), the phrase “definitional equality” (and sometimes “judgmental equality”) is also used for the reflexive-transitive-symmetric closure of reduction.

In particular, I don’t agree that these uses of = are “all the same thing”. The common use of the same symbol for them, in addition to causing confusion among students, may lead many working mathematicians to

thinkthat they are all the same — which, in addition to perpetuating the problem withtheirstudents, causes difficulty for them when they try to learn type theory!Fair enough. The reason I think of the definitional use of equals as the same as the others is that it is also usually preceeded by explanatory text in well-written mathematics (“Let f(x) = x^2 + 2”, for example) and once the definition is made, the typical use of = applies. f(x) and x^2 + 2 are now indeed the exact same thing.

Nonetheless, a different symbol would be valuable. I’ve seen := used even by non-logicians, and it makes the exposition clearer.

Yes, of course once the definition is made, a definitional equality is an ordinary one. (Although in type theory, even then it’s not quite the same as a propositional equality, but that’s a peculiarity of type theory.) But the point is exactly that: a definitional equality is more than an ordinary equality

becauseit is the process of making a definition. In particular, it always simultaneously introduces a new symbol for the object being defined (here “f”).There was a similar post about equals at “Mathematics for Teaching”. Now they have a great post on variables that deserves mention: http://math4teaching.com/2013/08/25/the-multiple-meanings-of-letter-symbols-part-2-of-x/

Now to call out another great blog about this issue: https://betterqs.wordpress.com/2015/10/18/what-is-the-meaning-of-the-equal-sign/

What I love about this blog entry is that it provides some specific questions with which to engage students.

What I’d really like to see is a discussion of how to fix the problem. Starting points to engage students are good, and ideas for things to tell them once they are engaged (like your breaking down of equality into at least four kinds) are also good. But quite often it seems that no amount of

talkingat students can break through their ingrained misconceptions. What can we have themdothat will help them develop the correct intuitions?