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.