# The Evolution of Equals

High school math teachers spend a lot of time explaining to kids that all the things we told you before ain’t necessarily so. Turns out, for example, you can subtract a big number from a smaller one.  Fractions might be “improper” if the numerator is larger than the denominator, but they’re completely rational so long as both are integers. You can take a square root of a negative number.  And so on.

Other times, though, we have to deal with ambiguities that mathematicians yell at us about later. Which really isn’t fair. For example, consider the definition of variable and then tell me how to explain y=mx+b. Or function notation–if f(x) = 3x + 7,  and f(3) = 16, then what is f(a)? Answer: f(a) = 3a+7. What’s g(x)? Answer: A whole different function. So then you introduce “indeterminate”–just barely–and it takes a whole blog post to explain function notation.

Some math teachers don’t bother to explain this in class, much less in blogs. Books rarely deal with these confusing distinctions. But me, I soldier on. Solder? Which?

Did you ever think to wonder who invented the equal sign? I’m here to wonder for you:

Robert Recorde, a Welsh mathematician, created the equal sign while writing the wonderfully named Whetstone of Witte. He needed a shortcut. “However, for easy manipulation of equations, I will present a few examples in order that the extraction of roots may be more readily done. And to avoid the tedious repetition of these words “is equal to”, I will substitute, as I often do when working, a pair of parallels or twin lines of the same length, thus: = , because no two things can be more equal.”

First of his examples was: or 14x+15=71.

Over time, we shortened his shortcut.

Every so often, you read of a mathematician hyperventilating that our elementary school children are being fed a false concept of “equals”. Worksheets like this one, the complaint goes, are warping the children’s minds: I’m not terribly fussed. Yes, this worksheet from EngageNY is better. Yes, ideally, worksheets shouldn’t inadvertently give kids the idea that an equals signs means “do this operation and provide a number”. But it’s not a huge deal. Overteaching the issue in elementary school would be a bad idea.

Hung Hsi Wu, a Berkeley math professor who has spent a decade or more worrying about elementary school teachers and their math abilities, first got me thinking about the equals sign: I don’t think this is a fit topic for elementary school teachers, much less students. Simply advising them to use multiple formats is sufficient. But reading and thinking about the equals sign has given me a way to….evolve, if you will…my students’ conception of the equals sign.  And my own.

Reminder: I’m not a mathematician. I barely faked my way through one college math course over thirty years ago. But I’ve found that a few explanations have gone a long way to giving my students a clearer idea of the varied tasks the equals sign has. Later on, real mathematicians can polish it up.

### Define Current Understanding

First, I help them mentally define the concept of “equals” as they currently understand it. At some point early on in my algebra 2 class, I ask them what “equals” means, and let them have at it for a while. I’ll get offerings like “are the same” and “have the same value”, sometimes “congruent”.

After they chew on the offerings and realize their somewhat circular nature, I write:

8=5+2+1

8=7

and ask them if these equations are both valid uses of the equal signs.

This always catches their interest. Is it against the law to write a false equation using an equals sign? Is it like dividing by 0?

Ah, I usually respond, so one of these is false? Indeed. Doesn’t that mean that equations with an equals sign aren’t always true? So what makes the second one false?

I push and prod until I get someone to say mention counting or distance or something physical.

At this point, I give them the definition that they can already mentally see:

Two values are equal if they occupy the same point on a number line.

So if I write 8=4*2, it’s a true equation if  8 and 4*2 are found at the same point on the number line. If they aren’t, then it’s a false equation, at least in the context of standard arithmetic.

So if the students think “equals” means “do something”, this helps them out of that mold.

Then I’ll write something like this:

4x-7=2(2x+5)

Then we solve it down to:

0=17

By algebra 2, most students are familiar with this format. “No solution!”

I ask how they know there’s no solution, and wait for them all to get past “because someone told me”. Eventually, someone will point out that zero doesn’t in fact, equal 17.

So, I point out, we start with an equation that looks reasonable, but after applying the properties of equality, otherwise known as “doing the same thing to both sides”, we learn that the algebra leads to a false equation. In fact, I point out, we can even see it earlier in the process when we get to this point:

4x = 4x+17

This can’t possibly be true, even if it’s  not as instantly obvious as 0=17.

So I give them the new, expanded definition. Algebraic equations aren’t statements of fact. They are questions.

4x-7=2(2x+5) is not a statement of fact, but rather a question.

What value(s) of x will make this equation true?

• x= specific value(s)
• no value of x makes this true
• all values of x makes this true.

We can also define our question in such a way that we constrain the set of numbers from which we find an answer. That’s why, I tell them, they’ll be learning to say “no real solutions” when solving parabolas, rather than “no solution”. All parabolas have solutions, but not all have real solutions.

This sets me up very nicely for a dive back into linear systems, quadratics with complex solutions, and so on. The students are now primed to understand that an equation is not a statement of fact, that solutions aren’t a given, and that they can translate different outcomes into a verbal description of the solution.

An identity equation is one that is true for all values of x. In trigonometry, students are asked to prove many trigonometric identities,, and often find the constraints confusing. You can’t prove identities using the properties of equality. So in these classes,  I go through the previous material and then focus in on this next evolution.

Prove: tan2(x) + 1 = sec2(x)

(Or, if you’re not familiar with trig, an easier proof is:

Prove: (x-y)2 = x2-2xy+y2

Here, again, the “equals” sign and the statement represent a question, not a statement of fact. But the question is different. In algebraic equations, we hypothesize that the expressions are equal and proceed to identify a specific value of x unless we determine there isn’t one. In that pursuit,  we can use the properties of equality–again, known as “doing the same thing to both sides”.

But  in this case, the question is: are these expressions equal for all values of x?

Different question.

We can’t assume equality when working a proof. That means we can’t “do the same thing to both sides” to establish equality. Which means they can’t add, subtract, square, or do other arithmetic operations. They can combine fractions, expand binomials, use equivalent expressions, multiply by 1 in various forms. The goal is to transform one side and prove that  both sides of the equation occupy the same point on a number line regardless of the value of x.

So students have a framework. These proofs aren’t systems. They can’t assume equality. They can only (as we say) “change one side”, not “do the same thing to both sides”.

I’ve been doing this for a couple years explicitly, and I do see it broadening my students’ conceptual understanding. First off, there’s the simple fact that I hold the room. I can tell when kids are interested. Done properly, you’re pushing at a symbol they took for granted and never bothered to think about. And they’ll be willing to think about it.

Then, I have seen some algebra 2 students say to each other, “remember, this is just a question. The answer may not be a number,” which is more than enough complexity for your average 16 year old.

Just the other day, in my trig class, a student said “oh, yeah, this is the equals sign where you just do things to one side.” I’ll grant you this isn’t necessarily academic language, but the awareness is more than enough for this teacher. #### 9 responses to “The Evolution of Equals”

• Jim

It is easy to take modern mathematical symbolism for granted but as you indicate it is the result of a long process of evolution and often pretty opaque to students.

• Roger Sweeny

Maybe (some) math came too easy to me but I just don’t see the problem. For me, the equal sign simply makes a statement. It is the verb in a common type of math sentence. (Other possible sentence verbs: is greater than, is less than, is a member of.) Like all statements, it may be true or it may not be. 8 = 5 + 2 + 1 is a true statement. 8 = 7 is a false statement. To make it true, change the sign to “is not equal to”, the equal sign with an oblique line through it.

In math class, unless otherwise told, you can assume the statement is true, or the problem is to find some value which makes the statement true. The latter is that worksheet with the boxes. Logically, it is the equivalent of a worksheet which has students look at a color picture up on the smart board and then “Fill in the box with the color word that makes the statement true.” or just “Fill in the box with the correct color.” “The ball is [box].” “The doll is [box].” Etc. I don’t see anything wrong with either worksheet.

Though you know better than I. Maybe students pick up the wrong idea that an equal sign always means do some operation on one side to get a number on the other. That misconception must indeed be corrected.

When I have students deal with a relation that may or may not be equal, I will very deliberately use an equal sign with a question mark over it. At the beginning of physics class, when I was trying to make them accept the idea that a correct physics formula always has the same unit on each side, I would have them recall the Ideal Gas Law from Chemistry the previous year: pressure times volume = amount of gas times gas constant times absolute temperature. Are the units the same on each side? They had to put in the units on each side with a question mark equal sign between the sides and simplify. Turns out they were and that the unit you get is the energy unit. A true equal sign can now–triumphantly!–be used. The Ideal Gas Law is actually an energy law.

(If they go far enough in math, they will find that an equal sign does not always mean “the same point on the number line.” For example, when two matrices are equal to each other.)

• educationrealist

I think the fuss about kids not knowing the equal sign properly early is overstated. I think it’s easy enough and good practice to correct it. The research finds that kids simply can’t cope with the equal signs in any other context. I haven’t noticed that at all.

However, the next one I mention–moving from “do the same thing to both sides” to “identity” is a common problem in advanced math.

• Jim

Everything is easy once you understand it.

• educationrealist

“(If they go far enough in math, they will find that an equal sign does not always mean “the same point on the number line.” For example, when two matrices are equal to each other.)”

Just was thinking about this. I think the matrix one is a bit different, in that you are solving things that do result in many different pairs being on the same thing on the numberline. But the matrix is tough.

• Roger Sweeny

For two matrices to be equal, every element of one matrix has to be equal to the corresponding element of the other matrix, which gets you back to simple equality. But hidden in that statement is that every element of one matrix has to have a corresponding element in the other, i.e., the matrices have to have the same number of rows and columns.

So since math is so wonderfully logical, the definition has to say both 🙂

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