And Equality for All

a brief biography of the equals sign

Like Roald Dahl and Catherine-Zeta Jones, the equals sign was born in Wales.

map of UK

It was 1557—not that long ago, in the scheme of things. Just a few years before the birth of Shakespeare. In fact, the Danish prince and the Scottish king captivated the public long before their humble Welsh neighbor reached wide renown.

The early equals sign was a lovely but ungainly thing, a long pair of parallels that its inventor called Gemowe Lines:

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Over the centuries, this stilt-legged creature shortened into the compact and tidy symbol we know today.

And before that? Well, mathematicians simply spelled out equalities with the phrase “is equal to.”

10 is equal to 7 + 3.

8 x 9 is equal to 72.

And of course, a2 + b2 is equal to c2.

The equals sign offered a way to avoid the tedious repetition of these words. Or, as Robert Recorde, the father of the symbol, put it: to auoide the tedioufe repetition of thefe woords.

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An equals sign, then, is a verb. It’s the mathematical equivalent of “to be”—just as common, just as concise, and just as powerful.

But that’s not what kids see, is it?

To them, ‘equals’ means something other than ‘equals.’

In their arithmetic years, kids almost always encounter equal signs in a single, limited context: to call for the result of an operation. They fill their days with questions like this:

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They get so used to statements of the form [number] [operation] [number] =

that they’re a little creeped out by statements like this:

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And totally deceived by statements like this:

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Asked what goes in the blank, kids choose 9, because 4 + 5 = 9.

Or, when asked to perform multiple operations—start with 7, multiply by 5, subtract 9, and divide by 2—I often find my students writing streams of ungrammatical gibberish like this:

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Of course, only the last of those equal signs makes any sense. The other two are bald lies. 7 x 5 doesn’t equal 35 – 9, and 35 – 9 doesn’t equal 26 / 2.

Poor old Robert Recorde would hang his head in sorrow.

Luckily, there’s a simple visualization of equality that can brush aside many of these misconceptions in one forceful sweep.

An equation is a statement of balance.

20141216074242_00009

See the two sides? I’ve got different weights on each but the total is the same.

I can take away the same weight—say, 15 pounds—from each side, and they’ll still be equal. They’re not the same as they were before, but still the same as each other.

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Similarly, I could add 5 pounds to each side, and they’d still be equal. Or I could double each side. Or halve each side.

So long as I do the same thing to each side, they’ll still be equal.

That familiar mantra—“Do the same thing to both sides of the equation”—is not an arbitrary dictate, cooked up by the Algebraic Rules and Regulations Committee in some air-conditioned boardroom. It’s a simple fact, which I capture in this rhyme:

If two things are equal
then do what you will
to both things at once;
they’ll be equal still.

Other symbols make sense in this light, too. The “>” symbol means “the thing on the left weighs more.”

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The “<” symbol means the opposite.

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And the “≠” symbol means “these two things aren’t equal, although I’m not telling you which one is bigger.”

I’ve had students ask me whether we can switch the sides of an equation, as if they need to consult the Bylaws of Algebra in some dusty legal library before making such a move. But understood with balance statements, it’s obviously true.

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My younger students can mostly solve linear equations in x. But they do so by wordless numerical intuition (“I just knew x had to be 7”) or by blindly executed procedures (“I subtracted 11, then I divided by 2, but I don’t know why that works”). With balances in mind, suddenly it all makes sense:

20141216074242_00013

Weirdly, my school’s textbooks teach “solving for y in terms of x” as a fundamentally different problem than merely “solving for x.” But in this light, they’re virtually identical.

The visualization can offer insight to older students, too. Students often solve simultaneous equations by “adding the two equations.”

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Of course, you can’t really “add equations.” That’s nonsense. What you can do is add the same thing to both sides of one equation.

And then, on the left, instead of “52” we write something that’s equal to 52:

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I don’t mean to say that this visualization is a magic pill, a cure for all misconceptions and fevers. No single key can unlock every door in mathematics. For that, you need flexible thinking, creativity, a healthy faith in your own abilities and a healthy skepticism of your own results.

Still, thinking properly about the equals sign sure helps.

I’ll confess: moving to the UK has heighted my affection for Wales. Sure, the English gave us a global language, the industrial revolution, and soccer. But the Welsh have gifted our planet a humble little symbol that compresses into two quick pen-strokes the far-reaching idea of equality itself.

Further Reading, for the Curious

FROM THE COMMENTS (i.e., the superior shadow-blog existing just below the surface of mine):

Nevin objects to my objection: “There are so many mathematical terms and symbols that are already overloaded to have multiple meanings, that [this use of the equals sign] really isn’t hurting anything.” The problem, to me is that students are not consciously adopting a different convention. Instead, they’re doing something unconventional, believing it’s conventional, and potentially missing an important concept in the process.

John Cowan points out, sensibly: “I think they use = that way because that’s what a calculator does; its = button means ‘Compute the (possibly intermediate) answer.'”

Howard worries about over-emphasizing notational issues: “It seems to me that there is an insane desire to make stuff look like math and force the mathematical language down like the production of foie gras

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18 thoughts on “And Equality for All

  1. About the “ungrammatical gibberish”…would you care if they used, for example, commas or semicolons to separate the the steps of their calculations? As in, “7×5; 35-9; 26÷2; 13”? What about a regular colon? Or a colon that’s stretched horizontally?

    I think it is perfectly obvious what is meant by the sequence of markings in your example, and indeed it is far more efficient to write it that way than to redundantly mark down each intermediate result twice, as in, “7×5=35; 35-9=26; 26÷2=13”.

    There are so many mathematical terms and symbols that are already overloaded to have multiple meanings, that it really isn’t hurting anything when someone overrides a symbol that happens to be two parallel lines to serve as shorthand for, “equals the next term and then that term is also used as the first input for the subsequent operator”.

    Math notation is all about being concise, and if anything I’d say people who come up with their own notation when it is convenient are advancing the cause. Personally, I use an arrow (→ or ⇒) for the purpose in question, but I don’t see any compelling a priori reason to prefer overloading one operator rather than another.

    • Hmm, interesting thoughts! I disagree on a few key points:

      1. First of all: Yes, I’d prefer arrows or semicolons. At their level of mathematics, those symbols have no fixed meaning to override. Makes more sense than overloading the equals sign.

      2. Notation is about more than concision; it’s about clarity and precision, too. Notation is a way of capturing mathematical thought on paper. A misuse of notation sometimes (as in this case, I believe ) suggests a problematic way of thinking.

      3. Yes, it’s clear what they mean, but for young students just learning how to communicate mathematically, that’s only half of the battle. They also need to learn the conventions of the language (just as English students must learn spelling and punctuation).

      4. Ideally, more than arrows or semicolons, I’d prefer something like this:

      (7 x 5 – 9) / 2
      = (35 – 9) / 2
      = 26 / 2
      = 13

      • It is all very well to overload a symbol if the original meaning is a good one, and the overloading makes the symbol more useful. But we have a situation where the original meaning (at least as understood by the pupil) of the “=” is “gives” or “is”, as in 2+3, is 5 is not the best one. Perhaps if the teacher used these words in this situation things might not be quite so muddied.

        It seems to me that there is an insane desire to make stuff look like math and force the mathematical language down like the production of foie gras, and leave the real world behind.

        What adult says “I have to add these two numbers, 6 and 9. Let me calculate the sum. They say “What’s 6 and 9 ? 15? Yes ! “.

        As well, the real world meaning of 3 x 10 = 30 is “three tens is 30”.

  2. Nice post. Love the visual with the balance. Regarding the systems of equations: perhaps this point could be communicated more powerfully by drawing *two* balances, one showing the equality of 2x-3y and 18, the other showing the equality of 8x+3y = 52. Then transfer the items from one balance onto the other!

  3. The position matters when the symbol = is overloaded with its imperative meaning, as in most programming languages. We are commanding the computer to make two things equal, and there are two ways to do that – changing the left, or the right side. By convention, we assume the left side.

    The “mathematician’s language” R actually supports assignment operators -> and is rarely used. Logic languages like Prolog will actually try all possible ways to make the two sides equal and output all solutions.

  4. Hi Ben,

    Sometimes having multiple equals signs can make things less confusing, and is perfectly valid in a mathematical statement. For example, I could say, “If there are only two possible outcomes, and one has a 70% chance of happening, the other has a 30% chance of happening”. Or, I could Probably term this more succinctly using math, with:

    Y = 0 or 1

    if P( y = 1 | x ; Θ ) = 0.7, then P( y = 0 | x ; Θ ) = 0.3 because P( y = 0 | x ; Θ ) + P( y = 1 | x ; Θ ) = 1

    I’m sure you can Probably make this better, I Probably made an assumption or two I shouldn’t have. Just part of my homework problem! My point is that you can definitely have multiple ‘equals’ signs in an expression. It is true that each equals sign is considered individually, i.e. I am stating that for the probability that y is equal to one, given x, parameterized by zero is equal to zero point seven, so I could just substitute y = 1 with something like ‘z’, and end up with P( z | x ; Θ ) = 0.7 for one, but z still represents y equaling one.

  5. I sometimes wonder if we should have kids write “is equal to” until they hit algebra. I don’t think they really get it. I’ve used the balance illustration many times, and a lot of kids still don’t seem to understand. I was reminded of that with my younger son this fall. He’s in prealgebra and while he gets the whole “what you do to one side, you must do to the other,” it took a couple years of drilling that phrase even after seeing the balance analogy. I also think it would help them to better understand what is meant in word problems.

  6. If they wrote “is equal to” it wouldn’t necessarily mean they processed the language. I work a whole lot with students one-on-one and oh, boy, the strategies they use — usually successfully — to get the right answer are amazing, and often based on manipulating the symbols to resemble the example they’ve figured out is the best model. They crash *hard* as soon as comprehension is necessary.
    Happily in the long run, our courses aren’t set up for that to really work… so, with varying degrees of assistance in the thought process, they almost always get a lot better…

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