## Notes on Notation – Equality

The humble little equal sign, $=$, is called on to do a large number of different tasks.  As teachers, we often assume that kids will keep these tasks straight.  To an amazing extent, kids actually do.  And I suggest this is amazing precisely because teachers are rarely clear that $=$ stands for so many distinct mathematical ideas.

From early on in school, where kids do $3 + 4 =$, all the way to adults using the $=$ sign common on calculators, the message that $=$ conveys is very clear: Do This Now.  Come up with a number, it insists, give me “the answer”.
In that world, it makes perfect sense for Jen to write the sum of four numbers $3+4+8+1$ as follows: $3 + 4 = 7 + 8 = 15 + 1 = 16$.  If ms. Jones marks this wrong, saying that $3 +4$ is not equal to $7+8$, ms. Jones lives in a different world.  The world ms. Jones lives in is one in which the $=$ marks equality of the left side of the $=$ to the right side.  If the left side is indeed equal to the right side, the $=$ is used correctly; if not, the sign is used incorrectly.

Later in school, $=$ can mean a hypothetical.  When the student is told $x - 2 = 7$, and the student concludes from this that $x = 5$, the student will be asked to check that result in the original equation.  So the student would write $5 - 2 = 7$ and proceed to discover that – no, $5 - 2$ is not equal to $7$.  In $x - 2 = 7$, it is clear we’re talking about a very special $x$.  For most numbers, when used in the place of $x$, would not at all yield $7$ when $2$ is subtracted from them.  The symbol $=$ in $x - 2 = 7$ is being offered up as a clue for finding this very special value of $x$.  When you check to see if $x = 5$ solved it, you try $5-2 \quad _=^? \quad 7$ and eventually decide, no.  In this situation, the $=$ directs us to check whether equality in fact obtains.

A very different situation, also indicated with $=$ is at work in the following situation: $x + 1 + 2 = x + 3$.  This is sometimes introduced with the dreaded word simplify, and known among mathematicians as an identity.  In this situation, the very thing that made $x - 2 = 7$ interesting is now no longer at play.  What’s at play instead is the claim that $x+1+2$, on the one hand, and $x+3$, on the other hand, always work out to the same number, regardless of the value that $x$ happens to have.  Not only don’t we care about any particular value of $x$, the whole point is that we don’t need to care.  Conversely, if we ever thought that $x+1+2=x+3$ might help us as a clue in finding out what special value $x$ has, we are going to be very disappointed.

With this summation, we still haven’t exhausted the way $=$ is used in math class, and outside of it.  When people write stuff like “money = power”, or even “1 box = 12 pencils,” they don’t usually mean that both sides are the same.  They mean something like: “one aspect of the thing on the left side matches one aspect of the thing on the right side.”  Or perhaps they think in terms of exchange and trade.  If you have any doubt that one box doesn’t really equal any number of pencils, try to write with a box, then sharpen a box in a pencil sharpener.  We can make all kinds of allowances for what people really mean, and we should.  Don’t be too surprised if somewhere along the way just about every child makes the following mistake.  “If every box contains 12 pencils, and we write “b” for the number of boxes, and “p” for the number of pencils, what is the formula that connects b and p?”  Watch all the children write the following equation: $b = 12 \times p$.