The Overloaded Equals Sign: “Do This Now”
In this series about mathematical notation and its impact on the learning of mathematics, we’ve looked so far at the raised minus sign for negative numbers, the various ways of denoting multiplication, and various ways to denote the sequencing of operations. In the next few posts, I’ll be looking at the different mathematical ideas that are expressed, in standard notation, through the equals sign “=”.
In this post, we’ll look at the most familiar use of the equals sign, the one learned first, and the one in common use on calculators. In school, we all know what we are supposed to do with:
5 + 4 =
7 + 8 =
5 – 2 =
The job here, as we all learned, is to put a number to the right of the “=” sign, and this number is called “the answer”. The number we put there is either the correct answer or the wrong answer, and it is the job of the teacher to decide which is which. This is such an important part of the teacher’s job that the teacher even has a special edition of the textbook, different from the students’ version, and the teacher’s edition includes something called an “answer key”. The answer key helps the teacher in this all-important task. Like a traditional country doctor who is instantly recognizable by the white coat and the stethoscope, the teacher is distinguishable by the answer key.
Any well-trained kid will feel an urge, whenever she sees an equal sign with nothing to the right of it. The same urge is felt, clearly, by a four-function calculator whenever the “=” key is pushed. Something needs to be done! Right now! We need a number, and we need it stat!
This means that, at this stage, the equals sign has a clear direction to it, it is not symmetric in its use. For a kid, seeing
5 + 4 = 9 would look normal: it would look like an addition problem that somebody else had already answered. Yet
9 = 5 + 4 would look positively strange, and 5 + 4 = 6 + 3 would look positively wrong. To adults these don’t look wrong, but that means we are used to the equals sign in different contexts, for other purposes. In upcoming posts, I’ll investigate those other uses.
Staying with the “do this now” meaning of the equals sign, let’s take inventory of other ways to indicate this meaning, both in school and out. From this, a reasonable proposal might emerge for notational schemes that teachers could use to help disambiguate the mathematical ideas for their students.
If you listen to how 5 +4 = 9 is pronounced in parts of the English speaking world: “five plus four makes nine”. Though a minority pronunciation (I assume “five plus four equals nine” is considerably more common), the “makes” pronunciation connotes a production, an action with a direction and an intention. As one very obvious way to indicate direction is by using an arrow, “→” (which we might pronounce as “makes”), let’s play with this notation and see how it looks. In an earlier post, I showed
75 + 10 = _______ × 1.09 = ______ + 4 = _____ as a way to show the sequence of operations in ((75+10)×1.09)+4, knowing that this use of the equals sign upsets many teachers. So here is an opportunity to see if it looks better using the “→” instead of “=”:
75 + 10 → _______ × 1.09 → _____ + 4 → _____. I do think it looks better, but that isn’t a strong enough argument for proposing the → symbol. Other notations may look even better, and also look more natural. Moreover, other notations may be better able to coexist with standard textbook conventions. We might take inspiration from what computer languages have long used for assignment operators, where it was desirable to have a notation for assignment that was distinct from a notation for equality. If you ever learned the Basic programming language, it may have looked strange that you saw constructs like X = X + 1 used there. Many programming languages use the construct “x := x + 1” instead. The symbol “:=” for assignment doesn’t deviate too far from the “=” sign but makes the directionality clear. But for our purposes in disambiguating 8+4=, the assignment operator is oriented the wrong way around.
In light of this, I propose “=:” as an informal notation for “makes”:
7 + 5 =: 12
75 + 10 =: _______ × 1.09 =: ______ + 4 =: _____
The “=:” is easy to use in a classroom, is easily rendered from a keyboard; the extra colon is relatively unobtrusive, and can gradually be dropped as students get older. It is something the teacher can use to disambiguate the various meanings of “=” in the standard notation, and the teacher can do this without ever requiring the students to use this notation themselves. The notation is easily explained to parents, though little harm is done if such explanation never reaches the parents.