Just forget my Dear Aunt Sally

The purpose of a mnemonic is to make something easier to remember. Roy G. Biv represents the major colors of the spectrum (Red, Orange, Yellow, Green, Blue, Indigo, Violet); it has the weakness that most people tend to think of the spectrum having six colors these days, instead of seven, with purple in place of indigo and violet. But it’s catchy anyway. FOILing refers to multiplying binomials by listing all the pairings: First, Outside, Inside, Last.

A mnemonic that takes as much mental energy to remember as the underlying information is counterproductive, because not only do people need to learn the mnemonic’s encoding, they also need to learn the relevance of the decoded information. “Dear King Philip Comes Over For Good Spaghetti” is perhaps easier to memorize than “Domain Kingdom Phyllum Class Order Family Genus Species”, but is the aid to memory outweighed by the additional burden of having to remember the connection?

This brings me to the common mnemonic for remembering the conventional order of mathematical operations: Please Excuse My Dear Aunt Sally. Sometimes it’s simply the acronym: PEMDAS. Or PEDMAS. Or BODMAS, in Europe. Or BIDMAS. At any rate, the order is: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

I have several beefs with “Please Excuse My Dear Aunt Sally”.

1. It’s difficult to remember the correspondences. “Exponents” in particular lacks transparency. We tend to actually call exponents “powers” or simply “to the”. 4⁴ is “four to the power of four” or “four to the fourth” more often than it’s “four to the exponent of four”. And as easy as it may be to remember that P is for Parentheses, that leads to confusion about cases such as \(4/3(2+1)\): People who have a vague memory of PEMDAS will get confused and think that the multiplication has to take place before the division because parentheses have precedence.
2. It implies a false hierarchy among the words. Dear King Philip refers to nesting levels, with each successive word relating to a smaller group. I’ve seen people consistently think that multiplication has precedence over division because PEMDAS says so. PEMDAS is really P E MD AS, but that doesn’t come through in the mnemonic.
3. It’s incomplete. PEMDAS doesn’t tell you what to do when there are operations of the same level. This is relevant because subtraction and division are not associative: \((3 – 2) – 1 = 0\), but \(3 – (2 – 1) = 2\); \((8 / 4) / 2 = 1\), but \(8 / (4 / 2) = 4\). Hence we need a convention for parsing 3 – 2 – 1 consistently. We need to add the additional restriction that subtraction and division have to be done from left to right.
4. It’s misleading. In a recent discussion on PEMDAS, someone referred to the operation of the parentheses. While that’s imprecise but arguably “close enough”, it’s only a skip away from thinking of parentheses as an operator.
5. The underlying information is easy to remember with some basic common sense. This is my biggest beef: We don’t need a helpful mnemonic for the order of operations if we understand the underlying operations. If the convention were something other than basic common sense, I could understand the need for a mnemonic. But the only sense I can see is that teachers think students need “cute tricks” to remember stuff about mathematics, which in turn communicates that mathematics is difficulty and hence in need of cute tricks.*

The underlying information

At its core, exponentiation is shorthand for multiplication, and multiplication is shorthand for addition. For example, \[4^3 = 4\cdot 4\cdot 4 = (4 + 4 + 4 + 4) + (4 + 4 + 4 + 4) + (4 + 4 + 4 + 4) + (4 + 4 + 4 + 4)\]

Subtraction is intimately related to addition. It’s admittedly difficult for many students to understand that subtraction is equivalent to the addition of the inverse, but students tend to have difficulty with negative numbers in general. This would be another, important opportunity to emphasize the notion of negative numbers being related to positives.

Division is intimately related to multiplication. This is admittedly the most challenging part of the order of operations. We could theoretically get rid of division nearly entirely. Consider imaginary numbers. There is a single symbol that is used for all imaginary numbers: i. We could likewise have a single symbol to indicate the multiplicative inverse, and then replace all other division with multiplication. Regardless, I think student do generally have a sense that multiplication and division are related, even if they don’t grasp the notion of the multiplicative inverse.

When the precedence is the same, we go by convention from left to right for addition/subtraction and multiplication/division. We could just as easily go from right to left, but for most people, going from left to right is as natural as reading (certain writing systems, such as Japanese, Hebrew, and Arabic, notwithstanding). In point of fact, we do go from right to left with exponents: \(a^{b^c} = a^{(b^c)}\).

So here is the order of operations: Unless explicitly told otherwise with parentheses, resolve powers (working from right to left), then multiplication and division (working from left to right), then addition and subtraction (working from left to right).

Far more words than “Please excuse my Dear Aunt Sally”, and more difficult to learn, but less difficult to implement in the long term. Which, in my view, should be a primary goal of mathematics education.

* Now, if the cute tricks are actually useful, that’s something else. For instance, when teaching how to read slope from a graph, I tell students to pick two points at intersections and move along the squares: Up or down first, then over. A slope of 3/2, for instance, involves going “up 3, over 2”, which contains “three over two”, which is how students are used to reading fractions anyway. The only add-on is teaching that “down” is translated to “negative”, which makes sense from the coordinate plane anyway.