I was helping my daughter with her homework the other day. Math. She had to convert an algebraic expression into a word problem (by which she was supposed to express the order of operations correctly so the reader could follow them.) Reviewing what she wrote, I pointed out that it wasn’t quite correct and urged her to try again.
“PEMA”, I said.
“PEMA”, I repeated: “Parentheses, Exponentiation, Multiplication, and Addition.”
“PEMDAS” she corrected me.
“Oh, really?” I remarked, somewhat incredulously.
Being a product of New MathYes, that New Math... the one that Tom Lehrer parodied so well. It’s funny, or would be, if it weren’t so painfully true., I wondered what the Board of Education had cooked up here, as it brought back memories of failed educational theory—or at least failed implementation—messing up kids for a generation. Moreover, messing with the well-established Order of Mathematical Operations seemed a recipe for trouble.
For those that don’t know, PEMDAS is just PEMA with Division and Subtraction tossed in. On the surface, that might seem reasonable, but there’s a dark underbelly to this.
|PEMDAS isn’t “Broken”, just “Misunderstood”|
|PEMDAS apologists are quick to point out that PEMDAS isn’t “broken” or “wrong” or “a lie”, just misunderstood.|
That, to me, is exactly the problem: it is so readily misunderstood, and for that reason alone, it is broken: it allows such misunderstanding to form and persist.
I can understand how primary school teachers may be outraged at this negativity over PEMDAS. Their hearts are clearly in the right place for wanting to teach mathematics to their students, and they see themselves as instilling valuable knowledge about the subject. They certainly don’t want to countenance teaching something “wrong”.
What they may not realize is that many of those students don’t remember everything they’re taught, and certainly don’t remember everything about what they’re taught. In the case of PEMDAS, they have this device—and an imperfect recollection of what it means—and get themselves into trouble using it in later years. Much of the backlash in fact comes from college instructors and university professors who are faced with doing remedial math to “unteach” the errant recollection of PEMDAS before they can begin teaching their own curriculum.
That alone should be a wake-up call. Perhaps proponents have a blind spot for PEMDAS, and they need to recognize devotion to the device is misplaced: it is the thing in need of remediation.
You see, neither PEMA nor PEMDAS are the Order of Mathematical Operations. Rather, they’re only mnemonic devices. Just like “Every Good Boy Does Fine” and “FACE” are supposed to help music students remember the notes on the musical staff, PEMA and PEMDAS are just a way of remembering the accepted order of operations in an arithmetic expression (or fact as they seem to be calling it these days.) In its entirety, the Order is:
- Parentheses: Everything inside matching parentheses before anything outside them, from the innermost out.
- Exponential operations: Raising things to a power in right-to-left order.
- Multiplicative operations: Multiplication and Division, in left-to-right order.
- Additive operations: Addition and Subtraction, in left-to-right order.
That’s it. Any time you have two operations to be evaluated (or simplified as some literature says), you pick the one closer to the top of the list. When the two operations tie, you do them in the order that row specifies. A multiplication next to a division? Which one is to the left, since that’s the one you do first. Addition next to a subtraction? Again, which is to the left?
A very few programming languages have no notion of operator precedence so expressions must be written with explicit order stated.
Microsoft’s Excel and MathWorks’ Matlab (neither really programming languages per se but commonly used for calculation) implemented their expression evaluators “wrong” (in that exponents aren’t evaluated right-to-left) but that’s pretty much it. use when there’s any ambiguity.
And you’ll notice, that’s PEMA.
“But...” I hear you exclaim, “...what about division and subtraction? Don’t they deserve equal time?” That is the rationale for the D and the S being added to PEMA.
If you’ve been taught math correctly, you are aware that division is just a kind of multiplicationThe term is “inverse operation”, and each of the operations have their inverses.
One of my “concerns” with PEMDAS is that it allows the curriculum to short-cut placing the necessary emphasis on these inverses. On the other hand, their very absence forces the issue, in order to answer the question “so what about division or subtraction?": dividing x by y is just multiplying x by ¹/y... and hopefully children are being taught this as well.
Similarly, subtraction is just a kind of addition. Subtracting 3 from x is the same as adding -3 to x.
Besides, if you want to be really nit-picky pedants, PEMDAS fails on the equal-opportunity front for neglecting roots: really, in such an “all-inclusive” world, it should be PERMDAS... but it’s not.
On the other hand, PEMA isn’t bothered, because taking the nᵗʰ root is just a kind of exponential operation, raising something to ¹/n. Voila... everything is covered. The square root of x is just x½, just like its square is x². This makes sense: the square root of the square of x (for positive numbers, anyway) is x½∙2.
Yeah, yeah, yeah... so why is PEMDAS so bad?
While the device is taught to represent P, E, (M and D), (A and S), this subtle detail is frequently forgotten, and there’s nothing in the acronym to force its retention. There are six letters, so it’s very easy for somebody to reach into their distant past and think there were six rules, not four. PEMDAS allows this. Don’t believe me? Just google “division before multiplication”
This is not an isolated incident, Math Forum’s Confusion over Interpretation of PEMDAS points out that “Students take PEMDAS letter by letter, tending to do all multiplication before any division, rather than working from left to right.” Students are one thing. University Math departments are another. There, the author, Harvard professor Dr. Knill, asks
What is x/3x?
It depends ... [the author submits the problem two ways to the program Mathematica, which presents results]
The computer has chosen to do the division first. This is natural if one looks as [sic] division and multiplication as operations on the same level. To play the devils advocate, note however that the famous PEMDAS (parenthesis, exponents, multiplication, division, addition and subtraction) has the M before the D so that we would have to accept the second one ...
to a programming language and of course it applies the only rules it has been instructed to follow. In this case, multiplication and division from left to right. So the expression is understood to be 2 times x, divided by 3, multiplied by y, with one finally subtracted from it. Not what you intended? Then you need more bananas...parentheses....
I thank those visionary math teachers—Helen Joseph and Al Bertie (both gone, but not forgotten!)—who spared me and my classmates ... by just teaching us all proper math, and the mnemonic PEMA.
In any event, Dr. Knill pontificates for a bit and concludes with a reference to ScienceBlogs which treats the matter rather more succinctly.
“So What’s the big deal?” I hear you ask.
Very simple: PEMDAS (and BODMAS) were meant be mnemonic devices to help remember the order of mathematical operations, but have instead served to confuse students. I thank those visionary math teachers—Helen Joseph and Al Bertie (both gone, but not forgotten!)—who spared me and my classmates from this by just teaching us all proper math, and the mnemonic PEMA. Now, if only all students could be set straight.
If a consumer product is defective—such as brakes on a car or the cord on an appliance—such that it is the source of repeated injury, such a product is recalled and replaced. In the same vein, PEMDAS—clearly demonstrated to be defective as a tool to understand math—should likewise be removed from the curriculum. In any event, I’m taking the liberty of teaching my daughters math done right. It will just take a while to undo the damage.