What is the pedagogical justification and history for using mnemonics to teach order of operations?

Question

There was previously a question/rant here on MESE about why so many are still using the PEMDAS/BODMAS/BIDMAS/BEDMAS mnemonics to teach order of operations. The question was deleted (still viewable by 10K+ users), but there were some comments and an answer that had a link to an argument against PEMDAS, from which an interesting and useful question can be extracted.

Like the author of the linked argument, I, who attended government-run schools in the suburban US in the 1970s, also had never heard of PEMDAS until I was an adult. In fact, I probably first encountered the mnemonic while reading edu-blog posts and/or MESE or MSE. I don't remember how we were taught, but I know that I have internalized the rules so I don't have to think about them, whereas I encounter adults who have to write down "PEMDAS" before they can begin.

I realize this may be several questions, but they are interrelated:

Why, and less importantly when and where, did mathematics educators begin to use mnemonics to teach order of operations?

Note: there is a related question about why the mnemonic rule works, but that doesn't get into the justification for teaching order of operations in this way.

Answer 1

One source I found is the following from Dr. Math at Drexel

I suspect that the concept, and especially the term "order of operations" and the "PEMDAS/BEDMAS" mnemonics, was formalized only in this century, or at least in the late 1800s, with the growth of the textbook industry. I think it has been more important to text authors than to mathematicians, who have just informally agreed without needing to state anything officially.

Another link from Mr. McIntosh discusses GEMDAS

At the aforementioned department meeting, Ms. Hertzog, a math teacher at Challenger, said something about GEMDAS being superior to PEMDAS because with PEMDAS some learners get it stuck in their heads that parentheses are the only grouping symbols that need to be taken into account, or else they get confused when some other grouping symbol is used instead of parentheses. She made no claim to inventing GEMDAS, but apparently heard about it at a workshop somewhere.

Additional history is cited in a paper from Harvard about the ambiguity of order of operations which in turn cites a Slate article by Tara Haelle

Internet rumors claim the American Mathematical Society has written "multiplication indicated by juxtaposition is carried out before division," but no original AMS source exists online anymore (if it ever did). Still, some early math textbooks also taught students to do all multiplications and then all divisions, but most, such as this 1907 high-school algebra textbook, this 1910 textbook, and this 1912 textbook, recommended performing all multiplications and divisions in the order they appear first, followed by additions and subtractions. (This convention makes sense as well with the Canadian and British versions of PEMDAS, such as BEDMAS, BIDMAS, and BODMAS, which all list division before multiplication in the acronym.) The most sensible advice, in a 1917 edition of Mathematical Gazette, recommended using parentheses to avoid ambiguity. (Duh!) But even noted math historian Florian Cajori wrote in A History of Mathematical Notations in 1928-29, "If an arithmetical or algebraical term contains $\div$ and $\times$, there is at present no agreement as to which sign shall be used first."

Answer 2

Based on published works, it appears that both PEMDAS (the US version of the acronym) and BODMAS (the version taught in the UK) began to appear in print only in the 1980s, and really began to spike in 1990. See the Google Ngram below:

In fact, searching for PEMDAS within Google Books in the range 1960-2009 suggests that the acronym first started appearing in print in a large-scale way in test preparation books.

This doesn't mean, of course, that PEMDAS and BODMAS were not part of the oral tradition of teaching mathematics for a long time before they began to appear in print. In fact if you dig around older hits for BODMAS you find lots of people referring (in print) to it as something they were taught in school. But the acronyms don't seem to have themselves made it into print until the test prep industry normalized them to a certain extent. I suspect something similar is true about other acronyms like SOHCAHTOA.

(By the way, in response to those who commented under the OP that mnemonics like this do more harm than good: I mostly agree with that sentiment, but the example of SOHCAHTOA is a good reminder that the names of things are arbitrary matters of convention. In any right triangle there are indeed six possible side ratios to consider, but there is no way to "figure out" which one is called "sine" and which one is called "cosine"; at some point you just have to memorize it.)

Answer 3

"Order of Operations" as commonly taught and tested is just a mess.

Here is a picture from a real standardized test in New York. It was quoted in one of Hung-Hsi Wu's essays.

The order of operations as generally taught says you must evaluate $4^2$ before evaluating $\frac{6}{2}$. Huh? Is that how ANYONE who knows mathematics thinks about it? Is the universe going to explode if you divide 6 by 2 first?

As for the actual question from the test, the next line of work should look like:

$3-16+3$

Now, if you did the exponent and division simultaneously and the universe did not explode - you rebellious daredevil! - we can then move on to trying to figure out who the Hell cares if the rest of the evaluation is

$3-16+3=-13+3=-10$

or

$3-16+3=3-13=-10$ .

Either addition or subtraction could be the final operation.

So... should the real mnemonic be "PEMDAS - EWIDTMAA"?

[PEMDAS - Except When It Doesn't Matter At All]

Perhaps it's OK to use the mnemonic to teach initial calculations, but the real goal of the PEMDAS mnemonic ought to be to show students what can or should be considered a single entity, whether there are brackets to emphasize it or not.

For example, many students consider both of the following to be factored expressions: $(x-7)(x+1)$ and $x-7(x+1)$.

Students should be able to look at the second one and be able to consider $7(x+1)$ as a single number. This will allow them to see that the second expression does not "end up" with two quantities being multiplied like the first one does, so it is not a factored expression. Students should be able to interpret that second expression as the difference between $x$ and $7(x+1)$.

Some of this may seem obvious to those who are steeped in math, but interpreting that second expression as a difference is surprisingly rare. Many students doing well in high school algebra don't see a difference until I have them evaluate the expression for $x=5$ in: $x-[7(x+1)]$ and $x-7(x+1)$ then ask them to compare/contrast. Sometimes this turn the light bulb on.

Algebra tiles can also help with this.

If you have other or better ways to do it, I am all ears!

EDIT: Sorry for the somewhat off-topic rant. Returning to the question, I believe that PEMDAS was basically invented to ensure that students learned at least one correct sequence of calculations that would always be correct.