Can We Save “Order of Operations”??

In one recent post, I looked at some basic flaws in the mnemonic “PEMDAS” (there are several fundamental flaws). In another recent post, I talked about how unimportant a ‘correct answer’ can be in a math class. Let’s examine the intersection of those thoughts, and deal with saving the important topic of ‘order of operations’.

The two most common statements about why “order of operations” is important are:

“The order of operations is just an agreement so we all get the same answer.”
“You need to follow the order of operations so that you will get the correct answer.”

Both of these miss the point; their implication is that we can change the correct answer just by changing the ‘agreement’ about order of operations … that we could declare subtraction is always done before multiplying, for example. The order of operations is not just some coincidence of the mathematical language which will evolve to be anything fundamentally different.

The reason the ‘order of operations’ is so important is that the meaning of a mathematical statement is based on understanding the order of operations. In natural languages, the presence of multiple verbs in a statement is unusual … in mathematics, this is commonplace. Multiple operations in a statement with nouns and adjectives provides an efficient method of communication, which is why scientific advances increased dramatically after the use of symbolic mathematics (as opposed to the original verbal forms).

Not only does “PEMDAS” have little to do with correct order of operations, the way ‘order of operations’ is typically taught has little to do with mathematics.

When we learn a computer programming language, we face the issue directly — what is the precedence order for operations? Although there are some minor differences in the details, almost all precedence orders are based on a fundamental mathematical idea:

The more advanced operations are done prior to simpler operations.

We teach students that exponents are repeated multiplications; what we don’t divulge is that this means that exponents are more advanced operations … and therefore are done prior to multiplying. We cover the procedures for multiplying and dividing fractions, but do not make sure that students know that these procedures are based on the fact that multiplying and dividing are at the same level of complexity, mathematically speaking.

The fundamental idea that “more advanced operations are done first” covers the majority of what we try to do with ‘order of operations’. The difference is this: order of operations is treated as a memorization issue, while ‘more advanced operations first’ is calling for understanding and communication. How students get to a ‘correct answer’ is more important than the fact that they got a correct answer.

In those computer programming languages, operations are categorized into binary and unary types, just as mathematicians do. The ‘more advanced first’ principle handles almost all cases in both types. Even the type some of us complain about:

-5²

Even though this ‘ambiguity’ is not encountered very often in real-world problems, this is a core issue in communication. How do we interpret:

-x²

We certainly don’t want people to apply the opposite operation prior to squaring, and we certainly don’t want the answer to change when given in variable notation. In both of these problems, the “-” means opposite … which is less advanced than squaring; therefore, square first, then apply the opposite.

The few places where ‘more advanced first’ fails are also places where ‘order of operations’ fails, and these are often due to our failures to maintain integrity in our language. Our notation for trig functions is sometimes bad, or even incorrect (when it creates an inconsistency with other operations or functions). Even if we don’t change our behavior in trig functions, students will be better off with ‘more advanced first’ than they are currently.

I’d be happier if we never used the phrase ‘order of operations’; the entire implication of this phrase is ‘memorize the rules, or else’. Our students would have a higher quality learning experience if we just focused on ‘more advanced operations first’. The emphasis this involves on the meaning of expressions helps novices reach a deeper understanding of our mathematical language.

Which of these is a better answer to the question “why did you multiply before you subtracted”:

I multiplied first because the order of operations says to multiply before subtracting.
I multiplied first because multiplying has a higher precedence because multiplying is more advanced.

As we strive to help our students understand and reason in mathematics, an ‘order of operations’ has no place in the curriculum. Knowing a structure for operations, including ‘more advanced’, is critical.

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Learning math | Jack Rotman | October 14, 2016 10:29 am

2 Comments

By schremmer, October 14, 2016 @ 11:05 am

As it happens, because I was given an arithmetic course a couple of weeks before the semester started and, as usual, could not resolve myself using “the” textbook, I have been writing one on the same table of contents, a chapter or two ahead of the class. All this to explain why, as I sat in front of my computer to write the chapter on … “order of operations”, I was very happy to see Rotman’s take on the issue. However, I will have to defer my opinion on the merits of his case until later today when I have written said chapter.
By schremmer, October 14, 2016 @ 9:54 pm

Again, I wholeheartedly agree with Rotman. No reservation and here are a few observations.

1. Rotman mentions “the meaning of a mathematical statement</em" and he even invokes what happens in "natural languages” and “computer languages“. In other words, he does distinguish between the real world and the paper world in which we represent what happens in the real world but he doesn’t go far enough. The general idea ought to be the systematic use of a “Model Theoretic” approach. (Nothing to do with”modeling”. (See Symbolic Systems.)

2. I don’t like the phrase “more advanced operation” too much because it seems to evoke “advanced mathematics”.

3. I don’t like the phrase “order of operations” either but it does make for a good chapter title. I can’t seem to find a good alternative.

4. Without getting into details, the difficulty with powers is that we introduce them with the coefficient 1. With, for instance, $3x^{\pm2}$ read it as “3 multiplied/divided by 2 copies of x” the necessary grouping becomes natural.

P.S. I hope this site uses MathJax. More generally, since this site has no menu bar, where can I find the coding for answers in WordPress?

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