1. ## Order of Operations

Originally Posted by goldenmonkey
This is driving me crazy:

Use the following formula to calculate a, if c = 20, d = 10 and e = 2:

. . . . .a = c – d/2e
Note: Opinion is very often very evenly split regarding whether the final term on the right-hand side should be interpreted as:

. . . . .$\mbox{1. }\, \dfrac{d}{2e}$

...or else as:

. . . . .$\mbox{2. }\, \left(\dfrac{d}{2}\right)(e)$

The "right" answer will depend upon the exercise author's intention:

. . . . .$\mbox{1. }\, c\, -\, \dfrac{d}{2e}\, =\, (20)\, -\, \dfrac{(10)}{2(2)}\, =\, 20\, -\, 2.5\, =\, 17.5$

. . . . .$\mbox{2. }\, c\, -\, \dfrac{d}{2}\,\cdot\;e =\, (20)\, -\, \left(\dfrac{(10)}{2}\right)(2)\, =\, 20\, -\, 10\, =\, 10$

2. Originally Posted by stapel
Opinion is very often very evenly split …
I'm not sure about "evenly split", but my view is that Order of Operations ought to interpreted according to its definition, instead of opinions. It's a shame, when a math school publishes an entrance exam containing math mistake(s).

3. Originally Posted by mmm4444bot
I'm not sure about "evenly split", but my view is that Order of Operations ought to interpreted according to its definition, instead of opinions. It's a shame, when a math school publishes an entrance exam containing math mistake(s).
The problem here is that the order of operations as we teach it originated as a formalization of the way mathematicians had informally agreed to interpret expressions, much as a grammar textbook tries to give formal rules for a language that developed organically, not from a-priori rules. (Usage determines the "rules", not vice-versa, because usage came first.)

When that happened, there was some disagreement; the rules some teachers had chosen didn't quite fit what real people did, where division (expressed as $a \div bc$ or as $a/bc$) is concerned. It just looks visually like the bc should be thought of as a unit, contrary to the oversimplified rules. But since mathematicians would almost always write $\dfrac{a}{bc}$ or $\dfrac{a}{b}c$, the issue didn't come up much, and was never fully resolved.

Now that people more often type math, the issue comes up more; some people want to follow the (arguably wrong) rules taught in most textbooks, while others (including some textbook and test authors) want to make the rules more natural. It is a genuinely arguable issue.

Of course, the only proper thing to do is never to write the disputed type of expression, since you can't be sure how the reader will interpret it; either use the horizontal bar, or use parentheses as much as you have to for clarity. The people I consider to be the worst offenders are authors who write this way because they want to make sure their students are good followers of arbitrary rules. So, ultimately, I agree: this shouldn't have been written.

4. Originally Posted by Dr.Peterson
… [a/bc] just looks visually like the bc should be thought of as a unit …
Not to me!

Originally Posted by Dr.Peterson
… Of course, the only proper thing to do is never to write the disputed type of expression …
Of course, that's never going to happen.

May I suggest that, perhaps, there exists another thing to do? Let's start training instructors to raise this issue prominently and frequently, throughout the introductory-level courses. Maybe the situation could improve through education.

I see using a hyphen to represent both a subtraction operator and a negation symbol as similar. We currently have no other choice, when typing on nearly every device. I understand why it came this to, but (based on my experiences working with students at the community-college level) the issue doesn't seem to be discussed enough in class. Perhaps it ought to be, now that we're in the 21st century.

5. Originally Posted by Dr.Peterson
Of course, the only proper thing to do is never to write the disputed type of expression...
Originally Posted by mmm4444bot
Of course, that's never going to happen.

May I suggest that, perhaps, there exists another thing to do? Let's start training instructors to raise this issue prominently and frequently, throughout the introductory-level courses....
Agreed!

I first realized this issue when my students came to class one night, armed for bear. They had discovered that they'd come up with two different answers for one exercise, and the class was fairly evenly split over which was the "right" interpretation. It had never occurred to me that the expression they'd been assigned to simplify could be interpreted either of two ways. Once they explained the problem, I agreed that the formatting was the issue, and told them that I'd accept either answer, as long as they'd shown all their steps (which the students found to be eminently reasonable).

Since then, I've made a point to show an example of this sort when I teach the topic (and have posted what seems to have become the go-to example online), explaining that the notation is the issue, not them, and sternly instructing them that this is exactly why notation matters, and why we need to be clear in what we say and write.

Facebook still won't listen, but what's a girl to do?

6. Originally Posted by stapel
Facebook still won't listen, but what's a girl to do?
Seriously. Seemingly every two weeks, some arithmetic problem makes the rounds on FB that is just an ambiguous expression, and people proceed to argue ad nauseam over whether the answer is -17 or +7 (for example). What frustrates me the most about such comment threads is the number of supercilious people saying things like "The answer is clearly ___, don't you remember PEMDAS/BODMAS/BEDMAS/(Pod Bay Doors)??! How can you be this bad at math?"

But very few of the commenters are actually so bad at arithmetic that they can't even add or multiply. It's not any actual math they don't know, it's an arbitrary convention that they forgot or misapplied. It should be emphasized that this is an arbitrary convention, and that real working scientists and mathematicians would likely just write an unambiguous expression by grouping using parentheses in the first place. This kind of thing further reinforces the impression people get from grade school that math is a sequence of arbitrary rules and recipes that you just have to learn by rote, and that is sad.

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