I have gotten slapped by younger PEMDAS people for years, all of whom are just sure that their interpretation of the conventions is the correct one.
I notice that her audience seems to largely agree with her.
In general, I respond better to "it used to be that way but we changed it" than "no, you're wrong."
13 comments:
After decades of telling stupid computers what I want, I automatically lard my equations with lots of parenthesis. Computers don't like ambiguity.
I was taught that you do parentheses first, but that you eliminate them: thus, the answer here would be 1 because 6/6 because 2(3) has to be evaluated to 6 in order to completely eliminate the parenthesis.
That is what I was taught as well: resolve the parentheses first. Then one day that was no longer true and I was being sternly correctly by young math teachers.
1. I agree with James. Ambiguity is not your friend.
2. In my experience, mathematical expressions generally come with a context, which helps one who is knowledgeable of its derivation interpret the desired meaning. For example, 1/2πr would rarely be interpreted as (1/2)*n*r.
3. Another rule I learned was that the fraction bar served as the equivalent of parentheses around the numerator and the denominator. Unfortunately, if the expression is written linearly, there is no fraction bar, and the parentheses must be explicit.
I wouldn't say that PEMDAS is "wrong", but rather that the only time it gets used is in elementary/middle school when teaching order of operations, and that's not how practical math is done. To echo the ambiguity comments, the equations she's using are not clear, and would only be seen in a school textbook teaching said order of operation. Those books (and calculators) are the only place the gazinta sign is used, I can't even type it here without pulling up special characters, it's not on keyboards, and it's less clear on how the terms are grouped. In programming, more parentheses would be used, as James said, to keep the computer from doing what you didn't want. In print, if it really mattered, it would be written as a complex fraction, so there's no doubt what's on top and what's on bottom.
When the presenter finally gets to the convention for how to write equations on single lines (to save printing costs on parentheses), that's how practical math, science, engineering, etc. texts do it. That's what I got used to from higher algebra on, to the point where I didn't even realize that this issue could even exist.
What these types of demonstrations, where people get different answers and argue about them, show, is not that people are being taught wrongly, but rather that they're being given a screwy equation that would not be written that way if one was being practical and concerned about clarity. GIGO.
So, she's generalizing from an edge case to claim everything that transpired before she was born is wrong. Seems typical.
I teach "PERMDAS," after introducing the Mystery of the Missing Operation. I'll be happy to see how this compares to the vid. ;'
Re: Parentheses, I teach that they are not operations, but an Emergency Override symbol that forces the execution of a lower-level operation out-of-order. But in simpler terms, of course, depending on my audience.
"Whaddya think sirs?"
Well, it allows me to keep my "parentheses first," but that probably just makes me prejudiced in favor of that. I do like it.
The guys using them to earn their daily bread have it about right, though. If you have enough parentheses so there's no question what you mean it's probably best. Your students will be known by their number of parentheses.
It's the difference between learning and later expertise, I think. She's not not wrong (heh).
I'd wager programming protocol (I can't recall right now) would be multiplication before division, but again I'd explain to the wee ones who wanted to know that that was just a way of turning many parentheses into implied ("invisible") ones.
Rather like the Invisible 1 that precedes all multiplication problems you decide to process, or the Invisible 0 before all addition processes. :') But that's another story.
SHe didn't touch on the Higher level operations at all-- sad!! The challenge at Level 3 is explaining to my students why there are THREE operations, powers, and the two inverses of that -- roots AND logs. BUt I got that worked out also.
You do something for 30 years, I tell them, and it makes you look smart today. ;'/
Oh and P.S: I noticed she did not use the typical TI-84 calculator to demonstrate the subscribed belief. Which would have been instructive, since TI-xx calculators are infamous (in my classes anyways) for not doing PEMDAS "correctly," and bollixing up solutions. THe easiest example: calculate -12^2. Mr. TI-84 will respond, "-144."
I assume your are being sarcastic, since 12^2 (twelve squared) is 144.
FORTRAN, BASIC and other programming languages have built-in priority rules:
https://www.tutorialspoint.com/fortran/fortran_operators_precedence.htm
They do, but sometimes in corner cases where there are side effects to an operation, there can be unexpected or inconsistent results depending on what compiler is used.
These are easily circumvented by application of the "Don't get cute" principle together with the "Make it easy on yourself" rule.
It is fun for the student to try to put the whole calculation in a single line, but when you're trying to decipher a pile of careless code (possibly your own) the temptation to lack of charity is overwhelming.
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