quote:Originally posted by mr_porteiro_head: Rabbit, did you mean to say something in the above post where you quoted me?
Woops, yes I meant to say that I'm so accustomed to applying the standard order of operations that it actually took me some time to figure out why you thought the second case was any different than the first.
I'm genuinely surprised to see how many math and computer literate adults find the standard order of operations to be confusing. To me, they seem incredibly straight forward and unambiguous. I've been working with them long enough that its quicker and easier for me to parse a formula that uses the standard rules than one with lots of parentheses.
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quote:I'm genuinely surprised to see how many math and computer literate adults find the standard order of operations to be confusing.
Speaking for myself, it's not the order of operations that's confusing, but the sheer volume of data. When you have different operations with different OOO priority scattered throughout a lengthy, complex equation, the mental stack necessary to keep everything in order can be confusing.
By adding parenthesis, you break it down into multiple smaller units that are easier to evaluate individually.
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posted
*nod* Since you resolve and design each unit separately, it's easier to "debug" through unit testing even if the overall picture is murkier.
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quote:Originally posted by The Rabbit: Its really hard to find someone who has become proficient with the HP notation that doesn't prefer it.
I'm one of those people. I had to become proficient at HP notation for advanced calculus, chemistry and physics classes. Once I finished with undergraduate and could chose my own resources, I happily left Reverse Polish behind.
Ditto. I was never a fan.
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posted
So, additional data, I just tried the problem in the OP on one of my bosses, a physics professor who got his PhD in an era when there were no desktop calculators. He got it wrong, too. So did a physics undergrad we've got working with us. But we all agree on what the correct answer is once the issue is pointed out. So, sod the order of operations, that problem without parens is just plain misleading.
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posted
Pfft. The fact that PhD'd mathematicians and physicists can't do basic arithmetic is a well-documented phenomenon.
(In the days of print catlogs) the AMS had to print a separate catalog for members, because many of them couldn't calculate 20% off consistently. Posts: 32919 | Registered: Mar 2003
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quote:So, sod the order of operations, that problem without parens is just plain misleading.
Obviously you are correct, though as I said before I find that a genuine surprise.
I wonder if there would be the same degree of confusion if the problem were written 48/2*(9+3) or 48/2(9+3) rather than 48÷2(9+3). Even though nearly everyone would agree that all three ways of writing it are identical, the visual spacing is different. I think when its written 48÷2(9+3), the spacing creates a visual break between the 48 and the 2 that makes one want to group the 2(9+3), when its written the other way 48/2*(9+3), the visual break works the opposite way.
When I first saw the problem, I wasn't wearing my glasses and got the answer 72 because I read it 48+2(9+3). I don't think I've seen the ÷ sign used in a problem since elementary school. I'm pretty sure I've never seen it in an expression that omitted using a 'x' sign for multiplication.
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quote:Originally posted by TomDavidson: *nod* Since you resolve and design each unit separately, it's easier to "debug" through unit testing even if the overall picture is murkier.
As a general rule I agree. When I am coding long formulas, I very commonly do something like SenojRetep suggested because its so much easier to "debug" each little piece.
quote:In this case, if I were writing the formula you provided, I would do it as:
f1 = (1-1/r); f2 = (c1-2*c2+c3)/(dr^2); f3 = 2/r*(c3-c2)/(2*dr); f4 = k*(a*r-b)*c2; y = D*f1*(f2+f3)+f4;
But it definitely makes the overall picture of what I'm doing murkier. If I do this when I'm trying to show less advanced students how to work a problem, they have a terrible time following what I'm doing.
For me, however, using lots nested brackets generally has the opposite effect.
Thanks to MPH and Senoj for answering my question. Senoj, your assessment that number 6
quote:6. y = D*(1-1/r)*((c1-2*c2+c3)/(dr^2)+2/r*(c3-c2)/(2*dr)+k*(a*r-b)*c2
was the optimal number of brackets, was what I expected. It has two more sets of parentheses than case number 1. One set is completely unnecessary (dr^2), the other (2*dr) can be eliminated if you change the operator. I agree that changing "/dr/2" to "/(2*dr)" improves the readability of the equation. Adding the brackets around dr^2 just seems to make things more cluttered to me, but I can see how some people would find the lack of brackets ambiguous. Adding more brackets than #6 definitely makes it harder for me to decipher rather than easier.
The problem I face is that students are much more likely to bring me something like
The formula gives the wrong answer because they've misplaced one of the brackets, but because they've used so freaking many unneeded brackets, it's a lot harder to find the mistake. (I'm not exaggerating, I've seen hundreds of students do this)
They do this because they've learned to use brackets instead of understanding and applying the order operations.
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quote:They do this because they've learned to use brackets instead of understanding and applying the order operations.
No, if they'd learned to properly use parentheses, they wouldn't be making the mistakes in the first place.
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quote:I wonder if there would be the same degree of confusion if the problem were written 48/2*(9+3) or 48/2(9+3) rather than 48÷2(9+3). Even though nearly everyone would agree that all three ways of writing it are identical, the visual spacing is different. I think when its written 48÷2(9+3), the spacing creates a visual break between the 48 and the 2 that makes one want to group the 2(9+3), when its written the other way 48/2*(9+3), the visual break works the opposite way.
Yeah, agreed. The main issues here seem to be:
1) People getting confused by the ÷ sign. It looks like a + sign, and adds a level of unfamiliarity with the syntax that makes previous practice with OOO not 100% applicable. I immediately answered 288, but when KoM and others were convinced it was 2, I started to wonder if ÷ and / were different after all (see my second post).
2) People thinking that 2(x) is somehow different from 2 * (x). To me, this was a surprise to see. I've never thought of juxtaposition as anything other than a shorthand for multiplication. To even have to argue this was an odd feeling.
3) The answer of "2" is cleaner than the answer of "288". I have used this test taking strategy myself to good effect. When one answer is simple and the other less simple, the simple answer is usually the correct one.
If the original problem had been written as:
quote:3/4(12) = ?
A) 9 B) 3/48 C) None of the above.
I doubt there'd be a huge debate over juxtaposition, but maybe I'd be wrong.
Is the above example unclear to anyone here? Would those arguing that juxtaposition is somehow special think the answer should be B?
Edit: I'm also curious if Porter and others think the above should have been written as (3/4)12 or something. To me it is perfectly clear as written, but maybe I got more practice at this than others.
Edit 2: I also am wondering if my bad hand writing is to thank for how this seems obvious to me. I typed up a lot of my math homework in college so my teachers could read it, and the "insert equation" functions had not been added to word processors yet. When typed out, 3/4x + 1/2x looks a lot better than (3/4)x + (1/2)x, at least to me.
[ April 11, 2011, 05:40 PM: Message edited by: Ghost of Xavier ]
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quote:Originally posted by rivka: Pfft. The fact that PhD'd mathematicians and physicists can't do basic arithmetic is a well-documented phenomenon.
Well yes, granted. But in this case we're not making a simple mistake. Nobody is multiplying 2 by 12 and getting 25. We're all getting the same wrong answer! So it's not a question of simple arithmetical mistakes, but of mentally inserting the wrong parens.
As a complete aside, I remember being taught as a child to do what was called mental arithmetic (`hoderegning'), where you were supposed to ignore the order of operations. That is, the teacher would list up some problem like "Two, plus four, times 2", and you were supposed to keep a running total and get twelve, rather than 10. This was explicitly explained to be an exception to the usual rules. I don't recall the rationale, but presumably it was considered that when instructions are being given manually, any sort of dependence on the order of operations was Doin It Rong; any such problem should not be constructed so that you need to hold the whole stack of instructions in your head at once, a running total should be sufficient. Or maybe it was just to practice doing arithmetic fast; if you were still calculating 2+4 when the teacher said "times 2", you'd get the wrong answer anyway.
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posted
Yes, the parenthesis make the difference on (3/4)12 versus the other possibilities.
Great discussion, by the way. If I needed any refresher on my understanding of order of operations, I certainly received it. Posts: 1813 | Registered: Apr 2001
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quote:Originally posted by rivka: Pfft. The fact that PhD'd mathematicians and physicists can't do basic arithmetic is a well-documented phenomenon.
Well yes, granted. But in this case we're not making a simple mistake.
Sure you are. The difference is of type, not degree, and it comes down to the rules of basic arithmetic.
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posted
No, the point is that we're all making the same mistake. If we made errors of multiplication, you'd expect us to be all over the place - one multiplies two by twelve and gets twenty-five, another gets twenty-eight. But we all multiply, add, and divide correctly, just in the wrong order.
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posted
The fact that there are fewer types of mistakes possible with OoO than with multiplication hardly seems to prove that it is less simple.
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posted
The fact that you are all making the same mistake, suggests you all have the same deficiency. I'd suggest you all be required to enroll in a basic refresher course in arithmetic.
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posted
I'd consider that unreasonably rude phrasing Rabbit, but KoM has been a jerk about math proficiency enough on this forum that he probably deserves that and much more .
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If telling Ph.D. physicists they need to take a course in remedial arithmetic isn't understood in the joking tone in which it is intended, I give up.
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posted
Eh, could have been just me. I'm usually pretty good at picking up tone, or at least I thought I was, but that doesn't mean I don't make mistakes.
Edit: Though to clarify, I knew it was a joke, I just seem to have misinterpreted the tone of the joke.
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quote:Originally posted by The Rabbit: The fact that you are all making the same mistake, suggests you all have the same deficiency. I'd suggest you all be required to enroll in a basic refresher course in arithmetic.
Well yes. We just did, by having our mistake pointed out to us.
As an incidental aside, my ancient graphing calculator from Casio makes the same mistake when the problem is given in the form in the OP; but if I insert an 'x' between the 2 and the paren, it gets it right. Looks like some Casio programmer didn't consider an edge case.
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