Mathematics and precedence rules

[Kd7sov]

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

Don't we? How else are we to keep up our tails perpendicular, or spread out our whiskers, or cherish our pride? How else should we keep our minds nimble and active?

Now, if we were talking about full Linnaean classification of species, I would have no problem agreeing that it's not an everyday thing. It even has a broadly similar mnemonic. But the very basics of math, on a level with how to divide?

[ForzaFiori]

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

I don't think about how Germany is in Europe everyday, or that it's capital is Berlin, but I still know it, because it's BASIC geography. The basics of any field of study should be something you learn once and KNOW. I know DC is America's capital, I know where Japan and China are, where Australia is, what their capitals are, the 7 continents, etc. I can tell you that America was an English colony, I can tell you roughly when the dark ages are, who the US's first president was, the name of the person that discovered america is, what a mammal and a reptile are, I can explain how punctuation is used, and I can do basic math. I don't use ANY of those (save for punctuation) every day, or even every week. but they're BASIC knowledge that I just KNOW.

[pendell]

Now, if we were talking about full Linnaean classification of species, I would have no problem agreeing that it's not an everyday thing. It even has a broadly similar mnemonic. But the very basics of math, on a level with how to divide?

I reposted the query on my facebook page and received responses from people I know are not trolls. Only one person got it right. Most of my respondants cited PEMDAS or DMAS, but they still came up with answers of 0,1, and 5 .

They remembered their precedence mnemonic, but they'd forgotten how it worked.

I think these folks would have got it right if there had been only one precedence clash (say, 1 * 5 + 2) or two precedence clashes (1 * 5 + 2 / 3) . But when you use all four operations in the same line, only a minority of people get it right.

So I don't think it's fair to say most people can't do basic math. Most people can do the four elementary operations, and when put on the spot most of them can remember the mneumonic they were taught in school. But when push comes to shove, when the operation is at all complex there's a good chance it will be goofed.

Respectfully,

Brian P.

[Heliomance]

It's an error on similar level to mixing up it's and its, or there, their and they're. No-one should make that mistake, but plenty do.

[Asta Kask]

I don't think about how Germany is in Europe everyday, or that it's capital is Berlin, but I still know it, because it's BASIC geography. The basics of any field of study should be something you learn once and KNOW. I know DC is America's capital, I know where Japan and China are, where Australia is, what their capitals are, the 7 continents, etc. I can tell you that America was an English colony, I can tell you roughly when the dark ages are, who the US's first president was, the name of the person that discovered america is, what a mammal and a reptile are, I can explain how punctuation is used, and I can do basic math. I don't use ANY of those (save for punctuation) every day, or even every week. but they're BASIC knowledge that I just KNOW.

But that's not skills, that's facts. That's an entirely different thing. If you don't practise skills they atrophy (that may, indeed, be the important thing that separates them from facts). I studied French for three years, but that was twenty years ago. I could not hold even a simple conversation in French today without taking considerable time putting sentences together.

Saying everyone SHOULD know something doesn't make it so. Would the payoff to learn and maintain the priority rules of arithmetic be worth the time and effort? For most people? Almost certainly not, because poorly written examples are not that common. It doesn't matter for most people.

And that's another thing - this example is poorly written. It is entirely possible for something to obey all the formal rules and still be unclear. Take sentences like "The horse raced past the barn fell", or "The fat people eat accumulates". Both are perfectly well-formed English sentences, according to the rules of grammar. They are nevertheless difficult to understand.

[Elder Tsofu]

But that's not skills, that's facts. That's an entirely different thing.

Aren't rules facts, at least as much as the name of Berlin is one?

[THAC0]

As a teacher, one important skill is to know whether your students are failing an exam because you wrote the exam (incorrectly, to be too difficult, etc) or because they are (not paying attention/studying, etc).

This equation is clearly a function of the latter rather than the former.

Guess what? My thirteen year old kids, who have known that they need to capitalize the first letter in a sentence for about 7 years, sill don't always do so.

In fact, if I were (as a non language arts teacher) to grade based on that SINGLE fact, 95% of my kids would FAIL.

Guess what again. Not my fault, or their other teacher's fault, for not teaching them to capitalize the first letter in a sentence.

[THAC0]

And that's another thing - this example is poorly written. It is entirely possible for something to obey all the formal rules and still be unclear.

How can one obey the formal rules and arrive at an incorrect answer?

[Heliomance]

"The horse raced past the barn fell"

I honestly cannot see a way to parse that that works under standard grammar. The closest two interpretations I can think of are either that it's two entirely separate clauses and should be written "the horse raced past; the barn fell", or it's a poetic reversal of adjective and noun, and really ought to be "the horse raced past the fell barn".

...actually, no, I can see a third that is grammatically correct, but still wouldn't likely come up. "The horse [that was] raced past the barn fell" works. Is that the correct interpretation? Because using a verb like that (sorry, I don't know the names for the verious tenses and such involved) without the implied "that was" being explicit isn't exactly common practise.

[noparlpf]

As a teacher, one important skill is to know whether your students are failing an exam because you wrote the exam (incorrectly, to be too difficult, etc) or because they are (not paying attention/studying, etc).

This equation is clearly a function of the latter rather than the former.

Guess what? My thirteen year old kids, who have known that they need to capitalize the first letter in a sentence for about 7 years, sill don't always do so.

In fact, if I were (as a non language arts teacher) to grade based on that SINGLE fact, 95% of my kids would FAIL.

Guess what again. Not my fault, or their other teacher's fault, for not teaching them to capitalize the first letter in a sentence.

My mum is an English teacher. My little brother adamantly refuses to capitalise words when he types, and my little sister keeps making the same mistakes that my mum corrects every other day. It's frustrating.

I honestly cannot see a way to parse that that works under standard grammar. The closest two interpretations I can think of are either that it's two entirely separate clauses and should be written "the horse raced past; the barn fell", or it's a poetic reversal of adjective and noun, and really ought to be "the horse raced past the fell barn".

...actually, no, I can see a third that is grammatically correct, but still wouldn't likely come up. "The horse [that was] raced past the barn fell" works. Is that the correct interpretation? Because using a verb like that (sorry, I don't know the names for the verious tenses and such involved) without the implied "that was" being explicit isn't exactly common practise.

Well, there are a few ways to interpret it, but just because one can decipher it doesn't mean it isn't incorrect. Yuo can raed tihs, rhigt?

[THAC0]

My mum is an English teacher. My little brother adamantly refuses to capitalise words when he types, and my little sister keeps making the same mistakes that my mum corrects every other day. It's frustrating.

Indeed. And that does not mean that they have not been taught proper (whatever), simply that they have chosen to forget/not implement it. Which, to bring it back to the OP, is not the fault of the equation.

[noparlpf]

Indeed. And that does not mean that they have not been taught proper (whatever), simply that they have chosen to forget/not implement it. Which, to bring it back to the OP, is not the fault of the equation.

Well, some people are simply incapable of learning certain things (or else, just had bad teachers/teachers incompatible with their learning style). However, I put it down to laziness, in most cases.

[Pie Guy]

Bah, precedence. I just make all a-b's into a + (-b). Same with a/b into a * (1/b) There, that's like half the work done if you can remember multiplication before addition.

Yes, negatives and fractions are easier for me than precedence. I said it.

[the_druid_droid]

Speaking as someone in a fairly math-heavy field, who does quite a bit of numerical programming, I can't really see the purpose in the equation presented in the OP beyond being a test of PEMDAS memorization.

Certainly its meaning is well-defined, and there is a single correct answer, but on a practical level I think it's rather silly. If I were actually making use of such an equation, either in my notes or in a program (or any other situation where I would likely need to go back and reference earlier work) I would always use brackets to divide it up. More generally, if you have a choice between a potentially confusing but formally correct expression and a much clearer and still correct, but slightly bulkier one, I think the choice should be obvious.

After all, mathematical symbols are a form of language and the purpose of language is communication. Formally correct but poorly-communicated expressions are really only fulfilling half their purpose.

[THAC0]

Well, some people are simply incapable of learning certain things (or else, just had bad teachers/teachers incompatible with their learning style). However, I put it down to laziness, in most cases.

And even if they can't learn, that doesn't mean the equation is bad.

[noparlpf]

Speaking as someone in a fairly math-heavy field, who does quite a bit of numerical programming, I can't really see the purpose in the equation presented in the OP beyond being a test of PEMDAS memorization.

Certainly its meaning is well-defined, and there is a single correct answer, but on a practical level I think it's rather silly. If I were actually making use of such an equation, either in my notes or in a program (or any other situation where I would likely need to go back and reference earlier work) I would always use brackets to divide it up. More generally, if you have a choice between a potentially confusing but formally correct expression and a much clearer and still correct, but slightly bulkier one, I think the choice should be obvious.

After all, mathematical symbols are a form of language and the purpose of language is communication. Formally correct but poorly-communicated expressions are really only fulfilling half their purpose.

So obviously the point of that equation is to test the general population (of facebook users) on their ability to use PEMDAS. In that case, it fills it purpose perfectly.

And even if they can't learn, that doesn't mean the equation is bad.

Yeah, in that case the problem is on the user's side. On the other hand, one can probably survive without PEMDAS in "real life". My cat doesn't seem to care.

[pendell]

Speaking as someone in a fairly math-heavy field, who does quite a bit of numerical programming, I can't really see the purpose in the equation presented in the OP beyond being a test of PEMDAS memorization.

Certainly its meaning is well-defined, and there is a single correct answer, but on a practical level I think it's rather silly. If I were actually making use of such an equation, either in my notes or in a program (or any other situation where I would likely need to go back and reference earlier work) I would always use brackets to divide it up. More generally, if you have a choice between a potentially confusing but formally correct expression and a much clearer and still correct, but slightly bulkier one, I think the choice should be obvious.

After all, mathematical symbols are a form of language and the purpose of language is communication. Formally correct but poorly-communicated expressions are really only fulfilling half their purpose.

Thank you, druid_droid. You have said in three paragraphs what I have been fumbling to say for three pages.

My work with mathematics is in building software algorithms, most recently to process sales taxes (and there are diferent kind of taxes for different kinds of products -- alcohol isn't taxed the way candy is. In different states and countries). I try to make the algorithms as clear as possible, which means that any possible confusion in the human reader must be minimized. The business logic and equations must be run by accountants, then coded, then run against sample data in a variety of situations and checked by hand. I don't need to explain the consequences of error.

Before that, it was riemann sums for visual/IR frequency applications. Same thing.

Because the equations must be checked and double-checked by multiple human users, clarity and clear understanding is of primary importance. And so I prefer a bulkier equation that is less likely to be misinterpreted over one that is formally correct but confusing.

Respectfully,

Brian P.

[Asta Kask]

I honestly cannot see a way to parse that that works under standard grammar. The closest two interpretations I can think of are either that it's two entirely separate clauses and should be written "the horse raced past; the barn fell", or it's a poetic reversal of adjective and noun, and really ought to be "the horse raced past the fell barn".

...actually, no, I can see a third that is grammatically correct, but still wouldn't likely come up. "The horse [that was] raced past the barn fell" works. Is that the correct interpretation? Because using a verb like that (sorry, I don't know the names for the verious tenses and such involved) without the implied "that was" being explicit isn't exactly common practise.

That's the intended reading. It's not common practise but it is according to the rules. More examples can be found here.

[Knaight]

The average math SAT score is not the average score of the population. It is the average score of those who take the SAT--high school students who plan on attending college. A substantial minority of people do not take the SAT, such as high school dropouts and high school students with absolutely no interest in college. Therefore, the actual average is likely lower.

That would be one of the reasons why I stated that the SAT is a bad measure for the population as a whole. Another is that it only tests people of a particular age, yet another would be the extent to which people study specifically for the SAT.

[Ashery]

ETA: I'm not familiar with error analysis. May I ask how you came to those conclusions?

While nedz already (sort of) responded to this, I'll fill in the blanks.

Nedz is simply answering the question that's been neglected throughout most of this discussion: How are people arriving at the wrong answer?

For the 56 that gave the incorrect answer of 1: "They" did the operations from left to right without regard to, well, anything. This mistake was blamed on the calculator because it's the exact answer some (many?) calculators will spit out if you enter the equation exactly as given and because most students, even if they don't remember everything correctly, won't simply go from left to right.

6-1*0+2/2 = 5*0+2/2 = 0+2/2 = 2/2 = 1

For the 20 that gave the incorrect answer of 5: They used the order of operations correctly but made, as nedz pointed out, what was essentially a sign error.

6-1*0+2/2 = 6-0+1 = 6-1 = 5

The primary cause for this error, I'd argue, ultimately lies with the placement of the subtraction operation. If the original question was 6+2/2-1*0, I doubt many people would've answered 5. This mistake can be mitigated, at least in this situation, by using a technique someone pointed out earlier: Replace all a - b with a + (-b).

6+(-1)*0+2/2 = 6+0+1 = 7

It should be observed that the replacement only makes it easier because we're multiplying by 0 and therefore removing the negative/subtraction operation entirely.

As for how I'd make the argument that the subtraction operation is the primary cause: It screws with the early intuitive understanding of associativity. That is:

a+(b+c) = (a+b)+c

The problem is that the subtraction operation throws people off.

(6-0)+1 ≠ 6-(0+1)

What the right hand side correctly translates to:

6-(0+1) = 6-0-1 = (6-0)-1

Or, if we had used the a - b = a + (-b) technique:

6+((-0)+1) = (6+(-0))+1

If this question had been written as 6+2/2-0*1, there wouldn't have been an issue as you're not subtracting a set of parenthesis.

(6+1)-0 = 6+(1-0)

For the 6 that answered 0: I'm guessing they saw the *0 part of the equation and automatically assumed that the answer was 0. Maybe because they genuinely screwed up or maybe because they saw a "trick" question earlier that broke down into something like (a+(b*c/(d+e))/(f*g/h))*0 and just jumped the gun (If you feel like being a pedant: d≠-e, f,g,h≠0).

For the 11 that made other errors: Nedz has no idea why and neither do I

Also, this is why you should always show your work. It makes the "How" part of the question so much easier to answer. It also makes partial credit possible

[Jay R]

The given expression is a problem in a math text. Its purpose is to test who does, and who does not, know how to apply the rules.

If people who don't know how to apply the precedence rule can get it right, it is a bad textbook question.

[Saposhiente]

You'd be surprised by how many people who don't work with math regularly don't know that zero is even.

[Coidzor]

You'd be surprised by how many people who don't work with math regularly don't know that zero is even.

Most of the textbooks I've run into demur on the subject, certainly.

[noparlpf]

You'd be surprised by how many people who don't work with math regularly don't know that zero is even.

I've actually never heard that, but I guess it makes sense because 0/2 isn't a fraction. (I don't know about "regularly" but I do calculus about once a week when I realise I have four hours until my P-Chem homework is due. That's a sort of math.)

[pendell]

You'd be surprised by how many people who don't work with math regularly don't know that zero is even.

*Ponders asking the question and looking stupid. Again.*

*Decides that looking stupid is as NOTHING compared to learning something *

So ... why is zero even? You can't divide by it. It seems an arbitrary classification.

Respectfully,

Brian P.

[Socratov]

[QUOTE=pendell;14033995]With respect, Rawhide, when 30,000+ out of 100,000 read that and come out with the wrong answer, I contend that is not , in fact, the case.

When thousands of people can copy-paste that equation into one of the most popular calculator programs and get the wrong answer , I contend it requires revision.

My idea of "clear and unambiguous" is that when 100,000 people read the equation, 99,000 come out with the right answer. 99,000 people of average intelligence and education.

If i write

2+2 = ?

How many wrong answers am I going to get from 100,000 average people?

snipQUOTE]

that depends on you expecting the caculation in integers.

For instance in integers it's the much awaited 4. However, to be overly correct one can assume that the '2's in the equation can be rounded numbers of anywhere between (for significance kept on 2 decimals) 1,50 and 2,49

this then results in a range of minimum 1,50+1,50=3,00 (or: 2+2=3 for minimal values of 2)and a maximum of 2,49+2,49=4,98 (or: 2+2=5 for maximal values of 2), resulting 2+2=y in a range of 3=<y<5 for any value of 2

When we apply rounding off again we shall see that 2+2 can be solved as 3, 4 or indeed 5.

Now if the above explanation is the correct one, the 2+2=4 is not correct (in only providing 1 solution while others exist) even though 99.000 out of 100.000 people will probably say 2+2=4.

note: this phenomenon becomes funnier if applied to slightly more complex problems. Imagine a calculation where the function is the between-2,50-and-not-quite-4-th root of (whatever). At some point you will encounter the euler root and the Pi root. have fun calculating

[noparlpf]

*Ponders asking the question and looking stupid. Again.*

*Decides that looking stupid is as NOTHING compared to learning something *

So ... why is zero even? You can't divide by it. It seems an arbitrary classification.

Respectfully,

Brian P.

"Even" means it can be divided by two and not yield a fraction. So technically zero is even, because 0/2=0 and zero isn't a fraction.

[Castaras]

And also 2 being even fits the pattern of odd-even-odd-even, which is very much liked.

[Socratov]

"Even" means it can be divided by two and not yield a fraction. So technically zero is even, because 0/2=0 and zero isn't a fraction.

well, 0 is the black sheep of the mathematical family anyway, so...

[noparlpf]

well, 0 is the black sheep of the mathematical family anyway, so...

I usually like zero. It means I can eliminate an entire piece of an equation, or reduce it to a one, or something. Except sometimes I hate zero. Like when the equation says ln(φ)=(1/RT)(integral from 0 to p (dp/p)). Because without that actually being explained in class, when you get to it on the homework it looks like you're supposed to have ln(0).