Mathematics and precedence rules

[DeusMortuusEst]

I haven't looked at the original facebook post, since I'm at work, but if it is true that 3/10 can't get this right then I'm shocked and a little bit upset.

This is not hard math. It is not badly written (apart from the x. I though that it was a variable, but from the rest of the thread I guess it is meant as a multiplication sign.) and failing to interpret this correctly does not depend on the equation itself.

In fact it is very straight forward. It is something that you learn how to do at a very young age and then practice for several years.

That so many are ignorant of how to do basic calculations is frightening.

[Coidzor]

One part calculator dependence and increased ubiquity of calculators at younger and younger ages...

Another part that math we need to actually do in our daily lives for most people has been greatly simplified, much like how most reading materials are at or below a 5tth grade reading level or so. So precedence rules generally don't come up and so if you've not used order of operations in over a decade or several decades in the case of older internet users, it makes sense that there would be some difficulty in recalling them.

I concur that it is rather frightening though.

[Brother Oni]

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.

I'd personally go for 95,000 (5% failure rate), since this is likely to follow a Gaussian distribution. I'm fairly sure that more than 1 in 100 people are going to get the answer wrong, either wilfully, lack of education, mis-comprehending or misreading of the question, or simple brain fart.

[Coidzor]

Plus there's being in a rush because it's facebook.

[nedz]

If you don't run Windows Calculator in Scientific mode, you will always get the wrong answer - with or without brackets.

Windows Calculator in Standard mode:
6-1*0+2/2=1
6-(1*0)+(2/2)=1

(Note, both are the wrong answer.)

Calculators tend to evaluate X<op>Y left to right.

One part calculator dependence and increased ubiquity of calculators at younger and younger ages...

Its possible they (mis-) used a calculator, well see above (several posts)

Regrettably, no one seems to have tabulated and I myself do not have time to do so by hand. Does anyone have a suggestion for a software solution to do this?

I don't use Facebook, is it possible to get an extract ?

Reverse Polish Notation clears everything up, but it's mostly for computer science, IIRC.

Or old HP style calculators.

The standard implementation (since the days of Fortran) is to convert infix to postfix, and then process the resultant stack.

Infix: 6,-,1,x,0,+,2,/,2,=
Postfix: 6,1,0,x,-,2,2,/,+,=

[The Extinguisher]

I honestly don't see what the harm of putting brackets in the equation here is? Really, other than making people look dumb what does it accomplish?

Also, I would like to say as someone who does a lot of maths, order of operations doesn't really come up as often as people here seem to think. It's good to know, I guess, but no mathematician is ever going to write 6-1*0+2/2, they're just going to write 7.

[AsteriskAmp]

I honestly don't see what the harm of putting brackets in the equation here is? Really, other than making people look dumb what does it accomplish?

It shouldn't be needed, even if to some it would be more comfortable to work with.

Also, I would like to say as someone who does a lot of maths, order of operations doesn't really come up as often as people here seem to think. It's good to know, I guess, but no mathematician is ever going to write 6-1*0+2/2, they're just going to write 7.

No, but you DO have things similar in form like
f(t)=6-v₀*t+v₁/v₃
Evaluate at t=0 if
and x=0 for t=0
v₀=1-x
v₁=2x+2
v₃=2

Not actual equation for anything; t isn't time, v isn't velocity, x is arbitrary variable.

[Neftren]

It shouldn't be needed, even if to some it would be more comfortable to work with.
No, but you DO have things similar in form like
f(t)=6-v₀*t+v₁/v₃
Evaluate at t=0 if
and x=0 for t=0
v₀=1-x
v₁=2x+2
v₃=2

Not actual equation for anything; t isn't time, v isn't velocity, x is arbitrary variable.

With the advent of computer software packages such as Mathematica and MATLAB, I don't think I've actually bothered to write out the work anymore. Pop formula into Mathematica, setup inputs, Solve[f(x) == y, 0] and so on. As The Extinguisher pointed out, people who frequently use mathematics would be more likely to either write "7" in the above example, or express things in terms of other things (e.g. 3∏/2), in which you avoid all the sticky bits of excusing our dear aunt Sally.

[The Extinguisher]

It shouldn't be needed, even if to some it would be more comfortable to work with.
No, but you DO have things similar in form like
f(t)=6-v₀*t+v₁/v₃
Evaluate at t=0 if
and x=0 for t=0
v₀=1-x
v₁=2x+2
v₃=2

Not actual equation for anything; t isn't time, v isn't velocity, x is arbitrary variable.

Why not though? There hasn't been a real reason why we shouldn't use brackets other than "people shouldn't be so dumb they don't know basic math." Not using brackets is great for drilling students on order of operations rules, but it's not useful for anything else. This does create confusion, don't kid yourself on that. Anything that creates confusion shouldn't be tolerated in math. It needs to strive to be as concise as possible.

[Murska]

Well, if you assume that the equation is trivial in nature, then writing additional brackets is not only more work and a waste of time and energy but can also make it more confusing, as people have to do a double-take and go "Why are there brackets here? Am I missing something?"

[AsteriskAmp]

With the advent of computer software packages such as Mathematica and MATLAB, I don't think I've actually bothered to write out the work anymore. Pop formula into Mathematica, setup inputs, Solve[f(x) == y, 0] and so on. As The Extinguisher pointed out, people who frequently use mathematics would be more likely to either write "7" in the above example, or express things in terms of other things (e.g. 3∏/2), in which you avoid all the sticky bits of excusing our dear aunt Sally.

That you can use software for making calculations faster isn't excuse to not know something as simple as Order of Operations. Why learn to read if there are text to sound programs? Or learn to read music if there are midi reading software which produce sound? Or learn another language since you have google translate?

Each time you do more than two operations you have Order of Operations, even if not so glaringly patent. As for representation, yes, someone would write seven in that case, but in the case of operations with name-worthy constants and variables you would need to use Order of Operations if you wanted to solve in relation to one or another, or just change the form to something you actually want, of which Mathematica and Matlab may not always do the last one.

Why not though? There hasn't been a real reason why we shouldn't use brackets other than "people shouldn't be so dumb they don't know basic math." Not using brackets is great for drilling students on order of operations rules, but it's not useful for anything else. This does create confusion, don't kid yourself on that. Anything that creates confusion shouldn't be tolerated in math. It needs to strive to be as concise as possible.

There is no real reason to use it either other than people could get confused about something rather trivial.

It may create confusion to some, but it IS concise, there is no multiple value answer or misinterpretation on representation, it's human error which leads to a wrong result. I acknowledge it is an order of operations designed problem and normally anyone would express it differently (starting with the fact that you almost never see / or ÷ but instead it would be seen as a fraction, and ending with the fact that it indeed is just 7) but that doesn't mean it is wrong the way it is, it's another perfectly valid representation, same thing saying three is 3, 4-1, d(x^2+x+1)/dx|x=1, 3sin(Pi/2), 2.999999999..., etc.

[Tengu_temp]

Why not though? There hasn't been a real reason why we shouldn't use brackets other than "people shouldn't be so dumb they don't know basic math."

And that's a bad reason? People shouldn't be so dumb that they don't know basic math.

[nedz]

It does come up sometimes, but normally not with something so trivial.

If you just blithely type this into a calculator you might not realise that it's order of evaluation is
6,-,1,=,x,0,=,+,2,=,/,2,=
which is probably what caught most people out.

[pendell]

Pendell, you did something very bad when your lack of basic math was pointed out - you got defensive about it. You tried to find fault in something else, namely the structure of the equation. You tried to downplay the importance of knowing it or that it's a matter of course in the first place. I've seen people with reasoning like that before. If you don't know something that's common knowledge, you're an idiot. I'm no idiot and I didn't know it, so people can't be expected to know it.

Not so. I made my best guess based on memory, then asked questions, did the research , found my mistake, reworked to get the correct answer, solved it while noting the previous error. This is what research is all about.

I'm also, so far as I know, the only person in this thread to admit being wrong.

The equation that I pointed out was deliberately written to make people look stupid by relying on obscure rules of precedence which most people do not recall. Yes, such things are taught in grade school alongside things such as split infinitives, the use of 'who' vs. 'whom', what a gerund is, and how to diagram a sentence. Most people forget most of that but retain enough skill to function in their chosen professions.

If I were the only person who had made that mistake, then I would shut up and take it in good grace. Since I am by no means the only person who made that mistake, then I state that the equation was deliberately written to trip people up as the intended effect and did so.

I'll also thank you not to criticize me, and especially not publicly. Critique my silly ideas and statements, yes, by all means. But when you cross the line from attacking ideas to personal attack, I do not take it at all well. Most especially since :checks the birthdates on the profiles: -- I do not take being patronized at all well from someone fourteen years younger than I am.

This does create confusion, don't kid yourself on that. Anything that creates confusion shouldn't be tolerated in math. It needs to strive to be as concise as possible.

Agreed.

Respectfully,

Brian P.

[Iruka]

The equation that I pointed out was deliberately written to make people look stupid by relying on obscure rules of precedence which most people do not recall.

Wait, multiplication/division before addition/substraction is obscure now?

[AsteriskAmp]

I'm also, so far as I know, the only person in this thread to admit being wrong.

To a mathematical certainty. While in all other cases the discussion has been about whether it is "necessary" to include non-essential parenthesis and

The equation that I pointed out was deliberately written to make people look stupid by relying on obscure rules of precedence which most people do not recall. Yes, such things are taught in grade school alongside things such as split infinitives, the use of 'who' vs. 'whom', what a gerund is, and how to diagram a sentence. Most people forget most of that but retain enough skill to function in their chosen professions.

They are not obscure, they are fairly common and required for every single mathematical operation ever (just to a different degree). What it relies on are the wrongly taught mnemonics for them which are something which should at the very least be corrected, what's wrong isn't the notation but the way the concepts are being taught.

If I were the only person who had made that mistake, then I would shut up and take it in good grace. Since I am by no means the only person who made that mistake, then I state that the equation was deliberately written to trip people up as the intended effect and did so.

Or you happen to have the same reasons for making the same mistake. Mathematics is not based on population consensus. 1+1 won't become 3 because everyone makes the same mistake, otherwise we'd be voting to see which values certain series take.

[Castaras]

Wait, multiplication/division before addition/substraction is obscure now?

Pretty much. There's been some examples on facebook which were deliberately misleading, but the example you posted isn't. It's basic knowledge that for those who're even slightly good at maths should be fine with doing automatically.

[TheFallenOne]

If I were the only person who had made that mistake, then I would shut up and take it in good grace. Since I am by no means the only person who made that mistake, then I state that the equation was deliberately written to trip people up as the intended effect and did so.

I'll also thank you not to criticize me, and especially not publicly. Critique my silly ideas and statements, yes, by all means. But when you cross the line from attacking ideas to personal attack, I do not take it at all well. Most especially since :checks the birthdates on the profiles: -- I do not take being patronized at all well from someone fourteen years younger than I am.

I went out of my way to point out you're NOT stupid as I think I remember you writing more agreeable things before. I'm willing to bet some people came to different conclusions in light of this thread, but don't say so for obvious reasons. I was trying to help you in dealing with this, and there is no way around it, embarrassing failure.

And despite your 'Not so', you are doing exactly what I said you do: trying to downplay the whole thing. This isn't an obscure piece of knowledge by any reasonable standard.

[pendell]

I don't use Facebook, is it possible to get an extract ?

I just checked. There are 129,953 comments. I pulled the last 156. Of which, 138 had answers

The raw data is in the spoilers.

Spoiler

Show

7
1
1
1
7
5
0
7
5
0
7
7
5
5
7
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7
3.5
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0
5
7
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7
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1
7
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7
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7
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7
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7
7
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7
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7
1
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1
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7
1
1
1
7
7
7
1
1
1

I summarize:

138 answers
6 answered 0
56 answered 1
2 answered 2
1 answered 3.5
4 answered 4
20 answered 5
3 answered 6
45 answered 7
1 answered 14

so ... 45/138 = 0.32 * 100 = 32% got the correct answer. If some kind person were to take a histogram , we should see a cluster around 5 and 7, another big spike around 1, and outliers everywhere else.

Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math? Well, the mean SAT score for mathematics is 501 out of a possible 800. This would imply (to me) that the average person is conversant with primary school math but at sea when it comes to secondary school mathematics.

So I conclude that the fault is with the writer of the equation. Yes, it may be technically clear and unambiguous. And if you're writing for an audience of mathematicians , that is probably good enough. But when only 1/3rd of your target audience is able to come to the correct answer, the logical conclusion is that either the person has failed to communicate clearly or that this is a deliberate trick question. Given the identity of the author, it is almost certainly the second.

1+1 won't become 3 because everyone makes the same mistake, otherwise we'd be voting to see which values certain series take.

I quite agree. Nonetheless, an equation can deliberately be written to be more obscure than strictly necessary.

1+1 = 2
is the same as
(3^0) + (natural log of e) = ((10/10) * (1000) ) / 500

but the first is much more clear. As I stated, there is an art to writing an equation such that it is as clear as possible to as many readers as possible. The equation in the OP could be immeasurably improved for general consumption by an audience of high school graduates by the addition of brackets.


Respectfully,

Brian P.

[noparlpf]

Recently I saw the following on facebook:



My solution uses the precedence rule I was taught in school -- My Dear Aunt Sally (MDAS) -- multiply first (1x0), then divide (2/2), then add (0+1) then subtract (6-1) = 5.

However, I noticed some other people were using a very different precedence rule -- BODMAS (open brackets, division, multiplication, addition, subtraction). It reverses the order of division and subtraction from what I was taught.

Who uses BODMAS? Where is it taught? Is there any likelihood for a mistake if, say, you're calculating a tax result using MDAS when the original author intended BODMAS? Are there any other precedence rules I should know of?

ETA: I'm not convinced my answer is right. There may have been a mistake. Different people have come up with 0, 1, 5, and 7. I'm not sure whether they are correctly applying alternate precedence rules, whether they made a mistake, or whether *I* made a mistake.

Respectfully,

Brian P.

Well, in the US it's always, always PEMDAS. BUT people forget that M and D are considered equivalent, and A and S are considered equivalent; when it comes down to them, just go left to right in order. More properly, it's PE(M/D)(A/S). It's just retained as PEMDAS because that's easy to say.
Seems like BODMAS (could equally well be written BOMDAS) is just the European way of saying it?
Anyway, I kind of hate these things floating around facebook that are about basic order of operations--one, they cause huge arguments, and people are asinine enough without provocation, and two, they're designed to make people feel stupid.
For your example, I get 6 - 1 x 0 + 2 / 2 = 6 - 0 + 1 (and here we just go left to right, so it's 6-0 first) = 6 + 1 = 7.
My TI-84 says it's 7, and WolframAlpha says it's 7.

[pendell]

I went out of my way to point out you're NOT stupid as I think I remember you writing more agreeable things before. I'm willing to bet some people came to different conclusions in light of this thread, but don't say so for obvious reasons. I was trying to help you in dealing with this, and there is no way around it, embarrassing failure.

Then I will accept your feedback in the spirit it was given and leave it at that. Thank you.

At any rate, now I know the answer is 7 and why and what I did wrong the first time. For my purposes, that is sufficient. As far as I'm concerned, if a little embarrassment leads to a more correct answer it is a price well worth paying.

Respectfully,

Brian P.

[TheFallenOne]

Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math?

If the Facebook posters there were a representative group of the general population, unable to see previous comments or use the internet before answering, and we knew they weren't drunk, high, in a hurry or trolling when answering, then yes, it would.

1+1 = 2
is the same as
(3^0) + (natural log of e) = ((10/10) * (1000) ) / 500

but the first is much more clear. As I stated, there is an art to writing an equation such that it is as clear as possible to as many readers as possible.

You do realize that reducing a complicated formula to a way simpler one is a big part of school mathematics? Of course 1+1=2 is easier to grasp at a glance. That doesn't make a more complicated equation illegitimate.

[AsteriskAmp]

I just checked. There are 129,953 comments. I pulled the last 156. Of which, 138 had answers

The raw data is in the spoilers.

Spoiler

Show

7
1
1
1
7
5
0
7
5
0
7
7
5
5
7
7
7
3.5
1
0
5
7
1
1
1
7
2
1
1
1
7
1
1
7
7
5
5
1
1
7
1
1
0
7
1
1
1
7
7
5
1
1
5
1
1
7
4
7
7
7
7
7
0
7
7
1
7
7
1
1
5
7
1
5
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1
7
5
5
1
5
7
1
2
7
1
5
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5
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7
7
7
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7
1
1
7
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7
1
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7
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5
1
1
6
6
14
7
1
1
1
7
7
1
1
1
1
1
1
1
1
7
1
1
1
7
7
7
1
1
1

I summarize:

138 answers
6 answered 0
56 answered 1
2 answered 2
1 answered 3.5
4 answered 4
20 answered 5
3 answered 6
45 answered 7
1 answered 14

so ... 45/138 = 0.32 * 100 = 32% got the correct answer. If some kind person were to take a histogram , we should see a cluster around 5 and 7, another big spike around 1, and outliers everywhere else.

Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math? Well, the mean SAT score for mathematics is 501 out of a possible 800. This would imply (to me) that the average person is conversant with primary school math but at sea when it comes to secondary school mathematics.

So I conclude that the fault is with the writer of the equation. Yes, it may be technically clear and unambiguous. And if you're writing for an audience of mathematicians , that is probably good enough. But when only 1/3rd of your target audience is able to come to the correct answer, the logical conclusion is that either the person has failed to communicate clearly or that this is a deliberate trick question. Given the identity of the author, it is almost certainly the second.

It means exactly that though. A large share of the population lacks a correct understanding of primary school math.

The SAT results (which are also not accessible to me for some reason) only prove of how well a population of United States High Schoolers can perform a standardized test most schools prepare them for, instead of actually teaching them the skills which would aid them in the test but also serve for different situations.

Or, a third interpretation, said people have learnt the concepts wrong and hence all are making mistakes. Don't blame the writer, blame the way it is almost universally taught through (incorrect) mnemonics.

I quite agree. Nonetheless, an equation can deliberately be written to be more obscure than strictly necessary.

1+1 = 2
is the same as
(3^0) + (natural log of e) = ((10/10) * (1000) ) / 500

but the first is much more clear. As I stated, there is an art to writing an equation such that it is as clear as possible to as many readers as possible. The equation in the OP could be immeasurably improved for general consumption by an audience of high school graduates by the addition of brackets.

Alternatively 3^0+ln(e) = 10÷10*1000÷500

The clearest way of writing that out would have been 7=?. The purpose of those expressions are to show how a large share of the population lacks basic mathematical knowledge. Even then. odds are you'd see a part of the population get it wrong anyway.

[pendell]

That doesn't make a more complicated equation illegitimate.

Agree. But "illegitimate" and "clear/easy to follow" are not the same thing.

An equation may be legitimate and yet still be hard to follow and badly written. Likewise , an equation may be simple, clear , easy to follow, and wrong.

I think that is the heart of our disagreement. I do not view mathematics simply as whether an equation is accurate or not. I believe mathematics is in part language -- a way for people to express ideas to each other. Consequently I look for ways not just to express accurate ideas, but to express them as clearly as possible with a minimum of interpretation errors. In the above case, I believe the equation was written to maximize reader error. It is apparently simple, yes, but when 68% of the readers get it wrong, that tells me it is deceptively simple. It is a trick question, which is possible both in English and in mathematics.

Respectfully,

Brian P.

[nedz]

138 answers
6 answered 0
56 answered 1
2 answered 2
1 answered 3.5
4 answered 4
20 answered 5
3 answered 6
45 answered 7
1 answered 14

so ... 45/138 = 0.32 * 100 = 32% got the correct answer. If some kind person were to take a histogram , we should see a cluster around 5 and 7, another big spike around 1, and outliers everywhere else.

Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math? Well, the mean SAT score for mathematics is 501 out of a possible 800. This would imply (to me) that the average person is conversant with primary school math but at sea when it comes to secondary school mathematics.

So I conclude that the fault is with the writer of the equation. Yes, it may be technically clear and unambiguous. And if you're writing for an audience of mathematicians , that is probably good enough. But when only 1/3rd of your target audience is able to come to the correct answer, the logical conclusion is that either the person has failed to communicate clearly or that this is a deliberate trick question. Given the identity of the author, it is almost certainly the second.

Looking at this from an error analysis point of view:

56 (40%) probably used a calculator, badly
20 (14.5%) made the sign error (sign errors are very common)
45 (23.5%) got the correct answer
11 (8%) some random error, either hopeless or just messing around.
6 (4%) entered 0 (I'm not sure why here)

I think its quite an interesting maths question in that it exposes several common errors. Is it a trick question ? Or is it trying to teach something ?

[noparlpf]

I think all the analysis of the facebook answers is flawed--you're forgetting the huge population of people who don't care and just put something random down to cause trouble, and the smaller population of people who do know but are putting down incorrect answers to cause trouble.

So we have a few groups.
1. People who can't do grade school-level math.
2. A few subgroups that boil down to people who are just being asses. (I don't like the word "trolling". Call it what it is--being a ****.)
3. People who misread the problem, or used a calculator and mistyped it.
4. People who can do grade school-level math.

[pendell]

I think its quite an interesting maths question in that it exposes several common errors. Is it a trick question ? Or is it trying to teach something ?

Most likely the first. Still, you've a good point about error analysis. If it is trying to teach something, it's probably "Don't just copy/paste an equation into a calculator and expect the right answer. THINK about it. "

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

Respectfully,

Brian P.

[nedz]

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

I'm just guessing, but it is sort of what I do for a living

[Knaight]

Not so. I made my best guess based on memory, then asked questions, did the research , found my mistake, reworked to get the correct answer, solved it while noting the previous error. This is what research is all about.

I'm also, so far as I know, the only person in this thread to admit being wrong.

You also appear to have immediately jumped to the conclusion that admitting being wrong somehow grants you a moral authority over those who have not admitted being wrong on account of not being wrong in the first place. It does no such thing, and certainly doesn't substitute for an actual argument.

I'll also thank you not to criticize me, and especially not publicly. Critique my silly ideas and statements, yes, by all means. But when you cross the line from attacking ideas to personal attack, I do not take it at all well. Most especially since :checks the birthdates on the profiles: -- I do not take being patronized at all well from someone fourteen years younger than I am.

Similarly, "I'm older than you" is not a preposition from which "I'm better than you" can be directly derived from reasoning that isn't ridden with flaws. Your demand for a raised discourse rings hollow when you implicitly insult everyone younger than you.

The SAT results (which are also not accessible to me for some reason) only prove of how well a population of United States High Schoolers can perform a standardized test most schools prepare them for, instead of actually teaching them the skills which would aid them in the test but also serve for different situations.

More to the point, a 514 out of 800 on the SAT math section is an indication of mathematical deficiency given the current difficulty of the SAT math section*. It's a mixture of fairly basic arithmetic, fairly basic geometry, and extremely trivial algebra. It also briefly touches upon trigonometric functions and matrices, though there were few enough of them as recently as 2010 to reduce a score more than 40 or 50 points were every single one of them failed. That doesn't explain the loss of 286 points on average, let alone the loss of 386-401 points for approximately 15.9 percent of the test takers, assuming a roughly normal distribution. This is consistent both with a grasp of elementary math and utter inability in higher math, or with a pattern of knowledge gaps throughout mathematical levels, the latter of which would also explain the results gathered from Facebook**, particularly once one takes into account how neither the SAT nor Facebook are measures that accurately evaluate the population, due to heavy selection biases in both.

*I found data that indicate a 514 mean in 2011, with a standard deviation of approximately 100 to 115, with different sources varying. I will thus be using these numbers.

**I have not personally confirmed these, so this is a very provisional statement.

[Felyndiira]

So I conclude that the fault is with the writer of the equation. Yes, it may be technically clear and unambiguous. And if you're writing for an audience of mathematicians , that is probably good enough. But when only 1/3rd of your target audience is able to come to the correct answer, the logical conclusion is that either the person has failed to communicate clearly or that this is a deliberate trick question. Given the identity of the author, it is almost certainly the second.

If you posted an equation with an integral on facebook and forced 100,000 people to respond to it with what they think the answer is, I would bet that a similar number of people would not be able to. This does not prove that integrals are misleading or that all calculus equations should be represented with a shaded graph - it just means that those people do not understand calculus.

If you were to ask 100,000 people what the capital of Zaire is, most of them would likely not know the answer. That does not mean that the capital of Zaire is somehow ambiguous - merely, that most people do not know the capital of Zaire for whatever reason. It does not mean that whenever you write, say, "The President went to Kinshasa" you have to write "The President went to Kinshasa, Capital of Zaire, a nation in Africa located at the middle of the continent bordering, intersecting the equator, etc. etc." even if a number of people on facebook cannot pinpoint the capital or the nation on a map.

The equation is not at all ambiguous simply because there is a very clearly defined, internationally-agreed upon procedure for the order of operations; one that is taught within elementary school. The fact that 30,000 out of 100,000 people on Facebook does not know PEMDAS does not negate the fact that it exists and is one of the basics of mathematics in the same way that x number of people not knowing the capital of Zaire doesn't make Kinshasa "obscure and unclear in communication."

Does this imply that 68% of the population is so deficient in math skills that they do not understand primary school math? Well, the mean SAT score for mathematics is 501 out of a possible 800. This would imply (to me) that the average person is conversant with primary school math but at sea when it comes to secondary school mathematics.

EDIT: Corrected a few things I did not remember correctly about the SATs. You can achieve a score of around 240 by leaving the entire test blank. This means that 501 as the average score means that on average, test takers answered around 50% of the math questions correctly. That does not indicate a national competency in Math by any means.


On a completely random note, I actually think that mathematicians are more likely to disregard trivial stuff and write in absolutely ambiguous manners. I can't count how many times I wrote

log x+2

to mean log (x+2) and

log x + 2

to mean (log x) + 2 on my exams for courses like probability theory.