A math problem.

[mmswm]

But the logic of my example was also left to right.  10 red plus 17 yellow is 27 plus 6 green is 33 plus 52 black is 85 times 11 boxes is 935 bracelets.  The order I'm doing my math has nothing to do with math rules or parenthesis and everything to do with common sense rules - first I add up how many in a box then I multiply it by number of boxes. I would never think "ok I have 11 boxes" without then thinking "ok first I need to figure out how many in a box."   My intuition is based on that kind of real life situation.

Which is why one uses parenthesis - to communicate.  Not using parenthesis is failing to communicate the order you want things done in.  Regardless of order of operations being a rule, the communication is simply not there - if you want to be clear, communicate clearly, instead of arguing "but you're supposed to know already!"  Clearly plenty of people don't know already.  If you fail to communicate properly its not fair to say someone else failed to understand correctly.

And again, I agree with you!  But your example had a context -- the real life situation came first, and you derived an equation in accordance with that situation.  The situation itself suggested the order that operations should be performed.

In "pure" mathematics, though, expressions and equations can and do exist without real-world context -- there isn't a real-life "word problem" from which we can know the order to perform operations.  In situations like that -- especially where mathematicians are communicating with each other -- it becomes necessary to have a "rule," even if that rule seems arbitrary (and it often is).

I'm sure people are now asking, why do we teach something that relates more to "pure" math than to real world applications?  The answer to that is because order of operations is necessary for higher math and science classes.

I also agree with you about parentheses -- for best clarity, they should always be used.  Except, of course, when teaching order of operations to students.   

P.S. to Aeris -- Excellent point about reverse Polish!  My husband said the same thing when I told him about this message thread.  I remember having to get used to reverse Polish about 25 years ago when I had (I think it was) an HP calculator.

I'm going to disagree with you.  The rules of mathematics developed over centuries in an effort to understand the world around us.  Throughout the ages, philosophers, scientists and mathematicians have observed physical situations and strove to reduce their observations into nice, neat equations or processes. In some cases, more than one algorithm is developed and one wins out to become "standard", usually because it is either easier, more intuitive, or a flaw, or limitation is found with the others. In the case of reverse Polish, it becomes unwieldy when we move from simple numerical equations and introduce polynomials.  The "standard" order of operations does not.  On that point alone it makes more sense to keep that one standard. 

Even higher, more complex mathematics have a foundation in real world problems, though it becomes increasingly difficult to find simple real world analogies as the math gets more complex. The rules of math are never arbitrary.  There's a reason for all of them.

[Slartibartfast]

How about this:

You are a small business owner strugglig to keep afloat in these economic times.  You order a box of bracelets as store stock.  Its labeled:
10 red
17 yellow
6 green
52 black

Your finger slips and you order 11 boxes instead.  They are non-returnable.  Your store sells about 200 bracelets a week. How long will it take for you to move this merchandise, so you can plan when to make your next inventory order?

First you figure out how many bracelets you have:

(10+17+6+52)x11=935

Doing the multiplication first would be wrong:

10+17+6+(52x11)=605

Ok we can rework to make the multiplication first, but its a LOT more work and a stupid way to figure it out:

(10x11)+(17x11)+(6x11)+(52x11)=935

But its MUCH easier and more logical to do the addition first and then multiply.  This way when you divide your number you get the correct timeline.  It will take almost 4.5 weeks to move that stock, whereas if you multiplied first you would think you'd be almost sold out in 3. Assuming a half week delivery timeline, here is a HUGE cash flow difference for a business owner between ordering your next inventory batch at 2.5 weeks versus at 4 weeks!

If you don't use parentheses and order of operation, though, your answer would change depending on what colors you count first.  11 boxes x (10 red + 17 yellow + 6 green + 52 black) = 935 bracelets, done correctly.  If you go left to right and ignore parentheses, you'd get 185 bracelets if you counted red first ((11x10)+17+6+52=185), or 255 bracelets if you counted yellow first ((11x17)+10+6+52=255), or 605 bracelets if you did black first ((11x52)+10+17+6=605).  Obviously the number of bracelets present shouldn't change depending on which order you count them in, so if you were the store owner and getting different numbers every time you'd know you really needed to do the parentheses first 

[Acadianna]

Even higher, more complex mathematics have a foundation in real world problems, though it becomes increasingly difficult to find simple real world analogies as the math gets more complex. The rules of math are never arbitrary.  There's a reason for all of them.

I didn't say higher math lacked real-world applications or foundations.  My point was merely that, for expressions and equations that are given without real-world context, a rule for order of operations is necessary.  And stand-alone expressions and equations become more and more common as students progress to higher math.  At least, that's how I remember it.  I don't recall very many real-world word problems when I took advanced calculus, for example.

ETA:  I'd also be wary of using the word "never" -- it only takes one counterexample to refute the contention, and I can think of several.  For example, the use of the word "googol" to represent 10^100.  Yes, there's a "reason" for the term (the person who named it asked his child to suggest something) but that doesn't make the term itself any less arbitrary.  It's used by agreement, and not because it's a "better" term than any other.

Even one of the most basic elements of math is arbitrary -- the use of "x" as a commonly-understood variable.  If I write "ax^2 + bx + c," then most people understand that x is a variable and a, b, and c are constants.  This is by arbitrary agreement, to foster precise communication, and not because "x" is somehow better for the purpose than any other letter of the alphabet.

[WillyNilly]

How about this:

You are a small business owner strugglig to keep afloat in these economic times.  You order a box of bracelets as store stock.  Its labeled:
10 red
17 yellow
6 green
52 black

Your finger slips and you order 11 boxes instead.  They are non-returnable.  Your store sells about 200 bracelets a week. How long will it take for you to move this merchandise, so you can plan when to make your next inventory order?

First you figure out how many bracelets you have:

(10+17+6+52)x11=935

Doing the multiplication first would be wrong:

10+17+6+(52x11)=605

Ok we can rework to make the multiplication first, but its a LOT more work and a stupid way to figure it out:

(10x11)+(17x11)+(6x11)+(52x11)=935

But its MUCH easier and more logical to do the addition first and then multiply.  This way when you divide your number you get the correct timeline.  It will take almost 4.5 weeks to move that stock, whereas if you multiplied first you would think you'd be almost sold out in 3. Assuming a half week delivery timeline, here is a HUGE cash flow difference for a business owner between ordering your next inventory batch at 2.5 weeks versus at 4 weeks!

If you don't use parentheses and order of operation, though, your answer would change depending on what colors you count first.  11 boxes x (10 red + 17 yellow + 6 green + 52 black) = 935 bracelets, done correctly.  If you go left to right and ignore parentheses, you'd get 185 bracelets if you counted red first ((11x10)+17+6+52=185), or 255 bracelets if you counted yellow first ((11x17)+10+6+52=255), or 605 bracelets if you did black first ((11x52)+10+17+6=605).  Obviously the number of bracelets present shouldn't change depending on which order you count them in, so if you were the store owner and getting different numbers every time you'd know you really needed to do the parentheses first 

Nope.     You reversed my numbers and changed my inventory style.  I had me adding up the box totals first and based on the already existing label, plus most importantly you put the 11 first.  I didn't think the logical equation was "I have 11 boxes of a combination of10, 17, 6, and 52 bracelets" but rather "ok lets see there are how many per box?  And I have how many boxes now?"
If one goes left to right in the absence of parenthesis my equation still works perfectly:

10+17+6+52x11=935

I just didn't write it like that because I think that's sloppy, like bad grammar.  I was taught that mathematical equations, when properly written, include parenthesis for groups and quite frankly I think this thread has shown how very important they are for clarity to the non-mathematician masses.

[AdakAK]

Which is why one uses parenthesis - to communicate.  Not using parenthesis is failing to communicate the order you want things done in.  Regardless of order of operations being a rule, the communication is simply not there - if you want to be clear, communicate clearly, instead of arguing "but you're supposed to know already!"  Clearly plenty of people don't know already.  If you fail to communicate properly its not fair to say someone else failed to understand correctly.

Many times people do not understand fairly clear directions.  That is not somehow the fault of the direction makers, if the rules are set out and followed then the failure is on those who do not understand, however many there are.  Strict L-R without considering order of operations is wrong.  It might get you the right answer, when rules of operations is the same as L-R but that makes it lucky (like me when I get the right answer), not right. 

[Iris]

Since I am a teacher, I'm very well aware of how much teachers get bashed.  As a secondary math teacher, I'm also very well aware of the major shortfalls of elementary math teachers.  This is not the time or place to discuss it at length, but there's a reason why we perform so poorly as a nation in math and science.  While it's not all the fault of the teachers, we do play a role.  My most frustrating experience as a secondary teacher is having to "unteach" concepts that were taught incorrectly by previous teachers.

I do agree that WillyNilly seems to be a well-spoken, intelligent individual.  That's why I keep coming back to the teachers.  If somebody as plainly intelligent as WillyNilly still harbors confusion on such a basic concept, that very clearly points the blame towards those responsible for teaching said basic concepts.  As for the "use it or lose it" comment, we're not talking about trigonometry.  We're talking about basic arithmetic.  There's a vast difference in the usability of those two topics in every day life.

But WillyNilly (Hi, WillyNilly! Sorry for speaking about you like this, please appreciate that I am using you as a handy example, NOT trying to pick apart your life. Feel free to tell me to stop.) CAN do basic arithmetic, including complex and multi-step problems that involve the distributive law, that's very clear from her posts. She just wasn't familiar with the rules of formal, written mathematics. My attitude is this: So what? The last time I studied grammar formally was 30 years ago and I couldn't tell you what a subjunctive clause is to save my life (if there even is such a thing) but I can still communicate clearly most of the time, teach students effectively and write professional letters, emails and reports as required. That's not an indictment on the teacher who taught me grammar - it's a reflection that in my own particular version of 'the real world' I don't need, or have a particular interest in, the strict, formal rules of written English.

Yes, I agree it is very frustrating when children arrive in 7th grade not knowing the basic facts of formal mathematics and that IS something to critisise a teacher for in some cases, but I strongly disagree that not remembering the formal rules of mathematics however many years after someone left school is the same thing at all.

Anyway, I suspect that we have a different philosophy of teaching so we may never agree on this. I won't say let's agree to disagree because it irritates the life out of me when people make a point and then say that, so I freely offer you the last word 

[Aeris]

How about this:

You are a small business owner strugglig to keep afloat in these economic times.  You order a box of bracelets as store stock.  Its labeled:
10 red
17 yellow
6 green
52 black

Your finger slips and you order 11 boxes instead.  They are non-returnable.  Your store sells about 200 bracelets a week. How long will it take for you to move this merchandise, so you can plan when to make your next inventory order?

First you figure out how many bracelets you have:

(10+17+6+52)x11=935

Doing the multiplication first would be wrong:

10+17+6+(52x11)=605

Ok we can rework to make the multiplication first, but its a LOT more work and a stupid way to figure it out:

(10x11)+(17x11)+(6x11)+(52x11)=935

But its MUCH easier and more logical to do the addition first and then multiply.  This way when you divide your number you get the correct timeline.  It will take almost 4.5 weeks to move that stock, whereas if you multiplied first you would think you'd be almost sold out in 3. Assuming a half week delivery timeline, here is a HUGE cash flow difference for a business owner between ordering your next inventory batch at 2.5 weeks versus at 4 weeks!

If you don't use parentheses and order of operation, though, your answer would change depending on what colors you count first.  11 boxes x (10 red + 17 yellow + 6 green + 52 black) = 935 bracelets, done correctly.  If you go left to right and ignore parentheses, you'd get 185 bracelets if you counted red first ((11x10)+17+6+52=185), or 255 bracelets if you counted yellow first ((11x17)+10+6+52=255), or 605 bracelets if you did black first ((11x52)+10+17+6=605).  Obviously the number of bracelets present shouldn't change depending on which order you count them in, so if you were the store owner and getting different numbers every time you'd know you really needed to do the parentheses first 

Nope.     You reversed my numbers and changed my inventory style.  I had me adding up the box totals first and based on the already existing label, plus most importantly you put the 11 first.  I didn't think the logical equation was "I have 11 boxes of a combination of10, 17, 6, and 52 bracelets" but rather "ok lets see there are how many per box?  And I have how many boxes now?"
If one goes left to right in the absence of parenthesis my equation still works perfectly:

10+17+6+52x11=935

I just didn't write it like that because I think that's sloppy, like bad grammar.  I was taught that mathematical equations, when properly written, include parenthesis for groups and quite frankly I think this thread has shown how very important they are for clarity to the non-mathematician masses.

How were you taught to solve something like: 5x² + 3x - 9, where x=3?

[Acadianna]

That's not an indictment on the teacher who taught me grammar - it's a reflection that in my own particular version of 'the real world' I don't need, or have a particular interest in, the strict, formal rules of written English.

I think we teachers sometimes forget that we remember the things we studied in school only because we now teach them year after year.  For people who have no particular use for the material, the learning fcan fade over time.

And sometimes the rules even change.  I was taught, for example, that when using commas to separate words in a series, one never puts a comma before "and," since the word "and" itself serves as a separator.  Apparently, the current rule is just the opposite (though there seems to be continuing argument about the change among grammarians).  I know this only because I've occasionally taught resource language arts; otherwise, I'd have had no clue that anything had changed.

Another one that came up in the last month -- I do special ed inclusion duties in 6th grade classes, including world cultures.  When the general ed teacher taught students about the oceans, to my surprise she mentioned five of them -- the four I knew (Atlantic, Pacific, Indian, and Arctic) and a new one (Southern).  When I commented on not having heard of that one before (and I've taught resource world cultures in the past), she said that it was a relatively recent addition to geography.

So even things we think of as rock-solid precedent can change from what we learned in school.

[WillyNilly]

How about this:

You are a small business owner strugglig to keep afloat in these economic times.  You order a box of bracelets as store stock.  Its labeled:
10 red
17 yellow
6 green
52 black

Your finger slips and you order 11 boxes instead.  They are non-returnable.  Your store sells about 200 bracelets a week. How long will it take for you to move this merchandise, so you can plan when to make your next inventory order?

First you figure out how many bracelets you have:

(10+17+6+52)x11=935

Doing the multiplication first would be wrong:

10+17+6+(52x11)=605

Ok we can rework to make the multiplication first, but its a LOT more work and a stupid way to figure it out:

(10x11)+(17x11)+(6x11)+(52x11)=935

But its MUCH easier and more logical to do the addition first and then multiply.  This way when you divide your number you get the correct timeline.  It will take almost 4.5 weeks to move that stock, whereas if you multiplied first you would think you'd be almost sold out in 3. Assuming a half week delivery timeline, here is a HUGE cash flow difference for a business owner between ordering your next inventory batch at 2.5 weeks versus at 4 weeks!

If you don't use parentheses and order of operation, though, your answer would change depending on what colors you count first.  11 boxes x (10 red + 17 yellow + 6 green + 52 black) = 935 bracelets, done correctly.  If you go left to right and ignore parentheses, you'd get 185 bracelets if you counted red first ((11x10)+17+6+52=185), or 255 bracelets if you counted yellow first ((11x17)+10+6+52=255), or 605 bracelets if you did black first ((11x52)+10+17+6=605).  Obviously the number of bracelets present shouldn't change depending on which order you count them in, so if you were the store owner and getting different numbers every time you'd know you really needed to do the parentheses first 

Nope.     You reversed my numbers and changed my inventory style.  I had me adding up the box totals first and based on the already existing label, plus most importantly you put the 11 first.  I didn't think the logical equation was "I have 11 boxes of a combination of10, 17, 6, and 52 bracelets" but rather "ok lets see there are how many per box?  And I have how many boxes now?"
If one goes left to right in the absence of parenthesis my equation still works perfectly:

10+17+6+52x11=935

I just didn't write it like that because I think that's sloppy, like bad grammar.  I was taught that mathematical equations, when properly written, include parenthesis for groups and quite frankly I think this thread has shown how very important they are for clarity to the non-mathematician masses.

How were you taught to solve something like: 5x² + 3x - 9, where x=3?

How was I taught to figure that out ^ ?
For the life of me I can't remember.  But logically the first step is to figure out what x is.  Since you are telling me its 3, the second step is to ask why on earth you would use x?  Just write your equations with numbers; what possible real world application does using x have if you know x=3?  I have no use in my life for theoretical, or as called up thread "pure" math, and reject it as impractical just as I would reject the idea that one should only consider using a professional stove or drive an industrial grade vehicle. Sure a teacher should go over pure math, but as Iris is saying, to me the sign of knowing math, for the average non-mathematician is knowing how to figure out the situation and break it down, not knowing the formal rules of written math.

In average life, all math can be broken down to a word problem.  If you give me the word problem to that equation I bet I can tell you the exact correct method of working it out. What are we trying to figure out?

[mmswm]

*sigh*

I'm afraid I'm not expressing myself well.  Iris, I think we could have an enjoyable discussion about this if we could talk in person.  My limitations in expressing myself through text are clearly coming through, because I am getting frustrated at the fact that what I'm trying to say is apparently not coming across well to you and several others.

I'm going to try one last time, not because I want the last word, but because I want to try to accurately convey what I'm trying to say.  Also, WillyNilly please also forgive me for using you as a convenient example.  You are obviously an intelligent, well-spoken individual and I don't want to make you feel insulted in any way.

My philosophy of math education is that throughout the years, generations of teachers have presented topics without any real sense of context.  They push "rules" and "procedures" without teaching why they work or how they are built up from topics that came beforehand.  This leads to students who either don't get it, or forget fairly basic concepts shortly after they're taught. Even in more recent generations, with the push to teach the "why", too many elementary teachers just don't know enough themselves to really lay a solid foundation. Occasionally, teachers even come up with "whys" that are completely wrong (like teaching students that multiplication is repeated addition), which leads to confusion and mental brick walls when students have to build on prior knowledge.

I see my job as a math teacher is to "make it real" to my students, while at the same time laying, or re-laying the necessary foundation for my students to build upon in higher level math. As such, I've spent a lot of time working out real-world analogies for everything from basic arithmetic to Calculus, Linear Algebra and beyond (remember I've also taught at the community college level, and have tutored upperclass students).  I feel like a failure if any of my students leave my class thinking that math has all these rules that are arbitrary and don't make sense. Every single rule or algorithm I've ever taught has been accompanied by either a real life example or a build up from something that came before it.

WillyNilly's case in particular frustrates me.  Here she is arguing that the "rule" is not intuitive, yet every single example she comes up with shows an intuitive understanding of numerical reasoning.  This frustrates me on two levels.  First, I don't like seeing an obviously intelligent person thinking she's "bad" at something, when clearly she's not. Somebody, somewhere along the lines failed to connect the dots for her, leaving her with this idea that something doesn't make sense.   It's a disconnect between the notation and the application.  Why should I care about this? This leads me to my second frustration.  In the grander scheme of things, much of the public is of the opinion that math is difficult, mysterious, arbitrary with no real connection to real life.  This attitude leads to the idea that being bad at math is acceptable, which leads to even more students giving up or not caring. Basic mathematical literacy should be just as important as basic reading.  No sane person would ever make small talk with a person at a party by saying "you know, I just can't read.  I've always been bad at reading."  We should, as a culture, put the same importance on basic mathematics, including notation.  That notation is analogous to the rules of phonics, not more complex grammar. We don't do that though. 

Every post I've made in this thread has been an attempt to show that this particular concept, the order of operations, doesn't have to seem arbitrary and confusing.  That's all I've wanted to show. 

[Aeris]

How about this:

You are a small business owner strugglig to keep afloat in these economic times.  You order a box of bracelets as store stock.  Its labeled:
10 red
17 yellow
6 green
52 black

Your finger slips and you order 11 boxes instead.  They are non-returnable.  Your store sells about 200 bracelets a week. How long will it take for you to move this merchandise, so you can plan when to make your next inventory order?

First you figure out how many bracelets you have:

(10+17+6+52)x11=935

Doing the multiplication first would be wrong:

10+17+6+(52x11)=605

Ok we can rework to make the multiplication first, but its a LOT more work and a stupid way to figure it out:

(10x11)+(17x11)+(6x11)+(52x11)=935

But its MUCH easier and more logical to do the addition first and then multiply.  This way when you divide your number you get the correct timeline.  It will take almost 4.5 weeks to move that stock, whereas if you multiplied first you would think you'd be almost sold out in 3. Assuming a half week delivery timeline, here is a HUGE cash flow difference for a business owner between ordering your next inventory batch at 2.5 weeks versus at 4 weeks!

If you don't use parentheses and order of operation, though, your answer would change depending on what colors you count first.  11 boxes x (10 red + 17 yellow + 6 green + 52 black) = 935 bracelets, done correctly.  If you go left to right and ignore parentheses, you'd get 185 bracelets if you counted red first ((11x10)+17+6+52=185), or 255 bracelets if you counted yellow first ((11x17)+10+6+52=255), or 605 bracelets if you did black first ((11x52)+10+17+6=605).  Obviously the number of bracelets present shouldn't change depending on which order you count them in, so if you were the store owner and getting different numbers every time you'd know you really needed to do the parentheses first 

Nope.     You reversed my numbers and changed my inventory style.  I had me adding up the box totals first and based on the already existing label, plus most importantly you put the 11 first.  I didn't think the logical equation was "I have 11 boxes of a combination of10, 17, 6, and 52 bracelets" but rather "ok lets see there are how many per box?  And I have how many boxes now?"
If one goes left to right in the absence of parenthesis my equation still works perfectly:

10+17+6+52x11=935

I just didn't write it like that because I think that's sloppy, like bad grammar.  I was taught that mathematical equations, when properly written, include parenthesis for groups and quite frankly I think this thread has shown how very important they are for clarity to the non-mathematician masses.

How were you taught to solve something like: 5x² + 3x - 9, where x=3?

How was I taught to figure that out ^ ?
For the life of me I can't remember.  But logically the first step is to figure out what x is.  Since you are telling me its 3, the second step is to ask why on earth you would use x?  Just write your equations with numbers; what possible real world application does using x have if you know x=3?  I have no use in my life for theoretical, or as called up thread "pure" math, and reject it as impractical just as I would reject the idea that one should only consider using a professional stove or drive an industrial grade vehicle. Sure a teacher should go over pure math, but as Iris is saying, to me the sign of knowing math, for the average non-mathematician is knowing how to figure out the situation and break it down, not knowing the formal rules of written math.

In average life, all math can be broken down to a word problem.  If you give me the word problem to that equation I bet I can tell you the exact correct method of working it out. What are we trying to figure out?

Well, I was trying to figure out how on earth you were taught that all properly written mathematical expressions had parentheses to indicate every grouping. That's extremely troubling, since that would require the expression be written:

5(x²) + (3x) - 9

in order for you to solve it correctly.

Real world applications of this type of math would be things like: you throw a ball horizontally out your window on the 6th floor at a rate of <some rate>. How far will it travel before it hits the ground? I'm happy to actually give you a complete high school physics style projectile motion question if you insist.

You seem to have a violent antipathy for 'pure math' (despite the fact that pure math is generally applicable to a variety of real world situations). Given your hatred for it, perhaps joining in the discussion about a pure math problem is not a great idea.

[MommyPenguin]

For the life of me I can't remember.  But logically the first step is to figure out what x is.  Since you are telling me its 3, the second step is to ask why on earth you would use x?  Just write your equations with numbers; what possible real world application does using x have if you know x=3?

Just to clarify this part, the reason one might write the equation with x but tell you that x=3 might be because the equation is being used for many instances of x, but 3 is just a sample.  So, for instance, you might be given the equation, and then asked, "So what happens when x = 3?" and you would solve for 3, and then you might be asked, "And then what happens when x = 17?"  You might need to try multiple values of x in the equation to see what happens.  Why would you need to do this?  Well, for instance, my husband has worked as a satellite engineer.  He has all sorts of complicated equations that would rot your brain if I tried to post them here (I can't look at them without my vision blurring), but he'd have a given equation for the refraction of something or other, and then he might want to try changing the value of some aspect of the lens in order to see what would happen if the lens were thicker, or thinner, or angled a different way, or had a different value of concavity, etc.  In other words, sometimes you might have a specific equation that explains something (like, perhaps, the distance a satellite can communicate), but it depends on several variables (the lens, the power of the satellite, its location, the amount of light it receives, how much interference from space objects, etc.).  You'd try to find the information for all those variables and put it in and get a solution, but when those values change, so would the solution, because the solution depends on all those different variables.  Just as what the weather will be like today depends on the cloud cover, the movement of the wind, the time of year, the proximity and angle of the sun, etc.

[Iris]

How about this:

You are a small business owner strugglig to keep afloat in these economic times.  You order a box of bracelets as store stock.  Its labeled:
10 red
17 yellow
6 green
52 black

Your finger slips and you order 11 boxes instead.  They are non-returnable.  Your store sells about 200 bracelets a week. How long will it take for you to move this merchandise, so you can plan when to make your next inventory order?

First you figure out how many bracelets you have:

(10+17+6+52)x11=935

Doing the multiplication first would be wrong:

10+17+6+(52x11)=605

Ok we can rework to make the multiplication first, but its a LOT more work and a stupid way to figure it out:

(10x11)+(17x11)+(6x11)+(52x11)=935

But its MUCH easier and more logical to do the addition first and then multiply.  This way when you divide your number you get the correct timeline.  It will take almost 4.5 weeks to move that stock, whereas if you multiplied first you would think you'd be almost sold out in 3. Assuming a half week delivery timeline, here is a HUGE cash flow difference for a business owner between ordering your next inventory batch at 2.5 weeks versus at 4 weeks!

If you don't use parentheses and order of operation, though, your answer would change depending on what colors you count first.  11 boxes x (10 red + 17 yellow + 6 green + 52 black) = 935 bracelets, done correctly.  If you go left to right and ignore parentheses, you'd get 185 bracelets if you counted red first ((11x10)+17+6+52=185), or 255 bracelets if you counted yellow first ((11x17)+10+6+52=255), or 605 bracelets if you did black first ((11x52)+10+17+6=605).  Obviously the number of bracelets present shouldn't change depending on which order you count them in, so if you were the store owner and getting different numbers every time you'd know you really needed to do the parentheses first 

Nope.     You reversed my numbers and changed my inventory style.  I had me adding up the box totals first and based on the already existing label, plus most importantly you put the 11 first.  I didn't think the logical equation was "I have 11 boxes of a combination of10, 17, 6, and 52 bracelets" but rather "ok lets see there are how many per box?  And I have how many boxes now?"
If one goes left to right in the absence of parenthesis my equation still works perfectly:

10+17+6+52x11=935

I just didn't write it like that because I think that's sloppy, like bad grammar.  I was taught that mathematical equations, when properly written, include parenthesis for groups and quite frankly I think this thread has shown how very important they are for clarity to the non-mathematician masses.

How were you taught to solve something like: 5x² + 3x - 9, where x=3?

How was I taught to figure that out ^ ?
For the life of me I can't remember.  But logically the first step is to figure out what x is.  Since you are telling me its 3, the second step is to ask why on earth you would use x?  Just write your equations with numbers; what possible real world application does using x have if you know x=3?  I have no use in my life for theoretical, or as called up thread "pure" math, and reject it as impractical just as I would reject the idea that one should only consider using a professional stove or drive an industrial grade vehicle. Sure a teacher should go over pure math, but as Iris is saying, to me the sign of knowing math, for the average non-mathematician is knowing how to figure out the situation and break it down, not knowing the formal rules of written math.

In average life, all math can be broken down to a word problem.  If you give me the word problem to that equation I bet I can tell you the exact correct method of working it out. What are we trying to figure out?

Well, I was trying to figure out how on earth you were taught that all properly written mathematical expressions had parentheses to indicate every grouping. That's extremely troubling, since that would require the expression be written:

5(x²) + (3x) - 9

in order for you to solve it correctly.

Real world applications of this type of math would be things like: you throw a ball horizontally out your window on the 6th floor at a rate of <some rate>. How far will it travel before it hits the ground? I'm happy to actually give you a complete high school physics style projectile motion question if you insist.

You seem to have a violent antipathy for 'pure math' (despite the fact that pure math is generally applicable to a variety of real world situations). Given your hatred for it, perhaps joining in the discussion about a pure math problem is not a great idea.

Well, hey, originally it was 'here's a cool radio quiz' and we've taken it way off track. I think WillyNilly has a right to her opinion that formal mathematics 'grammar' is counter-intuitive because clearly it is counter to *her* intuition.

WillyNilly - the equation given is the equation of a parabola. The most understandable and common real world application of a parabola is that any thrown object follows a parabolic path. So in a very real sense pretty much everyone can solve that type of equation because pretty much everyone can *catch a ball*. The equation becomes very useful and relevant if you are in a situation (say engineering or military) where you want to know where something is going to land before you throw it and look. The equation allows boffins to change parameters like angles and velocity and work out consequences without going to the effort of actually throwing, or firing, things over and over again. It's a little more complicated than that as I'm sure someone will point out but hopefully that gives you a rough idea.

Ironically I can solve really hard projectile motion equations but am *rubbish* at catching things. Go figure 

[Slartibartfast]

How about this:

You are a small business owner strugglig to keep afloat in these economic times.  You order a box of bracelets as store stock.  Its labeled:
10 red
17 yellow
6 green
52 black

Your finger slips and you order 11 boxes instead.  They are non-returnable.  Your store sells about 200 bracelets a week. How long will it take for you to move this merchandise, so you can plan when to make your next inventory order?

First you figure out how many bracelets you have:

(10+17+6+52)x11=935

Doing the multiplication first would be wrong:

10+17+6+(52x11)=605

Ok we can rework to make the multiplication first, but its a LOT more work and a stupid way to figure it out:

(10x11)+(17x11)+(6x11)+(52x11)=935

But its MUCH easier and more logical to do the addition first and then multiply.  This way when you divide your number you get the correct timeline.  It will take almost 4.5 weeks to move that stock, whereas if you multiplied first you would think you'd be almost sold out in 3. Assuming a half week delivery timeline, here is a HUGE cash flow difference for a business owner between ordering your next inventory batch at 2.5 weeks versus at 4 weeks!

If you don't use parentheses and order of operation, though, your answer would change depending on what colors you count first.  11 boxes x (10 red + 17 yellow + 6 green + 52 black) = 935 bracelets, done correctly.  If you go left to right and ignore parentheses, you'd get 185 bracelets if you counted red first ((11x10)+17+6+52=185), or 255 bracelets if you counted yellow first ((11x17)+10+6+52=255), or 605 bracelets if you did black first ((11x52)+10+17+6=605).  Obviously the number of bracelets present shouldn't change depending on which order you count them in, so if you were the store owner and getting different numbers every time you'd know you really needed to do the parentheses first 

Nope.     You reversed my numbers and changed my inventory style.  I had me adding up the box totals first and based on the already existing label, plus most importantly you put the 11 first.  I didn't think the logical equation was "I have 11 boxes of a combination of10, 17, 6, and 52 bracelets" but rather "ok lets see there are how many per box?  And I have how many boxes now?"
If one goes left to right in the absence of parenthesis my equation still works perfectly:

10+17+6+52x11=935

I just didn't write it like that because I think that's sloppy, like bad grammar.  I was taught that mathematical equations, when properly written, include parenthesis for groups and quite frankly I think this thread has shown how very important they are for clarity to the non-mathematician masses.

I did change the order it was written in for the purposes of example - but do you really see a difference between "I have eleven boxes containing 85 bracelets each" and "I have 85 bracelets in each of eleven boxes?"  You can do it either way, and (mathematically and grammatically) it comes out to be the same thing: 11 x 85 or 85 x 11.

So the way you wrote it without parentheses, 10+17+6+52x11 comes up with a different answer than 17+6+52+10x11, which would correspond to which order you count the colors in the boxes in (counting the 52 black bracelets last versus if you had counted the ten red ones last).  But the way you wrote the problem in text was conceptually correct - figure out how many are in each box (which would be represented by parentheses in the written equation), then multiply that total by the number of boxes.