These 2 are the same
56 - 24 / 3 + 12 = 60
56 - 24 ÷ 3 + 12

this is not allowed, you can't put a parentheses into an expression in any ole place that you want, in this case it does not change the answer though but it is only serandipity that the answer doesn't change
56 - (24 / 3 + 12) = 36

you can put parenthesis where it does not change the problem
56-(24/3) +12 is allowed, because you would need to do the division first anyway
56+(-24/3) +12 is also allowed, here you added a parenthesis that didn't change any thing and used the unary minus in place of the binary minus, also allowed, in the original, this does not change the problem, adding a negative is the same as subtracting the same positive . It does change what is possible as a next step. because now the minus is inside the parentheses leaving only addition outside

56+12+(-24/3)here the above expression which is the same as the original used the commutative property of addition to change the order of processing, the answer does not change. Only addition and multiplication(of the common binary operations) are commutative. Neither subtraction nor division is commutative.
(-24/3) +12+56 or 12+56+ (-24/3) are also possible rewrites because of CPA

This changes the meaning of the original and is not allowed as a transformation of the original, it may or may not have been what was intended, but this was a math test not a ESP test.
(56-24) / (3+12)
this also changes the meaning of the original, it is the same as the one above and is not allowed as a transformation of the original
56 - 24
3 + 12

The mnemonics PEMDAS, BODMAS, BEDMAS, and BIDMAS, and associated "rules" are just conventions to make writing and reading math equations possible without having to use an excessive number of parenthesis or brackets or having to explain the rules each and every time you write something down.

The problem with the simple mnemonics is that they lead to misunderstandings. Equal precedence is not inherently obvious to the layman. Even those that know about precedence can have a problem with writing clear equations. For example, what does this mean?

1/3x

Is it 1/3 of x or is it 1 divided by 3x? Many people that know the rules would write that expression when they meant to write:

(1/3)x

Even knowing the rules, people should strive for clarity in writing equations. So this equation:

56 - 24 / 3 + 12 = ?

Should be written:

56 - (24 / 3) + 12 = ?

Unless, of course, you want to post it up on Facebook and start an argument.

FYI, Noah did say that the equation was actually written as: 56 - 24 ÷ 3 + 12 =, but that he didn't know how to get the ÷ symbol, so he used the /.

Noah, just FYI, I know how to get the ÷, but I don't off of the top of my head know the number for the symbol, so to get it in my previous post, I just did a google search for division symbol and copy/pasted into my post.

The "÷" or "/" makes no difference.

So was the math teacher hot or just blonde?

Yep

56 - 24 / 3 + 12

56-8+12

=60

60

There is absolutely no room for interpretation in math.

I have seen some ****ty math teachers but that knocks it out of the park!

Of course there is. Why? Because there are no inviolable rules regarding math notation.

BS. There are international rules of math. I have studied abroad and literally worked with people from every continent. They all speak the same math. It is wonderful (compared to other non-science disciplines, i.e. law, which are very country specific).

Seriously, you could give that math "problem" to scientists across the globe and the answer will be 60. And I am afraid in places such as Europe or Asia, there would be zero discussion on "interpretation". This is 5th grade math man.

G

4th grade. My daughter's in fourth grade.

Not BS.

Yes, that math problem is easy because those basic conventions have been agreed upon for a long time. But not all "conventions" are so set in stone. Read these:

https://artofproblemsolving.com/wiki/index.php?title=Mathematical_convention
Conventions
https://math.berkeley.edu/~wu/order5.pdf
https://en.wikipedia.org/wiki/Ambiguity#Mathematical_notation
Ambiguous PEMDAS
Common Errors in College Math

Do you want more??

I didn't read past the second link but nowhere do I see ambiguity what the answer to the math problem posted is. It is 60 and it will be if you ask anyone past 4th grade (apparently!) in the world. Maybe I am misunderstanding what the discussion is about here?

The second article points out some silly rules on left to right and distinguishing between addition/subtraction and multiplication/division. This should not be taught and I don't recall being taught this myself. That said, sometimes getting kids going without understanding the deeper reasons is not a big issue, it will all come together by algebra 2.

What is a big issue, and I have seen this in my kid's school, is that elementary and middle school teachers often have no clue about math and memorize the rules, passing them straight to the kids. A simple brain fart will have it all come undone as seen in the OP's example.

G

Wow. Did I say there was any any answer other than 60 for the equation the OP posted? No I did not. I did say that it could be made more clear for the layman.

My "no inviolable rules regarding math notation" text was in response to your blanket statement quoted here:

This statement of yours is false.

We didn't learn the PREMDAS thing in school in England back in the 50's and 60's and I dont think my daughter did in the 90's; however the answer, 60, only needs to be obtained via careful reading of the question.

agreed with averybody that the answer is 60, however I think it is good form to write mathematical expressions to eliminate any chance of ambiguity

Scott, in all the links the authors are merely expressing the idea that not all readers have learned or use the mathematically correct interpretation of common expressions, not that there isn't a common inviolate set of rules.

If you study math at a deeper level than is common is HS or even undergraduate math the reasons become more obvious. There are symmetry, associative, commutative, distributive, identity and inverse axioms that make up the mathematical concept of a Field which includes a group of #s in this case Real #s, The Field Axioms define what can be done on a given set of data and in what order. You need to understand the rules for the data set and the field axioms that apply to the data set. It is boiled down to overly simplified mnemonics in HS and undergrad work. This is generally adequate for most purposes but can fail at surprising places to those that haven't fully integrated all of the rules, both the obvious overt ones and more subtle underlying ones..

Wong, in the second link notes that simplicity of presentation often precludes including all of the parentheses that would eliminate all of the ambiguity. He goes on to urge clear unambiguous notation. This is what all good math teachers do, and what many less gifted math teachers don't do and what causes no end of difficulty for students(and teachers) who know most but not all the rules. In the absence of parentheses there are rules that tell the solver what to do and in what order, some obviously don't know or don't follow them.

The fault in the original question was w/ the teacher he/she intended something other than what she wrote, and then compounded the fault by not acknowledging the mistake.

Bill,

Yes, I understand all of that. The issue is that the difference between the layman and the high level academic is huge. Also, conventions change over time as mathematics evolves. Regardless, what I said is true. The conventions are not set in stone. The math itself is inviolable, but the way we write it down and interpret that writing is not. aigel is putting to much faith in the conventions being absolute and not ambiguous.

Yes but it all comes down to either knowing and following all the rules or not, in this thread there is no higher level esoterica involved

there is and has been evolution but as far as the HS math discussed here

again the HS math discussed here is well defined and not subject to debate except by those that haven't fully integrated it.

It is true that other countries have different notations, these are entirely process/organization oriented and not presentation oriented(by that I mean the way they show things like the process of long division or manually calculating a root). They would evaluate the expression here using the same rules as us.

Bottom line is the teacher erred and wouldn't own up to it. It wasn't a matter of ambiguity except in his/her own mind. In every one of the controversial math questions i've seen here and at my job the error has always been that the presenter wrote something different from what was intended often because of a lack of understanding of concepts, and then refused to acknowledge the error. The defense often includes semantics like 'ambiguous' designed to put the burden on someone else.

perhaps you can present some notation that you feel is ambiguous, that can then be discussed.

Please look at this one again:

Ambiguous PEMDAS