The trouble with BEDMAS is it mixes apples and oranges, it's like saying words should be alphabetized by color first, and then by letter.
Parentheses "group quantities", addition, subtraction, etc. are operations, BEFORE one does an operation, one ought to be CLEAR what one is operating ON. There is a difference between EVALUATING and SYNTAX.
In ordinary language, faulty syntax can be "sorted out" from context; in math, expressions are often "stand-alone" (no context).
When one encounters:
1 - 1 + 1, it's not clear what is INTENDED: even using BEDMAS, one cannot be SURE that the writer of said expression was ALSO using BEDMAS.
The insistence of BEDMAS being taught in schools, is further evidence to me that the school system is deeply flawed. I pity the poor programmer, who, when trying to code such an expression in a program, has to decide if the syntax of the programming language s/he is using matches the syntax of the person instructing him/her to write the program (even if that person is himself/herself).
Subtraction is a "bad operation", it is not associative:
$a - (b - c) \neq (a - b) - c$.
Addition does not have this problem. The same problem occurs with division, often people on these forums want us to simplify expressions like:
x/x^2 + x
and we have NO IDEA if they mean:
(x/x^2) + x = (1/x) + x = (x^2 + 1)/x, or:
x/(x^2 + x) = 1/(x + 1).
The trouble with:
"x minus..."
or
"x divided by..."
is: are we supposed to subtract/divide by EVERYTHING that comes after, or just the very NEXT thing? The problem is further compounded by the fact that many people don't even realize there's a difference.
It's like saying:
I gave my father the hat with the bacon.
Did I give my father some bacon, and a hat; or did I give him a hat decorated with tasty cured pork?