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What is 4 x 4 plus 4 x 4 plus 4 - 4 x 4?

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Organized by the order of operations (here, multiplication before addition):

(4 x 4) + (4 x 4) + 4 - (4 x 4) = 20 (correct)

If order of operations are ignored and the operations are entered in the order given, the result would be different:
4 x 4 + 4 x 4 + 4 - 4 x 4 = 320 (incorrect)
- - -
Unless 20 = 320, one of the two answers must be wrong.
The order of operations rules are not optional and are implicitly assumed in mathematics, much the same way that subject verb object structures are assumed in the english language. Do away with subject verb object structure, and sentences become very difficult to understand. The same thing is true of mathematics.
The Order of Operations
1) Exponents
2) Parenthesis
3) Mutiplication and Division (considered equal, because dividing by 4 is the same as mutiplying by 1/4)
4) Addition and Subtraction (considered equal, because subtracting 2 is the same as adding -2 or negative 2)

The "order of operations" rules have been a part of mathematics since the development of algebraic notation and had existed in logical mathematics well before algebraic notation.

For example, consider the following word problem: "A brand new dog kennel opens. In the first week, four groups of four dogs entered the kennel. In the next week, four more groups of four dogs entered the kennel. In the third week, only four dogs entered the kennel. In the last week, four groups of four dogs were taken home to their owners. How many dogs were in the kennel at the end of the first month?"

Notice that this isn't an algebraic problem but instead implies a simple mathematical expression. In order to respond to the word problem logically, dogs entering the kennel should be added. Dogs leaving the kennel should be subtracted. And several groups of dogs would be represented by multiplication, so "four groups of four dogs" should be represented mathematically by "4 x 4".

To evaluate this mathematic expression, we evaluate using order of operations: parenthesis, exponents, multiplication/division, and addition/subtraction. The mathematical expression would evaluated like so:

(4 x 4) + (4 x 4) + 4 - (4 x 4) multiplications evaluated first
16 + 16 + 4 - 16 = 20 additions and subtractions are evaluated next

This answer accurately predicts number of dogs in the kennel after the first four weeks of being open to be 20 dogs.
Ignoring order of operations, the expression would be evaluated like so:

4 x 4 + 4 x 4 + 4 - 4 x 4
16 + 4 x 4 + 4 - 4 x 4 --- multiply 4 x 4
20 x 4 + 4 - 4 x 4 --- add 16 + 4
80 + 4 - 4 x 4 --- multiply 20 x 4
84 - 4 x 4 --- add 80 + 4
80 x 4 --- subtract 84 - 4
320 --- multiply 80 x 4

This answer implies that the kennel described above has 320 dogs within, but that is logically inconsistent with the problem and ultimately the established rules of mathematics.
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