> The order of operations, as we > know, is an algorithm that must be followed exactly.
I'm not sure whether you really mean what you're saying here. I would like to believe that you don't, because taken literally, what you've said is misleading.
The order of operations is merely a convention that makes it easier for people to communicate in mathematical writing. There is no fundamental property of numbers that demands that 3 + 4 * 5 - 1 is equal to 22, rather than 59, 48, or 19. But agreeing to the conventional order of operations allows us to avoid having to use parentheses for clarification. As with any convention or definition in mathematics, the order of operations is an organizational tool and should serve to make things simpler. It would be nice if we could help students to see it this way.
If we began by telling our students something like "This is how numbers work: You have to do exponentiation before multiplication, multiplication before addition, etc. etc., otherwise you get the wrong answer," then we would be, from the beginning, teaching them that mathematics is a set of arbitrary rules handed down from above.