One of my dc informed me today that the Order of Operations is totally illogical and useless. Umm... I have to agree. Am I just showing my mathematical ignorance? While I enjoy solving the puzzle of the problems, I don't know of any uses for that order in real life outside of math class. Do you?
ETA: Okay, after googling, I see that I do use the order of operations some in my life without really thinking about it (adding up shopping expenses while deducting discounts, for example), but I never use exponents or square roots. At least, I don't think I do!
Okay, new question: in real life, why would we need to use exponents and square roots in an order of operations problem?
Sometimes I find that it is difficult to look outside of myself. So, just because I as a stay at home mom do not use the order of operations doesn't mean my dc will not need to in their future. I have to be careful to teach my kids to excell outside my range of daily experiences, especially if I want them to do something other than be a stay at home mom (which is a high calling, yet not everyone's calling).
The entire purpose of higher level math is to support the sciences/engineering. If one of your children desires at some point in their life to delve into any of the sciences or engineering (which is applied science), math is essential, including the order of operations. I can testify that as an engineer, I needed such knowledge. (My previous life was as a CHE).
I face the same sort of questions when dc are studying the language arts. I don't know why we need to know what a predicate nominative is, but just because I don't use that in my every day life, I teach the kids because in their careers, that may be necessary.
Blessings and I hope this helps!