I suppose my beef with PEMDAS is that I was one of those kids that did math in his head — I still do — but how I do it isn’t PEMDAS. So for me it was always an “add-on”, to make the teacher happy, so I’d double calculate it: first my own way, and then I’d show my work to the satisfaction of the teacher (and the grade).
I found mental parenthesis worked well and in cases of poorly formed math problems, like 2+2×4 in real life, I’d ask for clarification but in school I’d do PEMDAS because for me, that was a “school thing”.
That said, once I thought of math as geometry, I could do a form of PEMDAS again, but just as EMDAS. Exponents form squares, cubes, hypercubes and beyond. Multiplication is repeated addition. Division creates fractions. Addition is simply increase and subtraction is simply decrease.
But the parenthesis was my “go to” for clarity. I could only parse in a PEMDAS way once I had a mental image of what I thought the other operations were doing.
Microsoft Excel, I took to immediately, as it doesn’t use PEMDAS at all. Rather, it reads left to right, unless you use parenthesis. I found that far more natural.
That makes sense. The history as presented in this Microsoft Blog is interesting as it shows full (more or less) standardizing coming in the 1960s.
I’d have thought it much earlier than that.
I seem to recall in some history of education thing I read once upon a time, that there was a period of time where RPN was being promoted, as it was in common use among engineers and many early computer scientists.
I guess without standards, things can get messy.
Looking at RPN again:
“The infix expression ((15 ÷ (7 − (1 + 1))) × 3) − (2 + (1 + 1)) can be written like this in reverse Polish notation:
15 7 1 1 + − ÷ 3 × 2 1 1 + + −
Yeah, my brain is happier looking at parenthesis.