, Blue Ink wrote

*snip*

The problem is that operator precedence is largely a convention; there's no real reason why 3 + 3 * 4 = 15 and not 24; it's just that we decided it has to be that way.

That's what makes it hard to teach: you can paraphrase it all you want, but the answer is essentially "just because" and that never flies too well with kids.

Sure there's a reason. If the operators applied based on order in the equation, then the operations couldn't be freely associated with each other and always would have to be put in place in a certain order. For example, lets say you had an equation x × 2 = y × 3. You want to be able to rewrite the equation as x × 2 - y × 3 = 0 and still have it mean the same thing without putting in parentheses. If it was based on the order in the equation you couldn't do that very well. Stacking operations isn't very intuitive for people, only useful for calculators that can only enter one operation at a time. And if you had addition/subtraction taking precedence before multiplication/division you would have something that was intuitively strange, too.

Its not absolutely necessary, of course, but there's a reason behind it; its not arbitrary that standard order of operations works like that.