As far as I know (and here I am refering to my own math education), operator priorities of $+$, $-$, $\cdot$, $\div$, power and parenthesis are taught via some simple phrases like
- "pointy" operators ($\cdot,\div$) before "dashy" operators ($+,-$),
- powers before both,
- parentheses before everything.
But for many students a formula like $1 + 3 \cdot 4 \div 7 - 2^2$ is nothing more than a confusing sequence of symbols. I (as a mathematician and partially computer scientist) think more in terms of trees when seeing something like this. My brain automatically inserts the parenthesis and I have a feeling for how to type this into my calculator to get the result that is implied by these symbols.
However, we all know that students often struggle with applying these rules. My conjecture here is (and please correct me) that they see the above formula as a linear object instead if a tree like structure:
- What do you think about this approach of teaching operator priorities, or the structure of a formula in general? Could it be helpful when student are taught to draw these trees like they were initially taught schemes for addition, multiplication, etc.?
- Is such an approach already in use? If not, do you think it would be useful? If yes, does it help?
- At which stage of learning math do you think it is important to start teaching this? Maybe elementary school is too early, maybe university is too late.