following up on Ken's computer observation-- A common mistake made by Middle school and high school student programmers is to ignore the order of operations in writing their formula for calculating the average of three numbers. They quickly see the difference between (A + B + C)/3 and A + B + C/3. With the intoduction of graphic calculators into the lower grades, younger students will run into this more and more as they enter the formulas for their calculations.
Order of operations is also important in adding fractions. I think that just about all kids see the relevance of adding fractions. Take the case of Adding 2/5 + 3/4 and expressing the sum as a decimal-- using a calculator is permitted. Normal order of operations: 2/5 = .4, 3/4=.75, then add = 1.15 . No order of operations: .4+3 = 3.4 then 3.4/4 = .85