Several notes on "order of operations" have appeared - and I'm getting a bit weary of this non-subject. So I have resent my reply below to similar messages on other mail lists. ------------------------------------------------------------- Whenever I see messages about "order of operations" my first reaction is that this topic doesn't belong on a school mathematics mail list. If you are spending time teaching about order of operations you are wasting the students time on a non-mathematical topic - no useful math ideas are being developed or reinforced. Order of Operations has only to do with various versions of conventional laziness when writing math expressions. Before high school, and maybe not even then, there are no situations where expressions get so long and complicated that it would be unwieldy to write them in correct standard form with all parentheses and multiplications appearing rather than being implied according to some calculator convention. Just because the calculator allows us to be lazy in writing expressions, that doesn't mean it's a good idea to do it, and especially not in elementary and middle schools.
By dropping parentheses and multiplication signs much is lost for beginners. The importance of multiplication is diminished by letting it disappear from expressions - a strange way to honor its significance! It also tends to hide from the user of a calculator the fact that a calculator cannot add or multiply more than 2 numbers at one time. When an expression in calculator shorthand is input into a calculator the calculator internally must essentially put in all those ommitted parentheses in order to parse the expression and so decide precisely in what order it will have to proceed to evaluate the expression. Dropping parentheses deemphasizes (until it disappears) the sense of the sequence of time-steps involved in the calculation. After all, the correct definition of a math expression is that it is a "description of how to calculate something" -- so it will take an orderly collection of steps in time to actually carry out the expression evaluation. An expression is not, as sometimes said, a number or a funny name for a number.
Another bad effect of dropping multiplication signs is that, if you do it, then in order to reduce the likelihood of confusion, all variables or names of math objects tend to be restricted to single letters (because AB is assumed to be a product, not a single name). This is a bad idea, especially for beginners. Names of variables or quantities should whenever possible be recognizable mnemonic shorthands so that someone can open the book to a page and immediately guess what the expressions stand for, e.g. ageDad, ageSon, Wt, HtA, HtB, LengRope, AreaRec, distPQ, vol, AveX, NumCows, PxTicket. (Math object names should not be exactly the same as the ordinary words used in those situations - that is confusing too.)
My pedagogical rule: Never introduce new shorthand notations until things begin to get complicated enough so that the shorthand might in fact be useful, and only after the student has thoroughly understood the ideas involved and has practiced enough to be fluent in the current notation usage.
Calculator shorthand (conventional order of operations) is exactly that, a sometimes convenient abbreviation of the correct and unambiguous standard method of writing math expressions. In education its use is more harmful than helpful. -- Ladnor Geissinger Math Dept, CB 3250 Phillips Hall Univ of North Carolina Chapel Hill NC 27599