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Topic: [math-learn] Order of Operations - again
Replies: 1   Last Post: Feb 5, 2001 7:03 AM

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 Ladnor Geissinger Posts: 313 From: University of North Carolina at Chapel Hill Registered: 12/4/04
[math-learn] Order of Operations - again
Posted: Jan 30, 2001 10:12 PM

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.
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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
--
Math Dept, CB 3250 Phillips Hall
Univ of North Carolina
Chapel Hill NC 27599

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