Date: May 16, 1995 8:59 AM
Subject: re: order of ops

>>4(3 + 2)   =   4 x 3 + 4 x 2
>> 4(5) = 12 + 4 x 2
>>20 = 16 x 2
>>20 = 32

>4(3 + 2) = (4 x 3) + (4x2)
> 4(5) = 12 + 8
> 20 = 20

It's a matter of semantics. The associative property (for example) still
holds; it's a property of the operations, not the notation. However, a
student may need clearer wording of the property. After all,'s solution (the first one) may not be how one student might
think of the associative property, although it's perfectly valid--because of
the order of ops conventions.'s (the second one)
is another way to think of it which will work as long as you know to
calculate the parentheticals first; but because of order of operations, both
will work. If we throw the conventions out (not that I think we should!),
we simply may have to reexamine how we have stated things.

Of course, if we removed the standard order of operations, it seems unlikely
that we would write something like 3A + 2, as I believe someone used in one
argument. It seems more likely that we would write 3 x A + 2; the closeness
of 3A makes it more natural to want to do that operation first. It seems to
me (although I have not done any research) that at least some of our notation
(such as exponents, removal of multiplication signs, and of course
parenthesis or lack thereof) is useful *because* of the order operations

Eric E. Karnowski
Mathematics Editor
Janson Publications, Inc.