Confusion over Interpretation of PEMDAS

Date: 03/24/2005 at 14:03:33
From: wrw
Subject: Order of Operations (multiplication and division)

In telling students to "do multiplication and division IN THE ORDER 
THEY APPEAR," it seems they want to always do multiplication first.  I
think they follow the PEMDAS rule BY THE LETTER, so they want to
multiply before dividing.

When doing multiplication first, 8 / 2 * 4 = 8 / 8 = 1 

When doing multipliation and division from left to right, 8 / 2 * 4 = 
4 * 4 = 16

Date: 03/24/2005 at 15:26:39
From: Doctor Peterson
Subject: Re: Order of Operations (multiplication and division)

Hi, WRW.

If you think that students have a tendency to misinterpret the rule, 
you're probably right; but I think the reason is that PEMDAS is a 
poorly stated version of the rule, and it is easy to misunderstand it 
as meaning you do Multiplication, then Division, then Addition, then 
Subtraction.  That's not what the rule is supposed to mean, but many 
students don't get past the letters and see the meaning!

It's really wiser to think of subtraction as addition of the opposite,
and division as multiplication by the reciprocal, and just leave D and
S out of PEMDAS entirely, rather than try to fit them into the rules.
But we make the rules for people who aren't ready to see things in a 
mathematically mature way!  (I myself prefer to avoid PEMDAS 
altogether, and teach the "rules" in a more natural way that leads 
into this mature perspective.)

Translating these ideas into the case of multiplication and division, 
when we write

  8 / 2 * 4

we really mean

  8 * 1/2 * 4

which we can do in any order, since multiplication is commutative; 
clearly, however you do it, it comes out to 16, not 1.  The problem 
here is that people tend to see this as if it said

    8
  -----
  2 * 4

which means something different.

If you have any further questions, feel free to write back. You sound 
like someone I'd enjoy discussing this with!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/25/2005 at 15:02:05
From: wrw
Subject: Thank you (Order of Operations (multiplication and division))

Thanks so much!  I really like the idea of thinking of division as 
multiplying by the reciprocal and turning the whole 
multiplication/division portion into just multiplication.  I'll try
that out with my students and see if it helps.  Thanks again!