Order of Operations Dispute

Date: 09/26/2000 at 16:37:55
From: Terre and Robert Reidy
Subject: Order of Operations Dispute

The problem reads: N divided by (division sign)ml where n=12, m=6, and 
l=3. I believe the correct answer should be .6666, as 12 divided by 18 
equals this.  My husband agrees with me.

My son came home very upset from school, with a note from his teacher 
that the answer was wrong. She indicated that I should have divided 
the 6 (m) into 12 (n) before I divided the 3 (l) into the equation.  
Her answer was 6.  

My son is very upset with me; his teacher told him I was doing "old 
fashioned math."  Do I need to go back to school?

Date: 09/26/2000 at 21:03:38
From: Doctor Peterson
Subject: Re: Order of Operations Dispute

Hi, Terre and Robert.

Let's write the problem as

    n / ml, with n=12, m=6, l=3

I can give you some good news and some bad news. First, the bad news: 
according to the usual order of operations rules now taught, your 
answer is wrong. You can read about the rules in the Dr. Math FAQ:

  http://mathforum.org/dr.math/faq/faq.order.operations.html   

Basically, you do all multiplications and divisions before all 
additions and subtractions; and when you have a string of 
multiplications and divisions to do, you go from left to right in the 
order they come. In your case, that means you divide n by m giving 2, 
then multiply by l to get 6.

BUT...

You are not alone in your opinion. This part of the rule - doing 
multiplication and division together - is probably the last rule to 
have stabilized; I know that in the 1920's, at least, there was no 
agreement. It seems that an agreement developed, but it is unraveling 
now, as I hear from many students whose texts answer questions like 
this the way you did. It appears that they are adding an unstated 
rule, which seems entirely reasonable in this context, that an implied 
multiplication (indicated by simply placing two variables or 
expressions together, as in "ml") should be done first. It certainly 
looks as if it should mean that. The problem is that, although I've 
heard of this rule being followed frequently, I've hardly ever heard 
of it being taught, so these texts are not following their own stated 
rules.

Since this type of expression is so ambiguous, with people disagreeing 
on the rules, and the rules being easy to overlook, my own opinion is 
that neither your answer nor the teacher's is right: the question is 
wrong. No responsible mathematician would write such an expression; we 
would just say

     n
    ---
    m l

so there would be no question about its meaning. After all, the 
purpose of rules is to allow us to communicate clearly, not to help us 
trick students and start fights among families.

So you may in fact be "old-fashioned"; or you may be on the cutting 
edge. In any case, I'm afraid you'll just have to learn how they are 
doing it in class, and follow along. There shouldn't be many more 
issues like this to worry about.

You can read more about these issues in the Dr. Math archives:

  Order of Operations
  http://mathforum.org/dr.math/problems/wuandheil.05.19.99.html   

  More on Order of Operations
  http://mathforum.org/dr.math/problems/breitenbach.2.13.00.html   

I hope this helps at least a little.

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