Is Subtraction Separation or Taking Away?

Date: 01/24/2002 at 00:51:43
From: Geeta Mahashabde
Subject: Subtraction

Dear Dr. Math,

Do we introduce subtraction as 

1) separation of a group into two parts and we know one part and are 
trying to find the other, or

2) taking away ?

We feel that there is a thin line between separation and taking away.
Please help us.

Thanks,
Geeta

Date: 01/24/2002 at 09:14:40
From: Doctor Ian
Subject: Re: Subtraction

Hi Geeta,

In the following diagram, 

        /\
       /__\


      +-----+
      |     |
      |     |
      +-----+

is the triangle above the square?  Or is the square below the 
triangle? 

They are really just two ways of expressing the same idea. Similarly 
for 'separation' and 'taking away'.  The only difference between the 
two concepts is that in separation, you still know where the 
subtracted items are, while in 'taking away', you might not.

In general, the more ways you understand something, the deeper your 
understanding of the thing.  So I would say that there is no reason to 
prefer one conception to the other; and it would be better for the 
students to be  exposed to both.  

My own preference is to unify subtraction with addition, by showing 
that addition and subtraction are just two ways of looking at the same 
fact.  That is, if 

  a + b = c

this is exactly the same thing as saying that

  c - a = b   and   c - b = a

I think this is a simpler idea than thinking of addition and 
subtraction as being different things. (Of the two approaches you 
mentioned, this corresponds more closely to the idea of separation.) 

This approach works for multiplication and division, as well.  That 
is, if 
   
  a * b = c

this is exactly the same thing as saying that

  c/a = b    and     c/b = a

Is the cat on the mat? Or is the mat under the cat? Why make a 
distinction? 

Does this help? 

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

Date: 01/29/2002 at 07:30:25
From: Geeta Mahashabde
Subject: Subtraction

Dear Dr. Ian,

Thanks a lot for the reply. Yes, really the cat is on the mat and the 
mat is under the cat.

Geeta