Teaching the Concept/Meaning of Sum

Date: 10/23/2001 at 19:32:14
From: Linda Atamian
Subject: Teaching the concept/meaning of sum

I am a teacher working on math curriculum documentation. Today the 
word "sum" came up. The discussion was about teaching the meaning of 
the word sum to students in grades 1-5. I was surprised by the 
definition used by my colleagues. And I was further surprised by the 
definition in the few math glossaries I found online. To me "the 
answer to an addition problem or equation" is not a true or 
necessarily useful definition. I define the sum as the whole or total. 
With that definition I can say, if the sum is 10 finish the equation: 
10 = 4 + __. 0r when we are writing a related subtraction fact I can 
say, begin with the sum: 2 + 3 = 5 so ___ -3 = 2 or ___ -2 = 3. 

I also find that if the equation begins with the sum and the students 
must find a missing addend, they need to know the sum is what they are 
beginning with, not the result of solving the equation: 5 = 2+ ___? 

Am I way off target here? As a math teacher I stress understanding, 
not correctly answering questions or problems, although my experience 
is that children are correct when they understand what they are doing.

Date: 10/23/2001 at 23:01:51
From: Doctor Peterson
Subject: Re: Teaching the concept/meaning of sum

Hi, Linda.

I fully agree that understanding is important, and that we should be 
clear in our use of words. But I think the issue in the cases you are 
talking about is not really the definition of the word "sum." It's the 
misuse of the word "problem," which I often see in questions we 
receive.

Here's a proper definition, from Merriam-Webster (m-w.com):

   5 a (1) : the result of adding numbers (sum of 5 and 7 is 12) 

The sum is the RESULT of an addition, not the ANSWER to an addition 
PROBLEM. This is what the authors of the definitions you found mean, 
but their wording can be confusing in just the way you describe. 
A "problem" is a challenge given to a student, and as you say, 
_ + 2 = 5 is a problem involving addition in which the answer is not 
the sum but an addend. An "addition" (or, if you prefer, an addition 
"equation") is a statement of the form 2 + 3 = 5, in which two numbers 
are added, producing the sum as the result.

Students too often fail to use clear words like "equation," calling it 
a "problem." This makes it hard to distinguish between the object they 
are working with and what they are to do with it. If we use these 
words carefully, then there is nothing wrong with saying the sum is a 
result - not the answer to the particular task (problem) they are 
working on, but the outcome of an addition operation that is implied 
in the problem.

On the other hand, it would be just as dangerous to define the sum in 
this context as "the whole," because there are many problems involving 
a whole that don't involve addition, (say, fraction problems), and 
many addition problems in which the sum is only part of something. 
That's why I don't agree with your solution to the "problem." You're 
just a little bit off-target.

What do you think?

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