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Defining Multiplication, Redux

Date: 10/09/2014 at 06:39:17
From: Damar
Subject: 5x6=6+6+6+6+6 so is it wrong when we write 5x6=5+5+5+5+5+5 ?

We know that

   5 x 6 = 6 + 6 + 6 + 6 + 6 

But we also know that 

   5 x 6 = 6 x 5

So is it wrong to write this instead?

   5 x 6 = 5 + 5 + 5 + 5 + 5 + 5

Date: 10/09/2014 at 08:16:56
From: Doctor Carter
Subject: Re: 5x6=6+6+6+6+6 so is it wrong when we write 5x6=5+5+5+5+5+5 ?

Cut 3 rows from the bottom of a chessboard, leaving a rectangle with 5 
rows and 8 columns. Then cut two columns from the right to get a rectangle 
with 5 rows and 6 columns.

Let's count the squares in this trimmed board. 

There are 5 squares in each column, so if we count the squares column by 
column, we'll get 5 + 5 + 5 + 5 + 5 + 5.  

On the other hand, there are 6 squares in each row. So if we count squares 
row by row, we'll get 6 + 6 + 6 + 6 + 6.

Both ways of counting achieve the same thing -- they count the number of 
squares in the 5 by 6 chessboard. So they must give the same result.

- Doctor Carter, The Math Forum 

Date: 10/09/2014 at 08:58:04
From: Doctor Peterson
Subject: Re: 5x6=6+6+6+6+6 so is it wrong when we write 5x6=5+5+5+5+5+5 ?

Hi, Damar.

Would you please tell us the context of your question? This may be 
important in understanding the distinction you're getting after.

As Doctor Carter showed, because multiplication is commutative, both 
statements are correct. Many educators, however, introduce the concept of 
multiplication by "defining" x*y as "x repetitions of y"; others introduce 
it as "x, repeated y times." It makes some sense temporarily to take only 
the one meaning in talking about it with children; but they should very 
soon learn about commutativity (perhaps through examples like Doctor 
Carter's) so that they can think of it in both ways. And later, they 
should learn that multiplication really IS something much broader than 
this. To that end, some educators argue that we should not teach that 
multiplication is repeated addition at all.

But is your context related to elementary education, or to something else?

- Doctor Peterson, The Math Forum 

Date: 10/10/2014 at 05:59:15
From: Damar
Subject: Re: 5x6=6+6+6+6+6 so is it wrong when we write 5x6=5+5+5+5+5+5 ?

Yes, this is related to elementary school, where the teacher finds fault 
in his students for writing

   2 x 3 = 2 + 2 + 2

I wanted to get a statement from an expert.

Date: 10/10/2014 at 09:16:57
From: Doctor Peterson
Subject: Re: 5x6=6+6+6+6+6 so is it wrong when we write 5x6=5+5+5+5+5+5 ?

Hi, Damar.

Here is a question and answer from just last month about the same issue:

> Dear Dr. Math,
> My 2nd grader failed her math quiz today because the teacher insists 
> that there is an order as to how a multiplication sentence should 
> be written. 
> Here's an example:
>    ##   ##   ## 
>       Addition Sentence: 2 + 2 + 2 = 6
> Multiplication Sentence: 3 x 2 = 6
> The teacher said that 2 x 3 = 6 would be wrong.
> Another example from the quiz shows a numbered scale with asterisks that
> represent jumps:
> <-0*--1---2---3*--4---5---6*--7---8---9*--10---11---12*--13---14---15*->
> Multiplication Sentence: 5 x 3 = 15
> The math teacher said that this is the only way that can be written, and
> that writing it as 3 x 5 = 15 is wrong.
> I have already spoken to the teacher and explained that these can be
> interpreted in two ways, but she insists that this is how 2nd graders 
> should be taught!
> Unless I can prove her wrong with a written source of some kind, she 
> will not consider my daughter's answers.
> Please help.

I agree fully with you. But there is some reason behind the teacher's
position; and a compromise is possible.

Multiplication is commutative, so your point is correct; 2*3 and 3*2 are 
equal, and either could be used to represent the same calculation or 
situation. Mature thinkers don't need to pay any attention to the order, 
but use whichever order is most useful.

On the other hand, it is reasonable for students to INITIALLY be 
introduced to multiplication using a single model, which in this 
teacher's curriculum is "this many of those" rather than "this repeated 
that many times." This simply allows everyone in the class to be on the 
same page when they give examples.

But they should soon learn the commutative property so that, for example, 
if they want to find the cost of 50 items that each cost $2, they could 
just add two 50's rather than add fifty 2's!

Interestingly, many years ago I answered a question about an article in a 
teacher's magazine that made the exact opposite claim: that 2*3 "really 
means" 2 multiplied BY 3 (2 taken 3 times):

  Defining Multiplication 

Some time later, each of the authors of the article cited (which I didn't 
have access to at the time) wrote to me complaining that my comments were 
wrong. They didn't change my mind, but I did then get to read their full
article, and it did clarify the point they were trying to make. My 
conclusion is that it does make sense to teach kids initially to read 
"2*3" as "2 multiplied by 3," and perhaps to ask them to use that 
interpretation in class to make examples of where a particular 
multiplication would be used. They should not, however, be taught that 
this is the only interpretation.

If nothing else, you can refer your daughter's teacher to that 2001 
article from the Phi Delta Kappan, if she has access to it (I had to get 
it online through my school library site), because it makes the opposite 
statement about the meaning of multiplication, showing that what your 
teacher does is not the ONLY way to teach the subject; and moreover that 
other educators have the opposite opinion. It is not the way students 
SHOULD be taught multiplication, just the way they ARE taught it in 
this curriculum.

So here's the compromise I suggest: the teacher can teach one 
interpretation of the application of multiplication, but make it clear
that it is not the ONLY answer, just the one they are using IN THIS CLASS, 
for now. Your daughter should be commended for seeing that it really 
doesn't make a difference; and rather than being marked wrong, can be 
encouraged to learn the way the teacher says to do it merely as a class 
convention, to help students who need more time to catch on.

Having said all that, it does appears that the Common Core curriculum 
supports the interpretation that 2 * 3 = 3 + 3. So in classrooms that 
have adopted those standards, the teachers are doing what they are told, 
at least to some extent.

- Doctor Peterson, The Math Forum 
Associated Topics:
Elementary Definitions
Elementary Multiplication

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