This topic came up today on a homeschooling forum I frequent. It began as the related question, "**Do you think teaching Division as "repeated subtraction" is confusing?"**** **but the discussion quickly grew to include multiplication. I added a very long comment ...

**The problem with slogans in teaching:** Whether we are the teacher or the student, once we accept a slogan as dogma, we stop thinking.

"____________ is simply ____________ ."

Fill in the blanks however you want, and you have a BAD statement for a teacher to make.

**"Division is simply repeated subtraction?**"

As to the original question: I agree with the other commenters that "repeated subtraction" is a poor way to introduce division. On the other hand, it IS a great mental math technique to have in your toolbox for solving certain math problems. And the slogan does encapsulate ONE way of looking at division.

Let's consider the types of story problem (or real life) situations our student might meet which require division. First, we might have some amount of stuff that must be shared evenly among a certain number of whatevers, and we need to find out how much stuff each whatever will receive. Second, we may have some amount of stuff that must be measured out in chunks of a certain size, and we need to find out how many chunks we can make.

The latter situation looks much like subtracting the size of the chunk over and over until we run out of stuff. For instance, we might need 3/4 yard of fabric to make a certain type of pillow cover -- so how many pillows could we cover with 6 yards of fabric?

Also, as another commenter has pointed out, this understanding of division is at the heart of the standard long-division process.

Conclusion: Don't teach with the slogan. But do, as a teacher, think about what might have inspired the slogan and how it might help you develop a deeper, more flexible understanding of division.

**"Multiplication is simply repeated addition?"**

This slogan has the SAME problem as the statement about division (it encapsulates ONE way of looking at one very limited application of multiplication), but because the multiplication slogan is so familiar to us, we teachers don't recognize the problem. We have a familiar, comfy slogan, and we don't think deeply enough to realize the problem this can cause for our students.

If we train our students to think "multiplication is repeated addition", then we have no cause to complain when those same students can't solve story problems or when they get confused trying to remember the fraction rules.

Consider:

(2/3) x (5/6) = (2 x 5) / (3 x 6)

but

(2/3) + (5/6) is NOT = (2 + 5) / (3 + 6)

Why not?

Isn't multiplication just a special type of addition? So WHY are the rules so different?

The Fibonacci Series is created by repeated addition of the two previous numbers. Is that multiplication? We can form the square numbers by adding up the odd numbers: 3^{2}=1+3+5, and 4^{2}=1+3+5+7, and 5^{2}=1+3+5+7+9. That's definitely repeated addition, and squaring a number is a sort of multiplication...

The problem with the definition "multiplication is repeated addition" is that it leave unstated the MOST IMPORTANT difference between the two operations. That's why so many students are reduced to staring blankly at a story problem, asking, "Do I add or multiply?" We haven't given them any way to recognize the difference.

For more examples of how not understanding the difference between addition and multiplication makes learning fraction rules difficult:

For a more thorough exploration of the "repeated addition" debate:

**Then how SHOULD we teach multiplication?**

If we accept this argument, if we agree to no longer define basic multiplication as "repeated addition", then what? How does that affect the way we teach?

Mainly, we need to change our focus from * how *to

*.*

**why**We can teach multiplication in much the same way that we do now, using manipulatives arranged in groups or rows, pictures of multiplication situations, and rectangular arrays of dots or blocks. But instead of drawing our student’s attention to the

*of adding up the answer, we want to focus on the fact that the items are arranged in*

**process***.*

**equal sized groups**In other words, we teach our students to recognize the multiplicand:

- Teach children the useful word “per” and how to recognize a “this per that” unit.
- Have them label the quantities in their workbook: 3 cookies per student, 5 flowers per vase, 1 eye per alien, or whatever.
- If your story problem has a "this per that" quantity, then it must be a multiplication or division problem. You may be able to solve it with an addition or subtraction approach (especially if the numbers are small), but the heart of the problem is multiplicative.