Photo by Song_sing via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

My apologies to those of you who dislike conflict. This week’s topic inevitably draws us into a simmering Internet controversy. Thinking my way through such disputes helps me to grow as a teacher, to re-think on a deeper level things I thought I understood. This is why I loved Liping Ma’s book when I first read it, and it’s why I thoroughly enjoyed Terezina Nunes and Peter Bryant’s book Children Doing Mathematics.

Multiplication of whole numbers is defined as repeated addition.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not… Adding numbers tells you how many things (or parts of things) you have when you combine collections. Multiplication is useful if you want to know the result of scaling some quantity.

— Keith Devlin

## Many Models of Multiplication

All of us, whether children or adults, cling to our first impression of anything until reality forces us to revise that impression — and we tend to resist such revision as long as possible. Therefore, as homeschool parent-teachers we want to make sure that our students’ first impression of a topic is worth hanging onto, that it will serve as a solid foundation for future learning.

For many of us, our first impression of multiplication was as “repeated addition.” Unfortunately, this is NOT a definition worth clinging to. I’ve written before about What’s Wrong with “Repeated Addition”?, so I won’t repeat that argument here. But let me point out several important mathematical situations where repeated addition is definitely not multiplication:

• Triangular Numbers
$1 + 2 + 3 + 4 + ... + n = T_{n}$
• Odd Numbers Make Perfect Squares
$1 + 3 + 5 + 7 + ... + \left ( 2n - 1 \right ) = n^{2}$
• Infinite Series
$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = \sum_{0}^{\infty} \left ( \frac{1}{2} \right )^{n} = 2$
• Harmonic Series
$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... = \infty \,$

Can you identify what is different about each of the above situations compared to what our textbook authors Parker & Baldridge meant when they defined multiplication as repeated addition? That difference is the key to helping our children understand multiplication!

Do not limit your children to a single model of multiplication, especially a model as fragile as “repeated addition.” Instead, explore the many real-life Multiplication Models collected at the Natural Math website. Point out to your children how almost every multiplicative situation has one significant feature: a ratio or this-per-that quantity.

• Sets $\to$ items per set
• Skip Counting $\to$ steps per skip
• Number Line Jumps $\to$ spaces per jump
• Rectangular Array $\to$ rows per column (or columns per row)
• Time and Money $\to$ dollars per payment
• Fractals $\to$ copies per iteration
• Combinations $\to$ choices per option

When we help our students learn to recognize this-per-that situations, we give them a tool that will serve them well as they deal with elementary word problems and will also prepare the way for proportional thinking in algebra and beyond.

## The Mad Scientist Model of Multiplication

Multiplication is like a mad scientist’s ray gun that can enlarge or shrink things according to which scale factor the scientist sets:

[number] $\times$ [number] = [scale factor] “times the size of” [original amount]

photo by SoulStealer via flickr

That is where we get the word times for the multiplication symbol, though I also teach my students to use the word of, especially when multiplying with fractions or percents. If the scientist sets the scale factor at 1.0, that will leave the item exactly the same size. Any number greater than one will make the item grow, while a number less than one will shrink it.

2.9 $\times$ 6 = 2.9 times the size of 6
5 $\times$ 8 = 5 of 8
$\frac{1}{3} \times$ 12 = $\frac{1}{3}$ of 12
55% $\times$ 90 = 55% of 90

The ray gun has another setting in addition to the scale factor. This setting controls the type of growth or shrinkage. The scientist can make something grow by resizing (changing the size) or by replication (copying). With his Resize setting, he can turn a cockroach into a monster 413⅞ times its original size. Then he can switch to the Replicate setting, which creates multiple copies of the monster, until he has a whole army of giant cockroaches ready to attack.

Similarly, the mad scientist can shrink things by resizing or by partitioning (cutting it down to a fractional part).

The Resize setting may be used with any scale factor, but the Replicate setting needs a whole number scale factor (how many copies?), and the Partition setting needs a simple fraction or percent scale factor.

## Teaching Multiplication with Cuisenaire Rods

This post is part of the Homeschooling with a Profound Understanding of Fundamental Mathematics Series. [Go to the previous post. Go to the next post. Or start at the beginning.]

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1. July 16, 2012 4:35 pm

But the Khan Academy *says* that multiplication is just repeated addition, so it must be so, right?

Thanks for an exploration that doesn’t just make it complicated.

2. July 16, 2012 9:00 pm

Lol!
Sal is in good company — lots of people say that. It makes sense at the elementary level, when we’re just working with whole numbers, and we don’t think far enough ahead to realize that it falls apart with fractions. Or even just with zero: How can 0×5 mean to add 5 to itself zero times? It’s nonsense.

Repeated addition a good mental math thinking strategy for multiplication, as long as it’s just one part of our arsenal, but it collapses too easily to make a definition out of it.

3. July 17, 2012 11:33 am

I really think you’re too hard on repeated addition. 0×5 is “zero fives”, which is hardly nonsense! Ask a child how much she can buy with zero nickels and she’ll probably give you the right answer. Ask her how she needs to scale or partition a nickel to buy a chocolate bar, and she might be stumped Fractions are also fine. 0.5×10 is half a ten. 2.5×10 is two and a half tens.

Repeated addition is an easy explanation for any number problem until maybe you hit about grade 10 or 11 and get matrices, but then matrices are abstract objects anyway. In the end, multiplication is an invented and abstract concept. Nobody ever wants to multiply in real life, they want to repeatedly add, or scale, or forecast, or whatever and multiplication is the tool they use.

Young kids who DON’T realize that multiplication is a way of organizing and adding groups of objects find multiplication mysterious in my experience. To me, the early math is mainly bringing concepts back to counting, so multiplication is not even repeated addition because addition is counting. There are many algorithms and tables so if the focus is on those then a disconnect often occurs. If the child can analyze an algorithm in terms of counting then it’ll be solid learning.

4. July 17, 2012 12:41 pm

Ah, but “zero fives” is not at all the same thing as saying “add five to itself zero times.” Nor is “half of a five” the same as saying “add five to itself half a time.” What you are describing is not repeated addition.

Even so, in my mind the biggest trouble with telling our children that multiplication is “just repeated addition” is that we have not given them any way to distinguish the two concepts. Children desperately need a way to identify which types of situations call for addition/subtraction as opposed to multiplication/division. Otherwise, they will never be able to work word problems — and in the world of real life, ALL problems are word problems!

5. July 17, 2012 12:54 pm

Also, the difficulty appears long before matrices. Students who can’t make a conceptual distinction between addition and multiplication (because one is “just” a special version of the other) have absolutely no way to make sense of the way fractions behave. The rules for working with fractions will be abstract and meaningless, handed down from on high and accepted on the teacher’s authority. What a terrible way to learn math!

July 17, 2012 1:09 pm

Mmm, I see what you’re saying (maybe . I tell my kids to think of one side of the multiplication as an object, like “five cents”, and then the other side is how many times to count that object. For me that is repeated addition and that’s how I get zero fives. You can even write it out, 3X5 = 5+5+5 and 0 X 5 is just nothing there (equals zero, because zero is a number). I would not say “added to itself” because that seems to presuppose at least one. Maybe I don’t get repeated addition. If you have to say “added to itself” then that could definitely be confusing and I would agree with you! But, I still would teach multiplication as counting objects, which is exactly how I teach addition (incrementing).

7. July 25, 2012 9:02 am

Actually, I’m really not concerned with whether multiplication “is” or “is not” repeated addition. That debate (which still rages off and on in various blogs) is merely my excuse for analyzing how we teach multiplication — and specifically, for pointing out that we must teach it in a way that distinguishes it from addition.

The saddest thing that I have seen in my math classes is the student who reads a simple, one-step word problem and gets a glazed look in her eyes. Then she looks at the teacher for guidance and asks, “Do I add or multiply?”

It breaks my heart that a child can learn to crank through mathematical steps and procedures and yet have so little understanding of what she is doing.

We must teach our students how to recognize a multiplicative situation when they meet it in a word problem. And the best way that I know to teach such recognition is to point out and help students notice the multiplicand, the this-per-that ratio.

8. July 25, 2012 9:45 am

Here here to that. It is very sad, generally, when children “learn” to deal with problems by simply matching up the data they see with the tools they have. See this in physics all the time. “Oh, I have a mass and a force in the problem, I will use my formula Mass X Force = Acceleration”. If they have two formulas memorized for the same variables then they are forced to guess because they truly have no idea what to do.