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.
Thinking my way through such disputes helps me to grow as a teacher, to re-think on a deeper level concepts that 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
It Ain’t No Repeated Addition
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”? (and Multiplication Is Not Repeated Addition: Update), 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
- Odd Numbers Make Perfect Squares
- Infinite Series
- Harmonic Series
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 items per set
- Skip Counting steps per skip
- Number Line Jumps spaces per jump
- Rectangular Array rows per column (or columns per row)
- Time and Money dollars per payment
- Fractals copies per iteration
- Combinations 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] [number] = [scale factor] “times the size of” [original amount]
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 6 = 2.9 times the size of 6
5 8 = 5 of 8
12 = of 12
55% 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.
The mad scientist’s Replicate setting is like the common textbook description of multiplication as repeatedly adding the same amount. If the mad scientist’s wife fixes six bowls of stew and lines them up in a row, she can use the Replicate setting to create row upon row of additional bowls, until she has more than enough stew to feed all the minions. Our students can use replication to model multiplication with rows of blocks, making an original row and then copying it again and again, until they have as many rows as the scale factor says.
Teaching Multiplication with Cuisenaire Rods
- Introducing Multiplication as Rectangles with Cuisenaire Rods at Education Unboxed
The rectangular array/area model of multiplication is a powerful way of thinking that will help our children understand many mathematical topics and real-life situations.
- Multiplication/Strategy Game using Cuisenaire Rods at Education Unboxed
A suggested modification: Start the players’ rectangles from opposite edges of the graph paper, letting their “conquered” territories grow toward the center.
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.]