[Photo “Micah and Multiplication” by notnef via Flickr (CC-BY 2.0).]
Some Internet topics are evergreen. I noticed that my old Multiplication Is Not Repeated Addition post has been getting new traffic lately, so I read through the article again. And realized that, even after all those words, I still had more to say.
So I added the following update to clarify what seemed to me the most important point.
I’d love to hear your thoughts! The comment section is open down below . . .
Language Does Matter
Addition: addend + addend = sum. The addends are interchangeable. This is represented by the fact that they have the same name.
Multiplication: multiplier × multiplicand = product. The multiplier and multiplicand have different names, even though many of us have trouble remembering which is which.
- multiplier = “how many or how much”
- multiplicand = the size of the “unit” or “group”
Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.
A New Type of Number
In multiplication, we introduce a totally new type of number: the multiplicand. A strange, new concept sits at the heart of multiplication, something students have never seen before.
The multiplicand is a this-per-that ratio.
A ratio is a not a counting number, but something new, much more abstract than anything the students have seen up to this point.
A ratio is a relationship number.
In addition and subtraction, numbers count how much stuff you have. If you get more stuff, the numbers get bigger. If you lose some of the stuff, the numbers get smaller. Numbers measure the amount of cookies, horses, dollars, gasoline, or whatever.
The multiplicand doesn’t count the number of dollars or measure the volume of gasoline. It tells the relationship between them, the dollars per gallon, which stays the same whether you buy a lot or a little.
By telling our students that “multiplication is repeated addition,” we dismiss the importance of the multiplicand. But until our students wrestle with and come to understand the concept of ratio, they can never fully understand multiplication.
For Further Investigation
If you’re interested in digging deeper into how children learn addition and multiplication, I highly recommend Terezina Nunes and Peter Bryant’s book Children Doing Mathematics.
To learn about modeling multiplication problems with bar diagrams, check out the Mad Scientist’s Ray Gun model of multiplication:
And here is an example of the multiplication bar diagram in action: