[Photo “Micah and Multiplication” by notnef via Flickr (CC BY 2.0, text added).]
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:
Denise Gaskins' Let's Play Math 
I’ve been kicking this issue around in my head and with others for a decade or more. Two things that always strike me that each suggest something different is going on with multiplication that is not the case with addition.
First, how we are able to multiply rational numbers, rational algebraic expressions, etc., without needing to worry about finding a common denominator (though we can save ourselves work by canceling common factors appropriately when possible). We cannot do that with addition, though students often want to do just that for many reasons. Were it the case that “multiplication IS repeated addition” or MIRA, as I and some others have come to abbreviate this notion, shouldn’t we be able to treat the operations similarly across the board? That we can’t should give people pause.
Of course, no one argues that you can’t correctly if rather tediously compute the correct answer to many arithmetic problems that call for multiplication by doing repeated addition. Clearly, that’s true, though it would be interesting to see how one computes pi*sqrt(2) that way.
The other point that always strikes me is that of units. It makes no sense to add different units. You have to convert to the same units, if possible, in order to meaningfully add numbers with units attached. 3 people + 4 hours isn’t 7 of anything.
But we multiply many different units together to get new units. 3 people working for 4 hours each yields 12 person-hours (or to be retro, “man-hours”) of work. No one bats an eye at that notion. Such examples abound in science and technology. Foot-pounds, light-years, etc.
My examples won’t move people who are deeply entrenched in defending MIRA. They may not even ultimately prove to be as important as I sense them to be. But they strike me as worthy of consideration in trying to help distinguish meaningfully between two operations that are certainly connected (the distributive property of multiplication over addition comes to mind) but not identical.
I think you are right, Michael. The two operations do seem to be “connected but not identical.”
I find your point about units particularly significant, but maybe that’s because as a writer, I am a language person. Words matter, and the behavior of units under the two operations signals an important difference between them, which we might miss if we focused only on abstract whole number calculations.
And it could even be said that the difference in fraction calculations is the same — that is, it’s a difference in the behavior of units under the two operations. The denominators are in a sense the units each fraction is counting. That is, 3 fourths + 3 fourths will give an answer that is also fourths, but when we multiply, the denominator-unit changes.
But as for pi*sqrt(2) … that’s difficult to compute under any worldview. 🙂
Thanks for the quick response, Denise. I think we see these issues similarly; I agree with your comment in your penultimate paragraph: the two points I made are different facets of a connected structural distinction.
“But as for pi*sqrt(2) … that’s difficult to compute under any worldview.”
Well, yes and no. You’ll never be able to compute it more exactly than writing pi*sqrt(2). 🙂 But we can certainly estimate it to as many decimal places as we need: I’d just prefer not to have to do it by hand to very many decimal places. 🙂 It’s like doing irrational exponents: you can compute, say, 2^pi by taking the sum of 2^3 + 2^.1 + 2^.04 + . . . + as far as you care to take it. On the other hand, thinks like pi^e, I’ll type into Desmos and be quite content seeing 22.459. . . pop out. Is it plausible? Rounding pi down to 3 and e up to 3 would give 27, so it doesn’t seem out of line. Getting kids to do things like that seems like a real challenge for many of them: the notion of estimating/ball-parking answers before or after the fact, particularly via mental math, strikes many of them as “not math,” where they expect to get an exact answer (which for many means a positive integer. Sigh. Convincing them that sqrt(5+11) can’t equal sqrt(5) + sqrt(11) because the former is sqrt(16) = 4, whereas the latter has to be a number greater than 2 + a number greater than 3 and hence a sum greater than 5 might be easier if I dressed in wizard robes when making that argument to them: seems like magic to many of them, probably “Dark Arts.”
The Dark Arts of Mathematics…
[From Crystal Ball Connection Patterns, photo by Sean McGrath via Flickr.]
LOL
Hi Denise..Thank you for posting all this yummy math stuff. I am a new middle school math teacher and hope to use some of your things in my classroom. Thx.
Matt
You’re welcome, Matt. Best wishes on the adventure of teaching!