Multiplication as many groups of the same size

It's been very good and educational for me to refine my thinking on multiplication vs. addition by reading some recent posts around the blogosphere, especially What's wrong with repeated addition by Denise and Devlin's Right Angle Finale at Text Savvy.

I feel that on some blogs people aren't even exactly talking about the same thing. The subjet we're dealing with - is multiplication repeated addition or not? - is subtle. Some people talk about how to define it - it is defined in some systems as repeated addition, and they feel that closes the issue.

BUT, I tend to agree with what Denise wrote: multiplication is a different operation from addition and somehow we need to get students to view it that way. I've always known that; I've never thought anything different. But yet how we present things to children is not always easy; we may understand the idea but not able to convey it right. Talking about multiplication as repeated addition MAY indeed leave the impression in children's minds that "multiplication reduces to addition" or, as Denise put it, it is a "subspecies of addition".

So, to try to summarize what I've been mulling over in my mind:

* Multiplication is simply a DIFFERENT operation from addition. It's not a "subspecies" of addition or some special kind of addition. AND, we need to stress that in our teaching.

Denise offers defining multiplication for kids as "counting by groups" - should be same-size groups, of course. In teaching that, we need to emphasize the meanings attached to the two factors: in M × N, M would be the amount of groups, and N would be the number of elements in one group. M is called the multiplier, and N is the multiplicand.

We need to emphasize the difference between additive and multiplicative situations in word problems:

"Mark has two baskets and each basket has five appples." => this is a multiplication situation
"Mark has five apples in one basket and five in another." => this is an addition situation.

Joshua says, "They are different ideas, fundamentally. The "processes" of finding a product and finding a repeated addition sum are the same for both problems, but the ideas involved--INCLUDING THE MATHEMATICAL IDEAS--are very, very different."

It does sound simple and clear, right? I hope it does. The IDEAS are different even though the way we find the answer may be the same. But the IDEAS match these situations:

"Mark has four baskets and each basket has three appples." => this is a multiplication situation
"Mark has five apples in one basket and seven in another." => this is an addition situation.

Let's keep going.

  • In multiplication, 1 is the special number so that if you multiply by it, "nothing" happens . Also called the identity element.

    In addition, zero has the similar role.

  • Multiplication: each (real) number except zero has its multiplicative inverse so that if you multiply the number and its inverse, you get 1.
    Addition: Each (real) number has its additive inverse so that if you add the number and its inverse, you get 0.

  • Multiplication has an opposite operation called division.
    Addition has an opposite operation called subtraction.

Pretty nice, eh? As we keep emphasizing these distinctions, hopefully we can develop in the students' minds the idea that they're different animals, not the same. One is not a special case of another. We can strive to define multiplication (initially) as "so many of the same-size groups" or counting by groups. Of course we have to FIND products by adding repeatedly, but we can treat them as different operations.

Then later, students will encounter multiplication of fractions and of decimals, and leave behind the idea that they solve multiplication by repeated addition. Yet, the basid properties of multiplication hold true:

  • Its identity element is 1.

  • Every number except zero has a multiplicative inverse (a.k.a. a reciprocal number)

I also found an interesting study quoted at Text Savvy. I quote:

Two alternative hypotheses have been offered to explain the origin of the concept of multiplication in children's reasoning. The first suggests that the concept of multiplication is grounded on the understanding of repeated addition, and the second proposes that repeated addition is only a calculation procedure and that the understanding of multiplication has its roots in the schema of correspondence. . . .

Pupils (mean age 6 years 7 months) from two primary schools in England, who had not been taught about multiplication in school, were pretested in additive and multiplicative reasoning problems. They were then randomly assigned to one of two treatment conditions: teaching of multiplication through repeated addition or teaching through correspondence. . . . At posttest, the correspondence group performed significantly better than the repeated addition group in multiplicative reasoning problems even after controlling for level of performance at pretest.

I should note here that, although it may read that way above, the ultimate aim of this study was not to compare the effectiveness of the correspondence and repeated addition treatments; it was to test two hypotheses about the "origin of the concept of multiplication in children's reasoning." Obviously, one of the hypotheses says that the origin is in repeated addition, and another says that it is in correspondence.

In other words, the "significantly better" performance of the correspondence group over the repeated addition group was taken by the researchers not as evidence of the superiority of the correspondence treatment, but as evidence of the fact that children begin to think about multiplication NOT as repeated addition but as a "one-to-many correspondence."

Well, I'm trying. Here's the way I changed one page in my Multiplication book to read for now. What do you think?

Click on the link to download the one page (PDF).


Anonymous said…
I like your new page on multiplication and addition. It ties the ideas together well, pointing out the difference in meaning AND the useful fact that they give the same answer.

One quibble with you and with Joshua at Text Savvy: "Mark has five apples in one basket and five in another." => this is an addition situation. (Joshua had a similar example about toys.)

This can actually be either an addition situation or a multiplication situation, depending on how the reader views it. One reader may add: "He has 5 apples and 5 more apples." But another might multiply: "Oh, I see. He has 2 baskets of 5 apples each."

It depends on whether the reader sees it as happenstance that the number of apples in each basket is the same, or whether he views it as a ratio of apples per basket (even though it is not stated that way). I tend to think in terms of multiplication automatically, when I see numbers that are the same, so the examples didn't make sense to me at first.
J.D.Fisher said…
Nice post, Maria. I'm not sure what the original lesson looked like, but it's nice.
Anonymous said…
I'm in the "disagree" camp, but I think the debate is stimulating in that it is asking all of us to think in a meta-cognitive way about addition and multiplication. Unfortunately, I think the sticking point is more one of language than of mathematics. For example, we can construe 1 as an "identity" in addition if we have only one thing. One set of 5 things is still 5.

One of the ways that children come to understand the traditional paper-and-pencil multiplication algorithm is by showing them a similar repeated addition operation: 3 x 27 is equivalent to 27 + 27 + 27. In both cases, we are finding the value of three sevens, placing the ones digit of that value in the ones place in the answer, and regrouping the tens. How can they be seen as anything other than two forms of the same task?

In terms of actual process, they are different. But in terms of what is going on in terms of quantities, they are exactly the same.

Repeated addition can be displayed as an array, just as multiplication can. Three times five is a 3 x 5 array, with 15 items in all. Five plus five plus five is drawn as three rows with 5 objects in each -- a 3 by 5 array.

I think this is, essentially, a non-question. If you observe children who are moving from basic conceptual levels to more efficient, strategy-based thinking, you will see that they are "inventing" multiplication as they have lots of addition experiences. Beginners will count to seven, and seven more, and seven more. But proficient, maturing learners will think, "I know that three 7s add up to 21."

I think the one difficulty with your page on multiplication is that it excludes the reversible nature of those equations. 5 x 2 can be represented as 5 things in two groups just as accurately as 2 things in five groups. Since this concept of "switch it around" often supports computational fluency, I'd suggest that you think about including "opposite" illustrations on your worksheets when possible.

And, yes -- I think division is repeated subtraction.
Maria Miller said…
"I think the one difficulty with your page on multiplication is that it excludes the reversible nature of those equations. 5 x 2 can be represented as 5 things in two groups just as accurately as 2 things in five groups."

I have a whole lesson in my book just about this idea. The free one-page download is just one page of the book that has 65 pages of material for teaching multiplication concept and the tables.
Anonymous said…
Two comments: First, it is good to NOT clutter up the definition of multiplication with discussion of the commutative property. Then the fact that multiplication is commutative can be an "Aha!" discovery that leads to deeper understanding: Not all mathematical operations are commutative, but isn't it neat that multiplication is?

Second, did you notice Joshua's interview with Devlin? I posted a relevant excerpt here. With that comment in mind, I thought you might want to consider adding another page to your definition, pointing out that multiplication can be read as "of", and that multiplication can give an answer that is larger, the same, or smaller than the original group. (Maybe show the same group times 3, times 1, times 1/2, times 0.) In some ways, multiplication is like a volume control that can turn things up or down.

Then have some exercises to let the students practice multiplying by whole numbers and by simple fractions, translating the times symbol as "of". Miquon does this from the very beginning, and I think it helps. Students can easily handle this sort of thing before they have mastered their multiplication tables, and it helps to keep their thinking straight from the start.
Maria Miller said…
I like the idea of including 1/2 times a group in my book. I'll have to do that when I get time. Thanks!
Anonymous said…
Be sure to include other simple fractions, too, not just a half.
1/3 x 6 is easy,
or 1/4 x 12... You get the idea.

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