### 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?