Misconception #2: “Multiplication and division are opposite operations.”

This is true in a sense, but the implicit message is that multiplication and division go in opposite directions. This becomes a problem when we could use the equivalence of

This is true in a sense, but the implicit message is that multiplication and division go in opposite directions. This becomes a problem when we could use the equivalence of

**m/3 and 1/3 m**. Both are algebraic representations of this relationship:At a recent conference, a presenter was talking about pretesting students so that she could provide differentiated instruction. She was looking for, among other things, students who could recognize unreasonable answers with the knowledge that division makes numbers smaller. But of course, when you divide by a number less than one, you get an answer larger than the dividend. As we introduce the concepts of multiplication and division to students, there must be a way to do it without embedding ideas that are dead ends, or incorrect. The equivalence of ÷ 3 and x 1/3 or ÷ 1/4 and x 4 is an important concept that we should be aiming at from the start.

Underlying both multiplication and division is the process of grouping quantities. We can make whole number groups, and we can also make fractions of groups; parts of groups. Are there are two different ways to represent the same event with symbols. If I have 9 objects and I separate them into three groups and focus on one of them, I can represent that action symbolically as 9 ÷ 3 or 9 x 1/3. Would it be possible to use both representations from the beginning? In fact, fractions themselves could be interpreted from the beginning as division problems with just a little tweaking I think. Instead of only focusing on 1/5 and 2/5 as one out of five and two out of five, could we highlight the concept of division when we introduce fractions? When you take a whole, or a group, divide it into 4 equal shares and focus on one of them, that is 1/4. When you take 3 wholes or groups, divide each into 4 equal shares and focus on one of them, that is 3/4. We certainly don't expect 3rd graders to master this concept, but they could explore simple examples.

The difference it would make in my class is that I would not have to unteach the clunky operations for 4/5 x 95 that many of my students cling to. And more importantly, I think it could help shrink the divide between the students who feel confident and may easily adopt mental math strategies and routines on their own, and the students who might understand strategies like dividing first with 4/5 x 95, but still feel more comfortable putting a 1 under the 95 and "multiplying straight across."

Underlying both multiplication and division is the process of grouping quantities. We can make whole number groups, and we can also make fractions of groups; parts of groups. Are there are two different ways to represent the same event with symbols. If I have 9 objects and I separate them into three groups and focus on one of them, I can represent that action symbolically as 9 ÷ 3 or 9 x 1/3. Would it be possible to use both representations from the beginning? In fact, fractions themselves could be interpreted from the beginning as division problems with just a little tweaking I think. Instead of only focusing on 1/5 and 2/5 as one out of five and two out of five, could we highlight the concept of division when we introduce fractions? When you take a whole, or a group, divide it into 4 equal shares and focus on one of them, that is 1/4. When you take 3 wholes or groups, divide each into 4 equal shares and focus on one of them, that is 3/4. We certainly don't expect 3rd graders to master this concept, but they could explore simple examples.

The difference it would make in my class is that I would not have to unteach the clunky operations for 4/5 x 95 that many of my students cling to. And more importantly, I think it could help shrink the divide between the students who feel confident and may easily adopt mental math strategies and routines on their own, and the students who might understand strategies like dividing first with 4/5 x 95, but still feel more comfortable putting a 1 under the 95 and "multiplying straight across."