Consider a problem like the following:
When children are asked what operation they would use to solve this problem, many will pick division. Some might think those children simply do not understand multiplication and division. However, that is not the case. Those children pick division as the operation because they know that "multiplication make bigger and division makes smaller." Research has shown that many children, and adults, hold this misconception.
Actually, calling it a "misconception" may be inappropriate. Rather, it is an overgeneralization children make based on their experiences. While students are working only with whole numbers, the only exception to this generalization is multiplication by 0 and 1. However, once they go beyond the "basic facts" stage of multiplication learning, practically all experiences involve multiplying by a number greater than one. The same can be said of division. The only time division does not result in a number less than the dividend is when it is divided by one. However, once again, practically all children's experiences before this point are division by a number greater than one.
Once the range of numbers is expanded to include decimal numbers and fractions, however, there are many cases where we do multiply or divide by numbers less than one. Therefore, it is an important goal of mathematics teaching that our students overcome this overgeneralization. A potentially powerful tool for this purpose is double number lines. If we represent the ribbon problem, it will look like this:
From this diagram, you can easily see that if the multiplier (represented on the bottom number line) is less than 1, the product (? mark) will be on the left of the multiplicand (1.80, i.e., amount corresponding to 1). On the other hand, if the multiplier is greater than 1, the product will be to the right of the multiplicand. Therefore, we can generalize:
If multiplier is greater than 1, multiplicand < product.
If multiplier is less than 1, product < multiplicand.
Similarly, you can use double number lines to contrast the situations when the divisors are less than 1 and those cases where the divisors are greater than 1.
However, the most difficult part for students (thus for teachers) is to help them understand that these situations are indeed situations where multiplication is the appropriate operation. For some students, double number line may not be sufficient. Another possible tool is to write mathematical expressions using words to describe the relationship among the quantities involved. In the ribbon problem, there are three quantities: cost of 1 meter of ribbon, total length of ribbon, and the price. The relationship among these three quantities can be expressed as
An implicit, yet very important, goal of teaching multiplication and division of fractions and decimal numbers is to expand students' understanding of these operations. In early elementary grades, these operations are considered in equal group situations. Thus, when the multiplier or the divisor (in the case of fair-sharing division, the quotient in the case of measurement division) becomes something other than whole numbers, students have difficulty interpreting what it means. Through teaching of multiplication and division of decimal numbers and fractions, we want students to develop more proportional understanding of these operations. For example, A x B = C, should be interpreted as "A is to 1, C is to B," or "C is B times as much as A." Although this is not an explicitly stated goal in the GPS, it is something all teachers must keep in mind.