In an earlier post (August, 2007), I discussed the relationship between the two division situations, fair sharing and measurement (discussed in element b) and multiplication situations. When students are first introduced to division, they must understand that the division is an operation needed when you are making equal groups with the given amount. We can use the division to find the number of groups given the number in each group (measurement division), or we can use the division to find the number in each group given the number of groups (fair sharing). We want students to develop a unified understanding of these two situations as "division."
In the introductory stage of division instruction, students focus on those division problems that are the inverse of the basic multiplication facts. Thus, they develop the strategy to find the quotient of division by looking for the related multiplication facts. For example, to solve 48 ÷ 6, students think about 6 and what multiplied together will equal 48. Since 6x8=48 (or 8x6=48, depending on the type of division), we can say the answer is 8. Thus, if we are giving each person 6 candies, 48 candies can be shared by 8 people.
What if we had 50 candies? There is no multiplication fact with 6 as a factor that will give the product of 50. Most children can solve this problem if they are allowed to use concrete materials to manipulate. As division with remainders is introduced, it is important that students initially use concrete materials to model the problem situation. They should compare and contrast the situation with earlier division situations (without remainder) to realize that these situations are also creating equal groups, thus it is appropriate to represent it with division sentences. Furthermore, implicit in division problem is to maximize the quotient (either the number of people sharing or the number of items for each person) - that is, the point isn't just to make equal groups but to use up as many of the given amount. The "remainder" is the amount left over when the maximum amount of the given amount is used up.
When students simply rely on computation (multiplication) to find the remainder, sometimes you see mistakes like 50÷6=9 remainder 4. This occurs because when you look for the quotient by checking the 6's multiplication facts in order, you recognize the quotient only after the product exceeds the total. Thus, some children mistakenly think that the quotient is 9.
If students do not understand the meaning of the remainder clearly, the opposite can also happen. Some students may say that 50÷6=7 remainder 8. Those students do not understand that we have to use up as many of the given amount. Since this is not a computation error (in the sense of getting an incorrect product or incorrect difference), the answer checking algorithm (Dividend = Divisor x Quotient + Remainder - Gr. 4 GPS) will not detect it. To avoid this error, students must clearly understand the meaning of remainders. Furthermore, since we are using up as many of the given amount, the remainder cannot be greater than the divisor - in either type of division situation. If the left over amount is greater than the divisor, that means we can give (at least) one more person his/her share (measurement division) or (at least) one more item to each person (fair sharing). But, this relationship, Remainder is less than Divisor, is the result of the often implicit requirement of division that we must use up as many of the given amount as possible.
By the way, in many Japanese elementary mathematics textbooks, the long division notation is introduced when students are learning division with remainders - after students understand the meaning of division, remainder, and the relationship between the remainder and the divisor. This notation is introduced to ease the mental demand involved in division with remainders. For example, in the case of 50÷6, children must identify the first multiplication facts with 6 as a factor that exceeds 50, 9, then subtract 1 from it to make 8 as the quotient, find the product of 6x8, then subtract the product, 48, from 50 to find the remainder. The long division notation can provide a way for children to record the intermediate steps and procedures more explicitly. However, it is important to remember that we are not really teaching children the long division algorithm here. The notation is simply introduced as a way to deal with simple division with remainders (that is, those division that requires the application of the multiplication facts only once). I will discuss the development of the long division algorithm, element e, in a separate post.