Understanding Multiplication Concepts: Lesson Study, a Technique from Japan Where Teachers Study Ways of Teaching One Lesson for Several Hours, Is Described by Judy Mousley as She Considers Concepts of Multiplication and Division
Mousley, Judy, Australian Primary Mathematics Classroom
Does 3 x 6 mean three groups of six or a group of three multiplied by six? Both? What about 30 x 6? Should we teach both ideas and aim make sure children understand that multiplication is commutative, or should we aim for consistent understandings?
Each year in teacher education courses Australia-wide, student teachers meet the idea of partition and quotition division. The difference is not hard to grasp, but the question of why both concepts need to be taught is always asked--or is at least preempted by lecturers.
In partition division, the action is sharing. The number of groups that need to be made is known from the start, and the aim is to find the size of the groups. It is like dealing out tickets, where the number of people is known but not how many tickets each person will receive. (To facilitate memory, I tell student teachers that the number of parts is known: 'parts-partition'.)
In quotition division, the action is partitioning. The size of groups that need to be made is known from the start, and the aim is to find the number of the groups resulting. It is like giving away bundles of tickets, where the number of tickets in each bundle is known but not how many people will receive some. (That is, the quota is known: 'quota-quotition'.) Quotition division can be likened to repeated subtraction in that one group is taken from the original amount, then another, and another, etc., until the original amount is exhausted.
This all seems rather theoretical, and not too important. We merely need to realise the need to teach the two division actions, and to give children problems of both types (see Figure 1).
Note that for both examples presented in Figure 1 the symbolic representation is
30 / 10 = 
Figure 1: Two types of division problems Typical partition problem: I have one large packet of Jaffas, containing 30 sweets. Ten children will be at the party. How many could each child get? Typical quotition problem: I have one large packet of Jaffas, containing 30 sweets. If each child is to get 10 Jaffas, how many children can be served from one packet?
Also, in both cases the answer is 3. However, different mental and/or physical actions are required to solve the two problems, so different language is likely to be used in discussing the problems. Essentially, two different division concepts need to be taught. Pupils do not have to be introduced to the terms 'partition' and 'quotition', but it would be interesting to have children of any age draw pictorial representations of the two problems in Figure 1 (or a similar set) and then discuss whether their drawings, and the thinking that led to them, displayed any differences.
Enough said about division. The difference between partition division and quotition division is quite straightforward. All that is needed is awareness by teachers and other people who prepare classroom resources, so that children get a range of experiences that give them a good understanding of each. We are left, however, with a question of whether to teach both ideas at once or to structure learning so that both ideas receive their share of attention by being taught separately or in a logical sequence.
Given the fact that most teacher education courses deal with the above, and primary teachers generally attend to both sharing and partitioning in teaching division, it is surprising that very few courses and classroom lessons take the equivalent forms of multiplication into account.
Consider, for example, 4 x 6. Does one of the numbers represent the size of the group and the other the number of groups? What would the children in your class say? How would they draw this problem?
In junior primary years, 4 x 6 is usually interpreted as four groups, with six in each group. Words like 'four lots of six', 'four groups of six' and, later 'four times six' are used. …