Early Multiplication and Division

The development of early multiplication and division knowledge includes the emergent notions of repeated equal groups and sharing, the development of skip counting and the use of arrays in teaching multiplication and division. Also explained are the ideas of numerical composite and abstract composite unit - important milestones in the development of numerical thinking, the idea of commutative, and the inverse relationship between multiplication and division. This begins with very simple notions of equal groups and sharing and learning number word sequences of multiples (for example, 3, 6, 9 ...), and extends to developing facile strategies to work out multiplication and division tasks in the range 1 to 100.

Repeated Equal Groups and Sharing

Children's initial ideas of addition arise from the idea of combining two groups of items. In similar vein, children's initial ideas of multiplication are linked to combining a number of groups, each of which contain an equal number of items, for example, combining six groups each of which contains three items. This situation is referred to as repeated equal groups

Finally, children's initial ideas of division are linked to sharing a collection of items into equal groups.

Children should be provided with experiences that involve the activity of making repeated equal groups where the number in each group and the number of groups is specified, for example: Make six groups of three, five groups of four, eight groups of two. Children should also be provided with experiences that involve the activity of sharing a given number of items into a specified number of equal groups, for example: Share eight items into four equal groups, ten items into two equal groups, fifteen items into three equal groups. Activities can also involve everyday materials and situations in which repeated equal groups or sharing arise naturally, for example: How many legs on four cows? How many fingers on six hands? For all the above activities, it is important to foster in children the development of the appropriate mathematical language, for example: three groups of four, eighteen shared into six equal groups.

Number Word Sequences of Multiples - Skip Counting

Number word sequences of multiples are sequences such as ‘two, four, six, eight, ....' - the sequence of multiples of two; and ‘five, ten, fifteen, twenty, ..." - the sequence of multiples of five. The activity of saying such sequences is typically referred to as counting-by-twos, counting-by-fives, and so on. This is also referred to as skip counting. It is important to keep in mind the distinction between, on the one hand, merely saying the sequence of multiples and on the other hand, using the sequence of multiples to count the items in repeated equal groups, for example, six groups of two. Learning the common sequences of multiples is important as a basis for multiplication and division. Typically, children's learning of these sequences proceeds in the following order: by twos, by tens, by fives and by threes. Of course, children begin to learn each new sequence before the previous sequence is known completely. The sequences by other numbers in the range 1 to 10, that is, by fours, sixes, sevens, eights and nines, generally arc not learned to the same extent as those stated earlier. Nevertheless, knowledge of these sequences will constitute an important basis for automatizing the basic facts of multiplication. Activities such as number rhymes can be useful for learning the common sequences (by twos, fives, tens and threes).

Counting Repeated Equal Groups

When children are learning the common sequences of multiples (twos, fives, and so on), they should also be given experiences in using these sequences in situations involving materials, that is, situations involving repeated equal groups of items. Teachers can demonstrate for children, counting by twos, by fives, and so on. For example, counting a collection of 20 red counters by twos involves moving a pair of counters together, in coordination with saying each word in the sequence - two, four, six, eight, and so on. It is tempting to regard activities such as counting by twos, by fives, and so on, as simple and straightforward activities. However, there is a tendency on the part of the teacher to imagine that the child, as well as imitating the teacher's actions also imitates the teacher's thinking. On the other hand, dose observation of children counting by twos and so on can reveal interesting limitations on their thinking, For example, a child who had been shown how to count by twos was asked to count a collection of counters by twos. The collection contained nine counters. The child counted by twos - two, four, six. eight, in coordination with moving a pair of counters for each number word. After saying 'eight' he moved the final counter and said 'ten'. That there was one counter rather than two remaining did not seem to be problematic for the child. To emphasize the point being made: as obvious as it might seem to an adult that, when counting a collection of counters by twos, each number word refers to two counters, this is not necessarily obvious to children! As another example, children might regard the number of items in a collection as being dependent on how it is counted (by ones, by twos, by threes, and so on).

Division as Sharing or Measuring

In an earlier section, division was explained as arising from situations involving sharing into equal groups. In this context there are essentially two different kinds of situations that give rise to division and, as a teacher, it is important to take account of these. Try the following situation for yourself. You will need two sets of 12 counters, for example, blue and red. Use the red counters to solve the following task: arrange 12 counters into 3 equal groups. Use the blue counters to solve the following task: arrange 12 counters into groups of 3. Draw a diagram corresponding to your solution of each of the two tasks. Each of these corresponds to 12 ÷3. Alternatively, one could say there are two different ways that the problem 12 ÷ 3 can be demonstrated using a collection of 12 counters and using the number 3. These are called sharing or partitive division and measuring or quotitive division.

To interpret 12 ÷ 3 in the sense of sharing is to regard the divisor (3) as indicating the number of equal groups. To interpret 12 ÷ 3 in the sense of measuring is to regard the divisor (3) as indicating the number in each of the equal groups, from the children's perspective, a division situation leads to a division expression 12 ÷ 3, rather than an expression leading to a situation. That is to say, it is usual for teachers to present children with a division situation before presenting the written expression.

The typical approach in teaching is to begin by working wholly or mainly with one only of the situations, typically the sharing situation. The thinking underlying this is that sharing into a given number of equal groups is typically a very common experience for young children. Ultimately children should be familiar with each of these two division situations. Given a number story relating to either situation, children should be able to generate the corresponding division expression. Try for yourself to write five number stories involving simple division in the sharing sense and similarly write five number stories involving simple division in the measuring sense. For each story, write the appropriate division expression.

Abstract Composite Unit

As children become more facile at working with repealed equal groups and sharing, and thinking in terms of multiples, they are able to think more abstractly about the numbers involved in multiplication and division situations. A major advancement is the ability to regard as a unit a number larger than one when it is appropriate to do so. Thinking of this kind is referred to as unitizing. Prior to this advancement, children who can think abstractly about numbers are able to think of numbers as composites but not as units. These two levels of thinking as a numerical composite and an abstract composite unit. Developing the idea of an abstract composite unit is fundamental for learning multiplication and division. For example with the idea of an abstract composite unit, a child who is asked how many threes in 18 can think abstractly in terms of repeated threes. Because the child can regard three as a unit (as well as a composite), they can think in terms of counting the units, that is, how many units of three make 18?

Using Arrays

As described above, experience with counters and so on organized into repeated equal groups can provide an important basis for the early development of multiplication and division ideas. As children's knowledge of multiplication and division develops, arrays can be very useful. We use the term 'multiplied by' to read a multiplication sentence (equation). Thus 5 x 4 is read as '5 multiplied by 4'. Thus you can regard 5 x 4 as meaning 5 + 5 + 5 + 5, and 4 x 5 as meaning 4 + 4 + 4 + 4 + 4. Related to this, we regard a 5 x 4 array, for example, as consisting of 4 rows of 5 dots. Thus, in a 5 x 4 array, the first number (5) indicates the number of dots in each row and the second number (4) indicates the number of rows.

Initially, children can explore the idea that there are the same number of dots in each row (similarly in each column). It is important to move children beyond the activity of counting the dots by ones.

In teaching situations, arrays can be used on an image projector in ways so that some or all of the rows are each screened by a strip. In this situation children are encouraged to count by multiples corresponding to the number in each row. This can be extended by screening the whole array, and encouraging children to visualize the array in order to count the dots in multiples. The array can then be unscreened so that each row is covered by a strip, and children can again count the dots in the array in multiples.

Commutativity and Inverse Operations

As children's knowledge of multiplication and division develops, two important ideas arise - commutativity and inverse. Previously commutativity was discussed in the context of addition. It also applies to multiplication (but not subtraction and division). In the case of multiplication, commutativity refers to the principle that when any two numbers are multiplied they can be multiplied in either order without affecting the answer, for example. 6 x 4 = 24 and 4 x 6 = 24. This is sometimes expressed as follows: for any two numbers a and b, a x b = b x a. Arrays are ideal for demonstrating the idea of commutativity. For example, the array for 6 x 4 can be turned through 90 degrees to show 4 x 6.

Multiplication and division art inverse operations, that is, division is the inverse of multiplication and multiplication is the inverse of division. The preceding statement refers to the principle that if a given number is multiplied by any number and the answer is then divided by the same number, then the answer is equal to the original number. This is demonstrated in Figure 10.7.

6 x 11 = 66 100 ÷ 4 = 25

66 ÷11 = 6 25 x 4 = 100

Children should develop sound knowledge of the principles of commutativity and inverse operations. This includes not only being implicitly aware of these principles but being able to use them flexibly. Examples of using commutativity are:

• using the known fact of 7 multiplied by 2 makes 14 (7 x 2 = 14), to work out 2 multiplied by 7; and
• working out 5 multiplied by 9 in order to work out 9 multiplied by 5

Examples of using the inverse operations principle are

• using the known fact 5 multiplied by 4 makes 20 (5 * 4 = 20) to work out 20 + 4; and
• using the known fact 6 multiplied by 7 makes 42 (6 * 7 = 42) to try to work out 42 + 7.

Extending Multiplication and Division Knowledge

Children's work in multiplication and division extends from the ideas of equal groups and sharing, to the development of abstract composite unit. These ideas are further extended to include the principles of commutativity and inverse operations. During this time most of the work in multiplication and division is in the range 1 to 100. Within this range, children will begin to habituate some of the simplest basic facts of multiplication such as doubles (4 x 2) and squares (6 x 6). As well, children can use doubling of known facts to work out other facts, for example 4 sixes is double 2 sixes, 8 fives is double 4 fives. When children have developed a range of facile multiplication and division strategies in the range 1 to 100, they are ready to work on habituating basic fact knowledge and extending multiplication and division to include 2-digit factors, for example, 30 x 2, 23 x 3.