Early multiplication and division

For young children, consideration of the concept of multiplication starts with putting objects into groups. These groups need to have the same amount of that object. Multiplication problems can also be represented by rates, scale and combinations. These are addressed in later primary years but the foundational understandings are laid in early conversations and opportunities to explore multiplication (McDonald & Rafferty, 2018).

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While children often have many informal lived experiences related to addition, subtraction, and division, it is unlikely they have had much need to group items (Siemon, et al., 2017). The concept of groups or grouping, therefore, won't necessarily be an intuitive one. Children may have been asked to get into a group, or walk in a group, but this is not generally connected to multiplicative thinking.

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Children will need scaffolded experiences to understand that grouping requires putting the same amount of objects into each collection. Application of this process relies on children being able to count collections and understand the concept of cardinality (the last number said is the total of the group). 

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Using materials that are close to hand and connected to a child's world will help them make sense of the early concept of grouping. This may include food, toys, natural materials, items of clothing (i.e. pairs of shoes), or themselves. Copying and creating repeating patterns with objects, sounds, and/or body movements provides an opportunity to talk about the group of elements that is repeating. Generating story problems from immediate or imaginative contexts will connect the concept of grouping with children's play. Asking and assisting children to get into "groups of 2" or "groups of 3" will reinforce the need for the same amount in each group.

Pairs are one of the first types of grouping children may be familiar with - they are groups of 2. Shoes and socks, then, could be a good place to start a conversation about groups.​ Finding a scaffold to help define a group of discrete items is helpful. This could be a plate, a hoop, a container - anything that helps define the boundary of the group. At these early stages, it is the creation of equal groups and reasoning about these that is important.

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How many groups? How many in each group?

Multiplication is counting based on composite units (putting parts together to make a new unit). To think multiplicatively we need to simultaneously consider how many groups, and how many in each group. 

Let's think about what that looks like.

This picture shows three groups of smiley faces. There are four smiley faces in each group. We would describe this as 3 groups of 4. The 3 tells us how many groups, the 4 tells us how many are in each group (or how much). This would later be described as three-4s.

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It is important to note that even when children are able to create the 'groups of', they will generally start solving the question of 'how many altogether' by counting in 1s to count all of the items. That is, they will count the first group of 4 in ones, then go on to the second group, adding by ones each time.

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Counting in composite units is more efficient than counting in 1s. For example, it is more efficient for me to count the smiley above faces in composite groups of 4. I could do this by repeated addition, which would be 4 + 4 + 4. Or I could solve it by skip counting 4, 8, 12. Both of these strategies are more efficient than counting in 1s. 

Eventually I may have some fluency with my number facts and use a more efficient strategy like doubling to work out the total of the first two groups above, i.e. 4 and 4, then count on in ones from this total to include the last group of 4 to get a total of 12. 

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