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Even though multiplication can be an abstract concept for many, the foundation for it can be nurtured in the early years. Multiplicative thinking for young children involves moving beyond counting when calculating quantity and beginning to think about groups, units, relationships, and patterns between numbers. There are many ways we can help build the foundation of multiplicative thinking early on.
Doubling
Doubling helps children see a number as a unit that can be doubled. For example, children can begin to recognize that 5 and 5 make 10, or that 6 and 6 make 12, without needing to count each object.
We can look for opportunities to use doubling throughout the day. If there are 4 children at one table, we might ask, “How many children would there be if we had another table with the same number of children?” Or, if we have 5 blocks, we could ask, “What would happen if we had another group of 5?”
Doubling also becomes a useful strategy for working with numbers that are close to a double. For example, a child might use 5 + 5 to help figure out 5 + 6. These kinds of connections help children begin to use what they already know to think about new quantities.
Unitizing
Unitizing is an important part of developing multiplicative thinking. It helps children understand that a group of objects can be thought of as one unit that can then be counted or repeated.
For example, instead of seeing 10 counters as 10 individual objects, children might see them as 1 group of 10. They are beginning to think about the structure of the quantity rather than just counting each object.
We can encourage unitizing by having children make equal groups with everyday classroom materials. We might ask:
- “Can you make groups of 3?”
- “How many groups did you make?”
- “How many are in each group?”
- “How many are there altogether?”
- “How did you figure it out?”
The goal is not simply for children to get the answer. We want them to start noticing that the same quantity can be organized and thought about in different ways.
Skip Counting
Skip counting by 2s, 5s, and 10s is an early way of helping children think about units and develop multiplicative thinking.
However, skip counting becomes more meaningful when children understand what they are actually counting. For example, counting 2, 4, 6, 8, 10 can represent counting groups of 2 rather than simply memorizing a sequence of numbers.
If each child gets 2 cubes, we might ask, “How many cubes will we need for 5 children?” Children can make groups of 2 and count them: 2, 4, 6, 8, 10.
This gives a purpose to skip counting and helps children connect the numbers they are saying to the groups they are seeing.
Repeated Addition
Repeated addition helps children make a connection between addition and multiplication. When children see 3 groups of 4, they might represent this as:
4 + 4 + 4
This is a useful way for children to begin thinking about equal groups. Multiplicative thinking involves noticing the groups and the relationship between the groups and the quantity in each group.
Educators can encourage this by asking open ended questions including:
- “How many groups do you see?”
- “How many are in each group?”
- “How could you count these?”
- “Could you count them without counting every object?”
- “What do you notice about the groups?”
These questions encourage children to think about the structure of the quantity and look for more efficient ways to determine how many there are.
Arrays
Arrays are another great way to help children visualize equal groups. Arranging objects in rows and columns helps children see equal groups and provides a visual representation that can later support their understanding of multiplication.
For example, 12 counters could be arranged in 3 rows of 4 or 4 rows of 3. Children can explore both arrangements and notice that the total stays the same even though the arrangement has changed.
Finding arrays in the world around us is an interesting way to “notice and wonder” about multiplication. We can find arrays in many places:
- windows on a building
- seats arranged in rows
- tiles on a floor
- eggs in a carton
- buttons on clothing
- objects arranged on a shelf
Look for Multiplicative Thinking in Everyday Situations
Multiplicative thinking can be developed through everyday classroom experiences. When children are sharing materials, setting tables, building with blocks, organizing collections, or lining up, there are many opportunities to think about equal groups. Educators can notice these moments of learning and facilitate thinking through responsive questioning. For example:
“There are 4 children at each table. How many children could sit at 3 tables?”
Or:
“We need 2 crayons for each student. How many crayons will we need for 6 students?”
Or:
“Here are 20 counters. Can you organize them into equal groups? How many different ways can you find?”
These questions encourage children to think about how quantities are related, rather than simply finding an answer.
Building the Foundation
Developing multiplicative thinking in young children is about providing experiences that help children see groups, units, and relationships.
When children double, make equal groups, skip count, organize arrays, and look for patterns, they are building ideas that will support their later understanding of multiplication and division. Responsive educators can use their observations of children's actions to help highlight this and invite them into the learning opportunities.
