Multiplication - what does your child need to learn?
Updated: Apr 21, 2020
What do you notice about this image of bananas? What mathematics do you see in it?
You might be wondering why anyone would even display bananas this way! Fair enough... I'll admit it's not a great example of math in the real world. Unless maybe you see it as a piece of modern pop art.
But it will help us unpack a few key mathematical ideas that children should be learning about multiplication:
the array model
properties of multiplication
Maybe you hear your child’s teacher talking about these ideas and want to understand them better. Or maybe you want to deepen your own understanding so you can help your child at home. In this post I explain what children need to learn about multiplication in order to multiply single and multi-digit numbers with understanding.
The bananas in this poster are organized in an array. Working with rectangular arrays helps students understand both multiplication and division.
As adults, we have lots of experience with arrays. We understand the structure of rows and columns and probably can use multiplication to figure out how many objects are in an array. For children, though, a thorough understanding of the array model takes lots of time to develop. Giving them this time and practice is critical. The array is extremely important for understanding multi-digit multiplication and division in later elementary school and algebra in high school.
Let's consider what a child could learn from doing a problem related to this banana picture...
First I ask students to estimate how many bananas they think there are in the whole picture - including the part that is out of view. It's always a lively discussion because some students argue that it could be any number - that the banana array could expand forever. Can you picture this array as the corner of a much larger array that has 100 or 1000 bananas in it? What might that look like? Students who are more grounded in reality will argue against this, as you can imagine. Who would make a poster that big they ask. Or, why would you want to? Both valid points. However, visualizing and describing a range of possible sizes of arrays is really good for kids.
Once we've had a bit of fun visualizing and debating, I let the students know that the whole picture of bananas is 15 bananas wide and 6 bananas tall. I ask them to figure out how many bananas there are in the whole picture.
Organizing quantities into equal groups is one of the first things students have to learn to make the shift from addition to multiplication. The array model already provides equal groups for children. Yet, some children don’t use the equal groups to help them. They draw the full array or build it with physical materials and then count by ones, like this:
Why might students use this inefficient method that's quite prone to error?
Some children don't yet understand that each row has the same quantity as all the other rows. Other children see the equal groups but don't yet know how to add them.
Digging deep to find out why children do the things they do is necessary for precise teaching to occur. A child who doesn't understand the equal group structure of arrays needs a different teaching approach than a child who isn’t yet able to add the equal groups.
Once children understand the equal group structure in arrays and have good addition skills, they will be able to use all sorts of strategies to find out how many bananas there are. I’ve shared a few common strategies below:
Early on, children will use repeated addition.
Some see that there are six rows of 15 and add 15 six times
Others see the columns of 6 and add 6 fifteen times.
You might be wondering why children would add 6 fifteen times when they could add 15 six times. Adding the fifteens seems a lot easier! But children don't start off with a robust understanding of the groupings in an array. Some see the columns as equal groups. Others see the rows as equal groups. Those who do see both rows and columns as equal groups might think it's easier to work with the sixes because six is smaller than 15.
In math class, where lots of students are solving this problem using different methods, we have the luxury of guiding them to compare methods. By looking at each others' methods, children learn how other people view the array. We guide them to notice that it doesn't matter if we use the rows or columns - both ways give the same answer.
In other words, it doesn't matter whether we count the 15 sixes or the 6 fifteens. Either way we get a total of 90 bananas.
This idea that 6 x 15 is the same as 15 x 6 is an important property of multiplication called the commutative property. This property is not obvious to most students. The array helps make it visible. And think about how useful this knowledge is when it comes to learning multiplication facts. If you know 6 x 8, you also know 8 x 6!
Combining groups to make bigger groups:
With encouragement, students will look for ways to make bigger groups so they can do less adding. Some students double rows of 15 to make 30's. This creates 3 groups of 30 which is much easier for most people to add. Some students can even skip count by 30's, so they might use skip counting.
While they won't realize it yet, essentially these students have changed this problem from 6 x 15 to 3 x 30. Eventually, with teacher guidance, they will come to understand that when they doubled the fifteen it resulted in half as many groups.
Later, this understanding of doubling and halving will become a powerful strategy for solving problems mentally! Think how nicely it could be used to solve the problem 14 x 4.5. By doubling the 4.5 and halving the 14, as shown below, we change the problem into 7 x9.
Using Relationships and Properties
A primary goal of teaching multiplication is to move students from using counting methods to using relationships and properties. There are a myriad of ways students use relationships to make this problem easier to solve. Doubling and halving was one example of using relationships.
Here's an example of a student who used their knowledge that 3 + 3 = 6.
They found the number of bananas in three rows and then doubled to find the total bananas in 6 rows.
This student saw that 15 could be split into three fives. They created three smaller arrays with five in each row. Then they just had to solve 6 x 5 = 3o and add the 30 three times.
This student split the rows of 15 into a section with 6 rows of ten and a section with 6 rows of five. This gave them numbers they could skip count with.
There are many other ways this banana array could have been broken down into smaller arrays. Some ways make the problem easier. Some don't.
When children break arrays into smaller parts, multiply the parts and add them back together like this, they are learning about another important property of multiplication: the distributive property.
Students don't need to be able to name the properties. They just need to be able to understand and use them.
Solving these kind of problems prepares students to understand the how the standard algorithm works when the time comes for learning about it. Once the students are comfortably splitting numbers by place value to multiply, they are likely ready to connect their method to a version of the standard algorithm.
So far we have talked about the development of early strategies for multiplying, some important properties of operations, and the array model.
Possibly the most significant milestone in learning about multiplication is the ability to unitize. This mathematical idea may be as important and challenging as the physical milestone of learning to walk! It requires a big shift in the way students think about number.
Unitizing is the ability to count equal sized groups as if each group is a single item or unit. The drawing below represents 12 cookies split into four groups of three. For a child to figure out how many groups of cookies there are, they have to be able to think of the 3 cookies as a single unit. and count that group as one instead of three.
Unitizing is so easy for us now that it's hard to fathom how it can be such a challenge for children. But up until this point, a child's only counting experience has been to count single items. Now we are asking them to think of a group as a single item (unit) and count those groups using the same counting numbers they usually use to count single items.
What do you know if your child is learning to unitize?
As your child develops the ability to unitize, you will likely notice changes in their language:
At first they count only the individual cookies.
Then they can count the groups using a specific context (plates).
Later they may use the phrase 'groups of'.
Finally the phrase "4 threes" makes sense to them and they will begin to use it.
Students who have fully developed the ability to unitize would be able to describe the banana array by saying there are "15 sixes" or "6 fifteens" and 90 bananas in total.
What else makes multiplication challenging for children?
There’s another reason that multiplication can be challenging for students. Students are often told that multiplication is just repeated addition. It is true that we can use repeated addition to solve a multiplication problem. But a multiplication problem is actually quite different in structure to an addition problem.
In addition problems, each number has the same role. The numbers join together to form a new quantity.
In multiplication, the two numbers don’t join together at all! The two numbers in a multiplication problem each have very different roles. It’s kind of like one number tells the other number what to do. Think about the diagram of 4 x 7 below, for example. Seven is an actual quantity. Four is a number that tells us how much to scale up or replicate the 7.
This is a big shift in thinking for students. Can you see how it's impossible to fully understand multiplication without unitizing? Students need to be able to count or think of four 7's to understand multiplication.
Describing the parts of an array and how they relate to the whole helps students develop an understanding of how multiplication works.
The importance of developing unitizing and a conceptual understanding of multiplication cannot be understated. In part this is why rote memorization of facts is a very limited approach to learning about multiplication.
To really make sense of what it means to multiply and divide, students need to solve single digit multiplication problems over and over, using concrete materials and mental mathematics. Instruction needs to be carefully sequenced so that children gradually develop conceptual understanding and fluency. With proper guidance and frequent opportunities to practice, students will become faster at solving multiplication problems and eventually able to call up many of them from memory.
Fluency and understanding are both important and can be developed together.
Feel free to leave your thoughts and questions in the discussion space.
Lawson, A. (2015). What to Look For: Understanding and Developing Student Thinking in Early Numeracy. Pearson Canada.
Fosnot, C. & Dolk, M. (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Heinemann.