How to Equip Your Students to Better Understand Multiplication, Part One

Common Teaching for MultiplicationAs I have coached and taught in the classroom, the three most popular ways to describe multiplication is showing ______ groups of ______, using repeated addition and making arrays. Now all of these methods have their place in a student’s understanding of multiplication, but if these methods are all they know, their understanding is limited. In this series of posts, I want to equip you to ensure that students have a deep understanding of multiplication.

If students only understand multiplication as repeated addition, they will be in real trouble when they try to find ½ x ¼. Some mathematicians would not appreciate defining multiplication as repeated addition, but I would argue that repeated addition has its place in a student’s understanding of the multiplication. Hung-His Wu, a mathematician who wrote Understanding Numbers in Elementary School Mathematics, uses the language of _____ copies of _____. I liked this because while coaching in a classroom I explained that multiplication is like a stamp.

Frog StampFrog Stamp  Frog Stamp

1st Stamp                                  2nd Stamp                         3rd Stamp

The objects on the stamp are what is in the group, and the stamp is the group. If the stamp has 3 frogs on it, and I stamped it 3 times, how many frogs are on the page?

Repeated addition becomes a springboard to solve multiplication problems, just like counting is a springboard to solve addition problems.

Let’s look at how the Counted Method works. Repeated addition and _____ groups of ______ would fit under this model of multiplication. I have created a simple student work mats you can download to help students think about multiplication through this model. Slide the work mat into a sheet protector and give students lots of practice talking about multiplication.

Multiplication Strategy Counting         Counting in Multiplication

Counted Model connecting Abstract        Counting in Multiplication with Mathematical Model

Multiplication Strategy abstract with counted representation        Counting Model in Multiplication Abstract

I created the second work mat because it is important that the mathematical model is connected to the concrete. (Mathematical Practice 4-Model with Mathematics). Students need to have mental images of repeated addition, so that when they are in the real world they can recognize a repeated addition situation as actually being multiplication.

Important: Wu emphasizes the importance of staying consistent when defining 3 X 5. Given the work mat I created, you would say 3 groups of 5. It is important we help students create a mental picture when they see the math model 3 X 5. There is a relationship between 3 groups of 5 and 5 groups of 3, but that relationship is better explored through the array model.

How have you taught multiplication using the counting method?

How to Equip Your Students to Better Understand Multiplication, Part Two

How to Equip Your Students to Better Understand Multiplication, Part Three

Beverly Ford

Beverly Ford has been working in education for the past twelve years as a teacher, coach, and staff developer. She is incredibly passionate about mathematics, brain research, and teaching techniques. She loves to do her part in education because she knows the power that can come from having a good education. She would love to hear from you at

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6 Responses to How to Equip Your Students to Better Understand Multiplication, Part One

    • Thanks, arrays and repeated addition are the two most common ways to teach multiplication. Students think about things in different ways. When a student sees 3×4, what picture comes to mind? It could be 3 groups of 4 or a rectangle with 3 rows of 4. Stay tuned for more on arrays in part two.

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