As 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.
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.
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?