If you clicked a link to this post, I am going to assume you teach multiplication.

Why does it seem to be such a sticky point for teachers, students, parents, and society at large!? Everybody has some story about multiplication - times tables, facts, flash cards. I get it, students need to internalize their facts. Internalized facts will help them as the grow into higher algebraic thinking with longer expressions and more equations. But internalizing is different than memorizing. Memorizing is difficult for most human brains, especially isolated information with no connection. So while most of us are beating our heads against the wall trying to “get” kids to memorize their facts, what if we could be helping them internalize them through use instead? People who use facts often enough commit them to memory with connection. This is the kind of connection that will last and transcend a certain activity, grade, age level, etc.

With this in mind, what activities can we do that help internalization of multiplication facts?

The answer is - activities that aid in multiplicative thinking. And that brings us to our blog post today:

What is multiplicative thinking?

For us to truly help students in their facts and conceptual understanding, we need a good grasp on what multiplicative thinking even is. Let’s start from the beginning and scale up.

Early numeracy is the ability to name, represent, and describe quantities. Subitizing, counting games, and manipulatives are what is needed for early numeracy.

Additive thinking is the ability to join and separate numbers. Subitizing, using manipulatives, story problems, balance equation activities, building numbers, and more are needed for additive thinking. Counting, even skip counting, is still additive thinking.

Both early numeracy and additive thinking must be strong for students to build up to multiplicative thinking. There is one more component that needs to be strong for multiplicative thinking - unitizing. Unitizing is the ability to see something as separate parts and also a unit.

Once students are strong additive thinkers and able to unitize, they are ready to think multiplicatively. When I describe multiplicative thinking to students and adults, I compare it to my computer. Whenever I create something on my computer and want to recreate it, what do I do? I copy and paste. Multiplicative thinking is similar to that. Your brain is able to see a unit and copy/paste to see multiples. One way I sense that students are moving into multiplicative thinking is when they change their language from “There is a four and a four” to “there’s 2 fours”. When they use language like sevens, threes, fours, they are seeing that number as a unit, or a group. When they can do that, they can begin doubling, tripling, quadrupling the groups. We want students to be able to do this and not rely on skip counting, which is keeping them in additive thinking.

So how does this relate to math facts?

If we can help students become strong unitizers and have a strong sense of number, we can begin with doubling. Doubling is x2. It’s very important that students have a lot of access to visuals or hands-on manipulatives to build this concept. Doubling is a great place to start for multiplicative thinking (just like doubles is a great place to start for additive thinking!) because it’s a small copy/paste. For example, if you can see and know that this is a three, then you can understand that two of these threes would make six. This is early multiplicative thinking.

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Let’s keep going. You know this is a three. 2 of the threes is 6. What about 4 of these threes? Well, that’s 2 threes and 2 threes and I know 2 threes is 6. This must be….12!

Now we are thinking multiplicatively! Multiplication strategies for the facts should rely heavily on connections. So instead of having to “count by” or draw out groups, students use their other well-known facts and strategies to reach the others. There is the deep connection piece that is needed for internalization.

Luckily, you can use similar strategies for additive thinking and multiplicative thinking with some small tweaks to the language and understanding. For example, quick dots is a brilliant and easy activity to help multiplicative thinking. Students can look quickly at a dot image and tell you how many “fours” or “sixes” or “twos” they saw! Even if they can’t figure out the total yet, getting in that copy/paste mode will be helpful when learning to multiply.

Starting with doubling will open up many opportunities for strategy-connection. For example, doubling can help students build a strategy for x4, x6, and x8. Multiplying by 5 and 10 is relatively natural because students use 5 and 10 as benchmarks from an early age (and should be!). x5 can be used to build strategies for x6, x7, etc. x10 can help with strategies for x9!

All that matters is students re doing things that help them think multiplicatively.
Here are some more examples.