Whenever I'm together with my sister (shout out to Lizzie!) and her fourth grader at the holidays, the conversation inevitably turns to memorizing multiplication facts. Students learn so many cool and amazing mathematical ideas in the elementary grades, but multiplication facts seem to always be on students' and parents' minds.
Learning multiplication facts is often a stressful process for elementary students. It is often the time when many kids first become anxious about math. As students become stressed, it can lead to arguments between children and their parents, and no one wants that!
It does not have to be this way. Learning multiplication facts is a wonderful opportunity for parents to have mathematical conversations with their kids! This post will help you learn how to help your child learn their facts faster and how to be happier doing it.
Math Fact Fluency
Many different researchers have developed different models for how children's brain learn multiplication facts. While some of them are complicated, it is worth giving you a quick overview here. In general, there are three levels of understanding:
Level 0 - The student cannot generate an answer to a fact. This implies that they do not yet understand how multiplication works.
Level 1 - The student can generate an answer to a fact using a laborious procedure, usually skip-counting. They know that multiplication can be thought of as repeated addition, so they use this strategy to get the solution. For example, if asked 5 x 4, they may count by fives four times (5, 10, 15, 20). This allows them to solve facts, but is slow and challenging, especially for larger facts.
Level 2 - The student can generate an answer to a fact using more efficient strategies. At this level, facts still take more than 2 seconds to solve because students are using a strategy to figure out their answer. Their strategies will often use already memorized facts. For example, they may solve 6 x 7 by using 5 x 7 -- "5 x 7 = 35 and then I need to add one more seven, so 42".
Level 3 - The student has the answer memorized. They can recall the fact in less than two seconds and are not doing any math in their heads. They are simply retrieving a fact that they have memorized.
Level 2 Vs. Level 3
It is easy to see that level 1 will be much too slow to be a successful level of knowledge, but many parents and children wonder why level 2 is not sufficient. Research shows that students who reach level 3 have more success in future math classes.To be clear, this is only a small part of what determines success in math class. But, we know that it matters. So, we want all math students to reach level 3 for the 0 to 10 facts.
So, why is memorization better? In other words, why aren't the strategies in level 1 sufficient? It comes down to the brain. When students at level 2 use multiplication facts while studying other concepts, they need to dedicate more of their brain power to answering math facts.
Take for example, learning how to multiply multi-digit numbers (ex: 123 x 45). If you multiply these two numbers, you are recalling six facts.5x3, 5x2, 5x1, 4x3, 4x2, and 4x1. Students who have their math facts memorized (level 3) can dedicate all of their thinking to understanding and learning the algorithm. Students at level 2 have to learn and understand the algorithm while ALSO solving the multiplication facts using their strategies. Having level 3 knowledge of math facts likely makes learning future ideas easier.
Wait...Don't Count Out Level 2!!!
Many people hear this and come to the conclusion that they should aim right for level 3. They pull out the flashcards and the Mad Minutes and start drilling away. But, this is not the way to help!
Level 2 is a crucial stage of development for students. It is in this stage where students develop their understanding of how multiplication works. Even though level 3 is a goal, trying to skip level 2 will weaken your child's understanding of mathematics. Because, it is at this level where students learn strategies that are useful beyond just memorizing multiplication facts.
As parents, the best thing you can do to help your child learn their multiplication facts is to have conversations about the strategies you can use to solve a multiplication fact. Through these conversations you will bolster your child's understanding of mathematics and help them move up the levels of understanding introduced above. So, here are the strategies you and your child can employ while solving multiplication facts:
Zeroes, Ones, Twos, Fives, and Tens - By the time your child starts studying multiplication, they will (hopefully) have a solid understanding of addition and have reached level 3 for most addition facts. This knowledge can serve as the foundation for the first learning multiplication. Multiplying by 0, 1, 2, 5, and 10 are the easiest facts to learn. For example, multiplying by 2 is the same as adding a number to itself. Students should already know this from their addition facts. Starting with these facts gives your child easy reference points to help with other facts. This is not technically a strategy, but these facts often serve as anchors for learning other facts.
Turn-around Facts - Multiplication is commutative. This means that you can rearrange the order of the factors and get the same result. For example, 5 x 4 = 20 and 4 x 5 = 20. This knowledge nearly halves the number of facts children need to know.
Doubling - Facts can be found by doubling known facts. For example, 4 x 7 can be solved by knowing 2 x 7 = 14 and then doubling 14. This strategy can be used anytime one of the factors is divisible by 2. That's a lot!
Skip-Counting Up - You can often skip-count up from a known fact to find an unknown one. This strategy is often used for 6s and 7s. For example, I know 5 x 8 is 40, and I can use this to figure out 6 x 8 by counting up one more eight from 40. This is an extremely important strategy because it introduces students to the Distributive Property of Multiplication Over Addition. You and your child don't need to be thinking of the Distributive Property when you use this strategy, but know that you are giving them experience with an idea that is crucial in Algebra. Here's what those steps look like more formally:
6 x 8
(5 + 1) x 8
5 x 8 + 1 x 8
40 + 8
Skip-Counting Down - If you can skip-count up, you can also skip-count down. This is a great strategy for 4s and 9s since you can count down from 5s and 10s respectively. For example, I can figure out 9 x 7 by calculating 10 x 7 - 7. This also gives students experience using distribution.
Halve - If you or your child is a good doubler, then you may also like using a halving strategy. This is great for 5s if you know your 10s, 3s if you know your 6s, etc. Example: I know that 6 x 8 = 48, so halving 48 helps me figure out that 3 x 6 is 24.
Combining Other Strategies - As you get experience with these strategies, you may find yourself using multiple strategies together. For example, if I wanted to figure out 23 x 7 I might use doubling to figure out 20 x 7 (7 x 10 x 2 = 2 x 70 = 140) and then skip-count up three more sevens (140 + 3 x 7 = 140 + 21 = 161).
If that feels overwhelming it is likely because you have not had much experience thinking through multiplication strategies. It is not caused by a difference in math ability. After teaching for nine years, I have no reason to believe that this level of thinking is out of reach for any child or adult.
Notice that ALL of these strategies work for facts beyond ten too. That's the power of level 2: these strategies extend beyond the zero to ten facts we expect our students to memorize. Spending a lot of time developing the level 2 strategies helps children become better mathematicians while also learning their facts.
Helping Your Child Reach Level 3 Through Discussions
Alright, now that we have the background knowledge we can figure out exactly how to help. The key idea here is that you want to move your child through level 2 and into level 3 not skip level 3. To accomplish this, you will want to have conversations with your child about facts. Here is an outline for how you might discuss 6 x 8:
"Do you remember the product of 6 x 8?" - Begin by asking them to try to recall a fact. This pushes them towards level 3. But, if they can't recall it that's okay!
"I don't remember"
"Can you figure it out using a strategy?" - Since they don't have the fact memorized, it is time for them to try to use a strategy to figure it out. You may have to wait a LONG time. That's okay. We already know they do not have the fact memorized, so give them as much time as they need to think through their strategy.
"How did you figure that out?" - Notice that I didn't say "wrong." They will figure out their mistake when they discuss their strategy.
"I know that 5 x 8 is 40 so I added one more 8. And, that is 48" - They may not even notice that their answer changed. That's okay here. I am focused on the logic of their strategy.
"That's an awesome strategy! So 6 x 8 is 48." - Notice that I praised their strategy since that is what I want them to focus on.
"Are there any other strategies you could use figure out 6 x 8?" - By asking for more strategies, I will reinforce the fact and also help them develop a wider variety of strategies. Again, I'll need to wait very patiently here.
"Well, I know that 6 x 10 is 60. So I could count backwards two more 6s. [long pause] 54 [long pause] 48. 6 x 8 = 48"
"That's another cool strategy! Are there anymore that work?"
"[Long pause] I can't think of another one"
"I thought of one! I know that 3 x 8 is 24. Since 6 is double 3, I can double 24 to figure out 6 x 8. That's how I figured out 6 x 8 = 48" - If there is an obvious strategy they missed, you should introduce your child to it. You might ask them to explain the strategy back to you if they like it. Hopefully, your child is being exposed to multiple strategies in their classroom too.
"That's a cool strategy. You are the awesomest parent ever." - You probably won't get this response but you deserve it!
"Now that we have talked about strategies, can you remember 6 x 8 without using a strategy?" - Now that I've helped solidify level 2 understanding, I want to give them practice recalling the fact to practice for level 3.
"Yeah! 6 x 8 = 48."
It takes TONS of these conversations to move children through level 2 to level 3 knowledge of their multiplication facts. Trust your instincts on where to begin. If your child is really struggling to learn their facts, start with facts you know they can do well and repeat those facts often.
Make sure your first few discussions are successful. And, never get frustrated. If they cannot do what you are asking them, then they cannot do it. If they can't skip-count up when multiplying by six, then talk about skip-counting up for their three facts.
Most importantly, enjoy these conversations with your child. It is amazing to watch children develop and then apply new strategies as they learn. You will be astonished at how sophisticated their thinking becomes after just a few conversations!