If you have ever tried to teach kids a way to think about a math problem but then later they go right back to using an inefficient method, then this video is for you. Yes, we all want kids to be fluent with math but fluency is more than just getting a correct answer.

I’m Christina Tondevold, the Recovering Traditionalist, and today we’re going to take a look at the root of math fluency in our quest to build our math minds so we can build the math minds of our kiddos.

Watch the video or read the transcript below:

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Watch all the videos in this Fluency Series:
Video 1: Fast Does Not Mean Fluent
Video 2: Helping Kids Learn Their Math Facts
Video 3: Math Experiences That Build Fluency, Not Memorization
Video 4: Students with Math Fluency
Video 5: The Importance of Visuals in Math
Video 6: The Root of Math Fluency (This Video)

The webinar is over but get your FREE Fluency starter kit below. Requesting the kit will also sign you up to be notified of any future webinars.

Now, as we move into wanting kids to be more flexible thinkers in math and not just kids who can calculate, we try teaching them ways to think about and solve problems. Let’s take, for example, 9 + 7. There’s a progression kids go through on their way to fluency, which I’ve outlined in other videos, so I’m not going to go deep into that here. 

But the general idea is that kids start out counting and eventually we want them to be able to just instantly know it. But if they don’t instantly know it, so many kids go right back to that counting stage. There is this in-between stage where some kids fall. These kids use things they know about the numbers to help them. So for 9 + 7, you might hear a kid say something like, “It’s just like being 10 + 6.” That is an amazing strategy and that kid has built amazing connections but textbooks try to have us teach that strategy to kids to help them become more fluent with their facts. 

The problem is that strategies can’t be taught to kids. There is so much that those kids need to know about numbers before they will ever use strategies like that. So in this video we’re going to take a look at what kids need to know.This is just one part of what helps kids become fluent in math. If you want more information, I have a free webinar coming up that gives you, The Three Keys to Building Elementary Students’ Math Fluency. Use the link to go get registered for it.

For right now let’s dive into looking at an addition problem and a multiplication problem to see what kids need to know about the numbers so that they can become fluent and use those cool strategies.

The Root of Addition

All right, so in the prior video in this video series, we talked about how 9 + 7, a lot of kids with this flexible thinking will start to see it as 10 + 6 and how much of an easier problem that is for a lot of kids but it’s not automatic that kids are going to see this. And so in the last video, we talked about how adding a visual to this helps kids be able to see those things. 

So if you haven’t seen the last video, make sure you click over and watch those other videos in this video series. But really the visual is just a piece of the bigger puzzle about how to build that flexibility and thus the fluency for your students. So let’s take a look at a different fact. We will come back to 9 + 7 here in a moment. But let’s investigate 6 +7 = 13. 

We’ve talked about this problem before in a prior video and about all the different connections that can be built. But I want to think about this for a moment as if this fact was like a flower and it is sprouted up, and what we get to see is just the beautiful flower and its petals. 

And that’s like seeing a kid who can say 6 + 7 = 13. That’s all we see. What we don’t see underneath is the root system of the flower. And the root system for these facts, or any way that a kid solves a problem, but we’re just looking at the facts for right now. When kids are solving problems, you see the answer but underneath of that there’s so much going on. 

So let’s take a look at what actually builds the root structure here. It starts off with kids counting. Counting is a fundamental piece of mathematics. And young kids, when they’re doing 6 + 7, will count out 6, count out 7 and then come back and recount it all. And what helps with that is of course having things to count, whether it’s their fingers, whether it’s blocks, whatever it might be.

This is a foundational piece that starts this root structure. It also then moves in to kids being able to hold one of the numbers in their head and then count on. So they’ll start at the 6 and then count on from there and then they’ll also do the 7 and count on from there. But this is dependent upon lots of work with context and understanding that it doesn’t matter what order they do it in, it will still get the same answer. 

So this is all a big part of the root system but a lot of this is still reliant upon counting. And it’s one of the complaints that we all have is that kids are still counting. Counting is not bad. Using fingers isn’t bad. But it depends where they are in their progression of understanding. 

You’re right, I don’t want a 5th-grader still counting on their fingers to add 6 + 7 but that is an okay thing for a 1st-grader. So as we’re progressing forward though, as we’re trying to get more fluency built for our students, we start to bring in these types of strategies where we want kids to break apart numbers and see how they can break 6 + 7, breaking that 6 into two 3s, so that they could put a 3 with the 7 and make a 10, and then they have 10 + 3. They also see doubles with this. There’s other ways that they can make a 10. 

So all of this is part of this root system. But as we’ve talked about in the prior video, having visuals helps kids understand those strategies. So it’s not just memorization of more and more ways to solve a problem. We want them to actually understand what they’re doing. So all of these pieces help build this root system that goes up and actually helps kids be able to show that beautiful flower of the knowledge of 6 + 7 = 13. 

But the real magic actually happens when we start building connections.

Did you see that slight change there? What I did here was draw all the lines connecting all the pieces of this. Like, the more kids can see connections, and even adults, the more connections we have, the greater our chance of recalling that information when we need it. So if we can connect all of these together, it helps build a more cohesive root system for our students with their mathematical understandings. 

Now I know this looks like a complicated mess here and you’re thinking, “How do I do all of this stuff?” Well, it comes down to basically building a sense of numbers for your students. 

There are 4 main components. There’s actually 8 components that make up all of number sense in my world, and it’s a compilation of things that I’ve learned throughout the years and throughout lots of research and books that I’ve read. And I talk more about them inside of the Building Math Fluency Starter Kit.

But let’s do a quick little overview of 4 of them that make a big difference in helping kids build these connections. 

So I pulled up one of the strategies here that’s pretty popular of seeing 6 + 7 as 6 + 6 and then just add an extra 1. It seems so simple but there is so much that kids have to understand about numbers in order for that strategy to make sense to them.

First of all, they need to have built a Spatial Relationship. They need a visual of 6 + 7 but also it needs to be a visual that helps build out the relationships, helps them see how 7 is just one more than 6 and how 6 + 7 is 6 + 6 plus 1 more. And they might see that using the rekenrek, that line could also possibly be drawn to the 10 frames. It also might come through lots of context that they’ve done, like maybe playing football and the scores that they get when they play football. So all of those connection points happen when they have built some kind of relationship and a big one happens when they have a visual, a spatial relationship. 

Another big piece here is that they have to be able to know what is One/Two More Or Less. They will not break that 7 apart if they don’t instantly know that if they take 1 from it, they’ll have 6 left over. And that’s part of this Part-Part-Whole understanding is that I have 7 but I can break it into a 6 and a 1. But how do they know they should break it into a 6 and a 1? Why not a 5 and a 2? Why not a 4 and a 3? It just depends upon the relationships they’ve built and the understandings that they have of the numbers. 

So let’s look at a different one here. This is a popular one where kids like to make 10. But they first, again, to help them with that, having those spatial relationships is key and it could be through a 10 frame, it could be through their fingers, it could be from the rekenrek. They need some kind of visual to help them understand that. 

The other piece here is that kids are using a Benchmark. And for young kids that benchmark is 5 and 10. If they don’t instantly know how much more to make 10, they will never use that strategy of make 10. They also still need to understand that they can break the 6 apart into pieces and which pieces should they break it into. In this one, they chose 3 and 3. But a lot of kids don’t understand why not a 4 and a 2? So that’s all parts of this. So basically let’s take a look at what those 4 were but let’s come back to our 9 + 7. So when a kid is trying to solve this 9 + 7 and think of it as 10 plus 6, they first have to ask themselves so many questions and they’re reliant upon their understanding of numbers. So the first thing is like how should I chunk that 7? Do I do a 6 and a 1, a 5 and a 2, a 3 and a 4? If they realize that, well, it would be helpful if I could make a 10 here. They also need to know how much more to make that 10. And if they chunk off the 1 from the 7, they have to instantly know what’s left over in order to make that 10 + 6. And then finally when they have the 10 + 6, they have to instantly know what 10 + 6 is. And we have so many kids that get to the point of what is 10 + 6 and they’re back counting on their fingers. So everything they did was kind of pointless to get them to that spot. 

So here’s what helps them. Again, visuals are key. But all of these things that they’re asking themselves, and using this visual, boils down to these 4 number relationships. That visual gives them the Spatial Relationships. They’re using Part-Part-Whole understanding by chunking things up. They have to use their knowledge of One/Two More or Less. They’re using the Benchmark of 10 a bunch because they’re trying to make 10 and then they’re adding on to 10.

These 4 things are so critical to helping kids build their number sense, which builds their flexibility. And as we’ve talked about, fluency is the byproduct of flexibility. 

So we’ve got to build their flexibility and how we do that is through number sense. And I have whole courses on number sense. We’re getting ready to open those up but before we open up the course for enrollment, I always offer free stuff as well because the course is paid and I know not everyone can pay for stuff right now or anytime, really. So we do offer lots of free stuff. And right now we are getting ready to do some free webinars that go into more detail about how to build fluency for the kids. So if you’re interested, come join me, sign up below this video and get on to the webinar.

The Root of Multiplication

All right, let’s move into multiplication. So multiplication, there’s 7 x 6. One of the fabulous ways to think about 7, which I didn’t learn until I was an adult was to chunk it and do 5 x whatever the number is and 2 x whatever that number is. But again, that is just like random numbers for so many kids. 

But if we show the visual that goes with it, this piece helps a ton for our kids.

If we can show the visual here, they start to be able to see the 5 groups of 6 on top and the 2 groups of 6 that’s on the bottom. It’s amazing that they can come up with that strategy just by using that visual. I don’t have to teach that strategy. But again this is just the tip of the iceberg or one little piece of the root system. 

So here’s a different fact, 4 x 6, and we’ll come back to the 7 x 6 here in a moment. But I just wanted to show you what the root system looks like for a fact for multiplication.

Again, we’ve got lots of visuals in here. Kids are still counting, that is the foundational way that kids will solve multiplication is they count out everything and then they will even skip count. 

There’s lots of strategies that kids come up with eventually. They start to see from those visuals how you could do 2 groups of 6 and then just double it to give you 4 groups of 6. A lot of kids will also start to see how multiplying by 4 relates to multiplying by 5 or even how multiplying by 6 relates to multiplying by 5. All of these help build that root system. And the more pieces we have down there in their root system, basically the more number sense they have built, the easier it will be for them to build their fluency. 

So let’s come back to this, 7 x 6. We want kids to understand that that can be solved by doing 5 x 6 and then 2 x 6 because there are 5s and there are 2s kids learn those fairly quickly and they start to have those instantly recalled faster than any other fact. So if they can use those types of facts that they already know to help them with something they don’t know, that is helping build their flexibility. But what actually goes into this? 

They first have to know, how are they going to chunk this? Are they going to chunk the 7? Are they going to chunk the 6? Once they decide how to chunk the 7, they have to know what is 5 x 6 and then what is left over after they do five times six. We have so many times that kids will say, “Well, I know it’s 5 x 6 “and then I just need 2 more.” But two more what? They sit there and they’re like, “Maybe just add 2. “Or do I add two 5s? Do I add two 7s? Do I add two 6s?” Like, they don’t know what to add. They know they need 2, but not 2 of what. So they have to know what’s left over. And once they actually do get 5 x 6 and 2 x 6, they have to know what 30 plus 12 actually is. 

So this visual again helps them see all of these. It helps make it a little more attainable but again all of these come down to those 4 number relationships. They have to have that Spatial Relationship. They’re using Part-Part-Whole by breaking it apart, which is basically the distributive property when we get into multiplication. They’re using the Benchmark of 5. Benchmarks change like as you work in with fractions you’ll get benchmarks of a half and a whole and one and a half and 2. 

So the benchmarks are different based upon different types of numbers but there are always benchmarks that make math problems easier. So multiplying by 5 can help make multiplying by 7 easier for kids. They also need to do One/Two More or Less but it’s not one and two, it’s one and two groups of something now that they’re using in multiplication. And then finally that benchmark of 30, like the 10s, multiples of 10s are still benchmarks. So we have so many kids who struggle with adding multiples of 10 to stuff. 30 plus 12 should not be a difficult task for a lot of our kids but it is. So there’s a lot of work around benchmarks that kids need to have before they will ever do these strategies. 

So we can’t just focus on the strategy. We have to focus on helping build their number sense. If we can build their number sense, that builds their flexibility. Again, as we’ve talked about, the byproduct of flexibility is that they build their fluency. So I want to encourage you through this whole series.

The way to get your kids to build their fluency is don’t focus on the strategies.

It’s not about having five different ways to solve a problem, it’s about the number sense that kids have that gives them the ability to create those strategies on their own. 

Focus on number sense, not strategies. 

I have a saying I say a lot that number sense is caught not taught. You can’t directly teach a kid and say, “9 is 1 less than 10. “Learn it, know it, memorize it, let’s use it.” It comes through experiences. 

In the webinar I’m going to be giving you my top 3 tips on how to build fluency and a lot of it comes by teaching them number sense. Again, I catch myself, I saw it on the screen, you can’t actually teach it, it’s about providing experiences for your students that allows them to catch these number sense concepts. 

So I hope you’ll come join me on that webinar so that we can work together to build your students’ fluency. And we’ll end with one of my favorite sayings here from Cathy Fosnot and Maarten Dolk that when kids have number sense, they naturally will have strategies at their disposal and we want them to look at the numbers first and choose their own strategy. Even with 6 + 7. You personally may like a double for that, you might like 7 + 7 and then subtract 1, but another person might like making a 10, it depends upon their own experiences, their own number sense, and the numbers that are in the problem, that’s what determines the strategy that we’re going to use. And the more we can build their number sense and the more we can build their flexibility, the more options they will have at their disposal. 

All right, I hope to see you on that upcoming webinar, 3 Keys to Building Elementary Students’ Math Fluency. I hope that this video has helped you build your math mind so you can build the math minds of your kiddos. Have a great day.

**The webinar mentioned in the video is over but get your FREE Fluency starter kit below. Requesting the kit will also sign you up to be notified of any future webinars.**

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