Well, math fact fluency is a big deal in elementary school. It’s one of those things that people cringe when a child does not have it. You’ll hear things like, “Kids these days don’t know their facts.” How many times have you heard that? But as always, I am not just about kids “knowing” their facts. I’m about kids understanding their facts. There is one big thing that helps kids understand the basic facts and not just know them.

I’m Christina Tondevold, The Recovering Traditionalist, and today we’re going to take a look at How Number Sense Impacts Math Fluency in our quest to build our math minds, so we can build the math minds of our students.

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Let’s jump right in to what fluency is. We’ll do a quick overview because I know I do a lot of videos on this and then we’ll jump into how number sense impacts developing that fluency. 

I’ve talked about it a lot, but I’m going to say it again: fluency is not just being able to get a correct answer quickly. Fluency has a third component and I like to say it like this: 

Flexibility is the missing piece to building fluency.

If all we have is getting correct answers and doing it quickly (Accuracy & Efficiency), we are not truly fluent. Flexibility is that missing piece for so many of our students and even us as adults. 

Now the other piece of this is we start to see this fluency show up in a lot of research. I first learned about it through Cognitively Guided Instruction (CGI), but there’s been lots of other researchers who have found the same thing, they just call it something slightly different. The researchers behind CGI saw kids go through this progression of how they solved problems. 

Down there in the yellow, that direct modeling and counting on, kids are modeling the problem, they’re often pretty accurate, but they aren’t very efficient.  So we quickly try to get kids out of those stages because we want them to be fast. 

We want them to be more efficient. We jump a lot of times to that fact stage, up at the very top, which is just knowing it. But if they don’t just know it, they drop down back to those yellow stages of counting on and direct modeling because they don’t have anything in between.

What cognitively guided instruction found was that in-between phase of derived facts. These are things that kids will use. They are something that they know and will help them with something they don’t know. 

I want you to think about a fact like this. Whether you teach single digit addition or not, you probably have some kids in your grade level who still struggle with this. So if they have a problem, 6 + 7. What we see is the end product of them saying 6 + 7 = 13. But underneath that beautiful flower should be a connected root system, but for a lot of our kids, it isn’t.

Before I show you this connected root system, think about it as if underneath this, all they have is that 6 + 7 = 13. That’s the only thing they know. That’s not a very strong root system and it isn’t for plants either. 

We want to build this connected root system for our kids, and what that might look like is pictured above. This looks a little overwhelming, but it’s not that kids need to know all of this stuff, it’s that these are things kids might know about 6 + 7.

They might have built a connection of 6 + 7 to a rekenrek or a 10-frame. They might see it as doubles. There’s all of these connections that kids might have made to 6 + 7. Research shows that the more connections we build around something, the easier it is to recall that information later when we want it.

If the only connection we have is that 6 + 7 = 13, there’s less things for us to be able to pull and build our understanding about that fact. Same is true when you get into multiplication.

For 4 X 6. If all they know is that 4 X 6 = 24, maybe they have some counting…they understand that they need to count by 6 four times. That’s one piece. To fully understand what 4 x 6 really means, we need to have lots of connections built. We want to help kids build these connections.

Let’s come back to the first one. We talked about 6 + 7. Let’s focus in on what derived facts might look like and sound like from kids when they see 6 + 7. A kid who’s in that derived fact stage won’t just say 6 + 7 = 13.

Actually let me take that back. Some do; some will just say 6 + 7 is 13. You don’t even know they’re using a derived fact unless you ask them. “How do you know 6 + 7 is 13?” 

You might hear things like this. “Well, 6 + 6 is 12, so you just add 1 more.” Or they might see it as 7 + 7 is 14 and then they subtract 1. 

Another one was one I never saw, but I heard it from a student. This is not the way I see it. But once I saw it through his lens, I couldn’t unsee it. He said, “Take 5 out of the 6 and 7 and you get 5 + 5. Then you have one and two more.” He’s finding 5s within the numbers.

Some kids like to make 10. They’ll say things like, “6 + 4 would make 10, so then I just have 3 more.” Or another kid might see how you can make 10 with a 7, and they’ll say, “7 + 3 makes 10 and then we just have 3 more.” 

Now, if you’re seeing that and that’s overwhelming to you and you’re thinking, “Oh my gosh, I have to teach them all of that stuff, they will never remember those. That’s a lot to have to teach a kid. Isn’t it easier to just teach them 6 + 7 = 13?” Remember: if all they know is  6 + 7 is 13, we’re not building their math mind. They’re becoming a memorizer. We want to help build math minds.

This really isn’t overwhelming at all because kids do it naturally when they start to see connections. All of these strategies that I just shared with you. 

If we take 6 + 7 and put it into a 10-frame, can you see every single one of those strategies? 

I will tell you: you can see every single one of those strategies just by putting this information into a 10-frame. Instead of just the digit 6 + 7, as soon as we show 6 things and 7 things, and in particular, if we put it in some kind of a structured format, like a 10-frame or a rekenrek, kids will notice.

They’ll notice that the picture of seven in a 10-frame is one more than the picture of 6. They’ll see the four that’s missing on the 6 and how we could move 4 of the dots from the 7 to fill that 10-frame. 

They see these things naturally. They are not things that we have to teach kids.

When we build their number sense, it builds their flexibility and thus it builds their fluency. 

When they become more flexible thinkers, when they can visualize and build a picture around a problem, they become more flexible thinkers. That helps them become more fluent by building their accuracy and their efficiency.

So let’s take a look at this back at our flower picture and how we build all those connections that built that root system. I’m just going to pull up one of those strategies. This is a strategy that is typically taught in a textbook. 

There’s lessons teaching our kids to use doubles plus one, doubles minus one. This is not something we can directly teach. In order for a child to be able to use that strategy and become a flexible thinker around numbers, they need to build their sense of numbers.

For a kid to be able to see that within 6 + 7 is a 6 + 6 + 1, they need to have some kind of a spatial relationship. They need to have a visual of 6 and 7 and be able to compare those two amounts. 

They also need to understand one and two more or less…if I take one out of the seven, how much is left? That’s building their part-part-whole understanding. We’re breaking apart the whole of 7 and we want them to understand that it stays the same.

We can chunk it into anything that 7 creates. It could be 4 and 3, 5 and 2. But in this case, I want to break it up into a 6 and a 1. Those number sense concepts are huge to be able to get kids into that derived fact stage and build their flexibility. 

Here’s another common one, this is a make a 10 strategy. 

Again, you probably have textbook lessons that will teach the kids to make a 10. But if they don’t have a sense of numbers, this strategy will not stick. They need to have spatial relationships. They need some kind of a visual that helps them see that 7 is just 3 away from 10.

They also need to understand the benchmark of 10. The benchmark of a 10 is super huge. So many of our kids don’t have that understanding of 10 being a benchmark. Even once they get to 10, they don’t know what 10 + 3 is. That’s all building their understanding of the benchmark numbers. 

Again, they’re breaking apart one of the numbers. They’re breaking apart the 6 and that comes through their part-part-whole understanding. All of these are number sense concepts. Without them, kids can’t build their flexibility and won’t ever use those derived facts.

When they see 6 + 7, they’re saying things to themselves like, “How should I chunk that 7?” Well, if you chunk off the 4 from the 7 to make a 10 with the 6, they have to know what’s left with the 7. Then once they make that 10 + 3, do they instantly know what 10 + 3 is?

All of that goes into understanding and building strategies for kids. It is not from a lesson that we can just teach once in the textbook and think they’re going to get it. It helps by layering in visuals. That builds that spatial relationship. All of those questions that they’re asking themselves are all things that come back to their sense of numbers.

Do they understand the benchmarks of 5 and 10? Do they understand part-part-whole? Do they understand one and two more or less? Do they have that visual picture to help them with all of that? 

Let’s come back to our one for multiplication. All of these connections are still tied to those number sense concepts. Here are a couple of strategies that kids might have if they’re working in that derived fact area that we saw in the CGI progression.

Again, you might think, “Oh my goodness, that seems like a lot. I don’t think my kids would ever come up with those.”

If they have a visual, would they? All those strategies that the kids saw, they see it when they have that sense of numbers, when they’ve built a spatial relationship.

When you see it like this, can you see that 4 x 6 is 5 groups of 6, but then I need to take away one of those groups of 6? Try every single one of those. It’s there. It happens when we allow kids to create those visuals and build a sense of numbers.

Let’s take a look at another one here. The crazy one; it seems crazy, but kids see it, that 4 x 6 is the same as having 4 x 5 and then 4 x 1. That visual picture can help us out, but here’s all the things that they’re asking themselves. 

“4 x 6, I don’t know what 4 x 6 is. So how could I chunk this to make it easier?”

“Okay, so once I chunked it and I’ve got 4 x 5, what is 4 x 5? And then once I’ve figured out 4 x 5, what’s actually left over? Is it just one that’s leftover? Is it 6, is it 4, is it 5?”

They have to think about all of that stuff. Then finally, when they get down to it and they have 20 + 4, do they know what 20 + 4 is?

These derived facts are not easy, but kids naturally develop them when they develop these number sense concepts. 

When they have a spatial relationship, when they understand how numbers relate to benchmarks, when they can break apart numbers and when they’ve built that part-part-whole understanding, when they understand one and two more or less. Even in multiplication, it’s one or two groups more or less, or one and two in each group more or less. All of these play into building that sense of number and number sense builds their flexibility. 

Remember, flexibility is the missing piece to building math fluency. The cool part is, it’s not just for their basic facts. How kids build that flexibility with single digits for addition and subtraction will play into how they think about multi-digit addition and subtraction. Kids who can break apart numbers, kids who can recognize how a number relates to a benchmark number will be able to break apart numbers in ways that you never thought were possible.

I want you to stop and think about what ways kids could break apart 286 + 67 and use their flexibility with numbers instead of just lining them up and adding the ones and adding the tens. I’m not saying that’s bad, but I’m saying that when we build their flexibility, they have options. They have connections built and they aren’t reliant upon just one way to solve a problem. 

Same is true when they get into multiplication.  If they have built a sense of numbers with 4 x 6, guess what’s going to happen when they do 4 x 57? They’re going to see different ways that they could break that apart and make the problem easier for them. It’s not just lining it up and going through the motions. We want to build their flexibility because that is the missing piece to getting them to be fluent with mathematics.

I hope that this video, as always, has helped you build your math mind so you can go build the math minds of your students. Have a great day.

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