So, we say we want kids to be fluent in math, but what does that actually look like?

I’m Christina Tondevold, the Recovering Traditionalist, and today we’re going to take a look at students with 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:

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 (This Video)
Video 5: The Importance of Visuals in Math
Video 6: The Root of Math Fluency

We think fluency looks like this,

but this can often be what my friend, Mike Flynn, calls an illusion of understanding. The kids can get the right answer, but are they really understanding what’s going on? And as soon as we extend that problem we see… that they only have a surface level of understanding.

Now, in prior videos in this series I’ve talked about how we define fluency, but in this video, I’d like you to actually see examples of kids with fluency. Now remember, fluency is not just being fast and accurate. It is also about being a flexible thinker. I go into more detail about that in the prior videos in this series. So if you haven’t watched those, make sure you go watch those. The link is right here where you’re watching it. Somewhere. In the description, it’s down below. You’ll see it. We also have a fluency starter kit that goes into a whole lot more detail to give you information about how to build this for your students. And the link is on the video series page. 

Now there are lots of ways that kids can show that they are flexible and fluent thinkers. But I’m going to share with you my top five. These 5 things are things that I personally look for in my own children to ensure that they are fluent in math. All the videos that I’m going to be showing are my own personal kids because it’s easy to get the parental permission to use their faces in videos.

They use what they know about small quantities to help them with larger quantities

So, our first key thing that I look for is that they use what they know about small quantities to help them with larger quantities. This video that I’m going to show you is of my oldest son when he was in 1st grade. I was working on a presentation of a number string, so it’s a string of related problems, and I just thought, he’s sitting next to me I know he could do the first one. And once I saw how he did the first one and then I showed him the next one, the second one, and I’m like, “oh I got to get this on camera!” So I grabbed my phone and I started recording how he would solve this third problem: 299+346.  So take a listen:

Below is a transcript of the video of the child.  If you’d like to watch it instead, go up to the video above and go to 2:52 in the video.

Christina – What do you think?

Son – I think it’s 646. 

Christina – How in the world did you get that?

Son – I added these two up and that’s 500 and then subtract 1 from the 6 and add it here, that’s 600 then all we have left- 

Son – No, it’s 45! 300- 645, because I forgot I took one away from that and put it over here and that would equal out of these three 599 and then we took one away from the 6, that’d be 5 for the 6 and that’d be 600 for the 599. And then I’d just have to add the 45.

So now just to remind everybody, that is not a problem that a 1st grader should be solving. Most 1st grade standards only talk about taking a two-digit plus another two-digit that’s a multiple of 10. So like doing 27 + 20 would be a 1st grade standard. 

That problem he solved is not a 1st grade standard. So it’s not something you would expect a 1st grader to be able to do. But what’s powerful there, is that kids who have this fluency and flexibility (the basis of fluency), can use how they solve problems with small numbers so they’ve built that fluency around their facts, which is one of the other videos that we talked about. How they solve those problems impacts how they solve bigger quantities as they go forward. That is a sign of students who are fluent with math.

They are able to solve problems they haven’t seen before

Our second one, which you kind of saw in that first video, but I want to show you another example. Our second one is that they are able to solve problems that they haven’t ever seen before. This video that I’m going to show you is my youngest son. It was just filmed here recently.  We were playing the game Farkle, and the video that you’re actually going to watch is a recreation. Because when we were actually playing Farkle, we don’t have our phones, I wasn’t filming stuff. We were just playing. But I was so amazed at what I saw happening during that game, I had to have him recreate it. 

So he wanted to be the score keeper, and if you’ve ever played Farkle, you are adding amounts up to 10,000. He’s in 1st grade. And he’s a young 1st grader. His birthday is in the summer, so he is a very young kid for his class. So this is not something that I thought should be easy for him. Having my 5th grader be the score keeper, no problem. But my 1st grader wanting to be the score keeper? I was a little hesitant. 

So I thought, I’d give him the paper, I’ll let him do it the first round where people are just getting, amounts in the hundreds. And all he has to do is just write down the number. He’s not adding hundreds and hundreds together. You know? 

So we start going, and he is adding the hundreds together. And he’s adding them when it’s like weird amounts. And then we get into where we’re working in the thousands and he added this amount together… and I was amazed. It was crazy. So let’s take a look and watch it in his own thinking here.

Below is a transcript of the video of the child.  If you’d like to watch it instead, go up to the video above and go to 6:28 in the video.

Christina – This was one of the problems. Somebody had 2,650 as their score. And they rolled 850 points. Can you tell me how you would figure that out?

Son – Uh… 400… 50. Wait, is it 450?

Christina – I don’t know.  You tell me. What are you trying to do?

Son – 350 and then I have… And then I have 500… 450 left. And then I add the one- the 350 to it.

Christina – Okay, so how did you get to the 3,000? (he had wrote down 3000 on his paper)

Son – 350.

Christina – Okay, and they scored 850.

Son – Now I need 500 more.

Christina – So how much would their score be now?

Son – 300 and 500. 300- 3,500.”

Christina – 3,500.  Nice thinking, dude! 

You’re a first grader!!  Like that should not be the way that kids in 1st grade- like that’s not what we expect 1st graders. I was telling my mom this story, I’m like in 1st grade we’re excited if a kid writes down 250 just correctly and not 20050. Right? Like that’s a big accomplishment sometimes in 1st grade is to just get them to write numbers in the hundreds and thousands correctly. Not be able to add and subtract those amounts. 

It’s not like he’s ever been taught that before. He’s using his understanding of numbers, his flexibility, his number sense and applying it to a new situation that he’s never seen before.  Again, that’s the power of building fluency for our kids.

They see math all around them

All right, number three. This one seems obvious to me, because I do it all the time. And there’s a lot of people who don’t. So number three is all about that kids will see math all around them.  They see the mathematical problems that appear in their daily lives. And not every child sees the math in their daily lives.

So this video was when we were at a drive through at my favorite coffee shop. Shout out to Dutch Brothers. But I had ordered a bunch of smoothies for our big family and at the time I was only buying stuff with cash. I carried cash with me to buy groceries, everything. So all I had when we were in the coffee drive-up was a hundred dollar bill. And my son saw this problem on his own and- so you’re going to watch the problem he came up with and how he figured it out.

Below is a transcript of the video of the child.  If you’d like to watch it instead, go up to the video above and go to 10:04 in the video.

Son – Um, my mom was buying some smoothies for $15 and she didn’t have anything less than a hundred so if they’ll take the 100 she would get $85 back and how I figured that out was I went a hundred 10 back from a hundred would be 90, then 5 more would be 15 then 5 more back would be 85.

Christina – Cool strategy, dude.

So again, this one is all about helping kids see math all around them, not just inside of a textbook. Fluent mathematicians don’t see math as a problem to solve. They see math as the vehicle to solve problems in their lives.

They use problems they know to help them with problems they don’t

Okay, number four. What I look for is that they use problems they know to help them with problems they don’t. So you’ve seen that kind of throughout all of these videos pretty much. But here’s another example of one of my other sons. He was working on a worksheet that was sent home, and he was seeing connections between problems and he was telling me about it. Well, his younger brother was a kindergartner at the time, and was sitting at the table and was catching on to what he was seeing. And so I wanted to video to see- cause he said, I’m going to solve this next one! So I videoed what he saw in these problems. Here you go.

Below is a transcript of the video of the child.  If you’d like to watch it instead, go up to the video above and go to 11:38 in the video.

Christina – What do you think this last one is?

Son – 68.

Christina – Why do you think it’s 68?

Son – Because this one has 20 and this one has 19.

Second son – He’s correct.

Christina – Well, so how does-

Second son – You’re correct.

Christina – What do you mean by that?

Son – Because this one only has 49 and this one has 49, but… this one is one less.

Christina – That one has one less? So the answer is going to be one less you think?

Son – Yes.

So in this one I just want to remind you that so many of our kids will see math problems as individual tasks to solve. Individual problems. They are not seeing connections between them. So one of the big parts of kids being truly fluent in mathematics is they see this interconnectedness.

They decompose and compose numbers

All right, our last one is that you’ve again, probably seen this all throughout, but I’m going to make the point: they decompose and compose numbers. So basically this means they’re breaking numbers apart to make them into friendlier amounts and they’re able to put them back together. 

This video was when we were shopping out at the grocery store, And we buy a lot of Gatorades. We have four kids, all of them are in sports, like all the time. And so we buy a lot of Gatorade. And you can see it there in the cart. So I had asked them all how they solved it. And my daughter was, I looked back, it was the spring of her 3rd grade year. So she was towards the end of her 3rd grade year and this is how she figured out how many Gatorades we were buying.

Below is a transcript of the video of the child.  If you’d like to watch it instead, go up to the video above and go to 13:26 in the video.

Daughter – So I know these are 10, so I’m going to leave those out. These are 10, so I’m going to leave those out. I’m going to leave those out because those are 10. So 2, 4, 6. Then, 10, 20, 30. So, 30 + 6 is 36.

Christina – So you broke it up so that there was 10 in each?

Daughter – Yes

Christina – And the 2 from each? 

So basically, that problem was 3 x 12. But she decomposed it and she saw 3 groups of 2, like the 2 on each of the ends of the Gatorades, and then the 3 groups of 10. That is such powerful thinking that moves into the distributive property for multiplication and will serve her greatly when she starts moving into algebraic reasoning stuff. 

So all of this stuff, the decomposing, it isn’t just about helping kids with their facts. It will transfer into bigger mathematical ideas as they progress through. And that’s really the string that ties it all together. So much of math is just about getting the right answer. But, the right answer is important, but how they get there is also very very important and will impact how they perceive mathematics throughout all that they do. 

I want to encourage you to go download the starter kit that I have for you. That gives you ideas about how to build this for students, what it actually is. It starts off with helping you decide is that flexibility really important to you, to help build the fluency, because you guys are bombarded with so much that you have to teach. Especially right now, you’re trying to making decisions about what is really really important. And so is building this flexibility for kids important? And if it is, I think it is, then the rest of that kit will really help you out. We talk about how to help build that for their basic facts and then activities and things that you can do to really help build this. All right, I hope that this video has helped you build your math mind, so you can go build the math minds of your kiddos. Have a great day.

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