Now, everyone inside and even outside of education thinks that kids need to be fluent in math, but our ideas of what that means are not always the same.

I’m Christina Tondevold, The Recovering Traditionalist and today I’m sharing The Missing Piece in Math Fluency in our quest to build our math minds so we can build the math minds of our students.

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog. (Some may be affiliate links which just means that if you do purchase using my link, the company you purchased from sends me some money. Find more info HERE about that.

The Flexibility Formula Courses are PD courses for elementary educators that help teachers learn how to build kids’ flexibility with numbers in order ot build their fluency with operations.

Download the Step-by-Step Guide to Increase Elementary Students’ Math Fluency here.

Now, as I said, if you asked any adult out on the street if they think that kids should be fluent in math, the answer would all be “Yes,!” But if you ask them, “what does fluency look like?” I bet there wouldn’t be a consistent answer. Or if there was a consistent answer, I can almost guarantee that it’s only a partial answer of what math fluency actually entails. 

What is math fluency?

Now, if you were to ask those people what fluency is, I can probably bet that the answer would be something like, “kids can give the correct answer quickly.” That’s what we tend to equate fluency with, is a correct answer and how quickly they can get that answer. 

According to Susan Jo Russell that is only part of it.

Yes, we do want kids to be accurate. They need to get correct answers. That is part of math. There are times when problems might have different answers based upon different things, but a lot of mathematical problems have a certain answer that we’re trying to get to. So kids need to be able to accurately get answers. 

The next part of this is they do need to be efficient. That’s how quickly they can get to an answer. It’s not that a kid has to have an answer in a specific amount of time to be considered fluent, it’s that they need to be efficient. Like it shouldn’t take them 15 minutes to solve a problem.

Those two pieces (Accurate & Efficient) are typically what people would respond with if we asked them “what does math fluency look like?” The missing piece is flexibility. 

Flexibility is that if we don’t immediately know the way to get to the answer, we have some ways to be able to figure it out. 

So let’s take a look at an example using the Cognitively Guided Instruction (CGI) progression levels. CGI was some research done about how young kids solve math problems. They’d give kids problems in a context, and then watch to see how they approach them. 

They saw kids who were in this stage that they called Direct Modeling. These were kids who would need to count out every part of the problem. So let’s say the problem was 3 + 4. I’m just going to do a small problem for us to have an example. A kid would count out “1, 2, 3” and then “1, 2, 3, 4,” and then come back and recount every single one of those to get to the answer of 7. That’s a direct modeler. 

Then they saw kids who were able to hold one of the numbers in their head and then count on from there, that’s the Counting On/Back stage. So it might be a kid who holds the 3 in their head and then counts on the 4. In this stage there’s also some variants where kids might start with the 4 and then count on the 3. But these two stages down here are what we would consider not fluent. 

In these stages they might be very accurate. In fact direct modelers are super, super accurate because they count and they double count, and sometimes they triple count because they want to make sure that they have the right counts. So direct modelers are very, very accurate but they’re not efficient, and they’re really not flexible. They do not have any flexibility at all. 

Kids in that counting stage, they are increasing their efficiency. They’re starting to become more flexible where we might see a kid who starts with the 4 first and then moves on and counts 3 instead of having to start with the 3 and count on 4. They sometimes aren’t as accurate. We will see kids in this stage that hold 3 and then they want to count on 4, they will go  “3”, but they’ll put up a finger for that 3 and then they’ll count on from there making them off by 1. But they’re starting to become more efficient, more flexible thinkers. But sometimes their accuracy might go down a little bit in that stage. 

These phases aren’t just for addition, that’s the example I’m giving here but kids who are working on subtraction go through these when they start first learning multiplication and division they go through these stages. 

Now, once they get into that Counting On phase, we tend to move them towards the Fact stage, just know it.  Just know what 3 + 4 is. If you see that on a piece of paper you should just know it’s 7. 

That’s the way I learned, and many of you probably did as well. The hard part of that is if a student is struggling, they don’t have it memorized yet, or they’re just not good memorizers. If they don’t instantly remember it, the only thing they have to fall back on is to go back down to that counting phase. 

So in that fact phase you will see them be accurate and efficient. But they don’t have flexibility a lot of times. They either just know it or they’re back down counting on their fingers. 

So what Cognitively Guided Instruction researchers found was that some kids found a stage in between called Derived Facts. They use something they know to help them derive a fact they don’t know. So in that 3 + 4 you might hear kids say something like 3 + 3 is 6 and one more is 7.  That is when you see that kids have built their flexibility. When they’re able to make connections between things you know that they have flexibility.

Flexibility is having connections that give you alternative routes to the same endpoint

Flexibility is all about building these connections. It gives you alternate routes to be able to get to the same endpoint. So let’s take for example, 6 + 7. If we want kids to know 6 + 7 is 13. Think of it like this flower. When the flower’s grown, this is what we see.

We see the end product that a kid knows 6 + 7 is 13, but what helped them get there is whatever is underneath the surface. If underneath that surface all they have is just being able to count, it doesn’t build enough connections. It doesn’t give them a solid root structure around that math fact. So we want to help build a scenario where kids are able to have lots of connection points.

They understand that 6 + 7 is related to 6 + 6. Maybe they used the visuals of a 10 frame or using the rekenrek to help them. All of these different connection points help build that solid foundation and build their flexibility around the numbers. 

Here’s one of the things I want you to understand about this. This picture might look a little overwhelming with all of these connection points, but it isn’t just about having lots of ways to solve a problem. The research on how brains learn and work is that the more connections that we have to something, the easier it is for our brains to recall that information when needed. So even though kids have all of these connection points, they may not use them. The connection points get stronger and stronger over time, and eventually the information just becomes instantly recalled when kids need it. 

The other thing to keep in mind when you’re looking at this picture, I don’t want to overwhelm you and think that kids need to build all of these connection points. I’m just showing them as an example of what kids might build. Instead of just making kids memorize 6 + 7 = 13. When we expose kids to 10 frames and to rekenreks and to problems in a contextual situation, they will start to build these understandings themselves. 

We don’t directly teach these connections. 

These are connection points that kids naturally develop when they have experiences around the numbers. It’s not that you directly teach these to the kiddos and it’s not just for addition. Here’s an example using multiplication. 

Again, we don’t want kids to just memorize 4 x 6 = 24. That’s only surface level understanding. They can give you the end product of 24, but what understanding do they really have? Do they truly have a full understanding and really are they fluent?  Because remember, without flexibility students are not truly fluent. 

Without flexibility, we are not truly fluent at anything

Your students may be able to get answers, and they will produce that end result that you are looking for, but without flexibility, they really are just following steps and procedures, or they’ve just memorized. 

It’s just like anything in your life. If you are truly fluent at something, you have flexibility. I want you to think about something in your life that you are fluent in.  If you feel like you are really fluent, I bet you have flexibility with it. 

I’m going to give you an example of something in my life that I do every single day but I’m not truly fluent at it. When I cook, I am very accurate and I am efficient. I can follow a recipe and I can be as efficient as possible at cooking that recipe. But I am not a fluent chef because I don’t have flexibility. 

If there’s a needed ingredient in that recipe, and I don’t have that ingredient, I can’t make the recipe. I’m also not the type of cook who can add a little bit of this and add a little bit of that to make it better. No, I have to follow the script. I have to follow the recipe “to a T” because I don’t have that flexibility. I don’t know if this thing makes it better or if I add this ingredient, it would make it worse. I do not have that flexibility. 

The hard part is that I look like I’m fluent because I’m producing the end result of a meal for my family, but I’m really not. I am not flexible, thus, I am not fully fluent. 

So one of the things I want you to take away from that example is that your students may be showing this illusion of fluency because they can get to those correct answers and they can do it quickly. 

Like I showed my illusion of being able to cook because I can follow a recipe, but I don’t have true fluency. Your students don’t have true fluency if they don’t also have flexibility. 

If you are wanting to build your students’ math flexibility, and thus their fluency, I have two things for you. 

First is something that’s coming in the future but I want to tell you about it so you can be thinking about it. I do a full online PD course for elementary teachers called The Flexibility Formula. That’s going to be opening soon.  We do a deep dive into what Flexibility looks like and how we build that flexibility around numbers for students. I only open up registration twice a year. So check down below this video so you can see the exact dates of when we’re getting ready to open up. 

Before we open up registration for that course I want to give you something for free to get you started. So down below this video, we link up my Step-by-Step Guide to Increase Elementary Students’ Math Fluency here.

All right, so I hope that this video 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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