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Research on strategy development

Don’t Teach Math Strategies

Now, I am not a fan of directly teaching math strategies to solve problems to kiddos. Now, instead, we should be helping kids build their number sense in order to help them become flexible thinkers. Instead, if we just directly teach the strategies, they just become rule followers and they have more rules to follow and more strategies to try to remember. 

So my philosophy, and it’s based on a lot of research that I can link up down below this video, is that when we build students’ number sense, we build their flexibility with numbers. The byproduct of that is that they naturally develop strategies without us having to directly teach those strategies to them. 

Now, how we do that is a big, big topic. My online courses, the Flexibility Formula, go in depth on that topic.  But in this video, I’m going to give you one of my favorite ways. 

One of my favorite ways is to ensure that your students have more than just a symbolic representation of numbers. Now, what do I mean by that? Well, if the only visual that they have of a number is just the symbol, the digit, it becomes very hard to think flexibly about numbers. 

Kids Need Visuals of Quantities

When they have those visuals, then they can group those quantities in ways that we’re trying to directly teach them with the strategies, but they can do it naturally on their own. 

So one of my favorite visuals to use is ten frames. 

No matter what grade level you’re working with, ten frames help kids build their understanding of numbers, and it allows kids to be able to visualize those amounts so that then they can perform those operations we’re trying to get them to perform. 

Now, typically, ten frames are only used in the early grades and the kids are shown exactly how to use the ten frames to solve a particular problem. But I’d like to show you how kids can use ten frames throughout all of elementary to help themselves develop their own strategies for addition and multiplication. 

There is no ‘one way’ to use a ten frame to solve a problem

Just because we put a ten frame in front of the kiddos doesn’t automatically mean that kids are going to use strategies to solve problems. They might still count one by one by one. But when it’s in a ten frame, kids now have the opportunity to group quantities.

 So let’s start off with single digit addition, and let’s take a look at how kiddos might think about 8 + 7 if they see 8 + 7 in ten frames. Typically, when we show 8 + 7 in a ten frame, it might look like this. 

Now, I’m saying typically, because this isn’t the only way to show an amount in a ten frame. I am just showing this one way as an example. But when we show it using a ten frame, our textbooks often use the ten frame to say, “You need to make a 10.” And typically they say, “Make a 10 by sliding some over to fill the ten frame.” Well, which ten frame? Typically, it’ll tell kids, “Fill the one that has the least empty spots.” So they’ll move them over there. 

But it doesn’t have to be this way. There are lots of ways that kids might be able to determine the total quantity here, and it doesn’t have to be the one way. 

For example, kids might want to make a 10, but they might want to move the three from the eight over with that seven and fill it the other way. 

That’s totally fine. We might also get kids who who do what I call finding the 5s. They see that the top row in each ten frame is full, and they know that two 5s make a 10. So they make a 10 that way. And then they just add in the 3 and the 2 extra from the bottom rows to make the 15 total. 

But some kids might not even want to make a 10. They might see it differently. Even in a ten frame, kids will often still see doubles and doubles plus or minus one. So in this one, we might have a kid who mentally takes 1 out of the 8, and now we have 7 and 7, and they know what 7 and 7 is and then they can just add 1 more. 

You might get a kid who does it the other way, and they want to add 1 over there with the 7, and now they’ve got 8 and 8, and then they can take away one. 

Again, these are just some examples. They are not the only ways kids will see them and they’re not the ways that we teach to kids. When we show the visuals, kids naturally start to see these groupings.

Ten Frames for Multi-digit addition

If we have ten frames that are showing 16 and 28, again, kids might see it in different ways. Here’s just an example of a couple. 

They might take two from the 16 and move it over to make the 28 now a 30.

That’s a typical strategy that our textbooks try to directly teach to kids. But when they have the visuals in front of them, they might naturally do that on their own. 

Another one is to do it the other way. Instead of making the 28, a nice friendly number, they might want to move some over from the 28 and give it to the 16 so that now we’ve got 20 and then the 24 to put them together that way. 

Some kids might not even make a 10. They might want to group all the full ten frames together first; add your 10, add your 20, and then work on the ones.  

Then when they get to those two ten frames showing the 6 and the 8, who knows how they’ll put those together? It might be similar to the ways that we showed on the previous ones with single digit addition. 

Another typical strategy that textbooks will try to teach kids is to round up, like, round one of those numbers to a friendly number. So instead of 16 + 28, make it 16 + 30, and then work from there. 

But a lot of kids get confused with that strategy. So instead, if those visuals are there, some kids might naturally want to add 2 into that other ten frame to make it full so they will naturally have the 30, and then they understand they need to subtract 2 because they were the ones that added that in. That was their idea to begin with.

Ten Frames for Multiplication

Let’s take a look at how ten frames might help kiddos with multiplication and developing strategies for single-digit multiplication. 8 x 6, first of all, it’s helpful to think about it as 8 groups of 6.

Now, in the United States, it is a convention that we say 8 groups of 6. Outside of the United States, you might say 8 x 6 in a different way and 8 x 6 might bring up a different visual for you. Here in the United States, typically we say it as 8 groups of 6. So here is a visual that helps kids see the 8 and see the 6 without needing to count. 

When we put those groups into a ten frame, I don’t have to count to figure out how many groups there are. I know that there are 8 groups, and I can see that it’s 6 in each group because of the way that those dots are arranged. Now, how you arrange this might determine a certain way for kids to look at it. 

A typical way that kids naturally will see this without me having to point it out is that kids will see that the top row is completely full.  They will say that is 5 groups of 6, and a lot of kids know their times 5s. They know that 5 groups of 6 is 30, and then they can just do the 3 groups of 6. 

That is a strategy that we want to help kids understand, is that if you know your times 4s, that helps you with your times 8s, it’s like having four groups of something and then doubling it. That is a strategy that textbooks will often try to teach kids for multiplying by 8. But once we put it into a ten frame, we can help kids be able to see those types of groupings. 

Now, another popular way for times 8 is to make a connection to 10. If I have 8 groups of something, how does that connect to 10 groups of something? Again, instead of trying to directly teach that, some kids will naturally want to do that. They will think, man, what if that ten frame was full, we would have 10 groups of 6.  I know 10 groups of 6 is 60, and that’s a whole lot easier, so then where do I go from there? 

That visual helps them understand that if they have 10 groups of 6, now they have to come back and subtract 2 groups of 6 from that answer. 

Now, some kids might not want to break apart the 8 groups, they might break apart what’s in each group. Here’s an example that I’ve heard kids say before, is that they see 8 groups of 5. They see the dot pattern for 5 in every one of those ten frame slots. 

So they do 8 groups of 5, and then they do 8 groups of 1.  That’s another way they might decompose that. 

Kids who are looking at breaking apart the amount in each group, I’ve seen this also, that they don’t see the 5, they see a two groups of 3.

So they see 3, and then they double that because there’s another group of 3 in every single one of those. 

Ten Frames Don’t Dictate A Certain Strategy

Visuals do not dictate a certain strategy. Even a ten frame. There’s no one right way to use a ten frame. They do not dictate a certain strategy.

They allow students the opportunity to find their own way to group the quantities and allow them to develop their own strategy.

As long as we step out of the way and let them. 

Remember, our textbooks try to show us “the way” to have kids use ten frames. But there’s no “right way.” 

Ten frames give kids a visual, and then it’s up to them to figure out how many are being shown in a particular quantity. 

Just because I showed all of those strategies doesn’t mean you need to directly teach your kids all of those strategies. Remember, all of those strategies were just possible ways that kids might see how to group those quantities and figure out the total for that particular problem. 

So if you liked this video, consider coming and registering for The Flexibility Formula. It’s my online PD course that dives deep into how kids build strategies on their own instead of us directly teaching them those strategies. 

One of the ways is to provide visuals that help kids develop their own way to see the solution to a problem. As I said, that’s just one of the ways. The whole course is all about how we build flexibility in their thinking about numbers. 

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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