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.)

• Number sense mini course for PreK-2 is available inside the Build Math Minds PD site for members of the site
• Number Sense mini course for 3rd-5th is available inside the Build Math Minds PD site for members of the site
• Clements & Sarama book
• Van de Walle Book

So, I’m not going to be talking too much about how we do this with kiddos. That is going to be in a mini-course that I’m gonna be doing all about how to build number sense. For this video, we’re gonna be taking a look at what number sense is. When you see a kid struggling and you say, they just aren’t getting it, they have no number sense, but you don’t really know what they’re not getting, what are those concepts?

That’s what we’re gonna be talking about in this post, is what are those concepts that those kids are lacking, or the reverse, those kids that you just see naturally get how numbers work together, they work fluidly and flexibly with numbers. They have these number sense concepts and those are what we’re gonna talk about today so that you know what to be looking for and then if you want to later you can join me for the how inside of the mini-course.

Early Numeracy Concepts (Subitizing, Verbal Counting, Object Counting & Cardinality)

Verbal Counting

One of the first things that kids start to do when it comes to building their number sense is just learning to count. This is known as verbal counting, just being able to say the counting numbers in the correct order. When kids are first starting out, you’ll hear them say things like one, two, 12, 11, 13, 24, like it’s just random numbers. They know the difference between numbers and letters but they aren’t saying it in the correct order.

Object Counting

Once they get the correct order down, that doesn’t really tell you that they understand numbers and understand counting because the whole purpose of being able to count is to count objects. So the next big concept that we need kids to understand is that you count for a purpose. There are things that you should be counting. So this is when kids start to attach their verbal counts to actual objects and they start touching objects as they’re counting.

Cardinality

The other piece in early numeracy still ties to counting and it’s called cardinality. It’s the understanding that once you’ve counted, you then know how many you have. You can answer the question of how many.

So, in this picture, if we asked how many apples that the girl has here, you may see a kid who can count one, two, three, four, five, six. But then when you ask, so how many apples does she have, they go right back and count, one, two, three, four, five, six. They don’t understand that six, that last number that they said tells the set.

Verbal Counting in the Upper Elementary Grades

Now we briefly talked about verbal counting for our young kids, this is just getting the counting sequence in order. As they progress through though, it’s not just counting by ones, it starts to become counting by tens, by hundreds, by twos, by fives, all of that skip counting stuff that we do is really building their number sense.

Now we do a lot of verbal counting with whole numbers but kids also need it around fractions and as we talked about, verbal counting really doesn’t mean that they understand it. So if we do a lot of skip counting, that doesn’t really mean they’re attaching meaning to this skip counting and that only happens when we start attaching objects and bringing in cardinality, ideas with it.

Object Counting & Cardinality in Upper Elementary Grades

Object counting and cardinality are intertwined together and they’re intertwined with verbal counting and subitizing. All four of these early numeracy concepts work hand in hand to help build a cohesive understanding of counting and subitizing. So we need to be doing them together.

When kids are skip counting, we want to have actual objects or at least visuals that they can skip count with. Now that idea connects to one of the biggest ideas in early numeracy because all the ideas that we just talked about right there had to do with counting and I cannot tell you how many times I have heard, “I can’t get my kids to get past counting.” Now there’s definitely a phase, a big idea around number sense is counting. Kids need that experience with counting. The problem is we don’t want them staying there forever.

Subitizing

The fourth area of early numeracy that helps us do that is this idea of subitizing. The idea is that they can tell how many without having to count. They can instantly tell how many without having to count.

In this image, you might have some kids who have to count to tell you how many apples but if you asked how many fingers is she showing, they know that it’s six instantly and they did not have to count. So in that situation, that child is subitizing. That helps kids not have to count one by one by one, they start seeing things as sets and as groups, and that is a big hinge point into helping your kids move past a counting phase, okay?

Let’s do a little activity here just to make that this is solidified because subitizing is really, really huge. We’re gonna look at what it looks like with young kiddos and then we’re gonna transition into how this impacts kids as they progress it through. I’m going to show some dots and then I want you to see if you can tell how many by looking for 2 seconds and then closing your eyes. Ready? Here we go.

Alright, could you tell how many? Most of us can’t unless you paused it in that moment but in the instant that it flashed, we can’t tell. Even as adults, we cannot subitize more than five unless it’s arranged in some kind of nice pattern for us, which is what I’m gonna do on the next, okay?

It’s the same amount of dots, I’m just organizing it in a pattern you might be familiar with, okay? Here it comes. Did you know that time? Yeah, right? I left it up there the same amount of time but this time you knew and you knew for sure because you were able to subitize using those groupings.

Now you really didn’t see nine, you saw a five and a four and then you put that together to help you determine that it was nine, okay? It’s not like you can just see that nine, you’re seeing the smaller pieces within. That’s a big concept of subitizing, alright? So, that’s where we start with in the early grades is helping kids see those small groups but how does this progress through in our upper grades in elementary?

Subitizing in the Upper Elementary Grades

Okay, so here we go again. I’m going to give you one that’s got a lot more dots on it and see if you can tell how many and I want you to think about what mathematical concept this could help develop for our upper grades kids, okay? So here it comes.

Now not just as adults, our instinct is to just tell how many we saw, but I really want you to think about what’s the mathematical concept that I was representing. Even though I’m helping my students build subitizing, the byproduct is we’re also working on multiplication.

Because multiplication is all about seeing groups and if students aren’t able to see groups, the small groups like that one group of five, if you grab one of those groups of five, we want kids to be able to subitize that because as we move into multiplication, if they can’t see one group of five, then they aren’t gonna be able to understand and visualize what five groups of five looks like. Because we can’t have them visualize 25 things as we talked about before, we can’t even visualize nine things. We have to see it in groupings. So as they move into multiplication, they really need to be able to have these small groupings so that we can use those to help them visualize what multiplication looks like, okay? And it’s not just with multiplication, you can do this with multi-digit numbers to really build number sense around those, so I’m gonna show another image here very quickly, here it is.

Alright, we tend to stop using visuals as kids start working into multi-digit numbers but kids still need them, right? They need to visualize that 36 is three groups of 10, that it’s 35 and one more, that 36 is four less than 40, right? Do you see all of those when I show you that visual? Kids need to be able to see those relationships and subitizing helps lay that foundation to allow us to have those conversations.

Now there is no work on subitizing around fractions but there are visuals, we want kids to instantly tell how much is there and as you work into fractions, you can still be doing subitizing around fractions.

We want kids to instantly know that that is 1/3.

We want them to instantly know that that’s 1/2. There’s definitely subitizing things that are easier like 1/3 and 1/2 just like four is easier to subitize than nine, okay? So you lay the foundation with the small amounts and then you work your way into those harder visuals to build for kids.

So in the early numeracy concepts, these are typically things that kids start developing before they ever come to school but they play such a huge role throughout all of the elementary grades that subitizing being able to instantly recognize, they need to be able to verbally count which is skip counting and counting one by one by one. They have to attach that meaning of the counts to actual objects and then once they’re done counting, they need to know how many. How many are actually there?

But this is not the ending piece of number sense. Because this is really focused on counting and seeing groups, but the biggest piece of helping kids build their number sense is by building relationships.

Number Relationships (Spatial Relationship, One/Two More & Less, Benchmarks, Part-Part-Whole)

I love this quote by Howden because it helps us see that you can’t directly teach it, remember I said that at the beginning? It develops gradually, you need to let kids explore numbers, visualize and start relating numbers.

So, the next pieces are all about building relationships. If we want to move kids past counting, they have to build their subitizing but then the other big piece is we need to help them see relationships. These come from a fabulous book by John Van de Walle and his colleagues called Teaching Student-Centered Mathematics.

But it actually came in the PreK-2 book. There are books for other grade levels by Van de Walle and his colleagues but these four concepts come from the pre-K to second. But again, they matter and they play a role all throughout the elementary grades.

We’re going to start with what it looks like with that small amounts and then I’m gonna give an example in each of these areas that are from the upper grades. I’m not going to show all the ways that it plays a role or else this video would be super long. So you’re just going to get one example of how it progresses through the upper grades.

Spatial Relationships

The first thing is that kids need a visual to go with a numeral because they can start to notice relationships. When I show these two images, it tells me a lot more about how three relates to four versus if I just have the digits.

When I have the digits here, I don’t know anything about how three compares to four but when I have the visual, I can see with my eyes that three is one less than four. I can see how they connect together. So the very first relationship that kids start building comes from the visuals and if the only visuals we’re providing are numerals, we’re doing a disservice to the students. So, start using visuals.

Spatial Relationships in the Upper Elementary Grades

As we work into multiplication, spatial relationships still apply. Kids need that visual. But it’s not just having the visual, it’s how does that visual relate?

This first visual is 10 but remember, as we work into multiplication, we start to see it as groups of. This is two groups of five but how does that visual relate, we’re looking for relationships here, how does that visual relate to four groups of five? What connections can I help the kids start to see here?

One/Two More & Less

Once kids have these visuals, then we can start using those to help build the idea of one, two, more and less. So this is of a rekenrek, if you’re not familiar with a rekenrek, it has red beads and white beads. The top rekenrek here is showing six, the bottom rekenrek is showing seven. When I have those visuals, I can then see this relationship of how six is one less than seven or seven is one more than six. But if I just have the digits where we’re comparing and I’m supposed to put the greater than, less than, or equal to sign in this like all the worksheets have on comparison. It doesn’t tell me anything, it doesn’t help me out, I can’t visualize it.

So we need those visuals to help kids see that relationship. If when they see the digit seven, they can connect it to a visual of a rekenrek or the visual in a 10 frame, or the dot patterns, I don’t care what visuals you’re using, I’m gonna use all of them in here because I want to encourage you to use them all, it’s helpful to have kids have lots of different visuals, okay? Because when they have those visuals, they can start seeing these relationships and the relationship of one and two, more and less is really, really huge.

One/Two More & Less in the Upper Elementary Grades

The same is true as we start working with numbers like 36. If I want kids to understand what numbers are one more, one less, two more, two less, do you see how having that visual of 36 could help me out?

I can’t just tell the kids 35 is one less than 36, I want them to see it. And do some of those visuals help point that out? Yeah! Some of them are more obvious than others but the visual piece of that helps build this one and two, more and less concept.

Benchmarks of 5 and 10

The other big relationship in the young grades is helping kids develop the Benchmarks of 5 and 10. Kids are often counting out manipulatives individually like the top row up there. But when I count out that top row, it doesn’t tell me anything about how many there are. I can count, I know that there’s nine but it doesn’t build anything about the number nine, except it’s one, two, three, four, five, six, seven, eight, nine.

It doesn’t tell me how it relates to five, it doesn’t tell me how it relates to 10 because those benchmark numbers, the five and 10 are so huge for our young kids, okay? And so, putting those manipulatives into a 10 frame, just draw a 10 frame on a piece of paper, I don’t care how you make your 10 frame, but letting kids put their manipulatives into a 10 frame gives them the opportunity to see how that amount relates to five and how it relates to 10.

It will not be obvious to all of your students, that’s why I say it gives them the opportunity to see that, you have to build that relationship and help them notice some of these things that they might not be noticed on their own.

Benchmarks in the Upper Elementary Grades

The benchmarks of 1/2 and 1 are super huge when it comes to fractions so your benchmarks don’t always have to be 5 and 10, it’s just whatever the benchmark numbers are, those always play a role no matter what grade level of kiddos you’re working with.

If you’re trying to think about where 3/4 would go, I don’t want kids to have to chunk that into fours to figure it out. I want them to understand that it’s more than a half and less than a whole. Because as we progress through like this, where would 12/13? I don’t want them cutting the whole into 13 pieces. Number one, it is very difficult to be accurate, and it’s cumbersome, it takes so long. I want them to understand that 12/13 is just a little bit under the one, okay? On that number line. And then where would 20/13 go? I want them to understand that it’s more than a whole.

These are big, big ideas that help with kids as they start trying to order fractions from least to greatest. When you see problems like this, it is not asking kids to find common denominators and compare them. Really all this is asking kids to do is be able to place it on a number line and use their knowledge of how those fractions relate to the benchmark numbers. If you do that, you’re able to put those fractions from least to greatest and I’m not gonna show you the answer, I want you to think about it. Can you use the benchmark numbers to compare those fractions and order them from least to greatest?

Part-Part-Whole

Alright, the last one is part-part-whole. This is the understanding that if I have seven things, if I have a whole amount, I can break that whole into its parts and I still have the original amount. So I can break seven into a one and a six, a two and a five, a three and a four. But again, if I just have the digits there, it’s not building a lot of these relationships for kids.

By incorporating the spatial relationships that we talked about to begin with, that visual helps kids see how I’m just moving the dots around, I still have the same amount of dots, I’m just moving them and now I have two over here and five over here. Three and four, they can visualize the moving of those dots and rearranging the dots to make that new but same amount, okay? We want them to understand that the parts come together to make that whole but we can break that whole into different parts and it still gives us the same amount.

Part-Part-Whole in the Upper Elementary Grades

In our young grades, we start to do things like decompose the number five. What are all the ways that you can break apart five? And kids will get zero and five, one and four, two and three, right, all of that stuff. Kids need to do that with fractional amounts as well. How can you decompose 5/8? It’s all the same ways. If it was a one and a four, it’s still a one and a four but it’s 1/8 plus 4/8. If it was a two and a three, it’s 2/8 plus 3/8, right? How does this help kids with addition?

If they’re able to decompose five, how would that help them when they’re adding nine plus five? If they did that with a fractional amount, could you decompose 5/8 in a way that would help you add it to 1/2? Can you break it apart? There’s a nice little way that I’ll give you a little hint, halves, using those benchmark numbers. 5/8 is really close to a benchmark number and if you can chunk it in a way that uses those friendly numbers, it makes addition and subtraction so much easier for our kids.

All of these number sense concepts, when kids finally start grasping them, it makes operating, whether it’s add, subtract, multiply or divide, all of those operations so much easier because kids are fluid, they’re flexible, they see connections, they start to be able to do all those strategies that we’re trained to help teach kids but we can’t directly teach that if they don’t have the number sense to grab onto it.

Again, remember that this is just about what makes up number sense. We did not delve into how to do it. The big thing about how is that you can’t directly teach it. Kids need to explore, they need to visualize, they need to play.

Remember, number sense can’t be taught, it’s caught. So if you want to learn more about these eight concepts and how to help your kiddos catch number sense, I would love for you to join me for the mini-course that’s happening, there’s one for PreK-2 and one that’s focused on how we help build these for our 3rd-5th grade kiddos. So if you’re interested in joining those, there are links below here that will take you to learn more about that.

But in the meantime, I hope that this has helped you build your math mind so that you can go out there and build the math minds of your students.

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