Components of Number Sense in 3rd-5th

Building Number Sense, Fact Fluency, Number Sense, Subitizing, Teaching Mathematics | 0 comments

We often hear about how number sense is such a big deal in the early grades of elementary. But it is so important for our upper elementary kiddos as well. Our textbooks, especially in the upper grades, tend to focus on just the skills around numbers. They want kids to multiply and divide, we’re also still working on addition and subtraction, but what they’re missing is the foundation of number sense. 

So last week we talked all about what number sense looks like in PreK – 2, if you missed that one, we’ll link to it down below. Today we’re investigating what it looks like in 3rd – 5th grade.  Now before all of you K-2 people head out, I want to remind you to get registered for those free webinars that I’m doing. It’s going to be live, it’s all free, and it digs way deeper into how to build these number sense concepts and connect them to number skills in a way that kids will see mathematics as more than just a worksheet. Down below this video there’s a link to get registered for those, it’s coming up quick so make sure you get registered.  

I’m Christina Tondevold, the Recovering Traditionalist.  Today, we are going to start diving into the components of number sense in 3rd through 5th grade as we try to build our math minds so we can build the math minds of our students.

Watch the video or read the transcript below:

The specific webinar mentioned in this post may have expired but you can catch Christina’s current webinar here.

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.)

Video series prior videos
#1: Why you shouldn’t teach math through a textbook
#2: How to Teach Elementary Math Without a Textbook
#3: Creating School Change When Others Don’t Want To
#4: Components of Number Sense in PreK-2
#5: Components of Number Sense in 3rd-5th
#6: Teaching Math through Story Problems
#7: Math Practice: Building Math Fluency through Games

 

The Number Sense courses have been revised and are now called The Flexibility Formula. Click here to learn more about the courses.
Now, one other quick reminder is that I have this guide that I put together to help you as you start to make your way to teaching mathematics without a textbook. 

There’s a link for that guide down below this video as well and it has information about these number sense concepts that we’re talking about today. I just want to remind you if you haven’t got that download, make sure you grab that. So I’m going to dive straight into looking at what number sense looks like, so I’m going to switch on over into my computer screen. 

4 Early Numeracy Concepts from Clements & Sarama

Okay, we are starting off with early numeracy concepts that come from the work of Clements and Sarama. Now bear with me for a moment because these are things that typically are associated with early elementary or even before kids get into elementary, but these still play a role with our kiddos in 3rd through 5th grade. 

So one of the first ones is the ability to Subitize. I’m going to briefly explain and then I’m going to show you what it looks like in 3rd through 5th grade. Subitizing is the ability to instantly recognize how many without having to count. Verbal Counting is being able to count, just say a count sequence in order. 

Object Counting is when you’re actually attaching those counts to something. So quick little example, for 3rd through 5th grade, verbal counting, we tend to do this a lot with multiplication where kids will learn to skip count, 3, 6, 9, 12, 18, so on. They’re just verbally counting, they aren’t associating those counts with things. They aren’t seeing sets and objects within those to see the 3, the 6, the 9, the 12. That’s a big difference for a lot of our kids. 

Verbal Counting is just when kids can verbally count but are they actually attaching the visuals that go with it, are they actually counting out sets to see how 3, then 6, then 9, then 12? That’s a big difference and it plays a huge role in helping kids develop multiplication. 

The last piece is what’s known as Cardinality and this is when they start to count and then can they answer that question, “how many?” We will get kids who can count out amounts but then they don’t understand that the last thing that they counted tells how many. So what’s this look like? 

Here’s an example for multiplication for subitizing as kids get into 3rd through 5th grade. I’m going to start you off with an easy one just to see subitizing in action. 

Did you know that was five? Most of us are familiar with that pattern and we do a lot of this work in the early grades. But we can use that to extend into multiplication. Watch your screen again for this one. 

Could you tell what there was there? 

Our abilities to subitize helps kids conjure up images of multiplication, of fractions, of decimals, whatever concept we’re working on, kids still need to be subitizing to help them have visuals to go along with it. 

Now Verbal Counting, we do a lot of this with whole numbers, right? We ask kids to count up to 18 by 2s but kids also need it for things like fractions. One of the coolest parts is having kids do this with whole numbers and do it with fractions and see the counts right next to each other. 

What do you notice that’s alike? What do you notice that’s different? And having those discussions about the similarities and differences is really huge. And on this they are just verbally counting, they’re not attaching anything to this, right? They’re just counting. 

So if we want kids to be working on Object Counting and Cardinality, we can bring this image back up and we aren’t subitizing anymore.  We are leaving it up and we’re asking them how many dots are there.

Are they counting, do they see the groups of 5s, and how are they counting? Do they know that once they finish counting, they can maybe tell you that it’s 25 but also associating that with what is it that we see here? We see 5 groups of 5, gives us that 25. That’s object counting and cardinality as it relates to multiplication, brief little quick overview. 

But even as they get into multiplication, this may seem like, “Oh that seems simple,” but as they get into work with fractions, here’s some cards that Graham Fletcher has put out about seeing amounts with fractions and a lot of kids struggle with this. Is this showing 3/12 or 3/4? 

And so we have a lot of conversations here about how it depends upon what the whole is. And this whole idea of object counting is dependent upon what is our whole. What’s our unit that we are counting? So Object Counting and Cardinality still apply as kids get into the upper grades.

4 Number Relationships

Now there’s also 4 number relationships. These relationships are basically what I consider this big web of understanding, it’s like a hammock that kids put all their other mathematical thinking inside of, it holds it all up. And the cool part about a hammock is even if part of it gets frayed, like if something is not connecting, they can make other connections to help them hold their mathematical thinking together. 

So the four number sense relationships are spatial relationships, one/two more and less, benchmarks of 5 and 10, and part-part-whole. These come from the book Teaching Students-Centered Mathematics by Van de Walle and his colleagues, and they actually come from the PreK to 2nd one. But they still apply as kids are working in 3rd through 5th grade. 

So spatial relationships is being able to look at how things relate. So it’s not just about how many are here.  I might have actually started out with a different version of this. I might have started with just a version of 2 groups of 5 and then showed 4 groups of 5, and then showing 5 groups of 5. And having kids build on relationships. What do you notice is the same and different from each visual? 

Spatial relationships is having visuals but looking at how those visuals can help build relationships between the concepts we’re trying to build. And it lays the foundation for the other relationships we’re going to talk about here. 

So One/Two More and Less, you still get that with multiplication, but I’m just trying to mix up the examples I give between multiplication and fractions because these are two big ideas in 3rd through 5th grade. 

One/Two More and Less is this idea of knowing instantly what is one or two more or less, and with whole numbers that’s fairly easy. But with fractions what’s that look like? With fractions, it’s like one or two more of the unit you’re working in. So what is 1/4 more than 3/4? One/Two More and Less is the counting sequence and it’s adding and subtracting but kids don’t make those relationships. 

One of the biggest parts of this relationship is helping kids connect the counting sequences to this idea of One/Two More and Less which then leads into addition and subtraction. It’s all interconnected. 

Another big one is the Benchmarks, and Benchmarks of 5 and 10 are one of the huge ones. There’s other benchmarks as you work with different types of numbers. With fractions the benchmarks aren’t going to be 5 and 10, they’re going to be 1/2 and a whole, but the idea is working with benchmarks. And when you’re doing multiplication, the facts that kids tend to know first are x5 and x10, so use that to your advantage. 

This is a visual using a rekenrek that shows how x9, when you have 9 in a group, how that connects to having 10 in a group. Man, what if we just had each one of those beads that are leftover over there pushed over.  How many would we have and can we use that relationship to figure out how many are actually pushed over? 

Now, the last idea is Part-Part-Whole. We do this a lot in the early grades with whole numbers. We ask kids all the time, “what are all the ways you can decompose 5?” And I will have you think about that for a moment. Not too long because we’ll sit out here forever but you could have 1 and 4, 2 and 3, 0 and 5. But how many times do we do that with fractions? What are all the ways that you can decompose 5/8? Can you break apart 5/8? 

One of the coolest pieces of Part-Part-Whole is not just can you do it but why should you? How does it help you in the grand scheme of mathematics? So I want you to think about this and I’m just going to leave this here to let you ponder.  

Maybe you’ll put it down in the comments below this video, but look at your ways that you decomposed 5/8. Do any of them help you add 1/2 plus 5/8? Is there any way to decompose 5/8 that makes it easier to add it to 1/2? Let me know in the comments of this video, okay? 

Connecting Number Sense to Number Skills

All right, so those are the eight number sense concepts but how do we actually develop these for kids? 

Well, unfortunately, that’s not a quick down and dirty easy thing for me to tell you.  That’s why I have full courses on how we develop it for students that I run all online. But for right now, I want to leave you with this quote by Howden and she says “It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.”

The thing I love about that last slide is it reminds us that math is not a worksheet.

Kids need to have experiences around mathematics.  They need to start visualizing. They need to see patterns. All of those pieces go into helping kids build this foundation of number sense. 

Now it is important that we still connect that to these number skills. Yeah, I still want kids to become fluent in addition, subtraction, multiplication and division, but how do we connect that with these number sense concepts to get our kids fluent and get them to meet the standards at their grade level? 

If you’d like to listen in on a more in-depth training you can watch the webinar training here.

So I hope that this video has helped you build your math mind so you can go build the math minds of your students, come join me on those free webinars. Have a great day.

Pin This To Pinterest for Later