Components of Number Sense in PreK-2

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

One of the hardest parts of teaching mathematics without a textbook is knowing and understanding the progression of mathematical thinking, but it is the most vital part.

Also, one of the areas that I talked about earlier in this video series is how essential it is to focus on the concept of number. But our textbooks tend to focus on number skills and not number sense. Yes, there is a big difference between those. For the next two weeks, I’ll be talking about what exactly is number sense in PreK – 2nd and 3rd – 5th#MathIsNOTaWorksheet.

I’m Christina Tondevold, the Recovering Traditionalist.  Today, we’re going to look at the components of number sense in PreK – 2nd Grade 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:

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

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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 I am going to get straight to it. I’m going to hop right over into my screen so that we can take a look at the eight components of number sense in PreK – 2nd grade.

4 Early Numeracy Concepts from Clements & Sarama

We are going to start off with the concepts of early numeracy. These come from the work of Clements and Sarama. The first one is this idea of subitizing. It’s the ability to instantly recognize how many. I’m going to go through each one of them and then I want to use the picture here to give a little bit more description. 

Verbal counting is the next one. That’s when kids are able to just say the number names in order. A lot of times, kids start out with like a singsong. It’s just like 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. They don’t really realize that they’re individual amounts. They aren’t paying attention to patterns, all that kind of stuff. Verbal counting is being able to say the numbers in order. 

Object counting is when the kids actually attach those words to things. This is when they start to realize that you’re counting for a purpose.

Then the final one in early numeracy is this idea of cardinality, that when you’re counting out an amount, the last amount you say, the last number you say tells you how many you have. 

This little girl here with her fingers and her apples here, let’s take a look at each of these 4 things. Kids who are able to subitize can look at this picture and if we were to ask them “how many apples does she have there?” a lot of the kids might not be able to subitize it. Subitize again is just instantly recognizing without having to count. 

However, they might be able to answer the question “how many fingers does she have up?” and they would instantly be able to tell that one. Subitizing is helpful, it’s easier I should say, to subitize if you can see nice groupings. A lot of kids will recognize that she has 6 fingers up because they know 5 and the 1. 

With the apples, if we ask them that, this is where kids might count. Verbal counting is when kids are just counting. But you can tell if they have object counting when you ask those questions like how many apples does she have, and you see a kid go 1, 2, 3, 4, 5, 6. You’ll also see kids who will double-count things, they’ll skip over apples, all of that happens in this phase of developing their object counting. Object counting is just being able to attach the name to the object. 

Now cardinality comes into play when you ask that question how many apples do you have. If you ask that question how many apples do you have, and the kid goes 1, 2, 3, 4, 5, 6, and that’s all they do. And then you say, so how many apples do you have, and they go 1, 2, 3, 4, 5, 6. They don’t get that when they’re done counting, that tells them how many they have. That’s a set. So, these 4 early numeracy concepts go hand in hand. They help develop each other. It’s not like you just work on subitizing, then you do verbal counting, then you do object counting. They all work together to build this comprehensive idea of early numeracy. 

4 Number Relationships

Now those four lay a foundation for helping kids be able to pay attention to number sense relationships. These four I discovered in the book, a lot of people call it Van de Walle book, but it’s Teaching Student Centered Mathematics. In that, there’s a small little part about the relationships that kids need to develop around numbers. Once I had read these, I’m like these are the things that so many kids are missing, they are: spatial relationships, one/two more and less, benchmarks of 5 and 10, and part-part-whole. 

I consider these four being the net of understanding. That’s why I’ve got the hammock there. I see them as a web of connectedness that holds together the mathematical understandings that our students have. If something gets missing, like if some part of that web gets broken, it’s okay because there’s lots of other connections in there that keeps their understanding secure just like in a hammock. Let’s take a look at each one of those fairly quickly.

The first one, spatial relationships, is having a visual of a numeral that goes along with a numeral and using those visuals to help them see relationships. In the early grades, if all I had up there was just the digit 3 and 4, it’s hard to talk about which one is more, which one’s less, how many more, how many less. But when kids have these visuals, the spatial image can come up, and then we can have much more deeper conversations about how these quantities relate to each other. Building relationships starts first with having lots of visual spatial images of these quantities so that we can use those to help us build the other relationships that we’re going to talk about.

The next one of one/two more and less is instantly knowing what is 1 and 2, more and less of any amount, but again having a visual picture helps with that. If I just had 6 and 7 on the screen and ask. Let’s just say I just had 6 and I ask them what’s 1 more than that, it’s harder for some kids to be able to do that. But if we have a visual of the rack on rack showing 6 on the top there, and then we ask them what’s 1 more, the visual that might come up for them is that visual of 7 that’s down below it. 

Those spatial relationships help lay a foundation for us to be able to build this relationship of one/two, more and less. It doesn’t just stop with one/two, more and less. As you start getting into multi-digit numbers, it becomes one 10 and two 10s more, so 10, 20 more or less. What’s 100 or 200 more and less? This idea of the one/two more or less is not just one or two, it’s one or two of those benchmark amounts more or less. 

Speaking of the benchmarks, the third relationship is the benchmarks of 5 and 10. In the early grades, it is so important that kids understand how numbers relate to the benchmark of 10.  However, they first need to be very solid with how numbers relate to the benchmark of 5. Our number system is built around the base 10 number system, but 2 5s make a 10. So, when kids are first starting out with numbers like 3, we don’t want to push them to see relationships to 10. We want to start out with how does it relate to 5 first. And then as we work into bigger amounts like 9, we can help them see how it relates to both 5 and 10. 

Now kids don’t see this if they just have random manipulatives that they’re working with. If we take those manipulatives and just put them into a 10 frame, it makes a world of differences, and kids can start to then have the opportunity to see how numbers relate to those benchmarks of 5 and 10. 

The last number sense concept is this idea of understanding part-part-whole. We’re very familiar with this because textbooks tend to do a lot of activities around this. You take a whole amount, and you break it up into all of the parts. I like to relate this a little bit to doing prime factorization. Do you guys remember having to do prime factorization in school? I was good at prime factorization. When I was teaching middle school, I had my kids do prime factorization. But I never knew why. Why are we doing prime factorization? What does it help me with? 

Even with this part-part-whole idea is like prime factorization for our older kids. It’s like they can do it, but they don’t know why they’re doing it. What’s the point in knowing all the ways that you can break apart 7? Yeah, kids learn what makes 7, but what is really the point? Why would I ever break a 7 into a 1 and a 6 versus a 2 and a 5? That is so important. 

Connecting Number Sense to Number Skills

Kids need to have that understanding about why we need to break numbers apart. It helps us as we move into addition and subtraction to make problems friendlier. That’s one of the biggest pieces. That’s a much deeper concept that I can do in just one video. That’s why I have full courses on how to develop number sense for kids. But for right now, I want you to consider this quote by Howden. 

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.” 

I say it all the time, number sense can’t be taught, it’s caught. 

I can’t directly teach a kid how 6 relates to 7. They need to explore those numbers. They need to visualize. They need to build it. They need to build these relationships in their own way. As that last slide reminded us, it does not happen through a worksheet. It happens through experiences, through seeing patterns, visualizing things. 

To go more in-depth on that topic, that’s why I’m doing those free webinars. So, make sure that you get registered below. We’re going to take a look at that progression of how we go from building kids’ number sense and how it impacts their ability to become fluent with those number skills, and we’ll also explore how to do all of that in the way that kids are supposed to, through experiences, visualizing, looking at patterns, and not just filling out worksheets. 

I hope that you can join me on one of those live webinars. Remember, it’s all free. So, make sure you sign up below. I hope that this video has given you a little bit of insight so that you can get back there and build the math minds of your students. Have a great day.

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