Number Sense: What Is It and Why Is It Important?
Two years ago, when I started teaching a new first grade class, I observed a student as she counted gems. I had just told a story about a king who needed help counting the jewels in his treasury.
She had her little pile of gems and was counting along beautifully. But then I noticed something. I paused to take a closer look. . . . She was counting out loud, but not matching each number up with a gem.
Sometimes she moved two gems while naming only one number. Other times she named two numbers as she moved only one gem.
This child was showing me she could count verbally, but she was not relating the gems with one-to-one correspondence to the numbers as she counted.
I’m often thankful to this little girl for this experience, because it taught me something very important for my own math teaching. Even after many years of teaching, I was assuming the children could count objects, based on their ability to verbally name numbers. I assumed they were one and the same, and if they could count out loud, they knew to match each number with one, and only one, object.
This child taught me the importance of the some vital elements of foundational number sense that I have since included with more intention in the learning experiences in my classroom.
What Is Number Sense?
Jamie York, in Making Math Meaningful: A Sourcebook for Teaching Math in Grades 1-5, urges teachers in the Waldorf classroom to spend the lower grades math classes establishing strong number sense. He goes on to encourage teachers to wait to introduce some of the standard algorithms for arithmetic, like vertical addition and subtraction with regrouping, until third grade in order to really focus on helping students develop strong number sense.
But, this recommendation always brought me back to the same question: What exactly is number sense? I had a vague gut feeling about it, but I was not able to articulate an explanation of it clearly in order to guide my teaching.
Christina Tondevold, of BuildMathMinds.com identifies eight number sense concepts.
Her website and videos have been very helpful in expanding my understanding of number sense, as well as inspiring me to try out some different approaches to teaching math in my classroom.
She cites the definition of number sense as “ . . . good intuition about numbers and their relationships. 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.” (Howden, 1989)
There are 8 number sense concepts that are identified by Tondevold:
First are the four foundational number sense capacities that we want to help children develop.
Verbal counting
Object counting
Cardinality
Subitizing
Once these foundational number sense concepts are secure, students are ready to explore the four number sense relationships, which include:
Spatial relationships
One and two more or less
Part-part-whole
Benchmarks of 5 and 10
I honestly had very little idea about what these eight number sense concepts even meant. This had not been a part of my teacher training.
However, as I’ve delved deeper into understanding each of these number sense concepts, I’m convinced that the Waldorf curriculum provides natural opportunities for developing these in our math lessons. And, as we become more familiar with the terminology, we can also begin to create space for children to have experiences that support developing number sense in a wide variety of ways.
Foundational Number Sense Concepts
Verbal counting is just that – naming numbers verbally as you count. This means the students know that 6 comes after 7, for example. Very young children learn to verbal count through imitation and language development.
Object counting is connecting each number to one specific object as it is counted. This is a capacity that can be modeled, practiced, and developed easily in first grade, if a child comes into the class without a firm grasp of it.
Cardinality is the understanding that the final number named in counting a group of objects is the actual number in the group. This may seem logical to us as adults, but a handful of my first graders did not understand this concept. If I asked them how many gems they had counted, they did not yet make the connection that the last number named was the number of gems in the pile.
Subitize is a term I only learned very recently. It is the ability to see a certain number of objects and know without counting, how many there are. The easiest example to illustrate what it means to subitize is probably what we all do when we look at dice – we automatically can see the dots and know we’ve rolled a 5 or a 3 or whatever comes up. We don’t have to count the dots. Children need to be able to subitize so they can hold a number in their minds and not have to start counting from scratch each time.
Number Sense Relationships
After the foundational number sense concepts, there are the concepts that involve numbers in relationship to others.
Spatial relationships mean that children can experience numbers in a visual way, not simply as a numeral. These visual patterns allow children to actually see what the number 10 can look like, for example. They allow children to understand relationships between numbers, so that they can easily see that 9 is one less than 10, and 11 is one more than 10.
One more and less are concepts that support the understanding that one more is the next number up, and one less is the number right before. Two more and less builds off of this concept. When the children can see this concept in a visual representation, it builds powerful understanding about number relationships and supports them in learning addition and subtraction facts.
Benchmarks of 5 and 10 are important because they are both important number bases in our number system. They quickly learn to skip count by 5s and 10s before almost any other number family. We have 5 fingers on one hand, 10 on both together. Counting on fingers is one of the first strategies that children naturally use to help them solve number problems.
Part-part-whole is the understanding that a number can be broken from the whole into parts, or that parts can be combined into a whole.
I highly recommend you find out more about number sense and how you can support children in developing it by checking out BuildMathMinds.com.
There is a wealth of information on this site, and professional development programs for math teaching are available.
What We Can Do in the Waldorf Classroom
When a child has a well-developed sense of numbers, number sense becomes “common sense”!
The Waldorf curriculum has many components of this list of number sense concepts already in place. Just by following the block rotations, telling number stories in our teaching practice, and incorporating main lesson book drawings of the number concepts we’re teaching, we support number sense.
Below are a few simple activities you can immediately incorporate into your daily rhythm to build number sense:
Stamping and counting footsteps in morning circle. This is a perfect first grade activity to reinforce object counting. Carefully observe the children to ensure they are all lifting their feet and chanting the number as they stamp. I’ve noticed some children have a tendency to to shuffle along and verbally count without making the relationship between their step and the number. Identifying the final number counted helps establish cardinality.
Using gems or other counting objects as manipulatives for daily math story problems helps students visualize the numbers represented. Counting objects are also very helpful for illustrating groups of objects when working with multiplication and division operations. I’ve even recently taken gems into my 6th grade math skills class to help students create visual representations of multiplication facts they struggle to remember.
Including hands-on activities to create visual number experiences. I’ve been incorporating a variety of hands-on activities into my math lessons, but my favorite is using 10-frames. 10-frames allow students to see visual representations of the numbers in relation to the number 10. They provide the opportunity to build number sense in so many ways! They can flexibly be incorporated into main lesson teaching time, or be used in a math skills practice class. And, they are an inexpensive math tool to incorporate into the classroom. All you need are copies of 10-frames and counting objects. I use glass gems, but you could use dried beans, popcorn kernels, or small stones . . . really anything you have that’s convenient.
I’ve included a free 10-frame PDF template that you can download and photocopy if you’d like to try it out with your students.
Click here to get your 10-Frame Template
There are two options you can use in this PDF — a single large 10-frame (in two colors for you to choose), or a smaller set of doubles so you can easily work with numbers larger than 10 (also in two colors).
It took some time and experimentation for me to figure out how to best weave a 10-frame into my math lessons, but it has become one of my favorite math tools for visualizing numbers and number relationships.
There are so many ways to use 10-frames. I give each of my students a template (or several) to have at their desk along with a handful of gems. We then work out number challenges according to my lesson objectives from there.
I especially like them for developing understanding around addition and subtraction, and counting by 10s in first and second grade.
Children need time, repetition, variations on the concepts, and lots and lots of practice.
If you’ve never used one before, I encourage you to experiment and see what works for you. Give it some time! At the beginning, some of the lessons I created with 10-frames were not too successful, but over time I was able to figure out how to use them as an effective learning tool in my Waldorf classroom!
If you’d like to learn more about number sense, check out this article by Christina Tondevold : What Is Number Sense . On the website there are many free resources that might benefit your own math teaching!
