# Understanding Number Sense: From Counting to Comprehension

Number sense is one of those things that I learned about after I started teaching. It’s been such a fun experience learning more about it and researching the different components of number sense. In this post, we will talk about the next parts of number sense. In part 1 of this series, we covered subitizing, magntiude, counting and one-to-one correspondence. If you want to learn more about those, head over to this post.

- Mastering Math: Exploring the Secrets of Number Sense Success
- Understanding Number Sense: From Counting to Comprehension

## What is Number Sense?

The simplest definition that I found for number sense is that it is the building blocks for all strands of math. This means a great deal to a preschool or kindergarten teacher. Number sense is extremely important for students to master. All strands of math going forward rely on students to have mastery of number sense. What exactly are the parts of number sense? There are 9 parts to number sense: subitizing, magnitude, counting, one-to-one correspondence, cardinality, hierarchical inclusion, part-part-whole, compensation, and unitizing.

## Cardinality

Cardinality refers to the understanding that the last count word used to represent a set of objects or events signifies the total quantity or size of that set. It involves grasping the concept of “how many” and recognizing that the final count in a sequence represents the cardinal number that describes the entire collection. Cardinality goes beyond mere counting and reflects an understanding of the numerical value associated with a set. It is an essential aspect of number sense as it enables individuals to comprehend and communicate the quantity of a group accurately.

The progression from counting to cardinality involves several stages of development. Initially, children engage in rote counting, reciting numbers in order without understanding their meaning or associating them with quantities. As they acquire one-to-one correspondence skills, they begin to match each object with a count word, realizing that counting should involve touching or pointing to objects in a set. This stage lays the groundwork for the concept of cardinality.

Through consistent practice, children develop an awareness that the last count word represents the total quantity. They start to understand that the final number spoken in a counting sequence indicates how many objects are in the set, demonstrating their comprehension of cardinality. With continued exposure to counting and meaningful interactions with quantities, children gradually progress from basic counting skills to a deeper understanding of cardinality, enabling them to accurately represent and describe the size of sets using cardinal numbers.

### Activities for Mastering Cardinality

There are many different activities that can be used to help children learn the concept of cardinality. Developing cardinality skills involves engaging in activities and using strategies that reinforce the understanding of the relationship between counting and the total quantity of a set. Here are some effective activities and strategies to foster cardinality skills:

1. Counting and Pointing: Encourage children to count objects while simultaneously pointing to each item. This activity helps them establish the connection between the count words and the objects being counted, reinforcing the concept that the last count word represents the total quantity.

2. Counting Collections: Provide opportunities for children to count various collections of objects, such as counting blocks, buttons, or toys. Encourage them to count aloud and associate the final count with the total quantity in the collection. For example, if they count a set of five blocks, emphasize that “five” represents the total number of blocks. You can get started with counting collections here. Grab it on TPT here.

3. Number Line Exploration: Introduce number lines and allow children to explore them. Start with simple number lines and gradually increase the range. Ask children to count along the number line and identify the last number, emphasizing that it represents the quantity up to that point.

4. Number Talks: Conduct regular number talks or discussions, focusing on the relationship between counting and cardinality. Ask open-ended questions like “How do you know how many there are?” or “How can you figure out the total quantity without counting each object?” These discussions help children reflect on and articulate their understanding of cardinality.

5. Visualization and Representation: Encourage children to represent quantities visually through drawings, diagrams, or manipulatives. For example, they can draw circles or use blocks to represent sets, and then count the items while associating the final count with the total number of representations.

6. Games and Songs: Incorporate games and songs that emphasize cardinality. For instance, play a game where children have to count objects and state the total quantity, reinforcing the connection between counting and cardinality. Singing counting songs that highlight the final count can also support their understanding.

## Hierarchical Inclusion

Hierarchical inclusion refers to the understanding that numbers increase or grow by one unit each time in a sequential order. It involves recognizing the incremental nature of numbers and their progression in a hierarchical manner. Individuals with a grasp of hierarchical inclusion comprehend that each successive number is obtained by adding one to the previous number, resulting in a steady and predictable pattern of numerical progression.

Activities and examples that demonstrate hierarchical inclusion can help individuals strengthen their understanding of this concept. Here are some strategies to foster the development of hierarchical inclusion skills:

1. Counting Sequences: Engage individuals in counting sequences, encouraging them to verbally recite numbers in the correct order. This activity reinforces the understanding that each subsequent number is obtained by adding one to the previous number.

2. Number Line Exploration: Introduce number lines and allow individuals to explore and interact with them. Encourage them to identify the pattern of numbers increasing by one unit as they move along the number line from left to right. This visual representation helps solidify the concept of hierarchical inclusion.

3. Skip Counting: Practice skip counting by multiples of one, such as counting by twos, fives, or tens. Emphasize that skip counting is an extension of hierarchical inclusion, as it involves increasing the numbers by a specific increment each time.

4. Number Patterns: Present individuals with number patterns and sequences where the pattern involves adding one to each successive number. Encourage them to identify the pattern and predict the next numbers in the sequence, reinforcing the concept of hierarchical inclusion.

5. Daily Life Examples: Connect hierarchical inclusion to real-life examples and situations. For instance, discussing how a calendar progresses day by day, or how the ages of family members increase by one each year.

## Part-Part Whole

Part-part-whole relationships refer to the understanding that numbers can be decomposed or broken down into smaller parts and combined to form new quantities. It involves recognizing the relationship between the individual parts and the whole, as well as understanding how they interact and contribute to the total value. The concept of breaking numbers into smaller parts and combining them to create a whole can be illustrated through various examples.

For instance, if we have the number 8, we can decompose it into the parts 5 and 3. These parts can then be added together to form the whole number 8. Similarly, the number 10 can be broken down into the parts 6 and 4, and when combined, they form the whole number 10.

Activities and models that facilitate the understanding of part-part-whole relationships include:

1. Number Bonds: Number bonds are visual representations that show the relationship between a whole number and its parts. They consist of a circle (or rectangle) representing the whole number, with lines or branches connecting the parts to the whole. For example, a number bond for 8 would have the number 8 in the center, with lines connecting 5 and 3 as the parts.

2. Ten Frames: Ten frames are rectangular grids with two rows of five boxes each. They provide a visual representation of numbers up to 10. By filling in the boxes with counters or drawing dots, children can see how numbers can be decomposed into smaller parts and combined to make a whole.

3. Manipulatives and Counting Objects: Engage children in hands-on activities using manipulatives like counting blocks, cubes, or coins. Encourage them to physically break apart numbers and combine them to explore the concept of part-part-whole relationships. For example, if they have five blocks and three blocks, they can combine them to form a group of eight blocks.

4. Story Problems and Real-Life Scenarios: Present story problems or real-life situations that require students to identify and use part-part-whole relationships. For instance, “Sara has 6 apples, and Mark has 2 apples. How many apples do they have together?” By applying their understanding of part-part-whole relationships, children can solve such problems by combining the parts to find the whole.

## Compensation

Compensation involves the ability to adjust one quantity to simplify calculations. By modifying one number while maintaining the overall value, strategic adjustments can make counting easier, faster and more accurate. In the context of compensation, when solving addition or subtraction problems, individuals may modify one or more numbers to simplify the computation. For example, in the addition problem 5 +7, a compensatory strategy would involve adjusting one of the numbers to a more convenient value while preserving the total sum. One might choose to increase 7 by 3, compensating for this change by subtracting 3 from the other number, resulting in 2 +10. This modification allows for easier mental calculation.

Similarly, in subtraction problems, compensation can be applied by adjusting the numbers involved. For instance, when subtracting 56 from 103, an individual might recognize that subtracting 50 instead of 56 would simplify the calculation. In this case, the subtraction problem can be reimagined as 103 – 50 + 6, where the 50 compensates for the adjustment made to the other number, and the additional 6 accounts for the difference between the adjusted number and the original value.

In the second post of our series on number sense, we delved into the key components that contribute to a deeper numerical understanding: cardinality, hierarchical inclusion, part-part-whole relationships, and compensation.

We discussed cardinality as the understanding of the total quantity of a set and how it ensures accurate counting. Hierarchical inclusion was explored as the knowledge that numbers grow by one unit each time, establishing a sequential and predictable pattern. Part-part-whole relationships were highlighted as the ability to break numbers into smaller parts and combine them to form a whole, fostering a comprehensive understanding of numerical composition. Lastly, compensation was explained as the skill to adjust numbers to facilitate mental calculations, simplifying computations while maintaining the overall value.

These components are crucial in developing a solid foundation of number sense. They enable individuals to grasp the fundamental concepts of numbers, their relationships, and their manipulations. By understanding cardinality, hierarchical inclusion, part-part-whole relationships, and compensation, individuals enhance their mathematical reasoning, problem-solving abilities, and overall fluency with numbers.

Stay tuned for the next post in our series, and continue your journey towards a strong foundation in number sense.

Want to grab what you need to get started? Get this math intervention set which is perfect for helping little learners master number sense.