Counting and Cardinality

One part of counting is being able to say numbers in their correct order. But it also involves assigning those numbers to individual items in a set and understanding that you only count each item once. (This skill is called one-to-one correspondence.) We often teach this concept by having kids touch each item in a set as they count. 

Cardinality is a way to describe how many things are in a specific set of items. For instance, imagine you ask a child “How many apples are in the basket?” They count the apples, and answer “five.” Even though they had to say the numbers one, two, three, four, five to count the apples, the cardinality (“How many?”) of the set is five. 

Number Recognition and Representation 

We’ve developed a variety of ways to represent numbers. Number words (one, two, three) and numerals (1, 2, 3) are two of the most common, along with methods like tally marks. It’s important for students to make the connection between a set of seven objects, the word “seven,” and the numeral “7,” understanding that they all represent the same thing. 

Order and Magnitude (Comparison)

One of the most basic number sense concepts is the idea of “bigger and smaller” or “greater than or lesser than.” When numbers are involved, this type of comparison is known as magnitude. Kids use number lines and other tools to help them compare numbers, understanding that 10 is greater than 3, or 5 is less than 7. 

As students learn specific numbers, they must be able to put them in order based on their magnitude. They also learn ordinal numbers like first, second, and third to describe the order of items in a set. 

Place Value and Pattern Recognition 

Learning place value also plays a role in developing number sense.

Place value is the value of each digit in a number based on its position in the number.

Understanding place value means children understand that the way a number is placed gives it a specific meaning. For instance, in the number 23, the numeral “2” means something different than it does in “42”.  

Patterns also play an important role in number sense, and place value is just one of many patterns used in math. Patterns in math can also show up as arithmetic or algebraic patterns, geometric patterns, and more. Pattern repetition is key to understanding arithmetic skills like multiplication and plays an even greater role in more advanced math. 

Estimating

Estimating in number sense means taking an educated guess about the number of something or answer to a problem. Kids use skills like rounding and estimating to reason out solutions and check to see if they make sense. 

Subitizing

Subitizing is the ability to recognize the number of items in a set without needing to count them.

Think of rolling a pair of dice. Once you’re familiar with the individual patterns on each side of a die, you don’t need to count the individual dots. You can quickly figure the total by subitizing what you see. Most young children learn to subitize up to about ten items, but it’s possible to subitize larger quantities than this. 

Number Relationships

This big category encompasses all sorts of relationships between numbers, including: 

• Greater than or less than 

• Equality and inequality 

• Number families (odds and evens, multiples of 3, etc.) 

• Number patterns (counting by fives, counting backwards by subtracting seven from each number, etc.) 

• Place value 

• Ratio and proportion 

Calculation

This more advanced part of number sense involves using arithmetic operations (addition, subtraction, multiplication, division) fluently. Students develop this sense throughout their elementary years. 

Mathematical Reasoning 

When people demonstrate mathematical reasoning, they can explain their thinking about a situation involving numbers. Here are some examples: 

• A preschool student says there are seven eggs in the refrigerator. When you ask them to explain how they know that, they count each egg one by one, demonstrating their counting and number recognition skills, as well as cardinality. 

• A fourth-grade student compares two fractions: 1/2 and 4/8 and declares that they are equal. To explain their thinking, they draw one circle divided into four parts, and another divided into eight parts. They color in the appropriate number of parts to represent each fraction and demonstrate that they’re equal. 

Mathematical reasoning is one of the skills we emphasize most when teaching kids math today.

Rote memorization of facts is helpful, but to be truly fluent in number sense, students must be able to explain not just what the correct answer is, but how they arrived at that answer.

There can be many ways to apply mathematical reasoning to one situation, so be sure to welcome all sorts of thinking from your child. 

Math Application 

Application involves taking concepts out of the classroom and using them in real life.

For young students, this might mean counting the number of plates needed for the dinner table or determining who has more milk in their glass. 

Older students apply arithmetic skills to the real world, figuring out how much three apples cost at the grocery store if each one costs $1.25, or determining how many more points they need to beat the high score in a video game.