Number sense is a critical concept in early learning that contributes to the building blocks of numeracy. The beauty of math is that it is an “international language”. You do not need a specific language to learn math. When students struggle with math later in school, it is often because early skills have not been fully developed. This can create gaps in conceptual understanding when tasks get more complex. There are several reasons why these gaps form. In this post, I will talk about the possible causes of these gaps. I will also suggest ways we can help early learners build a strong math foundation.

## Number Sense Development and Why Some Kids Struggle With Math

Do you have difficulty developing mathematical understanding in your students? Even though early numbers sense seems so logical, it can be difficult for students to learn. I have broken down the steps in this post to help you teach numbers sense with ease. Read on to get a solid understanding of how early math learning happens and how you can plan lessons in your classroom.

#### Kids can struggle with math because number sense is not strong. Here are some ways to encourage that learning:

- give more time and practice with counting
- encourage kids to see math around them
- assess as you go and move on to the next concept only when the one before it is mastered
- use manipulatives to build conceptual understanding first.
- reinforce the relationship between quantities, numbers, and other concepts
- talk about and scaffold all learning
- replace rote learning and speed drills with manipulatives and slow learning with rich tasks
- understand that sometimes fear or dislike of math has been modelled; model a positive math mentality

There are many more reasons why kids may struggle with math learning. The good news is that targeting learners early and with the right lessons and tasks will help kids start off on the right path to becoming numerate citizens! This will also help later primary learners that are struggling. You can create lessons for them to help fill those possible gaps they may be experiencing.

In an earlier post, I wrote about the importance of number sense. I targeted a few important concepts and steps to conceptual understanding. You can read that post here. This post has become one of my most popular posts so I have written this one to expand the information and lessons and introduce you to a resource that can help you teach early math skills. You will also be able to pick up some ideas here for early intervention.

Let’s take a look at where your learners start and some of the stages they move through as they are learning math. Remember, all kids will be at different stages of understanding. Small groups of children in your class will be similar in their learning and you can group them for instruction. The important thing is that you are aware of the stages and what they look like. When you assess your students using simple tasks, it will be easier to decide where to start instruction. This information is a summary of research and professional math resources by Kathy Richardson (2012), Alex Lawson (2015), Jo Boaler, Marian Small, NCTM.

## The Progression of Number Sense

### Number Sense and Counting

Kids need to have a load of practice counting. It is often surprising to see that counting is not part of the curriculum past kindergarten in some instances. Kids can be counting all the way through elementary! **What** they count and **how** they count will change but the learning established in this process serves to reinforce the concept of number and the relationship between numbers and quantities. (Even fractions and decimals can be counted too!)

In the book *Choral Counting and Counting Collections* Megan Franke et.al reinforce the need to be **teacher observers** as students count and then respond to the students’ needs. As you watch students count, you will see anything from counting without touching an object and really not understanding that each item represents a quantity, to kids sorting and skip counting. Your role is to notice these things and take notes. This formative assessment helps you to plan for instruction.

##### 1-1 Correspondence

Some math teacher/researchers suggest that we are bringing out the mathematical understanding that is already innately inside the child. Students will begin to understand that each object that they touch and count represents an amount. Give students lots of opportunities to count and tell a total in order to reinforce understanding.

Technically, 1-1 correspondence is the ability to match up an object to another object. Students have a set number of objects lined up and then they use counters to match each item counted. This helps to build a counting pattern and an understanding of quantity. We can use a number line for this.

##### Cardinality

Building number sense with a number line transfers to many other learning activities later, including measurement. Using a **number path** is a good tool to use with early learners or students needing extra reinforcement with cardinality.

Each increment is a little box so that students just learning can use this box as an “item” to count. Many students have a difficult time with a traditional number line because it is just a “space”. Having something tangible to count or refer to gives all kids access to the number line.

##### Subitizing

Students need to be able to recognize a quick image without counting. Being able to do this helps students to do mental math later. **Daily** subitizing practice is necessary for early learners to build this strong foundation. Students will be forced to count each item if they haven’t had enough practice subitizing..

You will notice the students that have not developed this ability because they are not able to recognize that 5 is five when rolling dice and will have to count each time. You can see how the ability to add will be affected. The child will struggle with counting-on because they have to count each set. Once one is counted, you may see the child recounting as they have “forgotten” the total. Students who can subitize can hold that number and count-on.

##### Magnitude

Magnitude determines whether an object (or pile of objects) is larger or smaller than objects of the same kind. The magnitude of a number is its distance from zero. (Vanderwalle, J, 2018) Looking at quantities, kids can usually determine who has the bigger pile of candy but they don’t always see this in other examples!

Is 10 large? That depends on what you are looking at. 10 grains of sand on the beach are not large. 10 people at a dinner table makes 10 look big. Take a look at this video to really understand the relevance of magnitude. Students learn to use the terms *more* or *less*, *greater* *than* or l*ess than* as they begin to understand this. Understanding magnitude sets the stage to understanding relationships between quantities and numbers.

### Number Sense and Number Relationships

##### Hierarchical Inclusion

The next number sense concept that develops is one that enables a child to see that there are smaller numbers within larger numbers. This is called hierarchical inclusion. Understanding that there are smaller numbers within larger ones helps students to be able to use numbers flexibly when beginning to do mental math.

Students are later able to see that 6 +5 can be added in your head by making a friendly number ==> I know that 6+4 is 10 and 1 more is 11. A child would not be able to do this if they did not have the understanding that 4 is nested within 5. Lots of practice counting and using number lines helps students to learn hierarchical inclusion.

Conservation

Conservation is the understanding that a quantity does not change if the arrangement changes. Piaget, A well-known psychologist, had a theory of conservation. (1954) Piaget thought that 5 and 6-year-old children were unable to understand that if you had two glasses with different shapes, and you poured water into one and then the other, the same amount of water was poured into both glasses. He determined that kids finally were able to understand this by age 7. It was later found that kids could do this much early than 7. You can read more about this here.

This same principle applies to math. If kids are given a set amount of objects to count and then these objects are moved around, they should be able to tell that the quantity hasn’t changed. If not, students need to return to such activities as counting and using number lines and rich tasks that get them thinking about numbers.

### Number Sense and Addition and Subtraction

#### Part-whole relationship

Numbers can be counted and then taken apart, They can be taken apart to create different sets. Part-whole relationships form the building blocks of addition and subtraction. When children understand how to decompose (break apart) numbers in different ways, they are ready to add and subtract.

Have you ever wondered why subtraction always seems so much harder than addition is for many kids? Part of this can be explained by the fact that they do not yet understand the part-whole relationship so they do not see the 4+5=9, therefore 9-5=4

Many of the common part-whole activities that students are doing break down numbers into boxes that are all equal such as number bonds. You may have seen these or used them in your classroom. It seems that maybe this may cause some confusion because if 9 is the whole and 4 and 5 make up the parts, 4 and 5 represent different quantities and therefore the boxes that represent them should be different sizes. That is why I will often use Cuisenaire rods to teach this concept.

#### Commutative and Associate Properties

The commutative property of addition (and multiplication) states that it doesn’t matter if you change the order of the addends, the sum is still the same. Associate properties refer to when you are adding (or multiplying) more than 2 numbers and you have the ability to group them any way you want and the sum will still be the same.

These properties help students understand that they can group numbers in flexible ways in order to make sense of the numbers or to calculate a sum in their heads. Imagine how useful this is when you are out shopping and you need to see if you have enough money to buy something.

#### Equivalence

When something is equal to or the same as something else means it is equivalent. The importance of this aspect of number sense is that understanding will help students see the relationships of numbers and quantities across the mathematical strands. For example, students will be able to see that 25+25+25+25=100, 4 quarters = $1.00 and when telling time we say “quarter past the hour” and there are 4 quarters (of 15 minutes) in an hour. There are also four 1/4’s in a whole.

### Number Sense and Place Value

#### Unitizing

Unitizing is understanding that you can count small, equal groups within a larger number or quantity. Understanding this will help students to begin sorting and grouping larger numbers to count. Larger quantities can be sorted into groups of 5 or 10 to count more efficiently. As students begin to work with larger numbers and group them into sets, they being to understand place value. Students have moved beyond just single digits and teen numbers to 2-digit numbers and beyond where each numeral has a value.

#### Place Value

To understand place value, students will need to know that a numeral’s value is dependent on the position it is in within a number. The building blocks of number sense lead to place value and understanding place value sets students up to understand concepts and relationships with larger numbers and parts of numbers less than one and below zero.

Lots of practice with manipulatives and making numbers, decomposing numbers and especially completing rich tasks and number talks are all great ways to develop place value understanding.

This overview of the early number sense learning progressions is a helpful tool for teachers as they work with children to develop a conceptual understanding of number sense. Your students can use these building blocks to acquire the math skills that will help them grow into competent, flexible thinkers and efficient problem solvers.

If you’d like a teaching resource to help you put these concepts into place in your early primary classroom, please see Building Blocks of Early Numeracy or click the picture below.

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