role of number sense in everyday problem solving and reasoning

How important is number sense? (Turns out, very)

Editor’s Note:

This is an updated version of a blog post published on September 14, 2017.

Number sense is an important construct that separates surface level understanding from subject mastery. Find out how it affects mathematical fluency and how your learners can develop it.

So, what is number sense and why is it important for learners to develop this skill?

The construct of number sense refers to a child’s fluidity and flexibility with numbers. It helps children understand what numbers mean, improving their performance of mental mathematics, and giving them the tools to look at maths in the outside world and make comparisons.

How to spot number sense

Children develop number sense gradually over time and at different rates through exploring numbers, visualising them in a variety of contexts, and relating them in ways that are not limited by formal written methods.

You can track their progress by checking for the following:

• An awareness of the relationship between number and quantity
• An understanding of number symbols, vocabulary, and meaning
• The ability to engage in systematic counting — including notions of cardinality and ordinality
• An awareness of magnitude and comparisons between different magnitudes
• An understanding of different representations of number
• Competence with simple mathematical operations
• An awareness of number patterns including recognising missing numbers

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Flexibility with number is key to number sense

Number sense is the ability to be flexible with numbers. It helps children understand both how our number system works, and how numbers relate to each other. Children who develop number sense have a range of mathematical strategies at their disposal. They know when to use them and how to adapt them to meet different situations.

What good number sense looks like

Good number sense helps children manipulate numbers to make calculations easier and gives them the confidence to be flexible in their approach to solving problems .

Children who develop number sense can assess how reasonable an answer is, and routinely estimate answers before calculating. They look for connections and readily spot patterns in numbers, which helps them predict future outcomes. They have several approaches to calculating and problem solving and can use and adapt these for new situations.

Children with good number sense enjoy playing with and exploring numbers and number relationships. As a result of these strategies, they can often find the most efficient solution to the problem.

What poor number sense looks like

Children with poor number sense tend to focus on procedure and will rely on methods that they feel secure with. They apply inefficient and immature strategies to calculations and fail to spot links and connections that could get them to the answer more quickly.

Often, children with poor number sense prefer to use pen and paper rather than working things out in their heads. They can be reluctant to estimate an answer before working it out and will generally accept whatever answer they get — without considering whether it is reasonable or not.

This was perfectly illustrated to me when a Year 5 child was trying to estimate the sum of two four-digit numbers before calculating the answer. She approached this task by calculating the answer and then giving an estimate. I asked her why she was doing it that way around and her reply was,

“It is much easier to find an estimate for the answer after you have worked out what the answer is.”

You have to admire her logic — if nothing else!

Children with poor number sense don’t enjoy maths and won’t spend time being creative with and exploring numbers. Ironically, they are doing a harder version of maths, that relies upon remembering and applying procedures, with little understanding of the underlying numerical concepts.

When does number sense develop?

Psychologists, Klein and Starkey (1988) found that we are born with a sense of number. They measured the focus time of babies looking at pictures of dots and discovered that when the number of dots changed the babies’ focus time changed.

These babies had appreciated a difference in numerical quantity.

Appreciating number quantity is a survival instinct. When our ancestors were out hunting and gathering they needed to be able to perceive danger. So, if one animal approached a couple of hunters, they saw this as an opportunity for a meal. However, if 10 animals approached them, they ran, or they became the meal!

We know that very young children can recognise the number of items in a group without having to count them. This is called subitising. Most people, but not all , can subitise up to six or seven items, when they are randomly arranged.

How learners can develop number sense

Number sense develops over time through opportunities to explore and play with numbers. Visualising numbers in different contexts, spotting relationships between numbers and predicting the patterns all contribute to good number sense.

Judy Hornigold

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Number sense

Here you will learn about number sense, including what it is and different ways to develop it.

Students will first learn about number sense from the beginning of their experiences with numbers and they continue to build on this knowledge throughout all experiences in math.

What is number sense?

Number sense is the ability to think flexibly and critically about numbers and their operations.

Someone with a strong number sense can…

• Solve or make reasonable estimates using mental math.
• Represent numbers or solve operations in more than one way.
• Make connections between solving strategies.

Number sense is not a “check the box” kind of skill. Each student lies somewhere on the spectrum of number sense, and with each mathematical experience there is an opportunity to build a deeper understanding or “sense of number.”

Number sense is embedded into any work with numbers and operations. This page will specifically cover whole numbers and the operations of addition and subtraction.

In young learners, building number sense around addition and subtraction may look like:

• Ask students to represent the number 8 in as many ways as they can. Then let students explain and compare their representations with others.

As students progress in their number sense and are ready to begin operating with numbers, activities may look like:

For example,

• Ask first grade students to compare the numbers 23 and 33 in more than one way. Then let students explain their comparisons with others. Then ask students how they could apply other students’ strategies to compare 33 and 43.
• Ask 2 nd grade students to subtract 83-59 mentally. Then ask students to share their thinking, while dictating their strategy on the board. Prompt students to make connections between the strategies they see being shared.
• Ask 3 rd grade students what number bond can help them solve 400-150. Then ask them how the same number bond could also help them solve 401-151 and 399-149. Encourage students to journal about their strategies or share them with other classmates.

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Common Core State Standards

How does this relate to 1 st grade math, 2 nd grade math and 3 rd grade math?

• Grade 1 – Numbers and Operations in Base 10 (1.NBT.B.2) Understand that the two digits of a two-digit number represent amounts of tens and ones.
• Grade 2 – Numbers and Operations in Base 10 (2.NBT.B.5) Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
• Grade 3 – Numbers and Operations in Base 10 (3.NBT.A.1) Use place value understanding to round whole numbers to the nearest 10 or 100.
• Grade 3 – Numbers and Operations in Base 10 (3.NBT.A.2) Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

How to develop number sense

In order to develop number sense:

Create mental strategies for thinking about numbers and operations.

Practice representing strategies in more than one way.

Share strategies and listen to the strategies of others, comparing and contrasting.

Number sense examples

Example 1: understanding tens and ones.

How many tens and ones are in the number 27?

• Create mental strategies for thinking about numbers and operations. 

Picture 27 hearts in your head for a moment… How do you see them?

Maybe you see a straight line of 27…

Maybe you see 27 in groups, like in five frames…

Maybe you see 27 in groups, like in ten frames…

There are many different ways to “see” 27 in your head.

There is not a “wrong” way to picture 27, but notice which ways make it easier to understand 27 as a quantity.

2 Practice representing strategies in more than one way.

Now, think about different ways to show 27 with base 10 blocks.

Maybe you show 27 ones…

Maybe you show 1 ten and 17 ones…

Maybe you show 2 tens and 7 ones…

There are many ways to group 27, but notice which ways make it easier to understand 27 as a quantity.

3 Share strategies and listen to the strategies of others, comparing and contrasting.

Looking at all the strategies above, what is similar? What is different? Did you think of 27 in another way?

Example 2: add within 100

Solve 33 + 48.

Combine 33 and 48 in your head… How did you add the numbers?

Maybe you started at 33 and counted up 48 \text{:}

34, \, 35, \, 36, \, 37, \, 38, \, 39, \, 40, \, 41, \, 42….

Maybe you grouped the tens and the ones:

30 + 40 = 70 , and 3 + 8 = 11 .

So now you can add 70 + 11.

Maybe you broke apart 33 to make a ten:

= (31 + 2) + 48

= 31 + (2 + 48)

Grouping the 2 + 48 together makes 50, so now you can solve 31 + 50.

There are many ways to mentally solve 33 + 48, but notice which makes the most sense to you.

One way to represent 33 + 48 is with base 10 blocks.

Another way is to use a hundreds chart.

You can also create hops on a number line:

There are many ways to represent 33 + 48, but notice which makes the most sense to you.

Looking at all the strategies above, what is similar? What is different? Would you solve 33 + 48 a different way?

Example 3: subtract within 100

Solve 84 - 68.

Take 68 away from 84 in your head… How did you subtract the numbers?

Maybe you started at 68 and counted up 84 \text{:}

69, \, 70, \, 71, \, 72, \, 73, \, 74, \, 75, \, 76, \, 77….

Maybe you started at 68 and counted up by groups of ones and tens:

From 68 to 70 is 2.

From 70 to 84 is 14 more.

So the distance from 68 to 84 is 16.

Maybe you broke apart 68 to subtract each place value:

84 - 60 = 24

24 - 8 = 16

Something to think about: There are many ways to mentally solve 84 - 68, but notice which makes the most sense to you.

One way to represent 84 - 68 is with base 10 blocks.

There are many ways to represent 84 - 68, but notice which makes the most sense to you.

Looking at all the strategies above, what is similar? What is different? Would you solve 84 - 68 a different way?

How to develop specific number sense strategies

In order to develop specific number sense strategies:

Decide if making \textbf{10} or using number bonds can help you solve.

Solve with your strategy and explain why it works.

Example 4: subtract within 20

Solve 16 - 7.

Number bonds help you use what you know about addition to solve subtraction.

Think about what number plus 7 is equal to 16 to complete the number bond.

Since 9 completes the number bond, it is the difference between 16 and 7.

16 - 7 = 9.

Example 5: add within 100

Solve 34 + 37.

It is not always easy to remember larger number bonds, but you can make 10.

Think about how you can regroup part of 34 with 37 to make a multiple of 10.

= (31 + 3) + 37

= 31 + (3 + 37) \quad *You can regroup 3 to go with 37.

So, 34 + 37 = 71.

Example 6: subtract within 1,000

Solve 300 - 150.

Sometimes you can use smaller number bonds, to help solve operations with larger numbers.

Think about what number plus 15 is equal to 30 to complete the number bond.

Since 15 completes the number bond, the difference between 30 and 15 is 15. Since 300 and 150 are 10 times larger, their difference is also 10 times larger.

So, 300 - 150 = 150.

Teaching tips for number sense

• Do your best to embed number sense activities into all math lessons and through all math centers, math skills and math problems. This does not require extensive extra planning – instead always look for ways for students to solve problems in multiple ways, explain their problem solving (written or orally) and critique the strategies of others.
• Many activities will naturally lend themselves to building number sense, particularly activities with real-life contexts, the use of hands-on manipulatives, and a classroom emphasis on problem-solving. While not always appropriate, worksheets that encourage students to solve in more than one way or analyze the thinking of others can also be useful.

Easy mistakes to make

• Thinking that children need to be a certain year old to develop number sense Even before students can formally use numerals or other number symbols, they can develop their sense of number. Some activities for pre-k students might include subitizing (recognizing the number of objects without counting), identifying more and less when comparing two groups of objects or learning to count using number words.
• Teaching algorithms too quickly Introducing algorithms before students have had time to explore a topic and grapple with their own ideas can eliminate a student’s motivation, creativity, and ownership and encourage memorization of rules over understanding. While there is no hard and fast rule as to how to progress a topic, be mindful in giving students time to develop ideas and remember that building foundational understanding takes time.
• Requiring students to use specific number sense strategies Unless directed by your state standards to do so, it is not necessary to insist that students use a certain strategy or ask students to memorize a strategy. While this is often done with good intentions, it is similar to asking students to memorize or use an algorithm too quickly. The best way to promote the use of number sense within the classroom is to use activities that allow students to solve in more than one way and consistently ask students to talk about their strategies. It is also helpful to promote a growth mindset and help students see the value in admitting to and learning from their mistakes.

Related arithmetic lessons

• Skip counting
• Inverse operations
• Two step word problems
• Money word problems
• Calculator skills

Practice number sense questions

1) Which choice is NOT equal to 36?

3 tens and 6 ones

26 ones and 1 ten

6  ones and 30 tens

2 tens and 16 ones

The model above shows 6 ones and 30 tens, which is NOT equal to 36.

It is equal to 306.

2) Solve 18-11.

There are many ways to solve 18-11. Two ways are with a model and by using a number bond.

Show the tens and ones in 18 with a model and then subtract 11 \text{:}

Use a number bond to solve.

Both ways show that 18-11 = 7.

3) Solve 46 + 19.

There are many ways to solve 46 + 19. Two ways are with a model and by making 10.

Show the tens and ones with a model and then combine them:

Regroup 46 to make a multiple of 10.

= (45 + 1) + 19

= 45 + (1 + 19)

4) Which strategy does NOT show 18 + 27?

“18 = 3 + 15, so I add 27 + 3 = 30 and then 30 + 15.”

“I started at 27 and counted up 18.”

This model shows 1 + 8 + 2 + 7 which is NOT the same as 18 + 27.

A correct model for 18 + 27 is shown below.

*Note: Other models can also be used to show 18 + 27, but all correct models show a total of 45.

5) Which strategy does NOT show 35 + 26?

“26 = 5 + 21, so I add 35 + 5 = 30 and then 30 + 21.”

“I started at 35 and counted up 26.”

There is a mistake in this explanation:

35 + 5 = 40, so the correct strategy is:

= 35 + (5 + 21)

= (35 + 5) + 21

*Note that this strategy, making 10, can also be used with different numbers.

6) Solve 600-200.

There are many ways to solve 600-200. Two ways are with a model and by using a number bond.

Show the hundreds in 600 with a model and then subtract 200 \text{:}

2 + 4 = 6, so 200 + 400 = 600, since the numbers in the bond are 100 times larger.

Both ways show that 600-200 = 400.

Number sense FAQs

Success in mathematics depends on a deep understanding of numbers. How students learn math can impact the level of this understanding. A focus on developing number sense in elementary school promotes flexible thinking around whole numbers, fractions and decimals. This type of knowledge helps students understand concepts more deeply and encourages creative approaches to problem solving. This is particularly important as math topics become more abstract in middle school and high school. Students who have greater number sense are often more successful at applying what they know to new and more complex mathematics.

For younger students, much of their development of number sense comes from activities that involve math facts. This includes (but is not limited to) opportunities to solve with models and drawings, solving real-world problems that involve basic math facts, solving math facts mentally and sharing and critiquing solving strategies with others. For older students, math facts can be a tool utilized to solve complex problems more efficiently.

The next lessons are

• Properties of equality
• Addition and subtraction
• Multiplication and division

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Number Sense in Math – Definition, Examples, Facts

What is number sense in math, essential elements in number sense, how to teach number sense, solved examples on number sense, practice problems on number sense, frequently asked questions on number sense.

Number sense refers to a person’s ability to work with numbers, understand their quantities, and use them in meaningful ways. It encompasses the understanding of concepts like comparing numbers, determining their values, and recognizing their significance in various contexts.

In early childhood, number sense starts developing naturally as children make simple comparisons, such as choosing a larger piece of cake or understanding when something is taken away. These experiences lay the foundation for understanding addition and subtraction.

As educators and caregivers, we play a crucial role in helping children build a strong number sense by connecting these early experiences to a deeper understanding of numbers. By guiding children to comprehend what numbers represent, how they relate to one another, and their relevance in everyday life, we can support their overall mathematical development.

So, let’s dive into the fascinating world of number sense and explore how it shapes a child’s mathematical understanding and growth.

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Definition of Number Sense

Number sense, as defined by Gersten and Chard (prominent researchers in the field of Education) refers to a child’s fluidity and adaptability with numbers. It involves understanding the meaning of numbers, performing mental arithmetic operations, and making comparisons to comprehend the world around them.

Numbers can be represented using symbols, such as 1, 2, 3, or written in words, like one, two, three, and so on. Developing number sense begins with recognizing these symbols and understanding the corresponding numerical values.

To gain a strong number sense, children need to grasp the components that contribute to their understanding of numbers. Let’s explore these components in detail and uncover the building blocks of number sense.

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Number sense refers to a wide range of math skills. Here are some important components of number sense:

• Counting and Cardinality : Understanding counting, number sequences, and assigning numbers to objects.
• Quantity and Magnitude : Grasping quantity concepts and comparing/estimating quantities.
• Number Relationships : Recognizing patterns, understanding place value, and identifying number families.
• Operations and Computation : Performing mental calculations, solving arithmetic problems, and mastering basic operations.
• Estimation and Approximation : Making reasonable guesses and rounding numbers to specific values.
• Spatial Sense and Number Patterns : Identifying patterns in numbers, shapes, and spatial relationships.
• Real-World Connections : Applying number sense to everyday situations like money, time, measurement, and data.

These components foster a solid foundation in number sense, enhancing a child’s mathematical understanding and problem-solving abilities.

Why Is Number Sense Important?

• Number sense forms a crucial foundation for future math mastery. Understanding number sense helps individuals manage personal finances, such as budgeting and understanding interest rates on loans.
• Strong number sense improves understanding and confidence in working with numbers. Having number sense enables individuals to make informed decisions when comparing prices at the grocery store or calculating discounts during shopping. 
• Children with strong number sense can manipulate numbers and employ flexible problem-solving strategies. Being able to estimate tips at a restaurant or mentally calculate sale prices during shopping showcases the practical application of number sense.
• Number sense allows individuals to recognize patterns and connections in numbers, enhancing problem-solving skills. Analyzing trends in data, such as tracking monthly expenses or understanding population growth, requires strong number sense.
• Poor number sense leads to dependence on fixed procedures and inefficient computation methods. Struggling to calculate sale prices or discounts accurately can be a consequence of weak number sense.
• Developing number sense equips individuals with critical thinking and logical reasoning skills applicable in various academic disciplines and everyday life.Interpreting statistical information in news reports or understanding measurements in recipes require number sense skills.

We now understand the value of number sense and the effect it can have on the young students in our classrooms. I firmly think that a student can benefit from having a solid grasp of numbers in every area of mathematics. Early focus on number sense builds a solid foundation for later grades when it comes to math and problem-solving that is more complex. Thus, it’s necessary to teach number sense at an early age. 

You can teach number sense in the following way:

1. Concentrating on base 

The foundation or base is very important while teaching any concept. Give it some time. Before moving on, make sure students are familiar with each idea. Having a strong understanding of place value and how the number system works from the very beginning will help students as they progress through their learning journey in math.

2. Clear Teaching

Each skill must be explicitly taught in a logical sequence. A critical error we can make is assuming a student understands a concept from years ago. The best opportunity to support students in making connections between concepts and ideas is during explicit teaching. It can be done in a variety of ways, such as whole-class teaching and modeling, facilitated groups, small-group work, or one-on-one interaction.

3. Practical Experience

Children learn by using concrete materials. They also enjoy hands-on activities and games – and enjoyment promotes the best kind of learning. It keeps them intrigued and focused and they learn better compared to just board writing and explanations.

4. Review and Revise!

Start with a week devoted to number skills at the beginning of EVERY SINGLE TERM. It aids in concept reinforcement, prepares students for the term, and also enables them to take up new abilities, concepts, and ideas that they might not have been prepared for in the past. Another method to keep reviewing your number concepts and engage your students’ minds is through your daily math warm up.

Facts on Number Sense

• Number sense refers to a child’s fluidity and flexibility with numbers.
• Children gradually and at varying rates acquire number sense through exploration, visualizing numbers in various contexts, and connecting them in ways that aren’t limited by formal written methods.
• Children who have a strong sense of numbers enjoy exploring and playing with numbers and their connections.
• A strong sense of numbers can be developed by seeing numbers in various situations, recognising patterns, and identifying relationships between numbers.
• Number sense is a key component to building a solid foundation for mathematical understanding.

In this article, we learned what number sense is and how it is necessary to have a good number sense. Children benefit from this by better comprehending the meaning of numbers, developing their mental math skills, and gaining the ability to make connections between numbers and arithmetic in the real world. Now let’s solve some examples and practice problems.

Example 1: Amy is comparing prices at a grocery store to find the best deal. What number sense skill will this activity help Amy? Give some examples of comparing numbers.

Solution:  

This activity will help Amy develop her skill of comparing and understanding numerical values, allowing them to make informed decisions based on prices.

We compare numbers using the symbols >, < , or =.

• 100 < 102
• 545 < 554
• 124  > 121

Example 2: Suppose that you are planning a party and deciding the number of snacks needed for the guests. What number sense skills do you need?

To plan a party and the snacks, you need to estimate the number of guests and the appropriate amount of snacks required. You need the math skill of estimation to make reasonable guesses about quantities and plan accordingly.

Example 3: How does solving a math puzzle involving number patterns help children?

Solving a math puzzle with number patterns involves recognizing and analyzing number patterns. It enhances their ability to identify relationships and predict future numbers.

Example 4: Give a real life example where you use number sense?

Measuring ingredients while following a recipe is a great day-to-day example that requires number sense. This activity will help a child develop their skill of measurement and understanding quantities, enabling them to accurately measure ingredients and follow the recipe’s instructions.

Number Sense in Math - Definition, Examples, Facts

Attend this quiz & Test your knowledge.

Which number sense skill involves understanding the concept of more or less?

An item costs $\$9.5$, but anna assumed it to be $\$10$ for finding the approximate price of 7 suchitems. which skill did she use, write the correct number for the following. 2000 + 200 + 20 + 4.

At what age do kids learn numbers and start counting ?

Children develop the ability to understand the actual concept of counting generally around the ages of two and four. By the age of four, children usually can count up to 10 and/or beyond.

What is subitising?

Subitising is instantly recognizing the number of objects in a small group, without counting. For example: knowing there are 5 coins here (without counting them).

How and when does number sense begin?

An intuitive sense of number begins at a very early age. Children as young as two years of age can confidently identify one, two or three objects before they can actually count with understanding (Gelman & Gellistel, 1978) But they understand the actual concept of counting generally around the ages of two and four.

Can 2 year olds recognize numbers?

By age 2, a child can count to two (“one, two”), and by 3, he can count to three, but if he can make it all the way up to 10, he’s probably reciting from rote memory. Kids this age don’t yet actually understand, and can’t identify, the quantities they’re naming.

Is number sense a skill?

Number sense refers to a group of skills. It can be learned or improved upon with time, practice, and determination.

What does number sense mean? What are examples of number sense fluency?

Understanding numbers and how they interact is known as having number sense. Children in the primary grades, for instance, learn how to separate and combine numbers when they experiment with and develop fluency with ideas like how to make 10 and how to break up 12 into 10 and 2.

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What is number sense?

By Bob Cunningham, EdM

At a glance

Number sense refers to a group of key math abilities.

It includes the ability to understand quantities and concepts like more and less .

Some people have stronger number sense than others.

Number sense is a group of skills that allow people to work with numbers. These skills are key to doing math — and many other tasks.

Number sense involves:

Understanding quantities

Grasping concepts like more and less , and larger and smaller

Understanding the order of numbers in a list: 1st, 2nd, 3rd, etc.

Understanding symbols that represent quantities ( 7 means the same thing as seven )

Making number comparisons (12 is greater than 10)

Recognizing relationships between single items and groups of items ( seven means one group of seven items)

Some people have stronger number sense than others. Kids and adults with poor number sense may struggle with basic math operations like addition and multiplication. They can also have trouble with everyday tasks and skills like measuring, handling money, and judging time.

Trouble with number sense often shows up early, as kids learn math. For some people, the difficulty lasts into adulthood. But with time and practice, these skills can improve.

Dive deeper

Examples of trouble with number sense.

Here are examples of what it looks like when people struggle with number sense.

Adding and subtracting. Imagine a pile of seven beads. Then take away two of them. People with poor number sense might not realize that:

The number of beads has shrunk

Subtracting the two beads means the group of seven is now a group of five 

Now imagine adding three beads to the pile. If someone struggles with number sense, they might not recognize that:

The group of beads has gotten larger

Adding three beads to the pile of seven makes it a pile of 10

Multiplying and dividing. When people need to combine items from several groups, they might go through the trouble of adding them. They may not grasp that it’s simpler to multiply them.

Likewise, they might not recognize that division is the simplest way to break up groups into their component parts.

Not grasping these concepts makes learning math and using it in everyday life a challenge. Learn more about math challenges in kids .

How schools can help

When kids struggle with math, schools often focus first on reteaching the specific math skills being taught in class. But this approach often doesn’t work for kids who struggle with number sense.

In that case, schools usually turn to an intervention process, where kids typically:

Work with “manipulatives” like blocks and rods to understand the relationships between amounts

Do exercises that involve matching number symbols to quantities

Get a lot of practice estimating

Learn strategies for checking an answer to see whether it’s reasonable

Talk with their teacher about the strategies they use to solve problems

Get help correcting mistakes they make along the way

For many kids with weak number sense, intervention is enough to catch up. But some kids need more support. They may need to be evaluated for special education to get the help they need.

Learn about intervention systems like RTI or MTSS .

How parents and caregivers can help

It takes time for kids to build number sense skills. But there are many ways to help. Here are some examples:

Practice counting and grouping objects. Then add to, subtract from, or divide the groups into smaller groups to practice operations. Combine groups to show multiplication. 

Work on estimating. Build questions into everyday conversations, using phrases like “about how many” or “about how much.”

Talk about relationships between quantities. Ask kids to use words like more and less to compare things.

Build in opportunities to talk about time. Ask kids to keep track of how long it takes to drive or walk to the grocery store. Compare that with how long it takes to get to school. Ask which takes longer.

Learn more strategies to help kids with math .

Explore related topics

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• Guiding Principles

Number Sense: What it is, why it’s important, and how it develops

Success in early math has lifelong implications

Researchers have found that children arrive at school on the first day of Kindergarten with wildly different levels of math knowledge. As Clements and Sarama (2011) point out, for example, “some six-year-olds have not acquired mathematical knowledge that other children acquire at three years of age” (p. 968).  These differences in children’s initial understanding of math have long-term implications for their success in school and in life, as “preschool children’s knowledge of mathematics predicts their later school success into elementary and even high school” (Clements & Sarama, 2011, p. 968).

There is broad consensus on what math knowledge matters

The good news is that researchers have identified a specific, well-defined set of concepts and skills that can make the difference between children’s success and failure in mathematics in the early years (Griffin, Case, & Siegler, 1996).  This knowledge is anchored in three key insights ( Griffin, 2004 ):

1)   Numbers represent quantities

2)   Spoken number names (“one,” “two,” etc.) and formal written symbols (1, 2, 3, etc.) are just different ways of referring to the same underlying quantities

3)   The quantities represented by the symbols have inherent relationships to each other (7 is more than 5, for example) and it is this property of the quantities that allows us to use the symbolic number representations to solve certain kinds of problems (putting objects in order, counting to determine how many objects are in a set, etc.)

This network of concepts and skills constitute what is called Number Sense .  Happily, these research-based insights are embodied in the most recent guidelines for teaching mathematics in both the Common Core State Standards (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010) and the Principles and Standards for School Mathematics published by the National Council of Teachers of Mathematics (2000).

We know how to support children’s Number Sense development

Case, Griffin, and Siegler (1994) found that children who have a well-developed Number Sense are able to succeed in early math (and beyond), while children who don’t are at much greater risk of falling increasingly further behind.  They also demonstrated that virtually any child could develop Number Sense when given access to a well-designed, focused intervention that provides opportunities to explore and discuss key concepts, make connections between different concepts, and develop their understanding at an appropriate pace and following an appropriate conceptual and developmental sequence (Griffin, 2004).

But Number Sense doesn’t develop by accident!

Case, Griffin, and Siegler (1994) demonstrated that every child can develop Number Sense – that’s the good news.  As I mentioned previously, however, it has been found that many children don’t .  Why not?  Because (as other research has established) Number Sense does not develop by accident or even as a side effect of engaging in informal activities such as puzzles or songs that appear on the surface to be related to math.  As Clements and Sarama (2011) caution:

[People] often believe they are “doing mathematics” [with young children] when they provide puzzles, blocks, and songs. Even when they teach mathematics, that content is usually not the main focus, but is embedded in a fine-motor or reading activity. Unfortunately, evidence suggests that such an approach is ineffective, owing to a lack of explicit attention to mathematical concepts and procedures along with a lack of intentionality to engage in mathematical practices (p. 968).

In short, although every child can learn Number Sense, not every child will unless we intentionally and systematically support that learning on an individual basis.

Native Numbers is an adaptive, research-based Number Sense curriculum

Native Numbers is an adaptive, mastery-based Number Sense curriculum grounded in research and delivered in the form of an iPad app . The adaptive features provide a highly individualized learning experience, enabling learners to move quickly through material they already understand and to spend as much time as they need to develop emerging concepts and skills.  The fact that it is mastery-based means that a learner has to demonstrate a minimum level of competency on each concept or skill before being exposed to more complex activities that depend upon that understanding. (You can see a video overview and demonstration of the app here .)

The curriculum contains twenty-five activities that are organized into five subskills.  These subskills are defined based on the Number Sense research mentioned previously as well as the relevant Common Core State Standards and NCTM standards .  Specifically, the subskills developed in Native Numbers are:

• Number Concepts : Connect number words and numerals to the quantities they represent
• Number Relations: Develop a sense of whole numbers and their relations, across different representations (“one”, 1, one turtle, etc.)
• Number Ordering : Understand relative position and magnitude of whole numbers
• Understand ordinal and cardinal numbers and their connections
• Count with understanding and recognize “how many” in sets of objects  

Native Numbers has been well-received by children, teachers, and parents

Initial feedback on Native Numbers from teachers, reviewers, and individual parents has been positive.  TeachersWithApps.com , for example, is a review site that tests apps in classrooms with children and bases their reviews on those data.  Here is what they have to say about Native Numbers:

TWA spent over a week with severely limited students to experience this app to its fullest. We were wildly impressed with the progression of how and when new concepts are introduced. We loved witnessing the eureka moments when you could see light bulbs going off as the kids played away. I doubt the developers knew just how addicting the app would be. It is fast paced and repetitive and loads of fun!

And Dr. Karen Mahon, a learning scientist and instructional designer who reviews educational apps,  had this to say :

The learning tasks are engaging, with consistent feedback for correct and incorrect answers, and the program automatically levels up as the learner makes correct responses.  This makes it more fun and interesting for kids….and MUCH more interesting for a reviewer like me!  But fun and interesting aside, adapting to the performance of the learner allows every learner to be successful, wherever a learner falls along the continuum of skills.

We are eager to learn from our learners, and the people who support them

At Native Brain, we are committed to developing tools that help all children succeed in school and in life.  We draw on the best available research on learning and teaching to empower and support not only the learners but also all of the people who share that purpose, including parents, teachers, administrators, policymakers, and researchers.  We constantly seek evidence of impact and data that can help us refine our offerings to increase their efficacy.  To that end, we invite your feedback, suggestions, and ongoing dialogue.  Please feel free to contact us at http://www.nativebrain.com/contact .

Douglas H. Clements and Julie Sarama (2011). Early Childhood Mathematics Intervention. Science , 333(6045), pp. 968-970.  Digital version available online via: https://portfolio.du.edu/portfolio/getportfoliofile?uid=216781

Sharon A. Griffin, Robbie Case, Robert S. Siegler (1996). RightStart: Providing the Central Conceptual Prerequisites for First Formal Learning of Arithmetic to Students at Risk for School Failure.  In K. McGilly (Ed.) Classroom Lessons: Integrating Cognitive Theory and Classroom Practice , pp. 25-50. Cambridge, MA: MIT Press.

Sharon Griffin (2004). Teaching Number Sense. Educational Leadership , 61(5), pp. 39-42.  Digital version available online via: http://www.ascd.org/ASCD/pdf/journals/ed_lead/el200402_griffin.pdf

National Council of Teachers of Mathematics (2000). Math Standards and Expectations: Number and Operations (Pre-K – 2 Expectations).  In Principles and Standards for School Mathematics . published by the National Council of Teachers of Mathematics.  Retrieved from: http://www.nctm.org/standards/content.aspx?id=7564   See also Number and Operations Standard for Grades Pre-K – 2, retrieved from: http://www.nctm.org/standards/content.aspx?id=26848

National Governors Association Center for Best Practices, Council of Chief State School Officers (2010).  Common Core State Standards for Mathematics (see esp. Kindergarten standards related to number pp. 6-11).   Washington, D.C.: National Governors Association Center for Best Practices, Council of Chief State School Officers.  Digital version available online at: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

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Number Sense and Operations

You often wonder  why some people are naturally good at mathematics while others struggle. The answer could lie in something called number sense. But what is number sense? And how do operations fit into this picture? Let’s unravel these mysteries one by one.

What is Number Sense?

Picture this:  You’re at a grocery store, buying your favorite snack. It says “$0.99” on the price tag, but when you’re at the checkout, the cashier asks you to pay more than 99 cents. Why? That’s where number sense comes into play.

Number  sense  is a person’s ability to intuitively understand, relate, and connect with numbers and numerals. Someone with good number sense can use numbers effectively, comprehend their significance, and apply them to real scenarios, like noticing that tax is added to your grocery bill.

Developing your number sense benefits you in school and the grocery store. It equips you with problem-solving skills , helps you make sense of data, and can even make a difference in understanding personal finances.

Why are operations necessary in mathematics?

The good ol’ addition, subtraction, multiplication, and division – these operations are the functional heart of mathematics . These basic operations are necessary for the world of numbers to seem quite complex and inaccessible, wouldn’t they?

Mathematical operations  help us count, compare, and discover the unknown in many mathematics problems. They are tools to combine numbers (addition), split numbers (subtraction), group numbers (multiplication), and distribute numbers (division). They help us resolve everyday issues, like splitting the check or calculating how long it will take to get somewhere, and ensure that mathematics theorems and laws are upheld.

However, remember that simply knowing the rules isn’t enough. Applying operations requires number sense to understand when and how to use them in different circumstances.

So, there you have it. Now you understand that building your number sense and getting comfortable with operations are crucial steps to becoming more comfortable with mathematics. You might even fall in love with numbers!

Here’s a quick summary of number sense and operations in table form:

Whole Numbers

Picture this – your early mathematics lessons. You’re learning digits, and then, suddenly, these digits unite to form numbers, a concept called whole numbers. Fascinating.

Let’s delve deeper into that.

Understanding the concept of whole numbers

Whole numbers are an essential but fundamental concept in math. These are the numbers that you see and use every day. They are digits from 0 and all numbers that follow, excluding fractions and decimals. Essentially, whole numbers are the backbone of our numeral system , laying a foundation for further math topics you’ll encounter in your studies and daily life.

Now, what happens when these whole numbers interact with each other? That’s where the magic (Addition and subtraction) happens!

Performing addition and subtraction with whole numbers

Addition  played a significant role in your early life and continues ever since, especially in practical situations. Imagine you have 3 apples, and your friend gave you another 2 apples; now you have 5. Here, “3+2=5” is an example of adding whole numbers.

Subtraction , on the other hand, represents the “taking away” operation. When you had those 5 apples, your sibling asked you for 2. After giving away 2, you are left with 3. The operation “5-2=3” demonstrates whole number subtraction.

And that’s the beauty of number sense and operations. Mastery of these, specifically in whole numbers addition and subtraction, lays the foundation for more complex mathematical operations such as multiplication and division.

Below is a brief recap:

Remember, all intricate numbers operations you encounter are merely extensions of these basic concepts.

Fractions are the first big step you take into advanced math. Grasping the concept of fractions is essential for your mathematical endeavors and plays a crucial role in your everyday life . Now, let’s dive into fractions a bit more.

Exploring the concept of fractions

Imagine you bought a pizza. Think of it cut into four identical pieces, each representing a fraction of the original pizza. Each piece is not a whole pie but a fraction, a part of the whole. So, a fraction is a value that represents the division of one quantity by another. In this context, the quantity is the pizza, and the parts are the slices. It is represented by two numbers separated by a slash (/) – the numerator and the denominator.

The top part of the fraction – the number above the slash, is called the  numerator – represents how many parts you have. The bottom part – the  denominator  (under the slash) – represents the total number of equal parts into which the whole is divided.

Adding and Subtracting Fractions

Let’s now discuss adding and subtracting fractions . Doing this might initially seem complicated, but once you master it, it’s a piece of cake. Here is a quick guide for you.

Firstly, you must ensure that the fractions you add or subtract have the same denominator. It is called having a common denominator. Suppose they don’t have the same denominator. In that case, you’ll have to find a common one, which is often the easiest by multiplying the two denominators.

Then, the process is straightforward when you have fractions with identical denominators, for example, 2/5 and 3/5. You add or subtract the numerators (depending on the operation in question) and write the result over the common denominator. So, for our example, 2/5 + 3/5 equals 5/5 or 1.

However, when subtracting, it’s crucial to subtract the small fraction from the larger one. So, subtracting 2/5 from 3/5 would give you 1/5.

In conclusion, you’ve now learned the basics of fractions- the concept of fractions and how you add and subtract them. Although they might seem challenging initially, fractions can be managed efficiently and effectively with consistent practice and understanding. Happy calculating!

Numbers make the world go round, and  decimals are vital  in shaping how you interact with digital, financial, and even physical phenomena in your daily life. Decimals, fractions in disguise, help you understand and measure the world more precisely. But before diving deep, getting a handle on the basics is essential.

Understanding decimals and place value

Decimal numbers  are based on 10, where decimal fractions represent parts of 10 or 10 raised to a hostile power. The position of each digit to the left or right of the decimal point dictates its value.

For example, in the number 123.456, the digits 1, 2, and 3 are whole numbers, the digit 1 is in the hundred’s place, the 2 is in the ten’s place, and the 3 is in the one’s place. The number 4, sitting right to the decimal point, is in the tenth place, 5 in hundredths, and 6 in thousandths.

Place value listening  lets you know that each place value, much like the weights on a barbell, has a different weightage and presence. And these aren’t just abstract notions. You use decimals to count currency, use dimensions, or prepare that perfect coffee recipe.

Performing operations with decimals

From addition and subtraction to multiplication and division , decimals can be easily tamed.

Let’s say you want to add 2.3 and 1.5. It’s very similar to adding whole numbers. Write the numbers one below the other, aligning the decimal points. When you’ve done that, add them as if they were whole numbers. The 3 and the 5 give you 8 (in the tenth place), and the 2 and 1 equal 3 (in the one’s place). Thus, the total would be 3.8.

Pro tip:  Remember to keep the decimal points lined up when performing operations with decimals.

As with any topic in mathematics, the key to mastering decimals is learning the basic rules and practicing as much as possible. You’ve got this!

As a recap, here’s a table summarizing the key points:

Introduction to Integers and their Properties

A fundamental idea in mathematics called an integer comprises both positive and negative whole integers and zero. Understanding integers is essential because they are used in everyday situations, such as temperature changes, comparing numbers, and solving real-life problems.

Integers have some unique properties that make them different from other numbers. Some critical properties of integers include:

• Ordering:  Integers can be arranged from least to most lavish, or vice versa, based on their numerical values. This property allows us to compare and rank integers.
• Significance of the Sign:  The sign of an integer determines its positive or negative nature. Positive integers are more significant than zero, while negative integers are less than zero. The sign of an integer indicates its direction on a number line.
• Additive Inverse:  Each integer has an additive inverse, which means that when you add an integer to its additive inverse, the result is zero. For example, the additive inverse of 5 is -5, and the sum of 5 and -5 is zero.

Adding and Subtracting Integers

Adding and subtracting integers involves combining or taking away integers to find a final value. Here are some rules to keep in mind:

• Adding Integers:  When adding integers with the same sign, you add their absolute values and use the expected sign. For example, adding 3 and 5 would give you 8 because they have the same sign (both positive). When adding integers with different signs, subtract the smaller absolute value from the larger one and use the sign of the integer with the more considerable absolute value. For example, adding 3 and -5 would give you -2.
• Subtracting Integers:  Subtracting integers can be considered as adding the additive inverse. To subtract an integer, change the subtraction sign to addition and the sign of the integer you are subtracting. Apply the rules of adding integers to find the final value.

Remember to carefully consider the signs and apply the rules correctly when adding or subtracting integers.

A solid understanding of integers and how to operate with them is crucial in various areas of mathematics. Developing number sense and mastering operations with integers will enhance your problem-solving skills and mathematical proficiency. So, practice regularly and explore real-life scenarios to strengthen your understanding of integers and their operations.

In conclusion, developing a solid understanding of number sense and operations is crucial for navigating everyday life and achieving success in various fields. By mastering these skills, you can confidently handle mathematical problems, make informed decisions, and effectively communicate numerical information. Remember, number sense is not just about manipulating numbers; it involves developing a deep understanding of their relationships and using this knowledge to solve real-life problems.

Summary of number sense and operations concepts

Number sense encompasses various skills and concepts related to numbers and their operations. It involves recognizing and understanding numerical patterns, estimating quantities, using mental math strategies, and developing a sense of magnitude. Conversely, operations refer to the mathematical processes of addition, subtraction, multiplication, and division. Understanding the properties and relationships between numbers is essential for performing these operations correctly and efficiently.

Importance of mastering these skills in everyday life

Mastering number sense and operations is crucial for various aspects of everyday life. Here are some reasons why these skills are essential:

• Financial Literacy:  Understanding numbers and operations is essential for managing personal finances, budgeting, and making informed financial decisions. From calculating expenses and understanding interest rates to saving and investing, numerical skills significantly ensure financial stability.
• Career Advancement:  Numeracy is increasingly vital in today’s job market. Many professions require employees to have strong math skills , whether analyzing data, managing budgets, or problem-solving. By mastering number sense and operations, you can enhance your career prospects and excel in fields that rely on quantitative analysis.
• Everyday Problem-Solving:  From calculating measurements and cooking recipes to understanding discounts and sales, number sense and operations are essential for solving everyday problems. These skills enable you to make informed decisions and understand numerical information in various contexts.
• Critical Thinking:  Developing number sense and operations skills promotes critical thinking and logical reasoning. It enables you to analyze information, identify patterns, and make connections between mathematical concepts. These skills are valuable in mathematics and other areas that require analytical thinking and problem-solving.

By understanding the concepts of number sense and operations and mastering the associated skills, you can enhance your mathematical abilities and apply them effectively in real-life situations.

Remember, practice and perseverance are vital to improving your number sense and operations skills. Keep exploring mathematical concepts, seeking opportunities to apply them in your daily life, and never be afraid to ask for help or seek additional resources when needed. With dedication and effort, you can become a confident and skilled problem solver in numbers.

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Big Ideas of Number Sense

Number Sense

Number sense is the ability to understand the quantity of a set and the name associated with that quantity. Number is an abstract concept, and young children tend to think in concrete terms. Thanks to recent research, we now know that infants are sensitive to quantity and can make good quantity comparisons (more than, less than). The transition from this innate, informal number knowledge to a conventional understanding of number sense is a major cognitive development that takes place gradually. It gets to the heart of a complex question: What is number sense? And why is number sense important?

Copyright: Erikson Institute’s Early Math Collaborative. Reprinted from Big Ideas of Early Mathematics: What Teachers of Young Children Need to Know (2014), Pearson Education.

● The quantity of a small collection can be intuitively perceived without counting. ● Quantity is an attribute of a set of objects and we use numbers to name specific quantities. ● Numbers are used in many ways, some more mathematical than others.

Interested in Book Suggestions?

Browse our favorite children’s books that explore Number Sense.

A New Focus for Familiar Card Games

Here you can download cards and simple-to-learn game ideas to help young children build their understanding of early math concepts such as cardinality and composing and comparing numbers.

Dot Card Transition Activity

Transition time is a great time for mathematizing a daily routine. This dot card transition is a relatively simple routine that builds number sense in a concrete way.

Explore Estimation While Enjoying a Whopper of a Tale!

This book is a delightful way to start a discussion about estimation in the early grades. Is it reasonable that Hugh Thomas caught a million fish?

Attendance Routine to Build Number Sense

This Spanish-language example of a rekenrek attendance routine demonstrates how one teacher can mathematize an everyday activity.

Matching Quantity with Child 3

A child produces a small set of counters to match a shown quantity. Notice how she produces an equivalent set without copying the arrangement she was shown.

Browse all that match this in our Idea Library.

Big Ideas of Early Mathematics:

What teachers of young children need to know.

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Demystifying Math: What Is Number Sense?

Learn how to understand number sense, and why building this foundation is so important in your child's math development..

What is number sense? As a teacher, I’ve been asked this question over and over again by many parents. To answer, I talk about the importance of it and why your children need to build strong number sense, but many parents don’t feel comfortable with the topic. It's not a term they're familiar with or one used when they learned math. Plain and simple, number sense is a person’s ability to understand, relate, and connect numbers. 

Children with strong number sense think flexibly and fluently about numbers. They can: 

• Visualize and talk comfortably about numbers. Number bonds  are one tool to help them see the connections between numbers.
• Take numbers apart and put them back together in different ways — e.g. breaking the number five down several times (such as: 5+0=5; 4+1=5; 3+2=5; 2+3=5; 1+4=5; 0+5=5 and so on), which helps your children learn all the ways to make five.
• Compute mentally — solving problems in their heads instead of using a paper and pencil.
• Relate numbers to real-life problems by connecting them to their everyday world. For instance, asking how many apples they've picked at a farm. ("Andy picked 5 apples. Amanda picked 2. How many apples did they pick in all?") 

Number sense is so important for your young math learners because it promotes confidence and encourages flexible thinking. It allows your children to create a relationship with numbers and be able to talk about math as a language. I tell my young students, numbers are just like letters. Each letter has a sound and when you put them together they make words. Well, every digit has a value and when you put those digits together they make numbers!

Here are some ideas for promoting number sense in a first grader: 

• Estimating to bring math into your child’s everyday world.  Estimate the number of steps it takes to get from the car to the house or how many minutes you have to wait in line at the grocery store.
• Model numbers in different ways.  Seeing numbers in different contexts really helps your children connect with numbers. For example, looking at numbers in a deck of cards or identifying numbers on dice or dominoes without counting the dots.
• Visualize ways to see numbers.  Every day I ask my students to visualize a number and tell me what they see. Your child will see numbers in different ways. Celebrate all the different ways and encourage her to think outside of the box. An eight can look like a snake or a 10 can be thought of as a baseball and bat.
• Think about math with an open mind.  Instead of asking what is 6+4, ask, "What are some ways to make 10?" This allows for more flexible thinking and builds confidence with knowing more than one answer. Or, you can also ask “Can you make eight with three different numbers?” or “What is 10 more than 22?”
• Solve problems mentally.   Instead of relying on memorization, encourage your child to use mental math (calculating problems in his head). So, if you know 6+6=12, then you know 6+7=13. He can use his double fact (6+6) to help find a harder fact (6+7) and build on concepts he already knows to think about problems. 

Strong number sense helps build a foundation for mathematical understanding. Focusing on number sense in the younger grades helps build the foundation necessary to compute and solve more complex problems in older grades. Building a love for math in your children begins with building an understanding of numbers.

Featured Photo Credit: >© stray_cat /iStockphoto

Grasping What Is Number Sense: The Key to Early Numeric Fluency

• September 12, 2024

Introducing young minds to the fundamentals of mathematics goes beyond mere counting; it embeds a deeper understanding of what is number sense, a critical component in early education. Grasping what is number sense sets the foundation for numerical fluency, influencing a child’s ability to think logically and solve problems effectively. This concept, though seemingly straightforward, encompasses a wide array of skills, from recognizing and comparing numbers to understanding their value and the relationships between them. Our journey into the realm of early numeric fluency begins with uncovering the essence of number sense and its pivotal role in cognitive development.

What is Number Sense?

Number sense, at its core, is a child’s intuition about numbers and their operations. It’s much like a mathematical sixth sense, naturally allowing young learners to grasp concepts such as magnitude, relationships, and numeracy patterns. In early education, nurturing this sense is key to building a solid mathematical foundation , making it a crucial aspect of our teaching approach.

How a Child Demonstrates Strong Number Sense in Math?

Have you ever wondered how you can tell if a child really gets numbers? It’s not just about counting on fingers; it’s seeing them use and understand numbers in ways that surprise you. That’s the magic of a strong number sense in action.

• Seamless Counting: They easily count forwards and backward, showing flexible number sequence understanding.
• Number Relationships: They grasp how numbers relate, like seeing 5 is more than 3 without counting.
• Estimation Skills: Quickly guess quantities in small groups without counting each item.
• Solving Problems Mentally: Solve simple math problems in their head, showing deeper understanding.
• Using Numbers in Daily Life: Naturally, use numbers in activities like sharing snacks or pretending to cook.
• Creating Patterns: Recognize and create patterns, understanding repetition and sequences.
• Confident Questioning: Ask questions about numbers, showing curiosity and eagerness to understand.

Signs a Child Has a Poor Number Sense in Math

Then, how about those moments when you notice your child struggling to grasp basic math concepts? Recognizing the early signs of poor number sense is important to provide the support they need. 

• Reliance on Finger Counting: Continues to count on fingers for basic addition or subtraction, well beyond the expected age.
• Trouble with Number Patterns: Has difficulty recognizing simple patterns or sequences in numbers.
• Difficulty Understanding Value: Struggles to grasp how much a number is worth and its comparison to others.
• Avoids Number-Related Activities: Shows reluctance or frustration with games or activities that involve numbers.
• Inconsistent Counting: Counts inconsistently, missing numbers, or losing track frequently.

Why are Number Sense Skills so Vital?

Number sense lays the foundation for all future math learning—it gives kids the basics they need to tackle more complex problems. Understanding numbers deeply helps children understand how they interact, building confidence and curiosity about math.

This isn’t just useful in school; practical life skills and problem-solving often rely on number sense. It nurtures independent thinking and real-world application, making numbers a daily tool, not a challenge.

Components of Number Sense

Understanding the building blocks of number sense is key to helping children excel in math. Let’s explore the essential components that form the foundation of strong mathematical skills.

1. Counting Skills

Understanding the sequence of numbers involves recognizing the order in which numbers follow, like knowing “1, 2, 3, 4” without skipping. Concepts of one-to-one correspondence and cardinality mean matching objects to numbers and recognizing that the last number counted shows the total quantity. These skills build a solid foundation for more complex math concepts.

2. Number Recognition

Identifying written numerals means spotting and naming numbers, like recognizing “5” on a page. Recognizing quantities without counting or subitizing is about instantly seeing how many items are in a small group. These skills help kids swiftly understand and work with numbers in everyday situations.

3. Number Relationships

Understanding more than, less than, and equal to helps kids compare quantities, like knowing seven is more than 5. Basic concepts of addition and subtraction involve recognizing how numbers combine or separate. These foundational skills are essential for solving everyday math problems and building further knowledge.

4. Estimation and Magnitude

Grasping the size of numbers involves understanding their magnitude—knowing that 100 is significantly larger than 10. Making reasonable guesses and approximations means using intuitive judgment to estimate quantities or distances. These skills enhance everyday decision-making and problem-solving by allowing quick, reasonable assessments of numerical information.

Development of Number Sense in Early Childhood

Imagine number sense as a little seed planted in early childhood, growing stronger with care and play. Let’s see how this critical skill blossoms in youngsters’ curious minds.

Stages of Development

• Infants and toddlers (0-2 years) – start to explore numbers through play, like clapping and stacking toys. They begin to recognize small quantities and show early signs of counting routines. These experiences lay the groundwork for future number understanding in a fun, engaging manner.
• Preschoolers (3-5 years) – develop more structured counting skills and start recognizing written numbers. They begin understanding concepts more and less and can often count objects accurately. These foundational skills are nurtured through interactive games and activities, promoting their number sense growth.
• Early elementary children (6-8 years) expand their number sense by mastering addition and subtraction. They start understanding place value, which helps them handle larger numbers. Classroom activities and real-world applications, like counting money, support their growing mathematical thinking and problem-solving skills.

Factors Influencing Development

• Genetic predisposition – plays a role in how children innately grasp numerical concepts. Some children may naturally exhibit stronger number sense due to inherited cognitive strengths. Understanding these differences can help tailor learning experiences to each child’s unique abilities.
• Environmental influences and parental engagement – are crucial in shaping number sense. Engaging in number-related activities, playing counting games, and integrating numbers into daily routines can stimulate early numerical understanding. A supportive and numeracy-rich environment significantly enhances development.
• Educational materials and methodologies – like manipulatives and interactive games, foster number sense by providing hands-on learning experiences. Effective teaching strategies emphasizing understanding over memorization help children grasp foundational math concepts, enabling them to apply these skills in varied contexts.

6 Activities and Games to Foster Strong Number Sense in Math

Activity 1: quantity quick look.

This activity helps preschoolers quickly recognize quantities without counting, enhancing their understanding of quantities.

• Place a few items (e.g., blocks, buttons) in front of your child in different configurations (e.g., in a line, in clusters).
• Ask your child to quickly tell you how many items there are without counting them.
• Engage in a discussion about how they knew the quantity so quickly.
• Gradually increase the number of items as your child gets better at recognizing quantities.

Activity 2: Number Manipulation Fun

This game encourages flexibility with numbers and helps children understand how numbers can be manipulated in their heads.

• Give your child a small set of numbers (e.g., 1-10) and some objects like blocks or beads.
• Ask them to show you different ways to make the number 5 using these objects (e.g., 3+2, 4+1).
• Encourage them to find multiple combinations to make the same number.
• Vary the target number based on their skill level and have them explain their thinking process.

Activity 3: Pattern Play

This activity allows children to identify and use patterns in numbers, which is a key aspect of strong number sense.

• Create a simple pattern using objects (e.g., red, blue, red, blue).
• Ask your child to continue the pattern.
• Once they grasp basic patterns, introduce number patterns (e.g., 2, 4, 6, 8).
• Ask them to identify the pattern and predict the next number.
• Encourage them to create their patterns using objects or numbers.

Activity 4: Quick Math Challenge

Strengthen computational skills by encouraging preschoolers to perform basic arithmetic operations quickly and accurately.

• Use flashcards with simple addition or subtraction problems (e.g., 2 + 3, 5 – 1).
• Show a flashcard to your child and ask them to solve it as quickly as possible.
• Discuss different strategies they could use to find the answer (e.g., counting on fingers, visualizing objects).
• Gradually increase the difficulty level based on their progress.

Activity 5: Estimation Station

This game improves estimation skills by encouraging children to make reasonable guesses about quantities.

• Gather a variety of small objects (e.g., beads, stones) and place them in a jar or bowl.
• Ask your child to estimate how many objects are in the container by looking.
• Write down their estimate.
• Count the objects together to see how close the estimate was to the actual number.
• Repeat with different quantities and containers to practice and improve estimation skills.

Activity 6: Exploring Number Relationships

Help your child recognize relationships between numbers and understand concepts like fractions and equivalencies.

• Use visual aids like fraction circles or number cards.
• Show your child how ½ is the same as 0.5 or 50%.
• Use physical objects (e.g., dividing a pizza or a set of blocks) to demonstrate these relationships.
• Ask questions like, “If we split this pizza in half, how many pieces do we have?” or “How many blocks are in half of this pile?”
• Encourage your child to find and explain other number relationships they notice.

Assessing Number Sense

Peeking into a child’s numerical world is like a treasure hunt. Let’s explore how to uncover the gems of their number sense understanding.

Formal Methods

• Standardized Tests : Tailored assessments to benchmark skills against age-appropriate standards.
• Curriculum-Based Measures: Specific tasks aligned with what is being taught to determine mastery of concepts.
• Diagnostic Tests: Used to identify specific areas of need or gaps in understanding.

Informal Methods

• Observations: Watching children during play or structured activities to note how they engage with numbers and solve problems.
• Games and Activities: Interactive approaches that reveal how children apply number concepts in different scenarios.
• Discussions and Questions: Casual conversations that encourage children to explain their thinking and reasoning with numbers.

Common Challenges and Solutions

Cognitive limitations.

Addressing developmental delays involves creating customized learning strategies for each child’s needs. Teachers can implement personalized activities that foster growth and development by assessing their strengths and weaknesses, ensuring every child progresses at their own pace. Regular monitoring and adjustments to these approaches are vital for continual improvement.

Instructional Challenges

To ensure engagement and motivation, use a variety of instructional techniques that cater to different learning styles. Incorporating visual, auditory, and hands-on activities keeps lessons dynamic and appealing, helping students stay interested and make meaningful progress. Consistent feedback and encouragement also play a key role.

Home Environment Challenges

Encouraging parental involvement enhances a child’s learning experience. Providing resources such as activity guides, informative workshops, and regular communication helps parents and caregivers support their child’s education at home. Sharing easy-to-implement strategies ensures that learning continues beyond the classroom, fostering a collaborative and supportive environment.

Final Thoughts

Mastering number sense is vital for developing numeric fluency and laying the groundwork for a child’s mathematical journey. We can help children strengthen their numerical skills through ongoing observation and tailored support. Ensuring continuous support and development at school and home is crucial for their academic success.

When you need a reliable partner to support your child’s learning, Baby Steps is here for you. Our experts and dedicated teachers provide personalized, effective strategies tailored to individual needs. Contact us at 347-960-8334 for Forest Hills and 347-644-5528 for Rego Park

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How Number Talks Assist Students in Becoming Doers of Mathematics

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Number talks are discussions where teachers encourage their students to mentally solve mathematics problems and then come together as a class to share their mathematical reasoning. As students share, listen, and discuss their solution strategies, they begin to make connections between how procedures are the same, different, and/or more efficient. In this chapter, I explore how a teacher leverages number talks to support students in becoming doers of mathematics. Findings from this study reveal how the teacher supported students to (a) develop agency, (b) distribute authority, and (c) share mathematical reasoning. Further, it was found that mental computation played an important role since it supported students to discover ingenious, effective, and efficient ways of solving mathematical problems.

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Woods, D.M. (2023). How Number Talks Assist Students in Becoming Doers of Mathematics. In: Robinson, K.M., Kotsopoulos, D., Dubé, A.K. (eds) Mathematical Teaching and Learning. Springer, Cham. https://doi.org/10.1007/978-3-031-31848-1_8

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The Importance of Number Sense to Mathematics Achievement in First and Third Grades

Children's symbolic number sense was examined at the beginning of first grade with a short screen of competencies related to counting, number knowledge, and arithmetic operations. Conventional mathematics achievement was then assessed at the end of both first and third grades. Controlling for age and cognitive abilities (i.e., language, spatial, and memory), number sense made a unique and meaningful contribution to the variance in mathematics achievement at both first and third grades. Furthermore, the strength of the predictions did not weaken over time. Number sense was most strongly related to the ability to solve applied mathematics problems presented in various contexts. The number sense screen taps important intermediate skills that should be considered in the development of early mathematics assessments and interventions.

Mathematics achievement is a key educational concern in the United States. Competence in mathematics is critical to the workforce in STEM (science, technology, engineering, and mathematics) disciplines and to international leadership. Although there is an upward trend in average mathematics test scores in elementary and middle school (e.g., National Assessment of Educational Progress, 2008 ), U.S. students still lag behind their counterparts in many other industrialized nations ( National Mathematics Advisory Panel, 2008 ). Moreover, within the school population, there are large individual differences in mathematics achievement associated with socioeconomic status ( Lubienski, 2000 ), home experiences ( Blevins-Knabe & Musun-Miller, 1996 ), culture and language ( Miller & Stigler, 1987 ; Miura, 1987 ), and learning abilities ( Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007 ).

Although considerable attention has been devoted to mathematics achievement in elementary and secondary school, foundations for mathematics learning are established much earlier ( Clements & Sarama, 2007 ). There is good reason to believe that the screening of mathematics achievement can be used to provide early predictors and support for interventions, before children fall seriously behind in school ( Gersten, Jordan, & Flojo, 2005 ). In the area of reading, which has been studied more thoroughly than mathematics, reliable early screening measures with strong predictive validity have led to the development of effective support programs in kindergarten and first grade ( Schatschneider, Carlson, Francis, Foorman, & Fletcher, 2002 ). Intermediate measures most closely allied with actual reading (e.g., knowledge of letter sounds) are more predictive of reading achievement than are more general phonological or perceptual measures ( Schatschneider, Fletcher, Francis, Carlson, & Foorman, 2004 ). Similar to that for reading, the present study is concerned with screening key number competencies children acquire before first grade, which can serve as a ladder for learning mathematics in school.

Number Sense

Number sense that is relevant to learning mathematics takes root early in life, well before children enter school. Primary, or preverbal, number sense appears to develop without or with little verbal input or instruction, and it is present in infancy ( Dehaene, 1997 ). The development of number sense begins with precise representation of small numbers, whereas large quantities are initially captured through approximate representations ( Feigenson & Carey, 2003 ).

It has been argued that these primary abilities are the basis for developing secondary symbolic -- or verbal -- number competencies ( Feigenson et al., 2004 ). When children learn the verbal count list and understand cardinal values for numbers, they learn to represent larger numbers exactly and see that each number has a unique successor ( Le Corre & Carey, 2007 ; Sarnecka & Carey, 2008 ). Symbolic number sense is highly dependent on the input a child receives ( Clements & Sarama, 2007 ) and thus is secondary to primary preverbal number sense but intermediate to the conventional mathematics that is taught in school. Key areas include counting, number knowledge and arithmetic operations. Although the relation between nonverbal and verbal number competencies is not always clear, there is general agreement that early verbal number competencies are necessary for extending knowledge with small numbers to knowledge with larger numbers and for learning school-based mathematics.

Children first map number words onto small sets (i.e., sets of 3 or less) through subitization or instant recognition of a quantity (e.g., Le Corre & Carey, 2006 ). For larger sets, counting usually is needed to determine the cardinal value. During preschool and kindergarten, most children learn to enumerate sets in a stable order (e.g., 1, 2, 3, 4, 5) using one-to-one correspondence and come to realize that the last number indicates the number of items in a set ( Gelman & Gallistel, 1978 ). Comprehension of these “how to count” principles allows children to enumerate any object or entity (e.g., heterogeneous or homogeneous) in any direction (e.g., left to right or right to left and so forth).

Counting facility extends numerical understanding in important ways ( Baroody, 1987 ). It helps children see that numbers later in the count list have larger quantities than earlier ones (e.g., n; n + 1; (n + 1) +1, etc.) ( Sarnecka & Carey, 2008 ) and manipulate sets through addition and subtraction, with and without object representations ( Levine, Jordan, & Huttenlocher, 1992 ). Learning difficulties in mathematics have been traced to weaknesses in intermediate number competencies related to counting, number comparisons, and set transformations ( Geary, 1990 ; Mazzocco & Thompson 2005 ). These number abilities are highly sensitive to socioeconomic status, suggesting the importance of early input and instruction (Jordan, Huttenlocher, & Levine, 1992). For example, low-income kindergartners perform worse than their middle-income counterparts on oral number combinations and story problems involving addition and subtraction ( Jordan, Levine, & Huttenlocher, 1994 ); they also use counting strategies less adaptively (e.g., they do not use their fingers to count on from addends; Jordan, Kaplan, Ramineni, & Locuniak, 2008 ).

Measuring number sense

Key number competencies can be reliably measured in kindergarten and early elementary school. Jordan and colleagues ( Jordan, Kaplan, Olah, & Locuniak, 2006 ; Jordan, Kaplan, Locuniak, & Ramineni, 2007 ) developed a “core” number-sense battery for screening children in kindergarten and first grade. To assess counting , children are asked recite the count sequence, to count sets of different sizes, to recognize correct, incorrect (e.g., counting the first object twice), and correct but unusual counts (e.g., counting from right to left). To assess number knowledge , they are asked to make numerical magnitude judgments (e.g., indicating which of 2 numbers is bigger or smaller, what number comes one and two after another number). Children also are asked to perform simple addition and subtraction calculations presented in three contexts. On nonverbal problems , children are shown a set of chips, which is then covered. Chips are either added to or taken away from the cover. The child must indicate how many chips are under the cover after the addition or subtraction transformation. Story problems , which refer to objects, are orally phrased as “Sue has m pennies. Bill gives her n more pennies. How many pennies does Sue have now?” and “Sue has m pennies. Bill takes away n of her pennies. How many pennies does Sue have now?” Number combinations were orally phrased as “How much is n and m ?” and “How much is n take away m ?”. Developmental studies show that children can reliably solve simple nonverbal calculations (e.g., 2 + 1) as early as three years of age, while the ability to solve comparable story problems and number combinations develops later, starting around four years of age ( Levine, Jordan, & Huttenlocher, 1992 ).

Longitudinal assessment over multiple time points in kindergarten showed three empirically separate growth trajectories in overall number sense as well as in number subareas ( Jordan et al., 2006 ; 2007 ): (a) children who started with low number competence and stayed low; (b) children who started with high number competence and remained there; and (c) those who started with low number competence but made relatively good growth. Low-income kindergartners were much more likely to be in the low-flat growth class than were middle-income kindergartners, especially with respect to addition and subtraction story problems. Children's overall performance on the number sense battery and their growth rate between kindergarten and first grade predicted overall performance and the growth rate in general mathematics achievement between first through third grades ( Jordan, et al., 2007 ; Jordan, Kaplan, Ramineni, & Locuniak, 2009 ) Although all subareas were significantly related to each other and to achievement outcomes, early facility with addition and subtraction number combinations was most predictive of later achievement ( Jordan et al., 2007 ).

Although our number sense battery has good reliability and predictive validity, it has a relatively long administration time and thus may be of limited practical value to classroom teachers. To address this issue, Jordan, Glutting, and Ramineni (2008) developed a reliable but abbreviated screen (referred to as the Number Sense Brief or NSB) through Rasch item analyses as well as a more subjective review of issues related to item bias. Internal reliability for the screen was at least .80 in kindergarten and first grade. Although the number sense brief screen is positively correlated with mathematics achievement measures, its predictive validity has not been established.

The present study examined predictive validity of the NSB screening measure. Children were given the screening measure at the beginning of first grade and mathematics outcomes were obtained at end of both first and third grades. Outcomes included overall mathematics achievement, as well as subareas of written computation and applied problem solving. It was hypothesized that number sense proficiency may be more relevant to applied problem solving than written computation, which may be more dependent on learned algorithms. To examine the unique contribution of number sense (as measured by the number sense brief) to these later mathematics outcomes, we also added the common predictors of age, verbal and spatial abilities, and working memory skills in our analyses.

Participants

Participants were drawn from a multi-year longitudinal investigation of children's mathematics development. They all attended the same public school district in northern Delaware. Background characteristics of children in first grade ( n = 279) and in third grade ( n = 175) are presented in Table 1 . The first graders included children who completed all measures in first grade and the third graders were children who completed all measures in first and third grade. In the first grade sample, 55% of the children were boys, 52% were minority, and 28% came from low-income families. In the third grade sample, 54% of the children were boys, 42% were minority, and 22% came from low-income families. Income status was determined by participation in the free or reduced-price lunch program in school. Moreover, most of the low-income children lived in urban, low-income neighborhoods. The differential attrition from first to third grade by minority and income status may limit the generalizability of findings and should be kept in mind when interpreting the current results. Participant attrition was due primarily to children moving out of the school district, rather than withdrawal from the study or absence on the day of testing. Although we were not able to determine why children left the school district, attrition may reflect lower family stability. All children were taught mathematics with the same curricular content and approach in first through third grade.

Demographic Information for Participants at the End of First Grade (n=279) and the End of Third Grade (n=175)

The measures were given to children individually in school by one of several trained research assistants. The NSB items were given in October of first grade, the cognitive measures (Vocabulary, Matrix Reasoning, and Digit Span tests) in January of first grade and the math achievement measures in April of first grade and again in April of third grade.

Number Sense Brief Screen

The NSB is a shortened version of a longer number battery given to children (e.g., Jordan et al., 2006 ; Jordan et al., 2007 ; Locuniak & Jordan, 2008 ; Jordan et al., 2009 ). The NSB has 33 items, which are presented in the Appendix . The items assess counting knowledge and principles, number recognition, number comparisons, nonverbal calculation, story problems and number combinations. The measure is reliable, with a coefficient alpha of .84 at the beginning of first grade ( Jordan, Glutting, & Ramineni, 2008 ).

Cognitive tasks

The Wechsler Abbreviated Scale of Intelligence (Wechsler, 1999) was used to assess oral vocabulary and spatial reasoning. At age 7, internal reliability is .86 for the Vocabulary subtest and .94 for the Matrix Reasoning subtest. The correlation between the Vocabulary and overall verbal IQ is .93 and the correlation between Matrix Reasoning and overall performance (nonverbal) IQ is .87.

A digit span test ( Wechsler, 2003 ) was used to measure short-term and working memory. A series of single digits of varying lengths were read orally to each child. Children were asked to repeat digits verbatim on a Digit Span Forward section, and then, to repeat digits in reverse order on a Digit Span Backward section. Digit Span Forward is a measure of short-term recall and Digit Span Backward a measure of working memory or active recall ( Reynolds, 1997 ). At age 7, internal-consistency reliability is .79 for Digit Span Forward and .69 for Digit Span Backward.

Mathematics Achievement

Math achievement was assessed with the Woodcock-Johnson III (WJ-III; McGrew, Schrank, & Woodcock, 2007 ), which is normed through adulthood. The composite achievement score (Math Overall) was the combined raw scores for subtests assessing written calculation (written calculations in a using a paper and pencil format; Math Calculation) and applied problem solving (orally presented problems in various contexts; Math Applications). Internal-consistency reliability is above .80 between first and third grades for each subtest. The correlation between Math Calculation and Math Applications is .68 for ages 6 − 8.

Raw scores from the NSB were used for all analyses. Mean raw scores and standard deviations on all tasks are presented in Table 2 for both first- and third-grade samples. Bivariate correlations are presented in Table 3 between the NSB raw scores and raw scores on the cognitive measures at first- and third-grades, as well as between the NSB and age at the beginning of kindergarten. All of the correlations were positive and statistically significant (i.e., all p values ≤ .05), with the two lowest correlations being kindergarten start age (.19) and Digit Span Forward (.34) and the highest correlations being Math Applications in first and third grades (.73 and .74, respectively).

Raw Score Means (SD) for the Measures in First and Third Grade

Correlations Between First Grade Number Sense Brief and Control Variables

Note . All correlations are significant, p < 0.01

A primary purpose of the study was to determine the unique contribution of the NSB in predicting criterion mathematics performance. Specifically, the study examined the extent to which the NSB predicted mathematics performance above-and-beyond the contribution of the control (nuisance) variables of age and general cognition related to language (Vocabulary), spatial ability (Matrix Reasoning) and memory (Digit Span Forward and Digit Span Backward). To accomplish these goals, students’ scores on the NSB were regressed on a series of established mathematics achievement outcomes (Math Overall, Math Calculation, Math Applications) using the two-stage model recommended by Keith (2006) . This methodology is sometimes referred to as a variance partitioning analysis ( Pedhazur, 1997 ) and/or a sequential variance decomposition analysis ( Darlington, 1990 ). At step one (model 1), the control (nuisance) variables entered simultaneously into an analysis. Step 2 (model 2) comprised entry of the NSB. The analyses were used to predict mathematics achievement in first grade and then in third grade. The independent contributions of predictors were evaluated through the interpretation of squared partial coefficients ( Meyers, Gamst, & Guarina, 2006 ; Tabacnick & Fidell, 2007 ). Effect sizes were estimated for the predictors using Cohen's (1988) f 2 , where values of .02 equal a small effect, values of .15 equal a medium effect, and values of .35 a large effect.

Mathematics Overall

Table 4 presents the results for predicting criterion performance on the Woodcock-Johnson III mathematics composite score (Math Overall). Model 1 (age and general cognitive measures) accounted for 47% of the variance in math in first grade ( p < .01) (with Vocabulary, Matrix Reasoning, and Digit Span Backward reaching significance) and 45% of the variance in third grade ( p < .01) (with Vocabulary, Matrix Reasoning, Digit Span Forward, and Digit Span Backward reaching significance). Results showed that the NSB made statistically significant, unique contributions to the prediction at first grade ( p < .01) and third grade ( p < .01) outcomes in Math Overall. In each instance, the NSB accounted for about 12% more criterion variance than the control variables. More importantly, Cohen's (1988) f 2 represented a medium-to-large effect sizes for both first- and third-grade criterion performance (respectively, .29, .21).

Results of Block Entry Regression for the End of First Grade Math Overall and the End of Third Grade Math Overall: Regression Coefficients and Variance Explained by Each Block of Variables

Mathematics Calculation

Table 5 presents the results for predicting Mathematics Calculation. Model 1 (age and general cognitive measures) accounted for 35% of the variance in Math Calculation in first grade ( p < .01) with Vocabulary, Matrix Reasoning, and Digit Span Backward reaching significance, and 33% of the variance in third ( p < .01), with Vocabulary and Matrix Reasoning reaching significance. Model 2 accounted for 41% of the variance in first grade and indicating that the NSB measure accounted for 6% more variance than the control variables. Cohen's (1988) f 2 value for the NSB was .10, which represented a small-to-medium effect size. Results for third grade were more impressive. The NSB accounted for a 14% more variance of Math Calculation than the control variables and Cohen's (1988) f 2 (.26) represented a medium-to-large effect size.

Results of Block Entry Regression for the End of First Grade Math Calculation and the End of Third Grade Math Calculation: Regression Coefficients and Variance Explained by Each Block of Variables

Mathematics Applications

Table 6 presents the results for Mathematics Applications where the results were most impressive. Model 1 accounted for 44% of the variance in Math Applications in first grade ( p <.01) with Vocabulary, Matrix Reasoning, and Digit Span Backward reaching significance, and 45% of the variance in third ( p grade <.01), with Vocabulary and Matrix Reasoning reaching significance. Not only did the NSB make significant, unique contributions that accounted for 14% to 17% of the criterion's variance, Cohen's (1988) f 2 represented a large effect siz e in predicting first-grade NSB performance (.44) and third-grade NSB performance (.45).

Results of Block Entry Regression for the End of First Grade Math Applications and the End of Third Grade Math Applications: Regression Coefficients and Variance Explained by Each Block of Variables

Number sense, as assessed by our screening measure, is a powerful predictor of later mathematics outcomes – both at the end of first grade and the end of third grade. In terms of overall mathematics achievement, number sense made a significant and unique contribution to our regression models, over and above both age and cognitive factors. Its predictability was as strong in third grade as it was in first grade, contributing an additional 12% of the variance in mathematics achievement at both grades. Our findings are in keeping with those of other investigations suggesting that weaknesses in intermediate symbolic number sense, or number competencies related to counting, number relationships, and basic operations, underlie most mathematics learning difficulties (e.g., Gersten. Jordan & Flojo, 2005 ; Geary, Hoard, Byrd-Craven et al., 2007 ; Landerl, Bevan, & Butterworth, 2004 ).

Analysis of mathematics achievement outcomes by the subareas of calculation and applied problem solving was additionally revealing. Calculation, a paper and pencil task, assessed conventional operations and procedures, whereas applied problems required children to solve novel problems in everyday contexts. Although number sense was a unique predictor of both mathematics achievement subareas, it was more predictive of applied problem solving. Noticeably, the effect of number sense as a predictor was large and significant for both first and third grade. With general predictors included in our model, number sense contributed an additional 14% of the variance in first grade and an additional 17% of the variance in third grade. Most surprising was the sustained and even stronger relationship between earlier number sense and applied problem solving over time. That is, we expected number sense at the beginning of first grade to predict mathematics problem solving at the end of first grade, since the content of the two measures are closely allied during this period ( Jordan et al., 2009 ). Mathematics problem solving becomes more complex by third grade, requiring children to solve novel problems involving a range of numerosities and operations. Likewise, the effect of number sense as a predictor of mathematics calculation became greater between first and third grades. Our findings support the notion that children who bring foundational knowledge of numbers to first grade are more likely to benefit from mathematical experiences throughout the elementary grades than those who do not have this knowledge ( Baroody, Lai, & Mix, 2006 ) and that the effect of weak number sense may be cumulative. Knowledge of number concepts and skill with mathematics procedures appear to be mutually supportive, each facilitating the development of the other area ( Baroody & Ginsburg, 1986 ; Rittle-Johnson, Siegler, & Alibali, 2001 )

Previous findings, as well as the ones reported in the present investigation, establish that general verbal and spatial abilities are related to mathematics achievement (e.g., Donlan, Cowan, Newton, & Lloyd, 2007 ; Shea, Lubinksi, & Benbow, 2001 ). Studies of children with disabilities show that language impairments compromise the acquisition of spoken number sequences ( Donlan et al., 2007 ) while spatial impairments inhibit understanding cardinality concepts ( Ansari, Donlan, Thomas, Ewing, Peen, & Karmiloff-Smith, 2003 ). Moreover, weaknesses in working memory capacity are a characteristic of young children with math difficulties ( Geary, Brown, & Samaranayake, 1991 ; Koontz & Berch, 1996 ; Wilson & Swanson, 2001 ; Swanson & Beebe-Frankenberger, 2004 ). For example, working memory weaknesses make it difficult for a child to hold one term of the problem in memory while counting on the number in the other term to solve an addition problem ( Lefevre, Destefano, Coleman, & Shanahan, 2005 ). Despite the influence of general cognitive factors, however, the present findings show that number sense is uniquely and meaningfully related mathematical development. This observation supports the suggestion that number concepts and principles develop independently of other abilities and might represent a relatively separate cognitive system ( Donlan, et al. 2007 ; Landerl et al., 2004 ).

Our relatively brief number sense screen is a valid and powerful measure that can be used to predict which children at the beginning of school are likely to have trouble learning mathematics. In reading, similar screening measures have been devised to help schools provide additional support and interventions ( Gersten, Jordan, & Flojo, 2005 ). Not surprisingly, early literacy skills related to letter-sound knowledge are more predictive of subsequent reading achievement than are more general cognitive factors ( Schatschneider, Fletcher, Francis, Carlson, & Foorman, 2004 ). It has been suggested that number sense is an intermediate ability that is achievable through early instruction ( Ginsburg, Lee, & Boyd, 2008 ). Previous studies have shown that poor mathematics outcomes for low SES children are mediated by weak number sense ( Jordan et al., 2009 ). Many disadvantaged, low-income children come to school with fewer number experiences than their middle-income peers ( Clements & Sarama, 2008 ). It is likely that the lack of such experiences results in deficient symbolic number sense upon entry to elementary school. Number sense, which involves interrelated concepts of counting, number knowledge, and operations, has promise for guiding the development of early intervention programs. Future work should also consider children's strategy use on number tasks, especially on addition and subtraction problems. Understanding whether children can use efficient techniques such as counting on from a cardinal value might add to achievement predictability as well as inform instruction. Although the present findings suggest considerable stability in mathematical knowledge between kindergarten and third grade, there is good reason to believe that this knowledge can be improved with targeted interventions (e.g., Baroody, Eiland, & Thompson, 2009 ; Ramani & Siegler, 2008).

Author Note

This work is supported by a grant from the National Institute of Child Health and Human Development (R01HD036672). We wish to thank the participating children and teachers for their extremely generous cooperation.

Items (N = 33) in the Number Sense Brief Screener ( Jordan, Glutting & Ramineni, 2008 )

Give the child a picture with 5 stars in a line. Say: “Here are some stars. I want you to count each star. Touch each star as you count.” When the child is finished counting, ask, “ How many stars are on the paper?”

• 1 Enumerated 5
• 2 Indicated there were 5 stars were on the paper
• Say: “I want you to count as high as you can. But I bet you're a very good counter, so I'll stop you after you've counted high enough, OK?” Allow children to count up to 20. If
• 3 Counted to at least 10 without error.
• Show the child a line of 5 alternating blue and yellow dots printed on a paper. Say: “Here are some yellow and blue dots. This is Dino (show a finger puppet), and he would like you to help him play a game. Dino is going to count the dots on the paper, but he is just learning how to count and sometimes he makes mistakes. Sometimes he counts in ways that are OK but sometimes he counts in ways that are not OK and that are wrong. It is your job to tell him after he finishes if it was OK to count the way he did or not OK. So, remember you have to tell him if he counts in a way that is OK or in a way that is not OK and wrong. Do you have any questions?”
• 4 Counted Left to Right (correct)
• 5 Counted Right to Left (correct)
• 6 Counted Yellow then Blue (correct)
• 7 Counted first Dot twice (incorrect)
• [For items 8 through 11, point to each number that is printed on a separate card and say: “What number is this?”]
• 12 What number comes right after 7?
• 13 What number comes two numbers after 7?
• 14 Which is bigger: 5 or 4?
• 15 Which is bigger: 7 or 9?
• 16 Which is smaller: 8 or 6?
• 17 Which is smaller: 5 or 7?
• 18 Which number is closer to 5: 6 or 2?
• Say: “We are going to play a game with these chips. Watch carefully.” Place two chips on your mat. “See these, there are 2 chips.” Cover the chips and put out another chip . “Here is one more chip.” Before the transformation say, “Watch what I do. Now make yours just like mine or just tell me how many chips are hiding under the box.” Add/remove chips one at a time.
• 22 3 − 1
• Say: “I'm going to read you some number questions and you can do anything you want to help you find the answer. Some questions might be easy for you and others might be hard. Don't worry if you don't get them all right. Listen carefully to the question before you answer.”
• 23 Jill has 2 pennies. Jim gives her 1 more penny. How many pennies does Jill have now?
• 24 Sally has 4 crayons. Stan gives her 3 more crayons. How many crayons does Sally have now?
• 25 Jose has 3 cookies. Sarah gives him 2 more cookies. How many cookies does Jose have now?
• 26 Kisha has 6 pennies. Peter takes away 4 of her pennnies. How many pennies does Kisha have now?
• 27 Paul has 5 oranges. Maria takes away 2 of his oranges. How many oranges does Paul have now?
• 28 How much is 2 and 1?
• 29 How much is 3 and 2?
• 30 How much is 4 and 3?
• 31 How much is 2 and 4?
• 32 How much is 7 take away 3?
• 33 How much is 6 take away 4?

Items copyrighted © by Nancy C. Jordan 2009

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