Making (Number) Sense

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Number sense is a foundational understanding of how numbers work. Researchers have demonstrated that every child can develop number sense. Unfortunately, it has also been found that many children don’t. This is because number sense does not develop by accident. It is not a “side effect” of doing puzzles, songs, or other informal activities that may be superficially related to math. Teachers are increasingly working to build number sense, especially in the early grades. However, you can give your children a jump-start by developing number sense before they start school, or if they are already in school, you can help them strengthen their number sense to be better prepared for deep mathematical understanding and success.

How Can You Do It?

General Tips

imgEncourage your child to explain his reasoning. This not only gives you insight into how he thinks, but it also helps the child evaluate his thinking and “cement” it in place.

Be sure to ask your child to explain her thinking all the time, not just when she makes a mistake. Constantly asking sends several important messages: your ideas are valued; math is about reasoning; and there are always other ways to look at a problem.

Real-world links
As much as possible, link math to real-world situations. When shopping, compare prices. When walking around the neighborhood or in the park, estimate distances using steps and/or standard measurement (feet, miles). Allow your children to pay at the checkout and count their change to make sure it is correct (matching the receipt, not necessarily doing a subtraction problem).

When teaching your child mathematics, make her verify estimates through doing. Point out that measuring is important, even though it is always an approximation since it can always be made more precise. For example, a pencil that looks 5 inches long might be 4 3/4 using a ruler with 1/4 inch marks or 4 11/16 on a ruler with 1/16 marks.

Numbers versus numerals
As you work with your child, remember that “number” is the amount of the things you are counting or measuring. The “numeral” is the shape we have decided to use to record it. For example, five beans on a plate can also be represented as five fingers on a hand, the oral or written word “five”, five tally marks on a page, a drawing of five beans, or the numeral 5. Connecting the glyph “5” to the amount of beans on the plate should be the last thing to focus on. It is far more useful for children to understand that five can be represented all those other ways.


These activities are divided into two types, prenumber activities and number activities. Prenumber activities are important for developing concepts that support and eventually lead to understanding the concept of numbers. They also help in developing understanding of problem situations (problem solving and critical thinking) and so should not be ignored.

You can do activities for as long or short a time as is comfortable (you will be there working along with your child). But you should be sure to do the activities multiple times over the course of several days. Later on, revisit activities that you haven’t done together in a while.

Prenumber Activities



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– Include transformation patterns as the child gets more skilled at continuing patterns. Transformation patterns are those where the pattern changes by movement instead of color, shape, etc. For example, using toothpicks:

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Using everyday objects and situations, provide opportunities for your child to discover that change the size or organization of objects does not change the quantity of objects. (Don’t worry, you don’t have to teach your child the words “organization”, “conservation”, or “quantity”!)

Group Recognition (Subitizing)

The activities used to develop this skill may take a little longer to prepare, but do not skip them! This skill is highly correlated to better understanding and scores in math.


Number activities


– Practice counting out loud backwards, initially from 10 then later count backward from 20. After they are fluent with these, ask them to count backward from other numbers. Finally, ask them to tell you what number comes before a given number.
– Count larger numbers by rote starting from a multiple of ten, for example 40 or 70. Point out that there is a pattern when counting, that all they need to do is say the ten and then count by ones, for example “40, 1, 40, 2, 40, 3 …” for counting 41, 42, 43.
– Practice skip counting by 10s. After reaching fluency starting with 10, have your child start at a different ten, for example start at 40. Don’t forget to also count backwards!
– After reaching fluency with 10s, practice counting by 10s starting at any number other than a multiple of 10. Instead of just counting “10, 20, 30 …”, start at 27: “27, 37, 47, 57”.

Once your child is fluent with counting by ones to 20, you can begin doing the following activities with them. Don’t forget to continue oral counting frequently!

Odd and Even Numbers


Good estimation skills develop slowly. As you work on these activities, focus on having your child get “close enough” rather than exact answers. Have discussions when the estimates are not reasonable; explain that “reasonable” estimates have good reasons behind the number, for example we do not need to buy 50 Valentine cards because there are only four students at each table and 8 tables of four is a lot less than 50. Keep the number of objects used for estimating within a range of numbers familiar to your child. For many new first graders, numbers within the range of 10-30 are appropriate. When estimating and counting larger numbers, have your child put the items into groups of “about 10” to count by tens to find a number close to the actual amount.

Mental Math

Real life often requires mental computation. Our children need to be able to move numbers around in their heads and discuss their strategies. They also need to be able to do it fluently, though not necessarily “fast”. For these activities, you will be saying an extended sequence of operations to your child, one part at a time. Pause after each operation to give your child a chance to perform the operation in his head.

EXAMPLE: “Start with 4, (pause) add 3, (pause), add 2 more, (pause) then add 3 again. How much is that? (12)

After hearing the answer, ask your child how he got the answer. He may repeat the sequence to you and explain his answers along the way, or he may explain how he manipulated the numbers. For example, “You said to add 4, 3, 2, and 3. I know that 3 and 3 are 6, and 4 plus 2 is 6. So I have two 6s. 6 plus 6 is 12!” The manner of arriving at the answer is not so important as being able to recall/explain how.
- Start with a sequence using 3 numbers. Gradually move up to longer sequences as your child gains proficiency.
- Use smaller numbers (1-5) first. Use single operations to begin with (only adding or only subtracting). Combine operations later as your child shows fluency and accuracy in single operations.
- Include adding and subtracting 10, but avoid numbers higher than 10 unless/until your child is very good at adding and subtracting 2-digit numbers without confusing the tens and ones.
- You can start simply by using “plus” and “minus” or “take away”. Later, use “add” and “subtract”. Don’t be afraid to use “increase it by” and “decrease it by” later on!

Number Sequencing (number ordering)

Adding and Subtracting

Be sure to start with composing and decomposing activities. Composing is just putting numbers together to make new number. Decomposing, then, is taking a number and breaking it into two or more smaller numbers. Doing these activities with counters for a long time before introducing written number sentences (equations like “4 + 5 = ”) builds an understanding of how numbers work. Putting numbers together results in a larger number. A number can be broken into many different combinations of numbers that all make the same amount when recombined.

To do these activities, use counters like dry beans. (Beans work well because they are hard and cheap, but you can use beads, buttons, or any other small objects, though I would stay away from cereals for these activities.)

Base Ten Manipulation

When students trade with base ten manipulatives, they can demonstrate that the number 14 may be represented as 14 ones or as 1 ten and 4 ones. Being able to understand this concept of place-value is essential for understanding higher math concepts.


These are just a few ways to develop number sense in your child. By working on these fun, simple activities with your child, you will be building important concepts and skills in number sense that are crucial for your child’s mathematics success. And remember to enjoy your time together!

If you are interested in reading more about number sense, contact your child’s teacher or read this document online at

What is Number Sense?

Number sense, simply speaking, is an understanding of the relationships between real-world quantities, the counting numbers in the student's spoken language, and the glyphs (numerals) and symbols used to represent them. (Griffin, 2004)

Number sense develops gradually in children. It comes about as a result of exploring numbers/quantities, playing with them, and paying attention to the ways numbers/quantities relate to other numbers/quantities. As number sense develops, children develop flexibility in thinking about numbers including the ability to represent numbers in multiple ways. They also develop an understanding of number magnitude (size) and are able to predict the effects of operations on numbers (e.g. adding positive numbers together results in a larger number).

Number sense can develop in some children through careful observation and play beginning at a very young age. However, as mentioned above, it does not just "happen" as a side effect. Many children need specifically guided instruction or participation in particular activities to develop number sense.

Why is it important?

Number sense enables students to understand and express quantities in their world. For example, whole numbers describe the number of students in a class or the number of days until a special event. Decimal quantities relate to money or metric measures, fractional amounts describing ingredient measures or time increments, negative quantities conveying temperatures below zero or depths below sea level, or percent amounts describing test scores or sale prices.

The development of number sense in students is important for many reasons. As well as those listed below, number sense is paramount in providing understanding not only of how numbers work together but also of why we manipulate them the way we do, i.e. why the algorithms work.

Number sense is the basis for understanding any mathematical operation and being able to estimate and make a meaningful interpretation of its result. Getting the right answer is important, but so is understanding why it is the right answer. In a study by Jordan, Locuniak, Ramineni and Kaplan entitled “Predicting First-Grade Math Achievement” from Developmental Number Sense Trajectories, it was shown that “early sense of numbers (sic) is a reliable and powerful predictor of math achievement at the end of first grade.” This, of course, predicts further success as the student advances through the grade levels, provided they continue to develop and implement their number sense.

Knowing the why of how numbers work is of utmost importance, and children should not be shown or encouraged to discover the “how” until they understand the “why”. Techniques such as using ten frames to represent numbers and using concrete models to show place value concepts are daily necessities for young children. Inquiry-based approaches (such as math dice games) to teaching children mathematics should be utilized as primary teaching methods over paper-pencil practice using numerals and symbols. This is not to say that explicit teaching number skills is not essential, especially for those students from low socio-economic status. We absolutely need to do this. It is saying that teachers should provide multiple opportunities for students to experience numbers and make connections before putting the pencil to paper.

Components of number sense

Number sense begins very early and must be a focus of primary grades math instruction. This is the solid foundation in math that all kids need. A sense of numbers is critical for primary students to develop math problem solving skills. The National Council of Teachers of Mathematics increasingly calls for districts to give more attention to building this skill, and studies have found that number sense accounts for 66% of the variance in first grade math achievement. The National Council of Techers (USA. 1989) have also identified five critical components that are characteristic of students who have good number sense:

These components can be broadly split into prenumber concepts and number concepts.

Prenumber Concepts

Prenumber concepts involve those that provide a base for the understanding of quantities and how number systems work. These concepts develop understanding of number meaning, number relationships, and magnitude. These are generally developed using real-world physical objects.


When students practice classifying, they are building understanding of different ways items can be grouped. We see classifying when children sort items by size, color, shape, or another quality.


Recreating and continuing patterns helps students build pattern recognition. This patterning skill is essential in understanding and using number systems, which are based on patterns.


When playing with objects in groups, children can compare the sizes of the groups. Exploration of comparisons should include spatial (how closely or loosely packed are the items), dimensional (how big or small are the items in the group), and other attributes. Ultimately, students should arrive at an understanding of the concept of one-to-one comparison to determine which group is bigger or smaller, more or fewer (less).


The concept of conservation is the understanding that something stays the same in quantity even though its appearance changes. To be more technical (but you don’t have to be) conservation is the ability to understand that redistributing material does not affect its mass, number or volume.


Subitizing is the ability to recognize the amount in a group quickly and without counting. When a student briefly sees the two rows of three pips on a die and recognizes it as 6, they are subitizing.

Number Concepts

Number concepts are those ideas and understandings that most non-mathmeticians think of as “math”. They are generally practiced using oral or written representations (numerals) of quantities. However, teachers should take care to always connect the symbolic representation (“four” or 4) to the number amount (* * * *).


Cardinality is the concept that, when counting objects, the last number counted in one-to-one counting is the amount. This reflects the understanding that a number names an amount. A student that has not yet grasped this concept will simply count the quantity over again when asked, “How many?”

Number Meaning

Number meaning is closely related to cardinality. Students should be able to recognize and explain that a numeral or “number” represents an actual amount. (Unfortunately, too many primary grade educators still use “number” when they should be referring to 98 or 7 + 4 as numerals.) When given a number, a student should be able to represent it using physical objects, illustrations, or even using tools such as number lines (as long as they understand that the numeral on the line represents all the steps taken from 0 to arrive at that point).

Referents for Numbers/Quantities

Referents are simply the symbols or glyphs we write to represent numbers. Most of the time this means numerals such as 1, 5, or 396, but it can also mean word forms like “two” and “a million”. Students should also understand that the same number can be represented in multiple ways, e.g. 42, 40 + 2, 6 × 7, etc. Exploring and using the various forms a number can take also builds understanding of place value and how the numeral represention is “built”. Finally, it leads to the understanding of variables like y and π being numbers not letters.

Number Magnitude

Magnitude is the concept of the “big-ness” of a number/amount, especially in relation to other numbers. Students should have multiple experiences with numbers on number lines and with objects and illustrations to be able to reasonably estimate random quantities. For example, does the collection of orange balls look like 30, 60, or 100? Students should also understand that magnitude scales may change. When comparing 5 and 10, one is twice as much as the other and can be easily shown to be so, but when comparing 72 and 77, they are very “close” on the classroom number line, while 6341 and 6346 can be considered almost “the same” in certain situations. Yet, all three pairs are separated by the same amount, 5.

Understanding of magnitude helps students check for reasonableness in their answers and make estimations by understanding how to round numbers and when it is useful and appropriate to do so.

Relationships Between Numbers

Students must learn to identify and use the relationships between numbers. This includes using rules and patterns as well as using the activities of composing (making a new number by putting other numbers together) and decomposing (breaking a number into two or more of its parts). Students should be able to do things such as recognize that in our base-10 system, decades (multiples of 10) can be added to other numbers by counting up by ten. They should be able to group numbers based on attributes such as odd/even or prime/composite. Understanding these relationships helps build fluency and accuracy when students begin manipulating numbers through operations.

Operations Involving Numbers

Once students become fluent in identifying and using relationships between numbers, they will find traditional math operations easier to understand and complete. A problem such as 345 - 287 might be solved more easliy with a technique such as compensation by adding 13 to both numbers and subtracting 300. You can probably do it in your head! Instead of relying on paper and pencil and traditional algorithms, they will be able to flexibly apply their understanding to solve addition, subtraction, multiplication, and division equations. Strong number sense also leads to not only completing operations with whole numbers but also understanding and using partial numbers such as numbers using decimal points/places and fractions.

THE BEARS AND THE GRAPES Once upon a time, there were three bears. The bears were hungry, so each bear ate 2 apples. Then they were not hungry any more. How many apples did they eat?

Problem Solving

Problem solving (word or story problems, situational math) naturally derives from the previously mentioned number and prenumber concepts. However, an argument can be made that teachers should be having students explore and solve these types of problems even before students can write or recognize numeral representations. By interacting with language, students as young as 4 or 5 can solve simple problems using objects. They can even “do” multiplication. Using toy apples (or blocks or beans, etc.), they can make groups of two for each bear and count them all. This concrete manipulation exactly represents the expression 3 × 2, and understanding this relationship between the story problem and the number manipulation used to solve it is exactly the goal of developing number sense.


Finally, measurement is another number aspect of number sense that is useful to develop and explore. It is related to the prenumber concept of comparison as well as the number concepts of magnitude and referents. When developing the idea and practice of measurement, students should be given opportunities to explore with various units of measure. This will lead them to the understanding that measurement is always an approximation because it can always be made more precise. When measuring a book with paper clips, you get a result (number and unit) that is close but maybe not exact, but if you measure using 1cm counting cubes you will get a result that is likely closer to the actual book length.


As early as 1994, the National Council of Teachers of Mathematics has been encouraging the development of number sense in students. Unfortunately, few school districts and education systems have prioritized this until very recently. There is much research into the benefits of explicitly developing number sense in our beginning and intermediate grade students. However, the wheels of beaurocracy are slow to turn and number sense "instruction” has just begun (as of 2014) to appear in some math textbooks, though not necessarily in those put out by the nation-wide publishers that perennially get the most school district contracts. Hopefully, you can use the activities and information presented here to begin to develop number sense in your child or students. If you are a parent or teacher, please pass along this document to your child's teacher or your colleagues, so we can begin to help even more children in their journey through mathematics!

- There are more number systems than our base-10 (decimal) system, and many are used throughout the sciences. Understanding number relationships also works in those systems! For example, in octal (base-8), decimal 12 is represented as “(14)8”, one "eight” and four “ones”. Adding 8 to this number simply advances the “eights” place so it becomes “(24)8” which translates to decimal 20. (2 in the eights place is 2 × 8, or decimal 16, which is then added to the 4 to make 20!)


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Originally written in StackEdit.