Building the Foundation: Developing Students with Number Sense

Summary:

• Number sense is flexible, relational thinking about numbers — not just procedures.
• Strong numeracy develops when students reason, estimate, and connect ideas.
• Instruction that promotes multiple strategies builds true mathematical understanding.

What Number Sense Is, Why It Matters, and How to Strengthen Students’ Numeracy

You may have noticed that concerns surrounding student performance in mathematics continue to grow. High-stakes test scores often appear low, while school marks have not declined as much, leaving teachers, parents, and students caught off guard. This is understandably troubling.

Much of the conversation has been centered on the need for early screeners to identify students who may be struggling so that action can be taken. In fact, many states in the U.S. and provinces in Canada now require them. But what kind of action should follow if a screener reveals a problem? Do we return to the tests and tasks of 50 years ago, or recognize that today’s tools, standards, and workplace demands have changed?

It is no longer enough for students simply to follow procedures. We need students to be able to solve problems, reason, and work flexibly with numbers. In other words, we need students to have number sense. You have probably heard someone praised for having it, or criticized for lacking it, but what does that really mean?

What Is Number Sense in Math?

Number sense is a way of thinking about numbers that goes beyond memorized procedures. It is an understanding of how numbers relate to one another, how they can be composed and decomposed, and how they behave in different contexts. Students with number sense recognize benchmarks, estimate reasonably, choose efficient strategies, and notice when an answer makes sense — or does not.

Examples of Number Sense

Before we can help students develop number sense, it is crucial to know how to identify when it is present, or absent, in a student. Here are some examples of when a student is using number sense to solve a problem.

One of our goals in elementary mathematics instruction is to equip students with number sense. That does not just mean students procedurally get right answers; it’s about equipping students who “see the big picture.”

Number sense develops over time as students build more and more connections between ideas they already know about numbers and new ideas they encounter about numbers.

Number Sense “Look-Fors”?

To decide whether a student is developing number sense, consider the questions you might ask yourself and the “look-fors,” or the actions students display that reveal their understanding. Here are some examples.

Ask yourself:

• Can students relate math to real-world situations they encounter?

What to look for:

• A five-year-old realizes what 5 candles on a birthday cake represent.
• An eight-year-old comes up with a problem that makes sense that would be solved by subtracting 210 from 425.
• A 12-year-old uses ratio thinking to decide which of the following is a better buy: A box of 10 granola bars priced at $5.25, or a box of 12 granola bars priced at $6.49.

Ask yourself:

• Do students realize there are usually many different approaches to a mathematics calculation or representation?

What to look for:

• An eight-year-old can show the number 24 to make it clear that it is even, but then show it a different way to make it clear that it’s a multiple of 3.

A 10-year-old realizes that 32 x 8 can be calculated by doubling 32, then doubling that result and that result; or the student might, instead, multiply 30 x 8 and add 2 x 8; or the student might, instead, multiply 32 by 10 and then subtract 2 x 32.

A 12-year-old realizes that the change in temperature depicted here could be described by either an addition or a subtraction.

Ask yourself:

• Do students make sense of why and how two calculations or representations are related?

What to look for:

• A six-year-old realizes why 6 + 6 and 5 + 7 must have the same answer and why. (By moving 1 from the first pile of 6 over to the second pile).
• A 10-year-old realizes why the digits in the answers to 23 – 9 and in 2.3 – 0.9 have to be the same without appealing to a rule (since 2.3 – 0.9 really is 23 tenths – 9 tenths).
• A student can explain why if 4 is a factor of a number, so is 2 (since 4n = 2 x (2n)).

Ask yourself:

• Do students often consider choices they have in a numerical situation and make thoughtful decisions?

What to look for:

• A seven-year-old realizes that to subtract 8 from 32, you could add up from 8 to 32 (2 to get to 10 + 22 more, so 24); or you could take 8 from 38 instead and then subtract the extra 6 you added on; or you could take 10 from 32 and add back 2.
• A 10-year-old realizes that to divide 810 by 7, it might be smart to rewrite 810 as 700 + 70 + 35 + 5 and then divide each part by 7, putting the quotients together.
• A 12-year-old realizes that to divide 2/3 by 2/15, that if 5/15 is equivalent to 1/3, then 1/3 contains exactly five "fifteenths.” Following that logic, two sets of 5/15 fit into two sets of 1/3. Alternatively, they may use inverse operations, identifying that 2/15 X 5/1 = 2/3, confirming the quotient is 5.

Ask yourself:

• Can students tell whether and/or how much more one calculation result is than another related one?

What to look for:

• A five-year-old realizes that 7 + 4 must be 1 more than 6 + 4 and why.
• A nine-year-old realizes that 8 x 32 is 8 more than 8 x 33 and why. (Since there are 8 groups, each with 1 more in it.)
• A 12-year-old realizes that 1/2 ÷ 3/5 is twice as much as 1/2 ÷ 3/10 and why. (Since it takes twice as many 3/10s to fit into something than 3/5s.)

Ask yourself:

• Do students seek to relate a new numerical situation to a familiar one?

What to look for:

• In figuring out 9 + 8, a student relates it to 10 + 8.
• An eight-year-old realizes that 7 x 9 must be 5 x 9 + 2 x 9.
• A 12-year-old realizes that 120% of 96 is the same as 60% of 2 x 96, or 192.

Ask yourself:

• Can students explain why the strategies they use make sense, beyond just stating the steps?

What to look for:

• A seven-year-old can explain that 12 – 8 is the same as 14 – 10 since the pairs of numbers are the same distance apart on a number line.

An eight-year-old can draw a picture to show why 8 x 5 is double 4 x 5. For example, the student might draw:

A 12-year-old can draw a picture to show why 60% of 50 must be 30. For example, the student might draw:

Ask yourself:

• Do students make appropriate generalizations?

What to look for:

• A six-year-old realizes that when you add 0 to any number, the result is the number you started with
• A nine-year-old realizes that for any fraction equivalent to 3/4, the numerator size is 3/4 of the denominator size.
• An 11-year-old realizes that when you multiply by a fraction greater than 1, the product is greater than the other number.

What Is Numeracy and How Do We Promote It?

Numeracy involves using math skills in daily life, such as interpreting data, managing money, or understanding measurements. It is the numerical equivalent of literacy.

In order to promote numeracy, we can ensure that we ask more questions that are about numerical relationships than questions that are about single situations.

For example, instead of only asking, What is 32 x 45?, we ask, How do you know that 32 x 45 is closer to 30 x 45 than 32 x 47 without even getting the answers?

We ensure that we focus students on a set of essential understandings that underpin lots of math students do instead of segregating each thing they learn. For example, we want students to regularly be asked:

• Why is this a subtraction (or multiplication or addition) situation?
• Why and how can you perform this division by thinking about multiplication?
• What kinds of decompositions make sense to use in this calculation? Why?
• What is easy to see about …?, (e.g., 2/3 with this representation?) What is harder to see? How might changing one of the values make it easier for you to calculate?
• How does knowing how the place value system works help you multiply by 100 (or add two three-digit numbers or work with decimals)?
• Why might you use a different strategy to compare …?, (e.g., 4/5 and 4/9 than to compare, e.g. 3/10 and 9/11?)
• What are some relationships you can think of between …?, (e.g., 40 and 400?
• What’s another way to calculate, e.g., 72 x 54?)
• Why does it always work to…?
• How do you know the answer will be …?, (e.g., 3 digits, even before you subtract?)
• How do you know the answer should be …?, (e.g., about 1000, even before you calculate?)

Often these sorts of questions might be taken up in number talks, but they could also come in other parts of lessons.

It is not enough that students can use rules to calculate to show they have number sense since often those rules are not sufficient, don’t lead to generalizations, and don’t allow for flexibility in different situations.

Can All Students Develop Number Sense?

Although some students may struggle more than others in making connections or viewing things from different perspectives, almost any student can get better at these things if they are regularly asked to think that way. It is less likely to happen for all students unless we lead them to focus on relationships and connections between numbers and calculations.

It is also less likely to happen if our screeners or diagnostic tools do not ask for the kinds of approaches described in the look-fors.

Ultimately, fostering number sense is crucial for building strong numeracy, preparing students for higher-level math and the numerical demands of life and work. By shifting instruction from rote memorization to relational understanding — encouraging strategy exploration, connection-making, and reasoning — educators can develop the flexible, conceptual grasp of numbers essential for problem-solving. Number sense is the foundational, non-optional skill for true mathematical competence.