Mental Math and Number Sense: Building Confident Mathematicians at Renbrook

By Julie Schlossinger, Ed.D., Associate Head of School and Head of Lower School

Sometimes a meaningful math lesson begins with a sheet of paper, a pair of scissors, and a yardstick.

Recently, I stopped by the Jane Shipp Terrace as our Grade Four students were finishing a collaborative challenge. Working in small groups, students had been given just one sheet of paper and asked to create the longest paper chain possible.

The purpose of the activity was teamwork and communication. Students needed to plan together, listen to one another, make decisions, and adapt their ideas as they worked. But when it came time to measure the finished chains, another learning opportunity appeared.

And that is often how learning happens at Renbrook.

From Paper Chains to Mental Math

As Grade Four teacher Mr. Altieri began measuring the chains with a yardstick, he invited students to calculate the total length in inches. At first, there was some hesitation. So, rather than simply supplying the answer, Mr. Altieri modeled how he was thinking about the numbers.

One chain measured two full yardsticks plus another 14 inches. He knew that two yardsticks equaled 72 inches because 36 + 36 = 72. To add the remaining 14 inches, he made the problem easier to work with mentally:

70 + 14 = 84, and then add back the 2, for a total of 86 inches.

He had not changed the mathematics. He had simply reorganized the numbers in a way that made the calculation easier.

When the next paper chain was measured, students began trying the same strategy themselves. With each new chain, more voices joined the conversation. By the time all of the groups’ chains had been measured, students who had initially been quiet were experimenting with strategies and sharing their thinking. The teamwork activity had naturally become a lesson in mental math and number sense.

What Is Mental Math?

Mental math means solving mathematical problems in your head rather than immediately relying on paper, a calculator, or a memorized written algorithm.

But mental math is not about answering faster than everyone else. It is about understanding numbers well enough to take them apart, put them back together, round them, compensate, estimate, and look for relationships. That flexibility is what educators often call number sense.

For example, a child solving 19 + 7 might think:

19 + 1 = 20
20 + 6 = 26

Another student might think:

19 + 7 = 19 + 5 + 2
24 + 2 = 26

Both students arrive at the same answer, but what matters is that they understand the numbers well enough to choose a strategy that makes sense to them.

This emphasis on flexible mathematical thinking is embedded throughout the Renbrook Lower School program. Renbrook’s Lower School mathematics curriculum incorporates concrete, pictorial, and abstract learning, number bonds, visual modeling, and mental math, all designed to help students develop strong conceptual understanding rather than simply memorize procedures.

Why Number Sense Matters

Research has long suggested that mathematical success is connected not simply to how many facts a child can recall, but to how flexibly the child can work with numbers.

In research by Eddie Gray and David Tall, children who were more successful in mathematics tended to manipulate and reorganize numbers flexibly, while less successful students relied more heavily on step-by-step counting procedures. One example highlighted in Stanford mathematics education researcher Jo Boaler’s Fluency Without Fear is 19 + 7. Students using number sense might transform the calculation into 20 + 6, making the relationship between the numbers easier to see.

That distinction matters. Knowing math facts has value. But mathematical fluency is bigger than memorization. It involves understanding how numbers relate to one another and being able to select efficient strategies for different problems.

Boaler’s review of the research also raises concerns about equating mathematical ability with speed. She notes research linking timed testing to math anxiety and explains how stress can interfere with working memory, making it harder for students to access facts they actually know. Math anxiety has even been documented in children as young as five.

The goal, then, is not to eliminate practice. It is to make that practice meaningful.

Short Conversations Can Have a Big Impact

The teacher might ask, “How did you figure that out?”

Another student may have used an entirely different strategy. Suddenly the class has several ways to look at the same problem.

A small 2020 study published in the International Journal of Instruction examined fifth graders who participated in Number Talks twice a week for six weeks. Students in the treatment group significantly improved the speed of their mental calculations, from an average of about 73 seconds to about 42 seconds. On an end-of-cycle assessment, they also generated an average of 3.14 strategies for solving problems compared with 1.22 strategies in the comparison group. Students reported greater confidence as well. Because the study was small, its findings should be viewed as one piece of a larger body of research, but they reinforce the value of giving children regular opportunities to talk about how numbers work.

This is an important distinction. The goal is not speed for its own sake.

Speed can develop naturally as students become more comfortable recognizing patterns and choosing efficient strategies. The deeper goal is helping children become flexible thinkers who understand why a strategy works.

How Families Can Practice Mental Math at Home

Parents do not need worksheets, flashcards, or a special math lesson to reinforce number sense. In fact, some of the best opportunities are already part of everyday family life.

In the car: Ask, “What’s 10 more than 47?” or “If we leave at 3:15 and the drive takes 20 minutes, what time will we arrive?”

At the grocery store: Try, “We have two items that cost about $3 each. About how much will that be?” or ask your child to round a price to the nearest dollar.

While cooking: Double or halve a recipe, compare measurements, or figure out how many cookies you will have if you make two batches of 12.

With cards or dice: Add or multiply the numbers drawn or rolled and encourage children to find the answer without counting one by one.

Play “Give Me a Number”: Ask, “Can you give me two numbers that add to 20?” Then ask for another combination.

While setting the table: Turn everyday counting into a quick problem. “There are four people, and everyone needs a plate and a cup. How many items do we need altogether?”

Practice patterns: Count by 2s, 5s, or 10s, then try starting from an unexpected number or counting backward.

Most importantly, ask: “How did you figure that out?” The explanation is often more valuable than how quickly your child gives you the answer.

When children explain their thinking, they begin noticing their own strategies. They also learn that there can be more than one legitimate route to a solution.

Making Math Feel More Like a Puzzle Than a Race

That morning on the Jane Shipp Terrace captured something important about learning. The paper-chain activity was not designed as a mental math lesson. Yet an observant teacher recognized an opportunity, modeled his thinking, invited students into the process, and gave them space to try. It is an example of what we mean when we talk about Bringing Learning to Life.

In Renbrook’s Grade Four program, students are developing increasingly sophisticated mathematical skills, including multiplication and division, fractions, decimals, measurement, area, perimeter, data, and probability. Just as importantly, they are learning to approach numbers with curiosity, confidence, and flexibility. A little mental math practice, offered often and without pressure, can go a long way.

We want children to know their math facts. But even more, we want them to understand numbers, trust their own thinking, consider different approaches, and have the confidence to tackle a problem that does not immediately reveal its answer.

Because mathematics should not feel like a race to see who gets there first. It should feel like a puzzle worth solving.

Frequently Asked Questions About Mental Math

What is mental math?
Mental math is the ability to solve calculations mentally by using number relationships, patterns, decomposition, estimation, and other strategies rather than immediately relying on a written algorithm or calculator.

Is mental math the same as memorizing math facts?
No. Knowing basic math facts can support mental math, but number sense also involves understanding how numbers relate to one another and being able to manipulate them flexibly.

Should children be encouraged to solve math problems quickly?
Efficiency can develop as children become more fluent, but speed should not be the primary goal. Giving children time to reason, try different approaches, and explain their thinking helps build deeper mathematical understanding.

How can parents build number sense at home?
Use ordinary situations involving time, money, cooking, games, travel, sharing, counting, and estimating. Asking “How did you figure that out?” encourages children to reflect on their mathematical thinking.