Memorizing vs. Understanding Math: How Can Parents Tell the Difference?

A good math grade is encouraging, but it doesn’t always tell you how well your child understood every topic it covered. They may have remembered enough steps and formulas to answer familiar questions correctly without fully understanding why those methods work.

So how can you tell whether your child is relying on memory or understands the math behind their answers?

We’re a team of math educators at Mathnasium of Woodbridge in Irvine, and we’re here to help you look beyond the score. Today, we’ll explain what memorizing and understanding look like in math. Then we’ll walk you through a few checks you can try with your child and ways to help them build understanding.

What Do Memorization and Understanding Look Like in Math?

Memorization means your child remembers a math fact, rule, or set of steps and repeats it. Understanding means they also know what the numbers and operations stand for, why a method works, and how the idea connects to other problems.

Researchers call this second piece conceptual understanding. The National Research Council's Adding It Up (2001) treats it as one strand of math proficiency, alongside procedural fluency. It means using procedures accurately, efficiently, flexibly, and appropriately.

So when your child follows a set of steps, that alone doesn't mean they're only memorizing. The report describes the two strands as interdependent, so children need both the meaning and reliable calculation skills.

The difference becomes easier to spot when you compare two explanations of the same problem. Here's how answers built on memory and on understanding might sound side by side:

Remembering that 6 × 8 = 48 is useful. The question is whether your child can also connect that fact to something meaningful, such as equal groups, a drawing, or a nearby fact like 5 × 8.

Which column sounds more like your child at the kitchen table?

In a study on mathematical equivalence, researcher Michelle Perry compared teaching children an underlying math principle with teaching them steps to follow. Children taught the principle were more successful at applying what they learned to problems presented in a different form 

At Mathnasium of Woodbridge, we often meet students who feel confident during class and then freeze on a test question worded a little differently. That moment can be confusing for kids and parents alike, and it usually points to a step that never connected to its meaning.

📕 You May Also Like: Is My Child 'Bad at Math' or Just Missing Foundational Skills?

How Can You Tell Whether Your Child Understands the Math?

You can tell by asking your child to explain, adjust, or compare their thinking, since a correct answer on its own shows only part of the picture. The three checks in this section take a few minutes each and fit easily into homework time or a conversation at dinner.

1. Ask Them to Show Why Their Answer Makes Sense

Your child's explanation reveals the thinking behind a correct answer, and a few open questions can bring it out. These prompts work across grade levels:

• "Can you show me with a drawing?"

• "What do the numbers in your answer represent?"

• "Why does that step work?"

With fractions, you might say, "You said \(\Large\frac{3}{4}\) is greater than \(\Large\frac{2}{3}\). Can you draw both fractions on two same-sized wholes to show why?" With place value, try, "In 352, what does the 5 stand for? Can you show that amount another way?"

Your child doesn't need a polished explanation. A rough sketch or a quick "because it's 5 tens" tells you they see the value behind the digit.

2. Change One Part of a Familiar Problem

Small changes to a problem your child has already solved show whether they see how its parts connect. For younger learners, ask, "If 29 + 16 = 45, what would 30 + 16 be? Can you work it out without starting over?"

In middle school, the same check works with equations. Ask, "You found that x = 4 in 2x + 3 = 11. What happens to x if the 11 becomes 13?"

Answers like "46, because 30 is one more than 29" or "x goes up by 1, since 2x has to grow by 2" show your child reasoning about the change. Starting the whole problem over is a common sign that your child is leaning on the steps, and it gives you a clear place to begin.

How do you think your child would handle that second question?

3. Ask Them to Compare Two Answers or Methods

Your child shows a different kind of understanding when they evaluate someone else's work. Here, they need to judge whether the math holds up, beyond following their own routine.

Imagine two classmates solving 63 − 29. One regroups, and the other subtracts 30 and then adds 1 back. Ask your child, "Do both methods work? Why?"

You can also share a claim to test: "Someone says 0.45 is greater than 0.5 because 45 is greater than 5. Do you agree? How could you check?" Notice whether your child reaches for place value, a number line, or money (45 cents vs. 50 cents) to settle it.

Keep these questions light and treat them as conversation starters. You're listening for your child's thinking, so a wrong answer with interesting reasoning gives you as much to work with as a right one, and it shows your child that mistakes are a normal part of figuring math out.

A few simple questions at home can show you whether your child is remembering math steps or understanding the ideas behind them.

How Can You Help Your Child Move Beyond Memorization?

You can help by giving your child more ways to see, handle, and talk about the math they already practice. These five habits work with the homework your child already brings home and add meaning to the steps they're learning in class.

1. Pair written practice with visual and hands-on activities. Alongside worksheets, give your child something to move, draw, or arrange. Fraction strips and a number line work well for fractions, while bundles of 10 straws make place value visible. Work through the activity together first, then connect it to the written calculation.

2. Switch roles. Let your child be the teacher while you play the student. Hand them a familiar example, such as \(\Large\frac{3}{4}\) + \(\Large\frac{1}{2}\), and let them choose how to present it, with a drawing, a story, or a step-by-step walkthrough. Ask a few curious "student" questions along the way and keep the mood playful.

3. Solve one example together first. Take turns completing the steps of a problem like 3(x − 2) = 15, and say your reasoning out loud on your turns: "I'll divide both sides by 3 so the equation stays balanced." Leave the finished example visible while your child tries a similar problem on their own.

4. Turn repeated practice into a game. Swap part of a worksheet for cards, dice, or a board game tied to the current topic. For multiplication, take turns rolling two dice and building the matching array with coins or beans. Work toward a shared goal, like filling a hundred chart together, so speed doesn't become the main challenge.

5. Connect new ideas to ones your child already knows. Before introducing a new rule, revisit a related idea. Build multiplication from equal groups, or introduce percentages through familiar fractions. Once your child knows that \(\Large\frac{1}{2}\) is 50% and \(\Large\frac{1}{4}\) is 25%, they can reason that \(\Large\frac{3}{4}\) must be 75%.

Which of these would fit most naturally into your family's routine?

If your support at home isn’t leading to progress and your child still relies on memorizing steps rather than thinking flexibly about math, it may be time to consider structured math tutoring.

While private tutoring can help with immediate questions or test preparation, a structured math learning program can provide ongoing support with longer-term learning goals. A program might include:

• Consistent sessions that give students time to build and revisit their understanding.

• Opportunities to explore different ways to solve a problem.

• Practice applying familiar ideas to unfamiliar problems.

• Feedback that encourages students to explain their reasoning and learn from mistakes.

Students in Irvine’s Woodbridge community and nearby neighborhoods can find math support at Mathnasium of Woodbridge. At our math-only learning center, students begin enrollment with a diagnostic assessment that helps us understand their current skills, learning gaps, and approach to math.

We use those insights to create a customized learning plan that helps students fill gaps and develop more flexible mathematical thinking. That means learning to recognize connections between concepts, choose an appropriate strategy, and explain why their solution makes sense, not simply repeat memorized steps.

Mathnasium of Woodbridge helps students in Irvine understand the ideas behind the math they learn.

📕 You May Also Like: 10 Fun Math Games to Play at Home with Your Child

How Mathnasium of Woodbridge Helps Students Build a Deep Understanding of Math

Mathnasium of Woodbridge is a math-only learning center in Irvine, CA. We're part of a network of more than 1,200 Mathnasium learning centers across North America.

When local families come to us for math support, we help their children connect the steps they remember to the ideas behind them. We shape instruction around what each student needs instead of handing everyone the same worksheets and drills.

The Mathnasium Method™, our proprietary teaching approach, guides that work. It helps students think flexibly, connect concepts, and approach unfamiliar problems with confidence.

Six elements shape how we build math understanding:

1. Personalization on a granular level: Each student begins with a diagnostic assessment that shows us their strengths, knowledge gaps, and how they approach math. With these insights, we create a learning plan built around their needs.

2. Teaching for understanding: We explain math concepts in natural, student-friendly language. We also combine verbal, visual, mental, tactile, and written techniques so students make sense of what they're learning.

3. Caring, trained tutors: Our tutors are specially trained in math and in the technical and emotional sides of teaching. They know when your child needs encouragement and when they're ready for a bigger challenge.

4. Problem-solving and critical thinking: In every session, students get time to work through problems on their own before our tutors rejoin them to check and correct their process. We guide them through both the how and the why behind each problem, so they build tools they'll use in math and beyond.

5. Singular focus on math: Our proprietary curriculum includes thousands of pages dedicated to math and nothing else, refined over more than 20 years. That focus lets us go deeper into how students learn, absorb, and retain math concepts.

6. A fun, confidence-building environment: Students often tell us our sessions don't feel like regular lessons. Game-based activities and rewards keep them motivated and engaged as they reach new levels.

And the results speak for themselves:

• 94% of parents report an improvement in their child's math skills and understanding

• 93% of parents report their child's improved attitude toward math after attending Mathnasium

• 90% of students saw an improvement in their school grades

To see how your child thinks about math beyond their grades, a free diagnostic assessment is a great place to start. From there, our local team will build a plan focused on the concepts your child needs next.

📅 Schedule a Free Assessment at Mathnasium of Woodbridge

Not near Woodbridge?

📍 Find a Mathnasium Learning Center Near You