In a 1999 study, Rittle-Johnson and Alibali found that children could learn a method for familiar math problems yet have trouble using it when the same underlying idea appeared in a new form.
We see the same pattern in our tutoring work at Mathnasium. Families often tell us that their child completes every problem on a worksheet correctly, then stalls when the same skill appears in a slightly different form.
From our experience, this usually means the procedure and the reasoning behind it have not fully connected yet. Your student may remember the steps without understanding why they work, which explains how homework can go smoothly while a quiz feels unfamiliar as soon as the wording changes.
Today, our education specialists dig into how memorization and understanding support each other and what you can do at home to help your learner connect the two.
Your child does not need to master understanding first and procedures second. The two usually grow through repeated back-and-forth. As they make sense of a math concept, the steps become easier to use. Practice can then reveal patterns that deepen that understanding.
Rittle-Johnson, Siegler, and Alibali documented this relationship in a 2001 study. Early conceptual knowledge predicted later growth in procedural skill, while practicing procedures also helped students develop a clearer understanding of the underlying ideas.
Another less obvious connection between math fluency and understanding is that quick recall leaves your student more mental space for harder work.
When they can recall a basic fact such as 7 × 8 without calculating it each time, they can focus more attention on multi-step problems, reasoning, and decision-making. The National Research Council’s Adding It Up report recognizes this kind of fluency as valuable in its own right.
Procedural fluency without understanding can leave your learner dependent on familiar steps. On the other hand, understanding without fluency can make even simple calculations slow and demanding.
Together, they help your child solve problems efficiently, adapt when the format changes, and explain why an answer makes sense.
To see whether your child has connected a procedure to the math behind it or is still relying on familiar steps, notice how they respond when the same concept appears in a new form.
Next time you review their homework or work through practice problems together, use this table to tell the difference between true understanding and memorization that has not yet become flexible.
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Signs of True Understanding |
Signs of Memorization Without Understanding |
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Your child can explain why a method works rather than just perform it. |
Your child goes quiet or guesses when asked to explain a step, even if the answer was right. |
|
They can solve a differently worded version of the same problem. |
They get stuck when a familiar problem is worded differently, even after acing the worksheet version. |
|
They can solve the same problem a different way, instead of relying only on the one method they were taught. |
They know only one way to get the answer and get lost if asked to switch the method. |
|
They catch their own mistake by checking if the answer makes sense. |
They may forget a method within days of a test. |
If you are still not sure whether your child understands the math or has simply learned the steps, structured support can make the difference clearer.
At Mathnasium, that support begins with a free diagnostic assessment, built to look past a right answer and identify exactly where each student's understanding is solid, and where it still needs to connect to the steps.
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You can help your child grasp the math behind a procedure without reteaching the entire topic from scratch. Our education specialists recommend a few simple strategies that help your student connect the procedure they use with the reasoning behind it.
Ask your child to talk through their thinking as they solve a problem, rather than just stating the final answer.
For example, if your learner is solving 3 × 24 and gets 72, ask them to explain each step. They might say, "I broke 24 into 20 and 4. 3 × 20 is 60, and 3 × 4 is 12. Then I added 60 and 12 to get 72.” You can follow up with, "Why does splitting 24 into 20 and 4 still give the same answer?”
According to a meta-analysis by Bisra et al., self-explanation supports math learning across a range of ages and topics.
At home, it can be as simple as asking, “Can you walk me through how you got that?”
As your child explains their thought process out loud, their understanding deepens, and they gain the confidence to problem-solve independently.
That’s the core of our tutoring approach at Mathnasium. Using Socratic questioning, our tutors prompt students to put concepts into their own words so they’re easier to retain and apply across math.
Ask your child to solve the same problem several different ways, then compare the results. Research by Rittle-Johnson and Star suggests that learners may develop a more flexible understanding when they compare different ways to solve a problem instead of repeating the same method each time.
Take 15 + 27. Your child might add the tens and ones separately, or round 27 up to 30, add 15, and subtract 3. Both methods lead to 42.
When your child solves a problem, ask them to try it again using another approach, such as objects, a drawing, or a different strategy. Then ask why both methods lead to the same answer.
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Before diving into the numbers, ask your child what the problem is describing. Try turning one of your student’s existing problems into a design decision they can test.
For instance, when they are working with area, give them 24 square tiles and ask: “How many different rectangles can you build with all 24 tiles?”
Your child might make a 1 × 24, 2 × 12, 3 × 8, or 4 × 6 rectangle. Then ask: “Which design has the shortest perimeter, and why?” This gives the numbers a concrete purpose. Your learner is using factors, area, and perimeter to compare real options, rather than completing a procedure with no clear reason behind it.
Jo Boaler’s research on contextualized math instruction suggests that meaningful problem settings can support conceptual understanding and help students see connections between mathematical ideas and their use.
This mirrors how Mathnasium tutors introduce new concepts, grounding a new topic in something concrete before connecting it to the abstract math behind it.
Before confirming whether an answer is correct, ask your child whether it seems reasonable for the problem.
Suppose a problem asks how many 6-inch pieces can be cut from a 48-inch ribbon, and your child answers 288 pieces. Instead of correcting them, ask: “Could one 48-inch ribbon really make 288 pieces that are each 6 inches long?”
That prompt helps them connect the answer to the size of the ribbon and recognize that multiplication does not fit the situation. They can then reconsider the operation and divide 48 by 6 to get 8 pieces.
If they rely mainly on memorized steps, they may follow the same process again and reach the same incorrect result.
If they understand the situation behind the problem, they are more likely to notice that the answer does not fit, even before they find the exact mistake.
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At Mathnasium, tutors help students understand why math works instead of relying on memorized steps
Mathnasium is a math-only learning centre dedicated to helping K–12 students of all skill levels excel in math.
In our tutoring work, we aim to help students build true conceptual understanding so they can explain their reasoning, check whether an answer makes sense, and apply familiar concepts to unfamiliar problems.
To develop that understanding, we use the Mathnasium Method™, our proprietary teaching approach, to meet students where they are and guide them forward step by step.
Each student begins with a diagnostic assessment that helps us understand their current skill level, knowledge gaps, learning goals, and how they think about math. It also shows whether they understand why a procedure works or are relying mainly on memorized steps.
Using these insights, we create a personalized learning plan focused on the skills the student needs most. The plan addresses gaps in the right order and connects conceptual understanding with accurate, efficient calculation as the student progresses.
Our specially trained tutors follow the plan closely and provide live, face-to-face instruction in a caring and fun group environment. They use mental, verbal, visual, tactile, and written techniques to make abstract ideas clear and connect procedures to their mathematical meaning.
Students also get room to think through problems before tutors step in. Our tutors ask questions, encourage students to explain their reasoning, and help them check whether an answer makes sense. This support develops problem-solving skills and greater independence in math.
Fun is an important part of our approach, too. We use game-based activities, rewards, and consistent encouragement to help students stay engaged as they improve their fluency and learn to reason through the how and why behind each method.
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
With over 1,100 learning centres across North America, there is likely a Mathnasium close to you.
For families in and around Brampton, Ontario, Mathnasium of Brampton Castlemore brings that same approach close to home, with specially trained tutors who help students connect procedures with lasting mathematical understanding.
If your child can follow steps but finds it hard to explain why they work or use them in unfamiliar problems, a free diagnostic assessment is a great place to start. From there, we create a personalized learning plan that helps your child develop the understanding and fluency they need next.
📅 Schedule a Free Assessment at Mathnasium of Brampton Castlemore
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Mathnasium of Brampton Castlemore is a math-only learning centre for K-12 students in Brampton, ON. Trusted by over a million parents, Mathnasium uses personalized learning plans and the proprietary Mathnasium Method™ to help students catch up, keep up, and get ahead on their math journey.
Our specially trained tutors deliver face-to-face instruction in a supportive and fun small-group environment, working with students to develop a deep understanding of math, build confidence, and improve academic performance.
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