Understanding Isn't Enough: What the Science of Learning Tells Us About Maths

Why meaningful practice, secure knowledge and flexible thinking belong together.

By Janey Ross | Fluency with Numbers

A child can explain a mathematical strategy beautifully on Tuesday and struggle to remember it on Friday.

They can build a number using counters, draw a diagram, talk through their reasoning and arrive at the correct answer.

And yet, when the numbers change or the familiar representation disappears, that confidence can disappear with it.

So here's a question worth asking:

If a child understands something today, does that mean they've learned it?

Not necessarily.

And that distinction matters enormously.

Because successful performance in a lesson isn't always evidence of learning that will last.

Understanding matters. Of course it does. But understanding at the point of teaching is only part of the story.

We also need children to remember what they've learned, retrieve it when they need it, recognise mathematical relationships and apply their knowledge in unfamiliar situations.

The goal isn't simply to help children get an answer today. It's to build mathematical knowledge they can still use tomorrow, next month and next year.

1. Learning isn't the same as performing

We've all seen it.

A lesson goes beautifully. Children are engaged. They're answering questions, using manipulatives and explaining their strategies.

Everything suggests that learning is happening.

But what happens a week later?

Can they retrieve the same knowledge without prompting? Can they recognise it in a different representation? Can they apply it to a problem they haven't rehearsed?

Soderstrom and Bjork (2015) make an important distinction between performance — what a learner can demonstrate during practice — and learning, which involves relatively lasting changes in knowledge and capability.

Immediate success can be encouraging, but it doesn't tell us everything about what has been retained.

This is why assessment shouldn't end when a lesson ends.

Sometimes the most revealing question isn't, “Can you do it?”

It's:

“Can you still do it, without my help, when it looks a little different?”

2. Rosenshine: Effective teaching is carefully constructed

Barak Rosenshine's Principles of Instruction (2012) offer a useful starting point.

His work brings together findings from cognitive science, research on effective teachers and studies of instructional support.

The principles aren't a rigid script or a fashionable teaching programme. They're a framework for thinking about how to make instruction more effective.

Several are particularly important in mathematics:

• Begin with a short review of previous learning.
• Present new material in small, manageable steps.
• Model processes clearly and provide worked examples.
• Ask plenty of questions and check the responses of all learners.
• Guide practice before expecting independence.
• Provide opportunities for successful independent practice.
• Revisit learning through weekly and monthly review.

Notice what's missing.

There's no suggestion that children should simply memorise procedures without understanding them.

Equally, there's no assumption that discovering a strategy once means a child is ready to use it independently.

Good instruction builds a bridge between what children already know, what they're learning now and what they'll eventually be able to do for themselves.

In mathematics, that might mean modelling how six can be seen as five and one, exploring that relationship with a ten-frame, discussing different ways of seeing the quantity and returning to it through carefully selected practice.

The representation supports the thinking. The questions reveal the thinking. The practice helps secure the learning.

Each has a purpose.

3. Cognitive load: Why secure knowledge matters

Working memory is limited.

When children are tackling unfamiliar mathematics, they may need to hold several pieces of information in mind at once.

Consider a child solving a multi-step problem involving multiplication.

If they're simultaneously trying to remember a multiplication fact, work out the operation, keep track of the numbers and interpret the question, working memory can quickly become overloaded.

Cognitive Load Theory, developed through the work of John Sweller and colleagues, helps explain why this matters (Sweller, van Merriënboer & Paas, 2019).

As knowledge becomes more securely organised in long-term memory, familiar information and procedures can be accessed with less conscious effort.

That can leave more capacity for attending to the unfamiliar parts of a problem.

But there's an important distinction here.

Reducing cognitive load doesn't mean removing mathematical challenge. It means designing learning so that unnecessary demands don't get in the way of the thinking that matters.

For a novice learner, explicit modelling, carefully chosen examples and guided practice may be particularly helpful.

As expertise develops, that support should change.

The aim isn't permanent scaffolding.

It's growing independence.

4. Retrieval and spacing: Why one successful lesson isn't enough

Here's where our everyday teaching routines can make an enormous difference.

Children don't necessarily retain knowledge simply because they've encountered it, understood it or practised it successfully once.

Learning needs to be revisited.

Research on retrieval practice shows that actively bringing information back to mind can strengthen later retention.

Research on spacing shows that distributing practice across time is often more effective for long-term learning than concentrating it into one extended session (Dunlosky et al., 2013).

The Education Endowment Foundation's review of cognitive science approaches (2021) also highlights the potential of these approaches while emphasising that their effectiveness depends on how they're implemented in classrooms.

For mathematics, this might look like:

• Briefly revisiting previously taught number relationships.
• Returning to a small set of facts over several days and weeks.
• Mixing familiar learning with carefully selected new material.
• Asking children to recall, explain or represent an idea before immediately showing them the answer.
• Checking whether knowledge remains accessible after a delay.

This doesn't mean endless worksheets or racing through flashcards.

Retrieval can be thoughtful, playful and mathematically rich.

A familiar image of six dots might invite children to recall the quantity, describe how they saw it, connect it to five and one, or represent it another way.

Importantly, not every dot-card activity is retrieval practice. A new representation might be used to develop subitising or reasoning. Retrieval happens when children actively bring previously learned knowledge to mind.

The important thing is that children aren't just encountering mathematics again. They're being supported to remember it, connect it and use it.

5. Representations: Helpful bridges, not permanent destinations

Counters, fingers, ten-frames, number lines and diagrams can be extraordinarily powerful.

They make mathematical relationships visible.

They give children something to notice, compare, discuss and manipulate.

The Education Endowment Foundation's early mathematics guidance (2020) recognises the importance of using manipulatives and representations purposefully to develop mathematical understanding.

Similarly, the What Works Clearinghouse mathematics intervention guidance (2021) recommends carefully chosen representations, mathematical language, number lines and systematic instruction.

But the purpose of a representation is not simply to make a lesson look engaging.

It is to reveal mathematical structure.

A ten-frame, for example, can help a child see seven as five and two, as three fewer than ten, or as a quantity that remains seven even when its arrangement changes.

Those connections matter.

The challenge for teachers is to help children move between representations, mathematical language, symbols and increasingly independent thinking.

Sometimes a child needs to return to a concrete representation. That's not failure.

And finger counting, in particular, shouldn't automatically be treated as a problem. Fingers can support early number understanding and mathematical relationships.

What matters is whether children's strategies are developing in flexibility, efficiency and appropriateness for the task.

We don't want children to abandon representations before they're ready. We want them to understand the mathematics well enough that they're not dependent on one representation forever.

6. Fluency: Not speed, but flexibility and freedom

Fluency is often misunderstood.

For some, it means recalling number facts as quickly as possible.

For others, the word raises concerns about pressure, timed tests and mechanical learning.

But mathematical fluency is richer than either interpretation.

It involves knowledge, accuracy, efficiency, flexibility and the ability to choose an appropriate approach.

Automatic recall is an important part of this picture. Secure number facts can reduce the demands on working memory and support more complex mathematical thinking.

But recall alone isn't enough.

A child might know that 6 × 4 = 24 without recognising how that fact connects to 3 × 4, 6 × 8 or 24 ÷ 6.

Equally, a child might understand those relationships but still need so much time and effort to reconstruct basic facts that more demanding tasks become difficult.

We need both.

Research-informed guidance also reminds us that fluency practice can take different forms. The What Works Clearinghouse (2021), for example, includes carefully designed timed activities as one possible approach to building fluency.

That doesn't mean speed should become the goal of every lesson or that all learners benefit from the same practice conditions.

Meaningful understanding gives number facts their structure. Purposeful practice helps make that knowledge secure and accessible.

That's the kind of fluency worth building.

Not hurried answers for their own sake.

Not memorisation disconnected from meaning.

But confident, flexible mathematical thinking supported by knowledge children can rely on.

7. What does this look like in practice?

It doesn't require a complete reinvention of the mathematics lesson.

It requires greater intentionality.

Start by identifying the mathematical idea children need to learn.

Find out what they already know.

Model the new learning clearly, using representations that expose the underlying structure.

Ask questions that make children's thinking visible.

Provide guided practice, with feedback, before expecting independence.

Return to important knowledge over time.

And gradually change the task, context or representation so children learn to recognise and apply the mathematics beyond the original example.

For example, when exploring a quantity of eight, we might ask:

What do you notice?

What do you see?

How did you see it?

What stayed the same? What's changed?

How do you know?

Show me another way.

Prove it!

These questions invite children to notice structure, explain relationships, make connections and justify their thinking.

But questioning alone isn't the whole lesson.

Children also need explicit teaching, successful practice, feedback and opportunities to retrieve what they've learned.

It isn't a choice between good explanations and secure knowledge. Effective mathematics teaching deliberately develops both.

8. A different question for maths education

Perhaps we've spent too long debating whether children should understand mathematics or remember it.

Whether they should use concrete materials or work abstractly.

Whether fluency means recall or reasoning.

These aren't helpful either-or choices.

The science of learning suggests a more productive set of questions.

What knowledge does this child need?

How can I make the mathematical structure clear?

How much support is appropriate right now?

What will help this learning become secure?

When will we revisit it?

And how will I know whether the child can use it independently?

Those questions take us beyond whether a lesson looked successful.

They ask whether our teaching is creating learning that lasts.

9. The thinking behind The Fluency Atlas

The Fluency Atlas grew from a simple question: What would mathematical practice look like if we designed it around how children learn?

Not more worksheets for the sake of it.

Not endless repetition.

And certainly not speed for speed's sake.

Instead, carefully sequenced opportunities to see, notice, connect, represent, explain, retrieve and apply mathematical ideas.

Beginning with Region 1: Seeing Quantity, the Atlas explores how children recognise quantities, notice numerical structure, conserve quantity, connect representations and build increasingly flexible number understanding.

The resources are designed to invite purposeful mathematical discussion, carefully selected practice and opportunities to revisit important ideas over time.

A dot pattern becomes more than something to count.

A ten-frame becomes more than a picture.

A familiar number becomes an opportunity to notice a new relationship.

The Atlas draws on research-informed principles of instruction, memory and mathematical learning. It is an approach to resource design, not a claim that any single activity or sequence has been independently proven to produce a particular outcome.

And it isn't about every child following an identical route.

It's about making the mathematical thinking visible, the practice purposeful and the learning increasingly secure.

Because meaningful practice isn't simply about doing more maths.

It's about making the maths we do count.

Understanding matters. But learning must last.

Children deserve more than lessons that look successful.

They deserve mathematics that makes sense.

Knowledge they can remember.

Relationships they can recognise.

Strategies they can select and use with confidence.

And opportunities to revisit, deepen and extend their thinking.

We don't need to choose between understanding and automaticity.

Between representations and recall.

Between reasoning and practice.

Understanding is essential. But understanding that can be remembered, retrieved, connected and applied is where learning becomes truly powerful.

Meaningful practice.

Strong foundations.

Flexible mathematical thinking.

That's what we're working towards.

Keep the Fluency Conversation Going

If you're a teacher, practitioner or parent who wants to understand not just what helps children learn mathematics, but why, I'd love you to join the conversation.

Not That, But This… is my email series exploring research-informed mathematics teaching, practical fluency routines and small changes that can make a meaningful difference.

You'll find:

• Research-informed ideas explained in accessible language.
• Purposeful fluency routines you can use in classrooms or at home.
• Thoughtful approaches to representations, retrieval and mathematical reasoning.
• Practical ways to build secure, flexible number understanding.

No gimmicks. No quick fixes.

Just thoughtful teaching, purposeful practice and ideas worth revisiting.

Explore Fluency with Numbers and join the conversation.

References and Further Reading

The following publications include foundational research and subsequent evidence reviews. Older sources are retained where they remain important to the theoretical basis of the article.

Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students' learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), 4–58.
https://doi.org/10.1177/1529100612453266

Education Endowment Foundation (2020). Improving Mathematics in the Early Years and Key Stage 1.
https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/early-maths/

Education Endowment Foundation (2021). Cognitive Science Approaches in the Classroom: A Review of the Evidence.
https://educationendowmentfoundation.org.uk/education-evidence/evidence-reviews/cognitive-Science-approaches-in-the-classroom

Rosenshine, B. (2012). Principles of instruction: Research-based strategies that all teachers should know. American Educator, 36(1), 12–19, 39.
https://www.aft.org/ae/spring2012/rosenshine

Soderstrom, N. C., & Bjork, R. A. (2015). Learning versus performance: An integrative review. Perspectives on Psychological Science, 10(2), 176–199.
https://doi.org/10.1177/1745691615569000

Sweller, J., van Merriënboer, J. J. G., & Paas, F. (2019). Cognitive architecture and instructional design: 20 years later. Educational Psychology Review, 31, 261–292.
https://doi.org/10.1007/s10648-019-09465-5

What Works Clearinghouse (2021). Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades. Institute of Education Sciences, US Department of Education.
https://ies.ed.gov/ncee/wwc/PracticeGuide/26