A Mathematician Is Closer to a Swimmer Than to a Librarian

It happened to me. And I’m sure it has happened to you too.

You studied for a math test the same way you had prepared for that history test the week before — and when the results came back, it felt like drowning.

What had gone wrong?

You had spent hours reading the manual. You understood the concepts. But in the exam room, doubt crept in. The questions seemed unfamiliar, as if they belonged to an entirely different topic. The examples you had studied no longer seemed useful.

The truth is simple: most of the work hadn’t been done at all.

You read the instruction manual, but never practiced with its tools. Your brain never had the chance to systematize procedures, nor to adapt and evolve through them.

And that’s the real difference between a good and a struggling math student.

The brain of an experienced math student isn’t “naturally talented” — it has been trained to organise new information within a growing mental framework. This allows them to easily access and combine different pieces of knowledge when solving problems.

But how is this framework created? Is it something we are born with? And if not — is there no hope for the rest of us?

The answer is clear: no, it is not innate — at least, not for the vast majority of people.

This framework begins to develop from a very early age and grows as we do. But it doesn’t appear by magic. It demands consistent, deliberate practice.

Just like a great swimmer trains in the pool every day, a great math student builds their skills through daily exercises. Each exercise strengthens the mental framework — much like each lap in the pool improves the swimmer’s strength and endurance.

Each repetition makes the next challenge easier and prepares the student for more complex tasks ahead.

And here lies the biggest trap: An unfortunate number of students fail to realise this.

They approach math the same way they study history or geography: by reading a textbook, memorising facts, and trying to link them in a meaningful way.

For math, that is merely the starting point — and by itself, almost meaningless.

Want to know if someone is really studying math? Look for the paper and the pencil.

Not all exercises are born the same, though. Each one fits its own purpose and thus they must be chosen and structured with care.
It is essential to start with questions that are simple enough to build confidence and establish core ideas — and then progress toward problems that demand deeper reasoning and integration of concepts.
This gradual increase in complexity is what truly strengthens the mental framework and prepares students to face the unexpected.

And here, another element becomes crucial: guidance.

Whether it comes from a teacher, a well-designed analogue resource, or a digital platform, students need an expert hand to curate these questions and guide their progression. Without this structure, even the most dedicated effort can become inefficient or frustrating.

In my own teaching, this is why I built the GHOST BUST Framework.
It was designed precisely to address this challenge — to give students a path where every step counts, where exercises are more than repetition: they are strategic, deliberate training sessions that gradually turn them into confident, independent problem-solvers.

Want to master math? Think like a swimmer, not a librarian.