The difference between
understanding Maths and
memorising Maths

A student can differentiate a function perfectly. Chain rule, product rule, quotient rule — flawless. Then you ask them what a derivative actually is, and there is nothing there. Not a wrong answer. Silence. They have been doing this for eighteen months and nobody ever told them what it meant.

That student is not lazy and they are not weak at Maths. They have done exactly what was asked of them. They learned the procedure, they practised it, they can execute it. By every measure the system uses week to week, they are doing fine.

And then June arrives, Section B puts a derivative inside a word problem about the rate a tank is filling, and they freeze completely. Not a wrong answer — a blank page. There is nothing to fall back on, because there was never anything underneath the procedure in the first place.

This is the single most common reason Irish students underperform in Leaving Cert Maths. It is worth understanding properly.

Two kinds of knowing

Maths education research draws a distinction between two types of knowledge, and the difference matters more than almost anything else in how a student experiences the subject.

Procedural knowledge is knowing how. The steps. The method. What to do when you see a particular kind of question. It is learned by repetition and it is demonstrated through action.

Conceptual knowledge is knowing why. One influential definition describes it as knowledge that is rich in relationships — a connected web in which the linking relationships run through the individual facts, so that all the pieces are joined to one another rather than sitting in isolation.

That phrase is the whole thing: a connected web. Procedural knowledge is a list. Conceptual knowledge is a network. And when you hit a question you have never seen before, a list gives you nowhere to go, while a network gives you dozens of routes in.

"Procedural knowledge is a list. Conceptual knowledge is a network. Under exam pressure, a list gives you nowhere to go."

What the research actually says

Here is where I want to be careful, because the honest picture is more interesting than the version usually presented.

The research does not say that memorisation is bad and understanding is good. It says both matter, and that they build on each other. Studies tracking students over time find that conceptual knowledge predicts later gains in procedural skill, and that procedural practice in turn feeds back into deeper conceptual understanding. The relationship runs in both directions.

The consensus among researchers is not conceptual knowledge or procedural knowledge. It is conceptual and procedural — with the question being when and where each is developed.

So this is not an argument against learning methods. You absolutely need to know how to differentiate, and you need to be able to do it quickly and accurately without thinking hard about it. Fluency matters. Practice matters.

The problem is what happens when procedure is the only thing a student has. And on that, the research is blunt: if procedural knowledge is the limit of a person's learning, there is no way to reconstruct a forgotten procedure. Conceptual understanding alongside procedural skill is far more powerful than procedural skill on its own.

Why students end up memorising anyway

There is evidence that students default to procedures wherever they can, and reach for concepts only when they must. That is not a character flaw — it is entirely rational. Procedures are faster, they feel safer, and in most week-to-week school assessment they are enough.

The Irish system reinforces this at every turn. There is a large syllabus and a fixed amount of time to cover it. A class test on differentiation will mostly ask you to differentiate things. A student who memorises the rules will do perfectly well in October, and will have no signal at all that anything is missing.

The signal never comes in first term. It comes in the mocks, or in June, when Section B asks them to recognise that a problem about a filling tank is a calculus problem in the first place. And by then it is late.

There is also something quieter going on. Memorising feels productive. You can sit down for two hours, work through twenty differentiation questions, get most of them right, and finish feeling you have studied hard. Building understanding feels slower and less comfortable — it involves sitting with confusion rather than accumulating ticks. Given the choice, most students choose the version that feels like progress.

What it looks like from the outside

The two are surprisingly hard to tell apart until pressure is applied. Both students get the same answers to standard questions. Both look fine in class. The difference only shows when the ground shifts.

Look at the last row. Section B is worth 150 marks on each paper — half the exam. It is specifically designed to test whether you can identify which mathematics a situation requires, before you do any mathematics at all. Memorised procedure cannot help you with that step, because the question does not tell you which procedure to reach for.

A test you can do on yourself

Pick a topic you feel reasonably confident about. Then try these, honestly.

• →Explain the topic out loud to somebody who does not do Higher Level Maths. If you can only recite steps, that is procedure. If you can say what it is for, that is understanding.
• →Take a standard question and change something about it. Different variable, different context, worded backwards. Can you still start?
• →Deliberately forget the formula. Can you rebuild it, or reason your way to the answer without it?
• →Ask yourself why the method works. Not what the steps are — why they get you the right answer.
• →Find the same topic inside a Section B question from a past paper. Could you have spotted it was that topic without being told?

Most students find they have understanding in some topics and pure procedure in others. That is normal and it is useful information — the topics where you freeze on these questions are exactly where your grade is currently sitting.

How to move from one to the other

The good news is that this is fixable, and it does not require starting the course again.

Ask why, every time. When you learn a method, do not stop at being able to do it. Ask what it is doing and why it works. If your teacher has not explained it, look it up. This single habit changes everything over a year.

Say it out loud. Explaining a topic to somebody else exposes gaps immediately. You will find yourself saying "and then you just..." — and that is precisely where the understanding stops and the memorising begins.

Work with the meaning, not just the symbols. A derivative is a rate of change. An integral is an accumulation. A logarithm answers "what power?" Attach language and meaning to the notation, and the notation stops being arbitrary.

Do Section B early and often. Most students leave contextual questions until the spring because they are harder. They are harder precisely because they test the thing that matters. Start them in first term, when there is still time for it to change how you learn everything else.

Sit with confusion for slightly longer. When something does not make sense, the instinct is to skip to the worked example and copy the method. Give it two more minutes first. That discomfort is what understanding feels like while it is being built.

"Understanding is uncomfortable while it is happening. Memorising is comfortable and then catastrophic. Choose your discomfort."

Why this is the whole of what we do

Equanimity means mental calmness and composure, especially in difficult situations. People sometimes assume that is about relaxation techniques or breathing exercises before an exam. It is not.

Calm in an exam hall comes from having somewhere to go. The student who freezes on Section B is not panicking because they are an anxious person. They are panicking because they have genuinely run out of options — the one tool they brought does not fit, and there is nothing else in the bag. That is a rational response to a real problem.

The student who understands has a network to move around in. When the obvious route is blocked they can try another. They may still find the question hard, but hard is not the same as hopeless, and their body knows the difference.

That is why we teach the way we do. Understanding first, then fluency built on top of it. Small groups of five, so that when a student says "I don't get why that works," there is time to actually answer them rather than move on. And no student sitting quietly at the back deciding that Maths is not for them, because in a group of five there is no back.

Memorising will get a student through October. Understanding is what is still standing in June.