Understand Math Instead of Memorising It: How to Do It

Understanding math means grasping the core of a rule – why it holds and when to apply it – instead of blindly running through a sequence of steps. If you understand, you can reconstruct a forgotten formula when you need to; if you only memorised it, you are stuck without it. The good news: understanding is not a gift you are born with, it can be built on purpose.

What does „understanding math" instead of memorising it mean?

The difference between the two is subtle but decisive. Memorising means storing a procedure – say, „to divide fractions, flip the second one and multiply" – as a pure instruction. Understanding means knowing why that trick works and how you recognise that you need it in the first place.

The difference becomes visible the moment a problem looks slightly different from the ones in your workbook. A memorised procedure fits only the exact form you practised. Understanding, by contrast, carries over the variation, because you transfer the underlying idea to the new situation. That is precisely what exams test – which is why pure cramming so often collapses there.

This does not mean you have to derive every formula yourself. It means that meaning comes first: what does this quantity measure? What happens, intuitively, when I change it? Only once that is in place does the calculation procedure become a tool rather than a magic spell you merely hope to recite correctly.

Why does pure memorisation fail in math?

Mathematical thinking runs on working memory – that small mental scratchpad where we hold intermediate results while we calculate. Its capacity is tightly limited. If you have to assemble a problem from many half-understood individual steps, you fill that scratchpad instantly – and lose track in the middle of the calculation.

Understanding acts like compression here: one genuinely grasped relationship takes up a single slot where five loose rules would otherwise sit side by side. Then there is stress. In a study of 154 first- and second-graders, higher math anxiety was linked to weaker performance – and specifically among the children who used the more demanding, working-memory-intensive strategies, because anxiety eats up exactly the capacity those strategies need (Ramirez et al., 2013). In short: anything that blocks working memory – exam stress just as much as a tangle of unexplained steps – sabotages the calculation. Understanding relieves the load; blind cramming adds to it.

Does understanding come before the procedure – or the other way round?

This is where a look at the learning science pays off, because the question is an old one. Bethany Rittle-Johnson and colleagues summarise the state of research in a widely cited review (2015): conceptual and procedural knowledge are not a one-way street but reinforce each other both ways. Understanding a concept better makes you surer in the procedure – and mastering a procedure often deepens your grasp of the concept. The two develop iteratively, in small alternating steps.

For practice, though, the authors draw a clear recommendation: as a starting point, it is usually better to begin with the concept and build the procedure on top of it, not the reverse. That matches everyday experience – a formula whose meaning you understand before you compute with it sticks; a meaningless formula slips away within days. „Understanding instead of memorising" therefore does not mean pitting one against the other, but choosing the right order: grasp first, automate second – and then let both pull each other up.

How do I practise math so that it sticks?

Understanding alone is not enough – it has to become reliable skill through practice. But how you practise is what matters. The common approach is blocked practice: twenty problems of the same type in a row. It feels good, because you quickly become fluent – but it mostly trains you to repeat the same procedure on autopilot, without recognising when it is actually called for.

The more effective alternative is interleaving: different problem types mixed together. Doug Rohrer and colleagues had 126 seventh-graders work the same practice problems for three months – one half blocked, the other interleaved. On an unannounced test a month later, the interleaved group scored 74 percent correct, the blocked group only 42 percent – a large effect (Rohrer, Dedrick & Stershic, 2015). The reason: when you practise a mix, you first have to decide for each problem which method even fits. That matching – recognise the problem, choose the method – is the heart of understanding, and it is exactly what blocking skips. Interleaving feels harder while you practise; that „desirable difficulty" is the price you pay for the material staying retrievable later.

Three principles make practice work:

1. Mix, don't block. Solve problems of different types in random order as soon as you know the basic forms. It feels harder – and that is precisely why it works better.
2. Space it out, don't bundle. Four short practice sessions across the week beat one long block the evening before the exam. Learning statistics, for instance, thrives on this spacing, because many similar methods are easy to confuse there.
3. Work without the model answer. Cover the worked solution and wrestle for the path yourself. Active recall consolidates far more strongly than rereading – even though rereading feels safer.

How do I truly understand a mathematical concept?

Understanding is not a flash of insight but a set of concrete moves:

• Explain it in your own words. If you can explain a statement out loud without jargon, you have understood it. Where the explanation stalls is exactly the gap that still needs closing.
• Ask why, not just how. Why do you flip when dividing fractions? Why does minus times minus give plus? The answer anchors the rule instead of leaving it dangling.
• Connect multiple representations. Seeing a function as a formula, a graph, and a table links the same concept along several routes – which makes it more robustly retrievable.
• Tie it to what you already know. New material sticks better when it docks onto existing knowledge. Percentages, for example, are just another way of looking at fractions, not an entirely new topic.

This sets math fundamentally apart from classic fact learning. When you cram vocabulary or dates, you rightly reach for repetition techniques like those in memorising facts with mnemonics. In math, by contrast, a memory trick is at most a bonus – the foundation stays the grasped principle. For more ways to study a subject strategically rather than by rote, see our Subjects & Topics section.

So is memorisation useless in math?

No – and this honesty belongs in the picture. A minimum of memorised knowledge is in fact a prerequisite for understanding. Times tables, basic formulas, or the first few square numbers should be automatically retrievable. The reason is working memory again: if you have to recompute 7 × 8 every time, you have no capacity left for the actual problem. Automated basic facts are not a contradiction to understanding but its foundation.

The mistake, then, is not memorisation itself but putting it in the wrong place: cramming complex procedures blindly instead of penetrating them. The myth that „you either have a math brain or you don't" also fails to hold up against the research: understanding develops iteratively through the right order and the right practice – it is a trainable skill, not an innate talent. The useful rule of thumb: firmly automate the few basic facts, but understand every procedure.

Conclusion

Understanding math instead of memorising it does not mean avoiding practice – quite the opposite. It means going in the right order: first grasp the meaning of a concept, then make it secure through spaced, mixed practice, and only firmly automate the truly elementary facts. If you want to have material explained step by step and ask targeted follow-up questions, you can use an AI tutor like the one from LearnCastAI, which breaks concepts down in plain words rather than just spitting out answers. The difference shows up at the latest in the exam – when the problem looks different from what you practised, and you can still solve it.