Math Vale

What order to learn maths in, from counting to calculus

Maths is a dependency graph, not a list: each topic needs specific earlier ones, and almost every wall people hit is a missing prerequisite rather than a hard topic. This is the full order, and what each stage is really for.

Updated

Almost nobody is bad at algebra. What usually happens is that fractions were never finished, and algebra is where that shows up — because an equation with fractions in it needs both skills at once, and only one of them is missing.

That is worth saying plainly, because the usual advice is to work harder at the topic that is going wrong. The more useful move is to go back one step and check the thing it depends on. This page is that map.

The seven stages, in order

Each stage assumes the one before it. You can move quickly through a stage you already have, but skipping one does not work — the gap simply appears later, wearing a different topic’s clothes.

  1. Number sense: counting, comparing, and the four operations

    Reading and ordering whole numbers, then addition, subtraction, multiplication and division. The goal is not speed — it is that these stop taking attention, so there is attention left over for the actual problem.

  2. Place value, negatives, and the order of operations

    Why 3.07 and 3.7 are different, what a negative number does under each operation, and the convention that decides what 2 + 3 x 4 means. This stage is where arithmetic becomes reliable rather than lucky.

  3. Fractions, decimals, percents and ratios

    The stage people skip, and the one that costs the most later. A fraction is a division that has not been carried out, a decimal is a fraction with a fixed denominator, and a percent is a fraction out of a hundred. Understanding that they are three notations for one idea is the whole job.

  4. Algebra: letters, equations and factoring

    Naming an unknown and doing the same thing to both sides. Algebra is not a new kind of maths — it is arithmetic with a name for the number you do not know yet, which is why fraction gaps surface here so violently.

  5. Geometry: shape, measurement and the Pythagorean theorem

    Angles, area, volume, similarity, and the theorem that turns out to be a distance formula in disguise. Geometry can run partly in parallel with algebra, but similar triangles and coordinate work need ratios first.

  6. Functions and trigonometry

    A function is a rule with one output per input; a graph is a picture of that rule. Trigonometry then studies the functions that repeat. Both need fluent algebra, because every question here becomes an equation halfway through.

  7. Calculus

    Limits, derivatives and integrals — what a quantity does as something else approaches a value, how fast it changes, and how much of it accumulates. Calculus needs functions and algebra to be automatic, because the calculus part of a calculus problem is usually the short part.

How to find your actual gap

Work backwards from where it stops making sense, not forwards from where you are comfortable.

If a topic feels impossible, take its prerequisites one at a time and try a mid-difficulty problem in each. The one that is slow rather than wrong is usually the gap — slowness is the symptom that comes before errors, and it is the cheapest one to catch.

Every lesson on this site names its prerequisites at the foot, so you can walk backwards until something is easy and then walk forwards again.

Start at the beginning of each stage

One representative skill per stage, each with a full lesson and worked examples.

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