What order to teach arithmetic in

Counting looks like the thing that comes before arithmetic rather than part of it. It is the other way round: addition is counting on, subtraction is counting back or counting the gap, multiplication is counting in equal groups, and division counts how many groups fit. Nothing later replaces counting — it only makes it quicker.

That matters for teaching because it explains what a learner is doing when they are slow. A child adding by counting on their fingers is not doing a worse method than the one in the textbook; they are doing the method the textbook is a shortcut for. Speed comes from doing that method enough times that the answer arrives first.

Subtraction is addition read backwards. Twelve take away four can be answered by asking what has to be added to four to make twelve, which is the same question to a learner who knows their number bonds and a much harder one to a learner who does not.

So addition comes first, and specifically the bonds that make ten. A learner who knows that seven and three make ten can borrow and carry later without being told to; a learner who does not will treat borrowing as a rule with steps to remember.

Two readings of subtraction are worth teaching as soon as the facts are secure: taking away, which is what a counter in a twenty-frame shows, and finding the difference, which is the gap between two marks on a number line. Same answer, different picture, and the second reading is the one that turns into mental arithmetic later.

Multiplication is counting in equal groups, and the array is the picture that makes it work: five rows of four dots is twenty, and turning the rectangle sideways gives four rows of five, still twenty. That single movement is the whole of the commutative law, taught without a sentence about laws.

Division is that array read the other way. Twelve shared between four can be seen as how many fours fit into twelve, or as twelve split into four equal groups. Both readings are useful, and both are visible in the same picture, which is why they belong in the same rung rather than a year apart.

Multiplication tables are worth memorising, but after the array, not instead of it. A table learned from understanding has a place to be reconstructed from; a table learned only by repetition has no way back when a fact slips.

Everything before this rung can be done by counting, and everything after it depends on the fact that ten ones can be bundled into one ten and moved into a column. That bundle is what lets a number be written with two digits, then three, then any number of them, without inventing new symbols.

Bundling is best met physically before it is met as a rule: twenty squares in ten-column rows, where each completed row visibly becomes a rod, and the readout says one ten and six ones. A learner who has watched the tenth counter complete a row will not need to be told twice why 16 is written the way it is.

Rounding is the same number line used for a different purpose: instead of asking where a number is, it asks which landmark it is nearest. Thirty-eight rounds to forty because it sits closer to forty than to thirty.

It is taught here because it makes every later rung self-checking. An answer to thirty-eight plus thirty-one cannot be seventy-eight, because forty plus thirty is seventy. Estimation is not a separate topic to be fitted in; it is what the number line gives you for free once rounding is understood.

All three are ways of writing parts of a whole, and the order is the order of how much working is left to do. A fraction says how many equal parts were cut and how many are being counted, which is the whole idea and needs no arithmetic. A decimal is a fraction whose division has already been carried out: three quarters is 0.75. A percentage is a decimal written out of a hundred, so the bottom number is always 100 and only the top gets written.

The reason to teach them together rather than in separate years is that the same amount can be written three ways, and moving between the three notations is the skill. Half the confusion around percentages comes from treating them as a new kind of number rather than as a fraction with a fixed denominator.

Equal parts come first within the rung, because without them a fraction names nothing. A bar cut into one long piece and one short one is two pieces, but neither is a half.

Negative numbers are what the number line does when you keep going left past zero. Nothing new is required beyond the line itself: further left is smaller, the sign says which side of zero you are on, and adding moves right while subtracting moves left.

Teaching it on the line also disposes of the commonest error, which is treating a minus sign as an instruction to make a number smaller. On the line, subtracting a negative moves you right, and the picture agrees with the rule instead of contradicting it.

A fraction compares a part to its whole. A ratio compares two shares to each other, which is a genuinely different question: twelve parts split four and eight is a ratio of 4 : 8, which simplifies to 1 : 2 and describes the same division of the same whole without ever mentioning the whole.

It sits last because it needs the fraction idea, the division idea and the simplification habit all at once, and because the distinction between comparing to the whole and comparing to each other is the thing most often muddled. Taught after everything else, there is nothing left to introduce and the comparison can be the whole lesson.

The instinct when someone is struggling is to explain the current topic more slowly. It is usually more effective to check the rung below, because most stuck points are a missing foundation rather than a difficult new idea.

Someone struggling with subtraction facts usually needs their addition bonds quicker. Someone struggling with long multiplication usually needs the arrays at rung four. Someone struggling with percentages usually needs to see that a percentage is a fraction out of a hundred, which is rung ten. The gap is almost always down the path, not further along it.