What does it mean to “borrow”?

“One Thread” series, article 1

Working with teachers and tutors, we keep hearing the same question: how do you explain something so that every child understands? We have no ready-made recipes, but we do have observations and techniques we would like to share. We will try to share our experience regularly, one small example from school maths at a time, and to follow where it leads — from primary school all the way into adult life.

Remember how we ourselves learned to subtract in a column. Fifty-two minus seventeen. You can’t take seven from two — so we “borrow” from the five: we put a dot over it, and the two turns into twelve. Twelve minus seven is five. The five with the dot becomes a four, and four minus one is three. The answer: thirty-five.

All correct. But try asking a second-grader: what exactly did we borrow? And why did the two become twelve rather than three — we only borrowed one, didn’t we? Quite often we will hear “because that’s the rule.” And that is an honest answer: the child has memorised a sequence of steps — the dot, a one on the left, minus one from the neighbour — but does not know what stands behind it.

And what stands behind it is very simple. Fifty-two is five tens and two ones. If in first grade the child tied counting sticks into bundles of ten, they see it literally: five bundles and two loose sticks. You can’t hand over seven loose sticks — there are only two. So you have to untie one bundle. Now there are four bundles and twelve loose sticks. That is where the “twelve” comes from: we borrowed not a one but a ten, and it fell apart into ten ones. Now you can hand over seven sticks, and five are left; out of four bundles you hand over one — three are left. That dot over the five is simply a note that one bundle has been untied.

The difference between these two children — the one who memorised the dot and the one who untied the bundle — is invisible in second grade. Both solve the example correctly, both get top marks. The difference appears later — and grows the further their studies go.

Third grade: a thousand minus three

There is an example every primary teacher knows as a stumbling block: a thousand minus three, in a column. You can’t take three from zero — so you borrow from the neighbour. But the neighbour is zero too. And so is the next one. A child who only has the rule “put a dot over the neighbour” gets lost here: what do you put the dot over if there is nothing there? The answers vary widely — all the way up to 1003: since you can’t subtract from zero, the child simply subtracts zero from the three.

A child who has bundles sees a different picture. A thousand is one big bundle. We untie it and get ten hundreds. But a place can never hold more than nine: nine hundreds stay where they are, and we untie the tenth. We get ten tens. Nine of them stay in the tens place, we untie the tenth — and in front of us are ten ones. From those we take three, and seven are left. Nine hundreds, nine tens, seven ones: 997.

And if the number already had ones of its own — say, a thousand and two minus three — the ten untied ones are joined by two more, making twelve. Take away three — nine, and the answer is 999. It is exactly the same idea as in the fifty-two example, simply applied several times in a row. There is nothing new to memorise.

Fifth and sixth grade: fractions

Three minus zero point four eight. A fifth-grader is told: “add zeros and subtract as usual.” And once again one pupil memorises a new rule, while another recognises an old action. We untie one of the three wholes — ten tenths. We keep nine tenths and untie one — ten hundredths. Now we have two wholes, nine tenths and ten hundredths. From ten hundredths we take eight — two are left. From nine tenths we take four — five are left. The answer: two point five two. It is the same as a thousand minus three, only now the bundles are to the right of the decimal point. And the zeros the fifth-grader is asked to “add” are simply the places for the tenths and hundredths that appeared when we untied.

And here is an example with no tens at all: five minus two and three quarters. You can’t take three quarters from zero quarters. Sounds familiar? We untie one whole — but now it breaks not into ten parts but into four quarters. Five becomes four wholes and four quarters; subtract — and we get two and one quarter. For someone who understood the bundles this is a discovery: it turns out you can “borrow” more than just a ten. For someone who memorised the dot, it is one more rule unconnected to the previous ones. This is exactly how mathematics turns, for a child, into a heap of rules that is impossible to keep in one’s head.

And in real life?

The bus leaves at a quarter past three, and it is now twenty to two. How long is the wait? An adult makes the very same borrow without noticing. You can’t take forty minutes from fifteen — so we untie one hour. Only this “bundle” holds not ten minutes but sixty. Three hours fifteen minutes is two hours seventy-five minutes; take away one hour forty — and one hour thirty-five is left. A child who understood the bundles will handle this at once. A child who memorised “write a one on the left” will turn fifteen minutes into a hundred and fifteen, get the answer “one hour seventy-five minutes” — and stare at it for a long time without understanding where it went wrong.

There is another everyday method that understanding bundles makes natural. Fifty-two minus seventeen can be worked out like this: take away twenty at once — that is just two whole bundles — and thirty-two is left. But we took away three too many, so we give them back: thirty-five. This is how people who “are good with numbers” calculate in their heads, and there is no talent involved: they simply see seventeen not as two digits but as two tens minus three. In the same way a shop assistant counts out change from a thousand for a purchase of three hundred and seventy som: adds thirty to make four hundred, then six hundred to make a thousand — and never subtracts in a column.

And here is what matters. An adult who reaches for a phone to take three hundred and seventy from a thousand is no less clever than the shop assistant. It is just that in second grade they were shown the dot, not the bundle. And ever since, arithmetic has been for them a procedure to carry out on paper rather than a way of seeing numbers.

Humanity took centuries to get here

A dot over a digit seems a minor point of school technique. In fact it stands for one of humanity’s longest inventions. A Roman merchant could borrow too: on his counting board every line was a place, and borrowing was a physical action — take one pebble off one line and lay out ten on the next. But he could not do this in writing. In Roman notation, CCV, the value of a sign does not depend on its position, which means there is no place to borrow from. People calculated on the board and only wrote down the result on the tablet.

To calculate directly in writing, two things were needed: a notation in which a digit’s value depends on its position, and a sign for an empty place — zero. Such a notation took shape in India, reached Europe through Arab scholars, and did not take root there quickly: for a long time it was treated with suspicion. Our short video stories on Instagram tell about this journey: “How an empty place became a number” and “Outlawed digits”. When a child unties a bundle of sticks, in five minutes they travel a path on which humanity spent many centuries.

Why this is a question of teaching method

Borrowing in column subtraction, a thousand minus three, decimal fractions, mixed numbers, hours and minutes, change in a shop — in the textbook these are six different topics in different grades. In reality they are one and the same action: exchanging a large unit for smaller ones. A child who has mastered this action once, in second grade, from then on simply recognises it in new clothes. A child who memorised the dot learns six rules — and each new one is laid next to the old ones without resting on them.

This has a price, and it should be named honestly. Bundles take more lesson time than showing the dot on the board. But that time comes back: in third grade there is no need to deal with zeros separately, in fifth — no need to explain why you add zeros after the decimal point. And there is one trap: sticks by themselves guarantee nothing. If a child unties a bundle because they were told to, the sticks become the same ritual as the dot. What works is not the object in the hand but the moment when the child explains in their own words why they did it.

So our suggestion for tomorrow is very simple and suits any grade. When a pupil solves an example with borrowing — whether it is a column, fractions or hours — ask them: “What did we borrow?” If they answer “a ten”, “a whole”, “an hour” — the thread holds. If they answer “a one” or “because that’s the rule” — that is no reason for a bad mark, but a reason to get out the sticks, even if a sixth-grader is sitting in front of you. One five-second question will show what is in the child’s head: an idea or a rule.

The same idea — in other hands

Bundles of sticks are not the only way. Here are three more, from different schools, and they all show the same thing: a place value is a bundle, and borrowing is exchanging it.

What the thread rests on

• Grade 2. Borrowing in column subtraction means untying a bundle: a ten falls apart into ten ones.
• Grades 2–3. Hours and minutes: when subtracting, an hour is untied — and it holds sixty minutes, not ten.
• Grade 3. A thousand minus three: if the neighbouring place is empty, untie the bundle of the next one — the same exchange several times in a row.
• Grade 5. Mixed numbers: a whole is exchanged for four quarters, just as a ten is exchanged for ten ones.
• Grade 5. Decimals: 3 − 0.48 is the same exchange, only to the right of the point; the added zeros are the places for tenths and hundredths.

You can discuss the article, share your own techniques and suggest a question for the next issues in the “One Thread” Telegram channel @youraizone.