Start here: he is not bad at maths.
Researchers in Kolkata worked with about two hundred children who ran market stalls and also went to school. Doing sums for customers, they were right roughly nine times in ten. Given the same arithmetic on paper in school format, they dropped to about a third — and on bare abstract division, almost none of them succeeded. Same children, same maths, same day.
The competence was there the whole time. What defeated them was the written form — the columns, the digits, the assumption that everyone knows what the 3 in 34 is worth. That assumption is what this page is about.
What is actually missing
To multiply 34 by 6, a child has to do something nobody teaches explicitly: break 34 into thirty and four, multiply each part, and put the pieces back together. If the 3 in 34 is just "three" to him, the big block comes out as 18 instead of 180, and the answer is wrong by a mile — not by a little.
This is why more table practice does not help. He is not failing to recall 6 × 3. He is failing to see that the sum contains 6 × 30, and that thirty is a thing you can hold.
The hurdle has a name
Treating ten as one thing and ten things at the same time is the single hardest idea in early number, and everything else here rests on it. A bundle of ten matchsticks is one bundle. It is also ten sticks. Both are true at once, and a child has to hold both.
Until that lands, "carry the one" and "borrow" are magic words. After it lands, they are just descriptions of something he has already done with his hands.
Six questions to find out where he actually is
Each takes under a minute. Do them casually, not as a test. And ask him to explain — children often build the right thing while holding the wrong idea, and the explanation is where that shows.
1. Put out 25 sticks. He writes "25". Circle the 5 — "does this bit have anything to do with how many sticks?" Then circle the 2.
Secure: Points at five sticks, then at twenty sticks — two whole bundles.
Not yet: Points at two sticks for the 2. This is the big one: he reads each digit at face value, so the 2 is just two.
2. "What is ten more than 47?" Then, "ten more than 95?"
Secure: 57, then 105, straight away and without counting.
Not yet: Counts up in ones. Ten is not yet one thing he can add in a single move.
3. "Show me 34 with these sticks. Now show me a different way."
Secure: Three bundles and four. Then unties one: two bundles and fourteen.
Not yet: Cannot find a second way. His place value is rigid, and subtraction will break on it.
4. "Write down one hundred and two."
Secure: 102
Not yet: 1002 — he is writing what he hears, piece by piece, rather than placing digits in columns.
5. "What does the 3 mean in 35? And in 305? And in 350?"
Secure: Thirty. Three hundred. Three hundred.
Not yet: "Three" every time. Position is not yet driving value for him.
6. For ages 8+: "You know 6 × 3 = 18. So what is 6 × 30? And 34 × 6?"
Secure: 180, then 204 — reached by splitting 34 into 30 and 4.
Not yet: Stalls on 6 × 30, or answers 18. This is the exact child this page is for.
The four stages
Ten minutes a day. Do not move on until the "you will see" line is true — each stage is what the next one stands on.
- Stage 1
Make a ten with his own hands
Loose matchsticks, counted into tens, each ten held with a rubber band. He does the bundling — that act is the whole point. A bundle he tied himself is a ten he believes in; a ten handed to him ready-made is somebody else's claim.
You will see: Asked how many in a bundle, he says "ten" without untying it to check.
- Stage 2
Say what each digit is doing
Now the writing. 35 is three bundles and five loose. Point at the 3 and ask what it means — the answer you want is "thirty", or "three tens", never "three". Alongside the real number name, say the maths name out loud: "three tens and five". Hindi and Gujarati names will not tell him this, so you must.
You will see: He reads the 3 in 35 as thirty, and the 3 in 305 as three hundred.
- Stage 3
Break the same number two ways
35 is three bundles and five. It is also two bundles and fifteen. Both are 35, and he should be able to show you both with his hands. This flexibility is not a party trick — it is exactly what subtraction needs, and a child who cannot do it here cannot do it in a column later.
You will see: Given 52, he can show it as 5 bundles and 2, and as 4 bundles and 12.
- Stage 4
Exchange, in a shop and on paper
Only now bring in money, and only ₹1, ₹10 and ₹100. A hundred-rupee note is not physically a hundred times a one-rupee coin, so it can never be his first ten — but once tens are real to him, buying something and getting change is exchange in its natural home. Then the same move on paper, using the word exchange and never the word borrow.
You will see: Doing 52 − 28 he says "I exchange a ten for ten ones", not "I borrow one".
Why matchsticks beat money at the start
A bundle of ten matchsticks is visibly ten times one matchstick. He can count it, untie it, and check. A hundred-rupee note is not visibly a hundred times a one-rupee coin — its value is something he has to be told and believe. Children given materials where the ten is genuinely visible understand place value markedly better than children given materials where it is not.
Anything identical and cheap works: matchsticks, ice-cream sticks, rajma, buttons, bottle caps, tamarind seeds. What matters is that he ties the bundles himself — and that he can untie them again.
Three activities that do the work
One per stage, in order. Each is ten to fifteen minutes with things already in your kitchen.
The Bundle of Ten
A box of matchsticks and a few rubber bands. Your child ties his own tens — and stops seeing 35 as two unrelated digits.
The Exchange Shop
A kitchen-table shop using only ₹1, ₹10 and ₹100. Where tens stop being matchsticks and start being money — and giving change becomes regrouping.
Build It a Different Way
Show me 52. Now show me the same 52, but differently. The two-minute game that decides whether column subtraction will work or fall apart.
Two shortcuts worth teaching — and when
Two methods from the Vedic mathematics tradition are genuinely useful here, for a specific reason: they make the columns more visible rather than hiding them. Teach them after stage 3, once bundles are real — then they land as "oh, so that is why", instead of one more rule to remember.
A note on the name, because honesty is the point of this site: these sutras were published in 1965 by Bharati Krishna Tirtha, and historians dispute whether they come from the Vedas at all. The mathematics stands on its own merit either way — which is how it is taught here.
All from nine, and the last from ten
निखिलम् — nikhilam. For subtracting from 100, 1000 and so on. Take every digit from 9, and the final digit from 10. No borrowing, no crossing out, and — the part that matters — the child works place by place, saying "hundreds, tens, ones" as he goes.
Vertically and crosswise
ऊर्ध्वतिर्यग्भ्याम् — urdhva-tiryagbhyam. For multiplying two two-digit numbers. The units multiply together, the tens multiply together, and the cross-products land in the tens column. It is the same partitioning as the rectangle above — laid out so each piece falls into its own place.
Do not start here. A shortcut taught before the concept becomes the method he reaches for under pressure, and it will hide the gap rather than close it. What the evidence actually supports is that practice organised so the structure shows builds understanding, while the same problems drilled in random order build very little. These two methods qualify — as the reward for stage 3, not a replacement for it.
Four things not to do
Don't say "you can't take a bigger number from a smaller one"
This one sentence is a known cause of a specific, stubborn error: answering 52 − 28 = 36 by taking the smaller digit from the larger in each column. The child is obeying you. Say instead that you need more ones, so you will exchange a ten for ten of them.
Don't say "borrow"
Nothing is returned, so the word describes the wrong action. Exchange is what actually happens, and it matches the bundle he unties with his hands.
Don't move to hundreds while tens are shaky
Children secure ones long before tens, and unitising ten is the hard part. Hundreds on top of a wobbly ten does not add knowledge; it adds confusion with a bigger number attached.
Don't accept a right answer as proof
Children frequently build the correct thing while holding the wrong idea — they lay out 25 perfectly and still think the 2 means two. The blocks do not teach; the talking does. Always ask him to say why.
And what about lakhs and crores?
Not yet, and it matters less than parents expect. The Indian and international systems are identical up to ten thousand and diverge only at 1,00,000 — where the commas fall and what the groups are called. The columns underneath never change.
When he does meet both — Indian textbook, international YouTube — show him 10,00,000 and 1,000,000 side by side and say plainly that they are the same number, punctuated by two different conventions. Place value is the mathematics; grouping is only how we read it aloud.
One thing our languages do not help with
In Chinese, twenty-three is spoken as "two-ten-three" — the structure is audible, and children hear place value before they are taught it. English half-manages it. Hindi and Gujarati largely do not:teis and tihattar are close to words you simply learn, with no reliable "twenty plus three" inside them.
We should be careful here — no study has tested this in Indian children, so treat it as reasoning rather than proof. But the practical response costs nothing: if the number name will not carry the structure, say the structure alongside it. "Teis — two tens and three." The materials and your words have to do the job the language does not.
Common questions
›My child knows the times tables but cannot do 34 × 6. What is missing?
Almost always place value, not tables. To do 34 × 6 a child must see 34 as 30 and 4, work out 30 × 6 = 180 and 4 × 6 = 24, then add them. A child who reads the 3 in 34 as "three" rather than "thirty" computes 18 instead of 180, and the answer collapses. The tables are fine. What is missing is the ability to break a number into its tens and ones and hold the tens as a unit.
›What is place value, in plain words?
It is the idea that a digit means different things depending on where it sits. The 3 in 35 is thirty. The 3 in 305 is three hundred. The same symbol, a different value, decided purely by position. It sounds obvious to an adult and it is genuinely hard for a child, because it asks them to treat a bundle of ten as one thing and ten things at the same time.
›Why does my child write 1002 when I say "one hundred and two"?
Because he is writing exactly what he hears, in the order he hears it: "hundred" becomes 100, "two" becomes 2, so 1002. It is a sensible mistake, not a careless one. The fix is not to correct the spelling but to build the columns — hundreds, tens, ones — so he places digits rather than transcribing sounds.
›Should I use rupee notes to teach place value?
Not first. Money is convenient and motivating, and ₹1, ₹10 and ₹100 map neatly onto ones, tens and hundreds. But a ₹100 note is not physically a hundred times a ₹1 coin, so its value is a convention a child has to be told rather than something he can see and check. Start with something where a ten is visibly ten — matchsticks bundled with a rubber band. Bring money in afterwards, when it becomes the natural place to practise exchange.
›Is it wrong to say "borrow" in subtraction?
It is worth avoiding, because nothing is ever given back. Say exchange instead: you exchange one ten for ten ones. That is literally what happens, and it matches what the child does with his hands when he unties a bundle. And never say "you cannot take a bigger number from a smaller one" — that sentence is one of the known causes of children answering 52 − 28 = 36, taking the smaller digit from the larger in each column because they were told the other way round is impossible.
›When do lakhs and crores matter?
Later than most parents fear, and less than they fear. The Indian and international systems are identical up to ten thousand and only differ from 1,00,000 onwards, and even then only in where the commas go and what the groups are called. The place value underneath is exactly the same. For a child working within hundreds and thousands, it simply does not arise. When it does, teach it as a reading convention rather than as new mathematics.
Once this is solid
Place value is what multiplication stands on. With tens secure, the times tables stop being 144 things to memorise and become a small set of facts he can rebuild — which is exactly what the maths plan teaches.
Times tables that actually stick →