A child sees:
52 − 27.
Two ones cannot take away seven ones.
The teacher says, “Borrow from the five.”
The 5 becomes 4.
The 2 becomes 12.
The calculation proceeds.
Now ask the learner:
What did we borrow, and when do we give it back?
If the child cannot answer, that is not surprising.
Nothing was actually borrowed.
The number 52 was renamed.
Five tens and two ones can be represented as four tens and twelve ones because one ten is equivalent to ten ones.
The total value remains 52.
Subtraction regrouping is an exchange inside the same number, not a loan from somewhere else.
This distinction is the key to understanding the standard subtraction algorithm.
The current Singapore Primary Mathematics syllabus places addition and subtraction algorithms up to three digits in Primary 2 and explicitly recommends base-ten sets or play money to illustrate those algorithms. That recommendation points directly to the underlying mathematics: subtraction needs place-value units that can be exchanged and renamed.
When that structure is secure, regrouping is logical.
When it is not, “cross this out, write this, put a 1 here” becomes a fragile ritual.
The quick answer: what happens when we regroup in subtraction?
Subtraction regrouping uses the same base-ten exchange relationships as addition, but often in the opposite direction:
- 1 ten can be exchanged for 10 ones;
- 1 hundred can be exchanged for 10 tens;
- 1 thousand can be exchanged for 10 hundreds.
Consider 52 − 27.
52 is:
5 tens + 2 ones.
To subtract 7 ones, rename one ten:
5 tens + 2 ones = 4 tens + 12 ones.
Now subtract:
12 ones − 7 ones = 5 ones.
4 tens − 2 tens = 2 tens.
So:
52 − 27 = 25.
The algorithm records the renamed minuend and then subtracts like units.
The number does not get smaller when we rename it
This is one of the most important ideas to make explicit.
When 52 becomes 4 tens and 12 ones, the number has not become 42 or 412.
It is still 52.
Why?
4 tens + 12 ones = 40 + 12 = 52.
The representation changed.
The value did not.
A learner who believes the top number changes during regrouping will have no stable quantity to subtract from.
So before teaching notation, build the same number in two equivalent forms.
52:
- 5 tens + 2 ones;
- 4 tens + 12 ones.
Ask the child to verify both totals.
This is the conceptual permission that makes the subtraction algorithm possible.
Why “borrow” can be misleading
The word “borrow” is familiar and many adults learned subtraction with it.
The problem is not the word itself.
The problem is the mental model it can create.
Borrowing usually means taking something temporarily and returning it later.
In subtraction regrouping, no unit is returned.
One ten is permanently renamed as ten ones inside the representation of the same number.
That is why words such as regroup, exchange or rename are often mathematically clearer.
The important thing is not policing terminology.
It is ensuring the child can explain the exchange.
Build 52 − 27 with base-ten materials
Represent 52 using:
- 5 ten rods;
- 2 one units.
Try to remove 7 one units.
There are not enough visible ones.
Exchange one ten rod for 10 one units.
Now the model contains:
- 4 tens;
- 12 ones.
Remove 7 ones.
5 ones remain.
Remove 2 tens.
2 tens remain.
Answer: 25.
Only after the learner can describe the physical exchange should the written crossing-out notation be introduced as shorthand.
The crossed-out digit is a record of renaming
In the written algorithm, 5 tens may be crossed out and replaced with 4 tens.
The 2 ones become 12 ones.
These marks should be read as a sentence:
I exchanged one of the five tens for ten ones, leaving four tens and creating twelve ones altogether.
If the marks cannot be translated into that meaning, they are being used procedurally rather than mathematically.
Worked example: 73 − 48
Standard form:
7 tens + 3 ones.
We need to subtract 8 ones from 3 ones.
Rename one ten:
73 = 6 tens + 13 ones.
Ones:
13 − 8 = 5.
Tens:
6 − 4 = 2 tens.
Answer:
25.
Check by addition:
48 + 25 = 73.
The inverse operation provides an independent structural check.
Worked example: 241 − 126
241 is:
2 hundreds + 4 tens + 1 one.
We need 6 ones.
Exchange one ten for ten ones:
241 = 2 hundreds + 3 tens + 11 ones.
Ones:
11 − 6 = 5.
Tens:
3 tens − 2 tens = 1 ten.
Hundreds:
2 hundreds − 1 hundred = 1 hundred.
Answer:
115.
The algorithm is straightforward because only one exchange is needed.
Zeros reveal whether regrouping is actually understood
Consider:
302 − 178.
The standard representation is:
- 3 hundreds;
- 0 tens;
- 2 ones.
We need to subtract 8 ones.
There are not enough ones.
We would normally exchange one ten, but there are zero tens.
So the renaming must begin at the hundreds place.
Exchange one hundred for ten tens:
302 = 2 hundreds + 10 tens + 2 ones.
Now exchange one of those tens for ten ones:
302 = 2 hundreds + 9 tens + 12 ones.
Now subtract:
- 12 ones − 8 ones = 4 ones;
- 9 tens − 7 tens = 2 tens;
- 2 hundreds − 1 hundred = 1 hundred.
Answer:
124.
This is the same exchange rule repeated across an empty place.
If the learner only memorised “borrow from the next digit”, a zero creates confusion.
If the learner understands place-value renaming, the path is longer but still logical.
Why crossing through zero is not a special magic trick
Teachers sometimes present subtraction through zero as an exceptional procedure.
Conceptually, it is not exceptional.
It is ordinary place-value exchange with an intermediate place containing no units.
To make ones available:
- find a larger place with a unit available;
- exchange one of those units downward;
- continue exchanging until the required smaller unit exists;
- subtract.
This explanation scales better than a special-case mnemonic.
Worked example: 500 − 268
500 is 5 hundreds, 0 tens and 0 ones.
We need to subtract 8 ones.
Rename:
500 = 4 hundreds + 10 tens + 0 ones.
Then:
500 = 4 hundreds + 9 tens + 10 ones.
Subtract ones:
10 − 8 = 2.
Subtract tens:
9 − 6 = 3 tens.
Subtract hundreds:
4 − 2 = 2 hundreds.
Answer:
232.
Check:
268 + 232 = 500.
The check is especially useful because zero-heavy subtraction is vulnerable to recording mistakes.
Do we always have to regroup?
No.
Consider 58 − 23.
8 ones − 3 ones is possible without exchange.
5 tens − 2 tens is possible without exchange.
Answer: 35.
The learner should not regroup automatically because subtraction is present.
The trigger is specific:
Regroup only when the current representation does not contain enough of the required unit to perform the subtraction directly.
That rule comes from quantities, not from visual patterns.
The standard algorithm is not the only subtraction strategy
For 63 − 29, the vertical algorithm works.
A mental strategy may be more elegant:
63 − 30 = 33.
Because 29 is one less than 30, add 1 back:
33 + 1 = 34.
Or think by missing addition:
29 + ? = 63.
29 + 1 = 30, +30 = 60, +3 = 63.
Total difference = 1 + 30 + 3 = 34.
Mathematical fluency includes knowing the standard algorithm and recognising when another structure is more efficient.
Common misconception 1: subtract the smaller digit from the larger digit in each column
For 52 − 27, a learner may calculate:
7 − 2 = 5 and 5 − 2 = 3, giving 35.
The child has ignored direction and treated each column as an unordered difference.
Repair: return to quantity.
The operation is 52 minus 27.
We are removing 27 from 52, not choosing a convenient digit order independently in each column.
Build the minuend and physically remove the subtrahend.
Common misconception 2: the crossed-out ten simply disappears
A learner reduces 5 tens to 4 tens but does not connect the lost ten to the new 10 ones.
The value appears to shrink before subtraction even starts.
Ask:
“Where did the missing ten go?”
It became ten ones.
Every reduction in a larger-place count must correspond to an increase in smaller-place units during renaming.
Common misconception 3: add ten to the ones but forget to reduce the tens
For 52, a learner writes 12 ones but leaves 5 tens unchanged.
The representation has become:
5 tens + 12 ones = 62.
The learner accidentally created ten extra ones.
Repair: model exchange physically. One ten must leave the tens collection when ten ones appear.
Common misconception 4: subtract across zeros with a special memorised chant
Rules such as “go next door, keep going until you find something” may produce answers but can collapse under stress.
Replace the chant with unit reasoning.
In 302:
there are no tens available to exchange.
Exchange one hundred for ten tens.
Then exchange one of those tens for ten ones.
The path is determined by the place-value structure.
Common misconception 5: borrowing changes the original number
A child may think 52 becomes 412 because the written digits now show a 4 and 12.
The learner is reading the regrouped annotations as a new ordinary numeral.
Explain that 4 tens and 12 ones is a mixed-unit description, not the positional numeral 412.
This is a powerful distinction:
4 tens + 12 ones = 52.
412 = 4 hundreds + 1 ten + 2 ones.
The written marks look similar but represent different unit structures.
Common misconception 6: subtraction must always start with the written algorithm
Some numbers invite easier reasoning.
100 − 48 can be viewed as:
48 + ? = 100.
48 needs 2 to make 50, then 50 more to make 100.
Total difference = 52.
The algorithm remains valid, but strategy flexibility reduces unnecessary complexity.
Children should learn a dependable general method without losing number sense.
Common misconception 7: a neat answer needs no check
Suppose 302 − 178 is incorrectly calculated as 234.
A rough magnitude check can detect trouble.
302 is about 300.
178 is about 180.
The difference should be about 120.
234 is far too large.
Then check exactly by addition:
178 + 124 = 302.
Reasonableness and inverse checking protect against procedural slips.
A diagnostic ladder for subtraction regrouping
Check 1: understand subtraction direction
Can the learner represent 52 − 27 as removing 27 from 52 or finding the difference?
Check 2: place value
Can the learner identify tens and ones in 52?
Check 3: exchange
Can 1 ten be exchanged for 10 ones while preserving value?
Check 4: rename the minuend
Can 52 become 4 tens + 12 ones?
Check 5: subtract like units
Can 12 ones − 7 ones and 4 tens − 2 tens be carried out accurately?
Check 6: record the exchange correctly
Can the learner explain every crossed-out and rewritten digit?
Check 7: cross a zero
Can the learner rename 302 as 2 hundreds, 9 tens and 12 ones?
Check 8: verify
Can the learner check with addition or estimation?
This ladder isolates quantity meaning, unit exchange, notation and checking.
The transfer test: give the same mathematics in another representation
Ask the learner to solve 63 − 28 in four ways:
- base-ten materials;
- vertical algorithm;
- number line or counting-up difference;
- mental compensation.
Possible mental route:
63 − 30 = 33, then add 2 back = 35.
If the learner can explain why all routes give 35, subtraction has become more than a layout-dependent procedure.
Subtraction as missing addition can reduce regrouping load
For some differences, counting up is efficient.
Take 83 − 78.
The vertical algorithm requires regrouping.
Missing addition is simpler:
78 + 2 = 80.
80 + 3 = 83.
Difference = 5.
This does not make the standard algorithm unnecessary.
It teaches method selection.
A mathematically fluent learner knows the general algorithm and recognises when the structure offers a shorter path.
Why subtraction errors often originate in place value
Teachers sometimes respond to regrouping errors with more subtraction practice.
But if the learner cannot explain:
1 ten = 10 ones,
then the subtraction algorithm is standing on a missing prerequisite.
Return to renaming numbers before returning to subtraction.
Ask the learner to show 64 as:
- 6 tens + 4 ones;
- 5 tens + 14 ones;
- 4 tens + 24 ones.
If those equivalences are secure, subtraction regrouping has a conceptual foundation.
What parents should listen for
- “I do not have enough ones, so I exchange one ten for ten ones.”
- “The number is still 52 because four tens and twelve ones is also 52.”
- “I reduced the tens by one because that ten became ten ones.”
- “There are no tens in 302, so I have to rename a hundred first.”
- “I checked because 178 plus 124 gives 302.”
These explanations show unit conservation and inverse reasoning.
What teachers and tutors should avoid
- Avoid teaching only “borrow from next door”. Explain the place-value exchange.
- Avoid treating zero-crossing as a special chant. Trace the exchange through the empty place.
- Avoid letting the top number silently change value. Verify the renamed form equals the original.
- Avoid the smaller-from-larger digit shortcut. Preserve subtraction direction.
- Avoid making vertical subtraction the only method. Use counting up, compensation and inverse reasoning where efficient.
- Avoid correcting the final digit without diagnosing the first broken step.
How this fits Singapore Primary 2 Mathematics
The current MOE Primary Mathematics syllabus includes addition and subtraction algorithms up to three digits at Primary 2. Its suggested learning experiences use base-ten sets and play money to illustrate those standard algorithms and reinforce the relationship between addition and subtraction.
The same level develops place value in hundreds, tens and ones and establishes 10 tens = 1 hundred.
Subtraction regrouping is therefore not an isolated calculation technique.
It is place-value renaming applied to subtraction.
The curriculum sequence becomes coherent when the learner can see that connection.
How do we know representations should support subtraction?
Evidence guidance from the Education Endowment Foundation emphasises explicit links between manipulatives, visual representations and abstract mathematics. Materials are most useful when they reveal relationships and are gradually faded as the learner internalises the structure.
The Institute of Education Sciences’ mathematics intervention guidance likewise recommends visual representations and explicit instruction that connects conceptual understanding to mathematical procedures.
The implication for regrouping is practical:
The crossed-out digits should be the end of an explanation, not the beginning of one.
First understand the exchange. Then compress it into notation.
A subtraction-regrouping readiness checkpoint
- Can the learner identify hundreds, tens and ones?
- Can 1 ten be exchanged for 10 ones without changing value?
- Can 1 hundred be exchanged for 10 tens?
- Can the learner rename 52 as 4 tens and 12 ones?
- Can the learner subtract like units accurately after renaming?
- Can every crossed-out digit be explained in words?
- Can the learner preserve subtraction direction rather than subtract smaller digit from larger digit?
- Can the learner regroup across a zero?
- Can the learner verify the result with addition?
- Can the learner estimate whether the answer is a reasonable size?
- Can the learner choose a mental strategy when it is more efficient?
The deeper lesson: subtraction does not need a new number system
It can feel to a child as though subtraction introduces a strange set of extra rules.
Cross things out.
Write smaller digits above.
Add a 1 in front of another digit.
But no new number system is needed.
The learner already knows the machinery:
- ten ones equal one ten;
- ten tens equal one hundred;
- a number can be renamed without changing value.
Subtraction simply uses that machinery when the current unit configuration is inconvenient.
The algorithm is not breaking the number apart. It is revealing another valid way the number was already made.
Where this leads next
Once addition and subtraction regrouping are understood, learners can focus more attention on method selection and mental calculation.
Not every two-digit or three-digit problem needs a vertical algorithm.
Some are easier by making a friendly ten or hundred.
Some are easier by compensation.
Some are easier by partitioning.
Some differences are easiest by counting up.
The next stage is not abandoning algorithms.
It is learning when each method is the right tool.
For the wider mathematics map, see eduKateSG’s How Mathematics Works resources.
Final thought
A child crosses out the 5 in 52 and writes 4.
The marks look destructive.
Mathematically, nothing has been destroyed.
One ten became ten ones.
The number remained 52.
The representation became more useful for the subtraction.
That is all regrouping is.
Borrowing is not taking something you owe back. It is exchanging one unit for an equal value in smaller units so the mathematics can continue.