The Borrowing Bug in Class 4
Class 4 in a Marathi-medium school in Kolhapur has 38 children. After a unit on subtraction, the teacher analyses the test item 503 − 268, which was taught with the standard algorithm and 'borrowing'. Nineteen children wrote 365, eight wrote 345, seven wrote the correct 235, and four wrote 265. In individual interviews, six children explain: 'In the ones column 3 minus 8 is not possible, so we do 8 minus 3 and write 5'; 'In the tens, 0 minus 6 is not possible, so we take from the 5.' They can recite 'one ten is borrowed' but cannot explain what 503 means as 5 hundreds, 0 tens and 3 ones; several say 503 is 'five hundred and three ones'. The school has two sets of base-ten blocks for the whole class, a blackboard, and 25 minutes for reteaching before the next unit. The head teacher asks whether to give more worksheets for practice.
1. Diagnose the error patterns 365 and 345, and decide what the next step should be.
Read the worked answer
365 (19 children). Each column has been solved by subtracting the smaller digit from the larger: 8 − 3 = 5 in the ones, 6 − 0 = 6 in the tens, 5 − 2 = 3 in the hundreds. This is the classic smaller-from-larger bug. The child is applying a rule that works when the top digit is larger, but has no place-value meaning to fall back on.
345 (8 children). The ones column shows the same smaller-from-larger move (8 − 3 = 5). In the tens, the child seems to have borrowed to make 10 − 6 = 4, but did not reduce the hundreds, so 5 − 2 = 3. This is a mixed pattern: a partial regrouping procedure with no conservation of quantity. It is more advanced than 365 but still not relational.
Next step. Do not give more worksheets. Nineteen children have the same systematic bug, so practice will entrench it. Reteach place value concretely: 503 is 5 hundreds, 0 tens and 3 ones, and one hundred must be decomposed into 10 tens before subtraction. Use the two block sets in rotation, with drawn place-value charts for the rest. Then return to the algorithm and compare 503 − 268 with 500 − 268 to expose the role of zero.
2. Plan the 25-minute reteaching lesson using only two sets of base-ten blocks.
Read the worked answer
Minutes 1–5: Number talk. Write 503. Ask 'How many hundreds, tens, ones?' Then 'Can you make 503 another way?' Elicit 4 hundreds, 9 tens, 13 ones and 4 hundreds, 10 tens, 3 ones. Record on the board. This prepares the decomposition.
Minutes 6–15: Two groups model, class records. Four children use block set A for 503 − 268; four use set B for 500 − 268. The rest draw squares (hundreds), lines (tens) and dots (ones) on a place-value mat. The key move is narrated: 'No tens, so I trade one hundred for ten tens; now I have 4 hundreds, 10 tens, 3 ones.' The teacher stops the groups at the trade and asks the class to predict the next step.
Minutes 16–20: Compare. 503 − 268 gives 235; 500 − 268 gives 232. Why the difference of 3? This connects the zero to the quantity.
Minutes 21–25: Error analysis. Write 365 and 345 on the board. In pairs, children explain what went wrong and correct them. Collect two explanations orally. Homework is two similar items, not a worksheet of twenty.
3. How would you answer the head teacher's suggestion of more worksheets?
Read the worked answer
Answer: decline, with reasons. Twenty-seven of 38 children made a systematic error, not a careless slip. More worksheets of the same algorithm give them more chances to rehearse 365, which makes the bug faster and harder to remove. Practice improves accuracy only when the underlying procedure is correct.
What to do instead.
- Two short daily tasks for a week: one 'explain the error' item and one contextual problem using money (₹503 − ₹268) or distance.
- A quick check after a week with a new item such as 604 − 357, to see whether the trade across zero transfers.
- Targeted support for the eight children with the 345 pattern, because they are close to the correct procedure and need the conservation-of-quantity idea, not a full reteach.
What to tell the head teacher. The class needs fewer items and more explanation this week; worksheets can return once the new check shows that the trade is understood. Offer the check data as evidence rather than arguing from preference.