The Answer Is Always Bigger: A Subtraction Bug in Class IV
In a Class IV mathematics class of 33 children in a government school in Khammam, the teacher gave a 12-item subtraction diagnostic. Lakshmi's paper shows: 68 – 42 = 26 (correct); 47 – 28 = 21; 84 – 57 = 33; 302 – 156 = 254; 500 – 78 = 578. On a place-value task with base-ten blocks she correctly built 302 as 3 hundreds, 0 tens and 2 ones, and correctly exchanged one hundred for ten tens when asked. On an oral word problem - 'A shop had 250 notebooks and sold 175, how many are left?' - she said 125, and when asked for the number sentence she wrote '175 – 250?'. Her addition with carrying is accurate. The teacher's comment on the paper was 'careless, do 30 sums daily for homework', and the parent has been told that Lakshmi is not paying attention.
1. Is this carelessness? State the rule that generates all five answers and explain what it reveals about her understanding.
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Test a rule against the items rather than judging the child. The rule 'in each column, subtract the smaller digit from the larger and write the result, with no borrowing' predicts every answer on the page:
- 68 – 42: 6−4 = 2, 8−2 = 6 → 26 ✓ (and correct, because no column needed regrouping)
- 47 – 28: 4−2 = 2, 8−7 = 1 → 21 ✓
- 84 – 57: 8−5 = 3, 7−4 = 3 → 33 ✓
- 302 – 156: 3−1 = 2, 5−0 = 5, 6−2 = 4 → 254 ✓
- 500 – 78: 5−0 = 5, 7−0 = 7, 8−0 = 8 → 578 ✓
A careless child's errors are random and inconsistent from item to item; here five items, including one right answer, fit one self-consistent procedure exactly. The probability of that by accident is negligible.
What it reveals: she has a coherent rule that avoids the hard part of the algorithm - regrouping. Her place-value knowledge is intact in isolation (she built 302 and exchanged a hundred for ten tens correctly), so the bug lives in the written algorithm, disconnected from the quantities it represents. 68 – 42 being correct is the most informative item on the page: the rule was never contradicted, so it survived. Her '175 – 250?' shows she knows subtraction is involved but cannot yet decide the order, which is a weak part–whole model of subtraction, not inattention. Calling this carelessness will send the intervention in exactly the wrong direction.
2. Why will 30 sums a day make things worse, and what would a three-day sequence look like?
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Why the homework fails: practice consolidates whatever procedure the child is using. Thirty daily repetitions of a buggy rule make it faster and more automatic and much harder to unlearn later. It also tells a child who has been reasoning consistently that she is careless, which is both untrue and demoralising.
Day 1 - create the conflict, concretely. She builds 47 and 28 with bundles of ten and loose ones, physically removes 28, and gets 19. The paper says 21. Ask: 'Both cannot be right - which one matches the sticks?' Do three more, each preceded by an estimate ('about 50 take away 30, so about 20 - 21 or 19?').
Day 2 - connect to what she already knows. Drawn tens-and-ones; expanded form: 47 = 40 + 7 = 30 + 17; then 30 + 17 − (20 + 8) = 10 + 9 = 19. Use her strength - she can already exchange a hundred for ten tens with blocks - and make the written 'borrow' the record of that exchange, not a separate trick.
Day 3 - strategies and checking. Number-line difference (count up from 28 to 47), friendly numbers, then the written algorithm alongside, with her explaining each step aloud. Keep only five varied problems a day, always with an estimate first and one word problem where she chooses the order and justifies it.
Diagnostic to separate bug from knowledge: give 68 – 42 again and ask her to explain why it worked. A child using the buggy rule cannot explain the role of regrouping; a child who understands will point to the columns needing exchange.
3. What would you put in front of the parent and the class teacher, and how would you know the intervention has worked?
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Put the item analysis in front of them, not the score. Show the five items side by side with the rule written out, and point to 68 – 42 as the key: it is correct on the page, and it is also exactly what the rule predicts. That single item shows this is a consistent method, not a lapse in attention. Add the base-ten block task, where she built 302 and exchanged a hundred for ten tens correctly - evidence that she is not weak in place value, only in connecting it to the written algorithm. Add the word problem, where she chose subtraction but reversed the order.
To the parent: explain that Lakshmi is reasoning, not daydreaming, and that 30 sums a day would train the wrong method faster. Ask for five problems a day with an estimate first, and for the word problems to be discussed orally before writing. Remove the 'not paying attention' message explicitly - she has been applying a rule that worked on the examples she met.
To the class teacher: raise the general point - the class diagnostic should be re-scanned for the same rule; children who score correctly on items like 68 – 42 may still hold the bug, so items that require regrouping must be the ones that count.
Evidence of success, four weeks on: she estimates before computing; she explains why a column needs regrouping; she scores correctly on new two- and three-digit items with zeros (500 – 78 is the acid test); and she writes the number sentence in the right order for a word problem. If those four appear, the bug is gone rather than hidden.