42 − 17 = 35: Diagnosing a Subtraction Bug Five Days Before the Test
A Class III teacher in a Tripura school has 34 children. She taught two-digit subtraction with borrowing for eight periods using bundles of ten sticks, but the bundles were demonstrated by her at the front and the children watched. On the class work, 26 of 34 wrote 42 − 17 = 35, 30 − 16 = 26 and 53 − 28 = 35. When she re-taught the same sums on the board and asked the class to repeat the steps aloud, the correct method was recited perfectly, and the next day the same 26 children made the same errors. Only 5 children are consistently correct. One child, Meghna, gets most sums right but counts on her fingers and takes about four minutes per item. The textbook unit test, set by the block resource centre, is in five days and will contain ten two-digit subtraction items, six of them requiring regrouping.
1. Name the misconception precisely, and design two questions that would confirm it rather than a general place-value weakness.
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The pattern - 42 − 17 = 35, 30 − 16 = 26, 53 − 28 = 35 - is the classic smaller-from-larger bug: at each column the child subtracts the smaller digit from the larger digit regardless of which is on top, and writes the digit. In 42 − 17: units 7 − 2 = 5, tens 4 − 1 = 3, giving 35. The child is not guessing; she is applying a consistent rule that ignores place value and regrouping.
Two confirming questions:
- A no-regrouping control item: 46 − 23. If she gets this right, basic column subtraction and place value are intact - the failure is specific to regrouping. If she gets it wrong, run a place-value check.
- A same-column item with explanation: 62 − 38. Ask her to say the steps aloud while doing it with bundles. If she says '8 minus 2 is 6' and writes 36, the bug is confirmed and is verbal, not accidental. Then ask her to build 62 with sticks and physically take away 38 - if she can do it with material and not in symbols, the difficulty is the written representation, not the concept.
Note: a child who recites the steps perfectly and still errs has a procedural gap, not a memory gap. Choral repetition will never fix it.
2. Plan the five days of instruction. Justify the sequence and say what the teacher must stop doing.
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Sequence: concrete → representational → symbolic, with the child doing the manipulation, not watching it.
- Day 1 – Children build and remove. Pairs get bundles and loose sticks; each pair makes 42, removes 17, and records what happened ('I opened one ten'). Teacher listens to the language; the key sentence is 'we broke a ten into ten ones', not 'borrow'.
- Day 2 – Draw the sticks. No materials: draw tens as lines and ones as dots for 53 − 28, cross out. This connects the action to a picture without symbols.
- Day 3 – Expanded form. 42 − 17 as (30 + 12) − (10 + 7), done with the drawings visible. This is where the rule 'never take a bigger digit from a smaller one' becomes visible.
- Day 4 – Error analysis, not drill. Put 42 − 17 = 35 on the board and ask the class to find and explain the mistake. Naming the bug is what displaces it.
- Day 5 – Mixed practice with two no-regrouping items and four regrouping items, plus one word problem.
Stop: demonstrating at the front with the teacher's hands, and whole-class choral recitation of steps. Both produce the appearance of learning seen here. Also stop giving ten identical items; give four and mark the errors by type.
3. Meghna gets the answers right by finger counting, in about four minutes per item. Is this a problem, and what should the teacher do in five days?
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It is not an error and must not be punished - but it is a ceiling that will hurt her.
- Finger counting is a legitimate, accurate strategy for a Class III child. Banning it produces anxiety and hidden counting; corrections that shame her will cost more than the four minutes.
- The real risk is efficiency and place value: finger counting works for 42 − 17 by counting on, but it will not scale to three-digit numbers, and it does not build the regrouping concept the unit is about.
- Five-day plan for her: do not make her repeat work she can already do. Give her two of the day's items and then a short derived-facts task - number bonds to 10 and 20, and 'make ten' jumps (17 + 3 = 20, then 20 + 22). Two minutes a day, not a separate syllabus.
- Ask her to explain her method once: 'how did you get 25?' A child who can explain has a strategy; a child who cannot is counting mechanically.
- Permission with a route out: tell her she may count if she needs to, and that the class is learning a faster way; let her compare her time with and without the drawing method on one item.
- In the test, give her the full time. Speed is not a Class III mathematics objective.