35 − 18 = 23
Mr. Naveen has finished teaching two-digit subtraction with regrouping to his Class 3 of 34 children in a government school in Gurugram. In Tuesday's worksheet, 19 children wrote 35 − 18 = 23. Eleven wrote 17. Four left it blank. In follow-up interviews, seven of the nineteen explained: '5 minus 8 is not possible, so we did 8 minus 5, that is 3, and 3 minus 1 is 2.' When Naveen asked one child to make 35 with bundles of ten sticks, she made three bundles and five loose sticks correctly; but when he said 'take away eighteen', she took one bundle away and then could not decide whether the loose sticks were five or eight. Two parents have asked him to teach the 'borrow and cut' shortcut so children can finish faster, and one offers to send a tutor who 'drills tables'.
1. Diagnose the error pattern. What does it reveal, and what should Naveen not conclude?
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What it reveals. The children are treating each digit as a separate small number rather than as a quantity in a place. 'Smaller from larger' is not a random slip; it is a sensible rule invented to avoid a result they believe is impossible, because they have no concept of a negative or of regrouping. Nineteen of thirty-four producing the identical answer means the source is the teaching, not the children.
The bundle interview is the key evidence. The child can represent 35 correctly - place value in the concrete is intact. What breaks is the link between the written algorithm and the physical action of unbundling. She took away a whole ten and then lost track of the ones, which shows she is not conserving the total: she does not know that 3 tens 5 ones and 2 tens 15 ones are the same amount.
What he should not conclude. Not carelessness; not low ability; not that the eleven correct answers prove understanding. Those eleven may be reproducing a procedure. Naveen should ask two of them to explain with bundles before assuming the class is split into 'got it' and 'didn't'.
The gap is conceptual (regrouping, conservation of quantity, part–whole), not primarily procedural. Teaching the procedure louder will not close it.
2. Sequence the remediation with base-ten material: what exactly does Naveen do, in what order, with what language, and what do children record?
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Stage 1 - Conservation (1 lesson). Child builds 35 (3 tens, 5 ones). 'Open one bundle. How many now?' - 2 tens, 15 ones. Count all: still 35. Repeat with 42, 51. The sentence to establish: 'Ek das ko das ikaiyon mein badalna' - the total does not change.
Stage 2 - The action of subtracting (1–2 lessons). From 35, 'take away 18'. The child hits the wall: only 5 ones. The move is to exchange one ten for ten ones, then remove 8 ones and 1 ten, leaving 17. Do it with materials five times, different numbers, some needing no exchange (46 − 21) so the child discriminates.
Stage 3 - Language. Replace 'borrow' with 'regroup' or 'exchange'. Borrow implies returning, and nothing is ever returned; the word is where the confusion starts.
Stage 4 - Recording alongside the material. Expanded form first: 30 + 5 → 20 + 15; 20 + 15 − 18. Then the column algorithm written next to the bundles, so the little '1' means 'ten ones I just opened'.
Stage 5 - Representational, then abstract. Tens as vertical lines, ones as dots, no materials; then digits only. Add a checking habit: 17 + 18 = 35.
3. How should Naveen answer the parents who want the shortcut and speed drills, and how should he assess whether the concept, not the trick, is learned?
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Answering the parents. Agree with the goal, not the method: fluency matters, and he wants their child fast and right. Then show the evidence: nineteen children in this class used a shortcut rule they invented because the meaning was missing, and produced 23. A 'cut and borrow' chant without understanding survives two-digit sums and collapses in Class 4 with three-digit numbers, with zeros (300 − 146) and in word problems where there is nothing to cut. The proof is in his own worksheet.
Offer a compromise, not a refusal. Five minutes a day of number-fact fluency - complements to 10 and 20 - is genuinely useful and separate from subtraction. The tutor's table drill is fine for tables; it must not replace the regrouping work. Invite both parents to a ten-minute demonstration with bundles at the next PTM; seeing a child unbundle one ten is more convincing than an argument.
Assessment. Do not score only answers. (1) A mixed diagnostic - 46 − 21, 52 − 27, 40 − 23, 35 − 18 - to see whether the child exchanges only when needed. (2) A 'draw and explain' exit ticket: a drawing of the exchange plus one sentence. (3) Interview six children, including two who answered correctly, asking them to show it with sticks. (4) Mark errors by type - no-exchange errors, reversal rule, computational slip - and track whether the reversal type disappears.