302 minus 168: two wrong answers that are coherent procedures
In a Class IV mathematics class of 29 children, the teacher gave the subtraction 302 minus 168. Fourteen children wrote 266, seven wrote 144, five wrote 134 and three gave other answers. A child who wrote 266 explained: 'In the ones place 8 is bigger, so I took 8 minus 2; in the tens 6 minus 0 is 6; in the hundreds 3 minus 1 is 2.' A child who wrote 144 explained: 'I borrowed from the 3, so the ones became 12 minus 8 is 4, the tens became 10 minus 6 is 4, and the 2 was left in the hundreds.' The school has place-value bundles, a hundred-square chart and an abacus, but the teacher has been teaching subtraction by writing the standard procedure on the board and asking children to repeat the steps in chorus. The next unit on multiplication is due in ten days.
1. Interpret the two wrong answers and the correct one. What does each reveal about what the children understand?
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Each wrong answer is a coherent procedure, not a random slip.
266 comes from treating every column as 'take the smaller digit from the bigger one'. The child's explanation confirms it: 8 minus 2, 6 minus 0, 3 minus 1. The underlying idea is that a digit carries its face value wherever it sits, so 302 is read as three separate numbers rather than as 3 hundreds, 0 tens and 2 ones. This is a place-value misconception, and more practice on the standard algorithm will not remove it.
144 comes from a partly learned algorithm: the child exchanges correctly at the ones place but leaves the 3 unchanged in the hundreds and treats the 0 tens as 10. The procedure is being followed as a chant without tracking what each exchange does to the neighbouring column.
134 shows that five children already have the concept. They can act as peer explainers, but they should justify with bundles, not merely show the answer.
2. Design the next two lessons so that the misconception is repaired rather than rehearsed.
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Begin from the quantity, not from the written sum. Give pairs a bundle of 3 hundreds, 0 tens and 2 ones and ask them to remove 168, physically. They will be stuck at the ones place, because two ones cannot give eight, and that stuck point is the lesson.
Then let them exchange: one hundred bundle becomes ten tens; take one ten and break it into ten ones, so the ones place holds twelve. Remove 8 ones, 6 tens and 1 hundred. Record each physical move on the board beside the written sum, line by line, so the symbol and the action stay together.
Add an estimation check: 302 is about 300 and 168 is about 170, so the answer is about 130. An answer of 266 fails the check, and children learn to self-correct.
Spend two periods on the zero case, one on a word problem, and use the five children who answered 134 as explainers with bundles in hand. Reasoning: the error is conceptual, so the remedy must restore the meaning of exchange before speed.
3. How should the teacher assess whether the misconception has shifted, given that the multiplication unit starts in ten days?
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Use four items in the same format, plus a short oral interview with six children.
- 405 minus 237, which forces an exchange across a zero.
- A word problem: a shop had 600 notebooks and sold 258; how many are left? This checks transfer and reading.
- A find-the-mistake item: a worked sum showing the smaller-from-larger error, with the question 'What did this child think? Explain.'
- An estimation item: without calculating, say whether 502 minus 149 is closer to 250 or to 350, and why.
Score reasoning, not only the answer: one mark for the correct result, one for a correctly shown exchange. Interview the children who wrote 266 and hear their explanation again.
Reasoning: item 1 tests the procedure, item 2 transfer, item 3 whether the concept is now explicit, item 4 number sense (the correct value, 353, is nearer 350). A single algorithm test would not show whether the misconception has moved.