Smaller-from-larger: diagnosing 503 − 278
In a Class 4 with 40 children, the teacher sets 503 − 278. Results: 10 children correct (225), 21 write 375, and 9 write 235. In short interviews the 375 group say 'you cannot take 8 from 3, so take 3 from 8'; the 235 group say 'you borrow from the 5'. Given base-ten material, 14 of the 30 wrong-answer children can bundle ten tens into a hundred correctly, but none can explain why the 5 becomes a 4. The teacher's plan for next week is twenty practice sums and a chant: 'if the top is smaller, borrow from the neighbour'. A parent tells her that borrowing should be taught as 'take one from next door and give it to the small number'. The class has ten minutes of mental maths a day and no worksheets beyond the textbook, and the unit test is in eight days.
1. Diagnose the two error patterns. What does each one show that the children do and do not understand?
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Two different faults, not one weak class.
- 375 (21 children): the smaller-from-larger rule. In every column the child subtracts the smaller digit from the larger, ignoring position: 8−3=5, 7−0=7, 5−2=3. The digits are treated as counters, not as 3 ones, 0 tens and 5 hundreds. Note that the answer is not flagged by the child as impossible - 278 was taken from 503 and the result grew in the tens place - so there is no estimate check either.
- 235 (9 children): an attempted regrouping, misapplied. They try to take from the 5 hundreds to make tens, but do not reduce consistently or do not carry the nine tens forward: tens become 10−7=3 instead of 9−7=2. They know 'borrow' as a ritual, not as conservation of total quantity.
- The base-ten interview pins it down: bundling is intact, the exchange idea is missing. A chant would train the ritual further, which is exactly what the 235 group already demonstrates.
2. Critique the 'borrow from the neighbour' language and design a two-day remediation sequence using materials and counting up.
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Critique: 'borrow' is wrong twice - nothing is borrowed and nothing is returned. 'Neighbour' hides the operation, which is an exchange: one hundred is traded for ten tens, and the total stays the same. A chant can generate 225 without understanding, as the 235 group shows.
Day 1 - exchange with material, no written sums. Build 503: 5 flats, 0 longs, 3 units. Ask the child to remove 8 units; they must exchange a flat for 10 longs, then a long for 10 units. Record in words: 'one hundred became ten tens; one ten became ten ones'. Repeat with three or four numbers, including one with zero in the tens. Ask each time: 'how many altogether now?' - it must still be 503.
Day 2 - two representations side by side. (a) Column method with the exchange written above the digits (9 and 13), not vague marks. (b) Counting up on an empty number line: 278 → 280 (+2) → 300 (+20) → 503 (+203), total 225. This gives a self-check that kills 375 instantly, since an answer larger than 503 is impossible when nothing was added. Only then, 6–8 practice items.
3. How would you assess whether the misconception is gone? Give the items and the criterion.
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Assess the concept, not the speed.
- Interview item, not a page of sums: 'Here is 402 − 156. Before you calculate, is the answer more or less than 300? Why? Now do it with the material and record each exchange.' A child with the concept predicts near 246 and says 'I had to break a hundred because there were no tens to break'.
- Error-hunt item: show 375 as another child's answer and ask 'what did this child do? Is it possible?' A response that names the smaller-from-larger rule is evidence of understanding, not recall.
- Transfer items with the zero in different positions: 500 − 268, 703 − 458, 630 − 254. Consistent success across positions is the criterion; one correct sum is not.
- Running record: tally the two error types on unseen items. The misconception is addressed when 375 no longer appears and the estimate check is used without prompting. The unit test in eight days should include one exchange explanation, not twenty sums.