403 minus 157: three different answers from one class
A Class IV teacher of 30 children ran a five-item subtraction diagnostic before starting the new unit. For 403 - 157, sixteen children wrote 354, six wrote 256, five wrote answers such as 346 or 254, and three wrote 250. The sixteen who wrote 354 also wrote 62 - 38 = 36 and 500 - 249 = 361. When asked, one of them explained: "In each line, small number minus big number is not possible, so big number minus small number." The textbook offers one worked example and then twenty practice sums. The teacher's plan is to work the example slowly on the board, say "borrow one from the next house" five times, and then give the twenty sums as classwork and forty as homework. The mathematics period is 40 minutes, there are no base-ten blocks in the school store, and the teacher has a bundle of broomsticks and a roll of string.
1. Interpret the answer 354. What rule are the children using, and why is a second board demonstration unlikely to change it?
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354 is not carelessness. It is a consistent, child-generated algorithm: subtract the smaller digit from the larger digit in every column, regardless of position. The child who said "big number minus small number" has stated the rule exactly.
Check it against the other two items. For 62 - 38: 8 from 2 gives 6, 3 from 6 gives 3, so 36. The correct answer is 24. For 500 - 249: 0 from 9 gives 9, 4 from 0 gives 4, 2 from 5 gives 3, so 349 by that rule, and the child wrote 361, which shows the same rule applied with a slip. The consistency across items is the evidence: the rule is the problem, not attention.
Re-demonstration cannot dislodge it, for four reasons:
- The teacher's demonstration is a procedure stated in words ("borrow one"), and the child already has a procedure that produces plausible three-digit answers.
- It creates no conflict. 354 looks like an ordinary answer; nothing in the child's experience says it is impossible.
- It is silent about the underlying concept: a hundred is ten tens, a ten is ten ones.
- The twenty sums that follow give sixteen children twenty more rehearsals of the wrong rule. That is how an error becomes automatic.
2. Design a 20-minute concrete-to-abstract sequence with the broomsticks and string.
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The target concept is regrouping: one hundred is ten tens, one ten is ten ones. Broomsticks and string are sufficient.
Step 1, build 403 (5 min). Give each pair 40 bundles of ten and 3 loose sticks. Ask them to show 403. They must notice that 40 tens and 3 ones is 403, so the zero in the tens place is a choice about grouping, not an absence.
Step 2, subtract 157 physically (5 min). "Take away seven ones." Impossible with 3. Untie one bundle: 3 ones become 13, and there are 39 bundles. Take 7. "Now take away five tens" - fine. "Now take away one hundred" - fine. Count what remains. The physical act is the concept, and the child who has untied the bundle will not ask which number is bigger.
Step 3, record alongside (5 min). Expanded form on the board: 400 + 0 + 3 becomes 300 + 90 + 13. Then the compact algorithm with struck-through digits. Children write the same sum in both notations.
Step 4, one contrasting pair (5 min). 403 - 157 beside 430 - 157, so the position of the zero is visible.
Avoid the word "borrow"; say "exchange" or "break one ten into ten ones". Borrowing implies a debt that is never repaid, and children who take the metaphor literally write the small extra digit in the wrong place.
3. What do you do with the six children who are already correct, and what formative record do you keep?
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The six should not receive forty more correct sums. They should get the error analysis, which is harder and builds the same concept from above.
- Error-analysis task for the six. Give them the three incorrect items: 403 - 157 = 354; 62 - 38 = 36; 500 - 249 = 361. Ask: what rule is this child using? Where does the rule break? Write the child a two-line explanation. Explaining an error demands more understanding than repeating a correct procedure.
- Extension. Three-digit subtraction with two zeros (400 - 236), then a missing-digit puzzle such as 5_2 - 2_7 = 275, which cannot be solved by the smaller-from-larger rule.
- Class error chart. A wall chart of the two or three error types found in the diagnostic, unnamed, each with a corrected example beside it. It is a record for the class, not a display of failure.
- Formative record. A simple grid: child's name against four concept steps - builds 403, exchanges a ten, exchanges a hundred, records in expanded form. Tick it from observation during Step 2, not from the homework. Re-run two items after a week and move to the next unit only when the zero case is secure; the five children who wrote 346 or 254 need the same regrouping work, while the three who wrote 250 need estimation checked separately.