The Rule That Works Until It Doesn't
In a third-grade math station, you watch Andre solve 52 − 27 on a whiteboard. He writes 35. His work shows 7 − 2 = 5 in the ones column and 5 − 2 = 3 in the tens column. When you ask him to explain, he says, “You take the little number from the big number, that's how you always do it.” He solved 81 − 36 the same way and got 55. He gets the correct answer on every problem that needs no regrouping, and he can explain that 52 is five tens and two ones when you hand him base-ten blocks. He has four problems left on the page, and the group has eight minutes before switching. He keeps glancing at the other students' boards.
1. What exactly is Andre's error, and why won't telling him “you have to borrow” fix it?
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The error: Andre applies “subtract the smaller digit from the larger digit in every column,” regardless of which number is on top. It gives the right answer whenever no column has a smaller top digit, so the rule has been reinforced many times. On 52 − 27 it gives 35; on 81 − 36 it gives 55.
Why “you have to borrow” fails: it is a second rule with no reason attached, and it doesn't contradict his rule - it just sits on top of it. Students who learn regrouping as a ritual often apply it where it doesn't belong, or revert under pressure.
Key evidence: he already knows 52 is five tens and two ones when the blocks are in front of him. The concept is there; what's broken is the link between the blocks and the written procedure.
What has to happen: he has to hit the wall himself - try to take seven ones from two ones, find there aren't enough, and be the one who decides a ten must be traded. The conflict has to come from the materials, not from me telling him his rule is wrong.
2. What do you do in the eight minutes before the group switches?
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One problem, one idea. Four problems in eight minutes is a rushed rehearsal of the same error.
- Build 52 with blocks (5 tens, 2 ones). Start where he is competent.
- Ask him to take away 27. He'll remove 2 ones and stop. Don't tell him - ask, “Can you take seven ones from two ones?”
- Let him trade. He breaks a ten into ten ones: 4 tens, 12 ones. Remove 2 tens and 7 ones → 25.
- Write beside the blocks, showing the 4 and the 12 matching what's on the table, and connect it to the written 52 − 27. Ask, “What did we do to the 5 to get 4?”
- One more problem with the blocks in view, then put the blocks aside.
What I skip: the other four problems and any drill. Speed now would just re-rehearse the wrong rule.
Tone: private and matter-of-fact. He's already glancing at other boards and knows something is off. “You've got a rule that works for some problems. Let's find the ones where it doesn't.”
3. What do you report to the teacher, and what should not be reported?
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Report evidence, not a label.
- Correct on all no-regrouping items; wrong in identical ways on every regrouping item - always subtracting smaller from larger. That is a consistent rule, not carelessness.
- Can represent 52 as 5 tens and 2 ones with blocks, so place-value knowledge is present; the written procedure is simply disconnected from it.
- He explained his rule aloud, which is how I know it's a rule and not a guess.
- What I tried: one problem with base-ten blocks, the trade, and writing the regrouped digits next to the blocks. He reached 25 with the blocks in view.
- What I did not do: finish the rest of the page for him or mark it.
Recommendation: re-teach regrouping concretely before more independent practice.
What not to report: a bare score or the word “careless.” And the four remaining items shouldn't be scored as independent work - that data would just re-measure the same misconception and look like a new failure.