Regrouping Errors in Second-Grade Subtraction
A second-grade teacher has 19 students midway through a 12-day unit on two-digit subtraction with regrouping. On a five-item exit ticket, 7 students answered 52 − 27 = 35, 5 answered correctly, 4 wrote 36 for 40 − 16, and 3 made unrelated counting errors. When interviewed, the 7 students consistently said they “took the smaller number from the bigger number” in each column. The teacher’s unit test is in six days, and the grade-level team expects all classes to give the same test on the same day. The teacher has base-ten blocks, place-value mats, and number lines available but has been teaching the standard algorithm with the phrase “borrow a ten.”
1. What does the pattern 52 − 27 = 35 reveal, and what is the most likely next instructional step?
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The answer 35 is not random. The students are applying a rule that each column must have the smaller digit subtracted from the larger digit, regardless of which number is on top. In 52 − 27, they subtract 2 from 5 in the tens column and 7 from 2 in the ones column, reversing the order in the ones place. This is a place-value and regrouping misconception, not simple carelessness. The phrase “borrow a ten” likely obscures the fact that a ten is decomposed into ten ones and remains part of the same number. The next step is to return to concrete decomposition: build 52 with base-ten blocks, attempt to remove 7 ones, physically trade one ten for ten ones, and record the written algorithm beside the blocks. Students should explain why the value has not changed. The teacher should also contrast “borrow” with “regroup/decompose,” because “borrow” implies taking something away and returning it later. Before the unit test, she should re-engage the whole class briefly, because the error is common enough that public reasoning will help the 4 students who mishandled the zero.
2. Design a 20-minute re-engagement lesson using a representation. Explain why it targets the error.
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Open with a number talk: “Which is larger, 52 or 27? Which column do we start with when subtracting, and why?” Then pose 40 − 16, because the zero forces decomposition and cannot be solved by reversing digits. Students build 40 on a place-value mat. Ask: “Can we take 6 ones?” No. Trade one ten for ten ones, leaving 3 tens and 10 ones. Remove 6 ones and 1 ten, leaving 24. Record the algorithm next to the blocks and connect each written step to the blocks. Then repeat with 52 − 27, this time asking students to explain why the ones digit becomes 12 and the tens digit becomes 4. Finish with a partner sentence frame: “I regrouped one ten into ten ones, so the number is still 52.” This targets the error because it confronts the incorrect rule with a physical constraint: you cannot remove 7 ones from 2 ones without changing a ten. It also gives the correct students a chance to explain rather than simply answer.
3. How should she assess whether the misconception is resolved, and what should she do for the five students who were correct?
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Use a new exit ticket with three contrasting items: one that requires regrouping, one with a zero in the minuend, and one that does not require regrouping. Include a prompt asking students to draw or explain one item. A correct answer alone is not enough if the student cannot explain the decomposition. She should look for the specific error of reversing digits in the ones column. If three or more students still reverse, she should reteach in a small group before the unit test and inform the team, not simply proceed. For the five who were correct, do not give them the same reteach as punishment. Give them a richer task: solve a three-digit subtraction problem with a zero, write a word problem that requires regrouping, or analyze a fictional student’s incorrect work and explain the error. That keeps them engaged and gives the teacher evidence of deeper understanding.