Borrowing that isn't borrowing: diagnosing 634 − 258
In a Std 4 class of 33, a five-sum diagnostic on subtraction produced this: 13 children correct on 634 − 258, 11 children wrote 424, 6 wrote 476, and 3 left it blank. The teacher's plan is to explain the borrowing procedure twice on the board and then set 20 sums of the same type as homework, because 'practice makes perfect'. She has one 40-minute maths period tomorrow, a bundle of 200 straws with rubber bands, and no base-ten blocks. The textbook's next lesson is the tables of 6 and 7, and the unit test is in nine days. Two of the children who wrote 424 finished the other four sums quickly and confidently.
1. What does each wrong answer reveal about the child's reasoning? Distinguish the two patterns precisely.
Read the worked answer
424 - digit-wise subtraction. In each column the child subtracted the smaller digit from the larger: 8 − 4 = 4, 5 − 3 = 2, 6 − 2 = 4. There is no borrowing because the child never compares the number on top with the number below; each column is an independent pair of digits. The speed and confidence of the two children who did this fits a rule that works for every sum they have previously met, where each top digit was larger.
476 - borrowing executed but not recorded. The child does exchange: 14 − 8 = 6 in the units, 12 − 5 = 7 in the tens. But the hundred is not reduced, so 6 − 2 = 4 instead of 5 − 2 = 3. The idea of exchange is present; the bookkeeping is not. This child needs the record of the exchange made visible, not the concept introduced.
The three blanks are a separate question - time, avoidance, or no strategy. Ask each child to do one sum aloud before assuming a misconception.
The instructional implication: 424 and 476 need different remedies on the same day, and 20 identical homework sums will consolidate 424 rather than repair it, because practice strengthens whatever rule is being used.
2. Reteach the algorithm or do something else in tomorrow's 40 minutes? Justify your plan using what is known about place-value learning.
Read the worked answer
Do something else first, then reteach. Repeating the procedure louder cannot dislodge a column-by-column rule, because the child has no reason to see the columns as quantities rather than as digit pairs. Place value and regrouping have to be experienced with material, then represented in pictures, then written as symbols.
Plan for the 40 minutes:
- 5 min: each pair builds 634 with straws - 6 bundles of 100, 3 bundles of 10, 4 loose - and confirms it against the written number.
- 10 min: they physically take away 258. They must open a hundred bundle and exchange it for ten tens, and open a ten for ten units; the teacher circulates and asks 'what did you just do?'
- 10 min: they write each move beside the straws as it happens, so the small '1' and the struck-out digit become a record of a real event.
- 10 min: whole class compares 424, 476 and the straw result, naming the rule: you may only take away like from like, and an exchange must be written down.
- 5 min: exit task (below).
On the tables and the nine days: keep a 5-minute oral table starter each day so the tables are not dropped, and use 10-minute subtraction starters for the rest of the week rather than 20 homework sums. One well-taught exchange is worth more than twenty repetitions of the error.
3. Give one five-minute exit task that shows whether the misconception has shifted, and explain how you would read the responses.
Read the worked answer
Exit task: one sum, 705 − 368, with the instruction: 'You may use the straws or draw. Write down what you did as well as the answer.'
Correct answer is 337. The zero in the tens is deliberate: it forces an exchange across a place, which neither 424 nor 476 can survive by accident.
Reading the responses:
- 337 with jottings or a drawing showing an exchange - the rule has shifted for this case; move on to word problems with regrouping.
- 463 - still digit-wise (8 − 5 = 3, 6 − 0 = 6, 7 − 3 = 4); the child needs more work with material, not more sums.
- 367 or 447, or a written remark like 'cannot take 6 from 0' - real progress: the child now knows an exchange is needed but cannot borrow across a zero. That is tomorrow's teaching point (take one hundred to make ten tens, then exchange one ten), and it should be named as progress, not failure.
- Blank - talk to the child individually; the issue may be confidence or time, not place value.
Rule for reading it: look at the jottings, not only at the answer. A right answer with no reasoning is not evidence that the exchange is understood, and a wrong answer with the right exchange drawn is evidence that it nearly is.