Class 4 Mathematics: The Persistent 46 − 28 = 22 Error
In a Class 4 mathematics class of 36 children, a diagnostic test on two-digit addition and subtraction shows: 29 of 36 solve 34 + 25 correctly; only 8 solve 46 − 28 correctly. Of the wrong answers, 17 write 22, 6 write 26, and 5 write 38. In a word problem - 'Ravi had 46 marbles. He gave 28 to a friend. How many are left?' - 19 children add 46 + 28, and 11 choose subtraction but then compute 22. The teacher's plan is to write the rule 'borrow one ten from the tens column and put it before the ones digit' on the board and assign 20 practice sums. The school has bundles of sticks, a place-value chart, and 40-minute periods; the chapter test is in one week.
1. Analyse the three wrong answers, 22, 26 and 38. What does each reveal about the children's place-value understanding?
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22 is not random: the child subtracts the smaller digit from the larger in each column, 8 − 6 = 2 and 4 − 2 = 2. This shows digits are being treated as separate quantities, not as 4 tens and 6 ones minus 2 tens and 8 ones. 26 is consistent with subtracting 20 from 46 and then ignoring the 8 ones, or treating 28 as 20; the tens operation is done but the ones are not. 38 is consistent with subtracting 8 from 46 and ignoring the 2 tens. Only 8 correct answers means the column procedure has been memorised as a set of marks without the underlying regroup. The word-problem errors confirm this: 19 children choose addition from the word 'gave' rather than acting out a decrease, and 11 choose subtraction but cannot carry it out. The teacher should not simply repeat the rule louder; the representation of ten as a bundle must be rebuilt.
2. Design a concrete-representational-abstract sequence for 46 − 28 in one 40-minute period.
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Concrete, 12 minutes: each pair gets 4 bundles of 10 sticks and 6 loose sticks. Ask them to remove 28. They cannot take 8 loose from 6, so they untie one bundle: now 3 bundles and 16 loose sticks. Remove 2 bundles and 8 loose; 1 bundle and 8 loose remain, 18. Representational, 12 minutes: draw the same on a place-value chart, cross out one ten, draw an arrow to the ones column and write 16 ones. Abstract, 10 minutes: set the column sum beside the chart and say 'regroup one ten as ten ones', not 'borrow', because nothing is returned. Do four examples together, including 40 − 28 and 46 − 29, so the rule is not tied to one pattern. Last 6 minutes: children solve two sums and explain one to a partner. The teacher listens for the phrase '3 tens and 16 ones', which is the evidence that the procedure is grounded.
3. How should the teacher address the word-problem errors without teaching keyword hunting?
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Act out the marble story with two children and a bowl: 46 marbles in the bowl, the child gives 28 away, and the class counts what remains. Then draw a bar model: a long bar for 46, a part labelled 28, and the unknown part. Compare this with a joining story so children see that the operation depends on the relationship, not on the word 'gave'. Ask them to write a story for 46 − 28 and to explain why addition would not answer it. Avoid a keyword chart, because 'gave' can appear in addition stories such as 'Ravi got 28 more'. Assessment should include one word problem with a drawing and a spoken explanation, not only the answer. In a large class, collect six explanations orally while the others write, and check the drawings for a part-whole bar rather than only the number sentence.