Fifty-two minus twenty-seven is thirty-five
In a Class 4 mathematics class of 35 children, the teacher has just marked the subtraction unit. Nineteen children gave 52 − 27 = 35, and the same children gave 63 − 28 = 45 and 40 − 26 = 26. When she asked one of them to explain, the child said, 'Seven take away two is five, five take away two is three - so thirty-five.' Nine children got all three items right; the remaining seven made unrelated errors, including 52 − 27 = 25 and 52 − 27 = 79. On word problems the picture was worse: only four children set up 'A shopkeeper had 52 pencils and sold 27; how many are left?' correctly, and six added. The teacher has covered borrowing on the board three times, using the word 'borrow' and saying 'take one from the tens place'. The school has bundles of sticks and a place-value mat, but they were used only in Class 2.
1. Analyse the error pattern. What exactly do the nineteen children understand and not understand?
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The pattern is not random - it is a consistent rule: subtract the smaller digit from the larger digit in each column, regardless of position. Check it: 52 − 27 gives 5 − 2 = 3 in the tens and 7 − 2 = 5 in the ones, hence 35; 63 − 28 gives 6 − 2 = 4 and 8 − 3 = 5, hence 45; 40 − 26 gives 4 − 2 = 2 and 6 − 0 = 6, hence 26 - three independent successes of the same rule, which is strong evidence. This rule works for every addition-without-carrying and every subtraction-without-borrowing they have met, so from the child's side it is a sensible generalisation; it breaks only when a column needs regrouping. So they do understand place value as columns and they understand subtraction as take-away within a column. What they do not have is the idea that a ten can be decomposed into ten ones - that 52 is not only '5 tens and 2 ones' but can be renamed '4 tens and 12 ones'. The seven children with 25 and 79 are a different group: 25 suggests a second subtraction step or a slip, 79 suggests they added, so they need separate diagnosis and should not be taught with the nineteen.
2. Plan the reteaching, starting from concrete material. What is the crucial experience the children are missing, and how should the vocabulary change?
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The missing experience is renaming - physically exchanging one ten for ten ones - not the algorithm. Sequence over two lessons. Lesson 1, concrete: each pair gets a place-value mat and bundles of ten plus loose sticks. Build 52 (5 bundles, 2 loose). Then set the problem 'take away 27' and let them hit the wall: only 2 loose sticks are there to take 7 from. Do not demonstrate; ask what can be done. Guide them to untie one bundle and see ten ones appear - the mat now shows 4 tens and 12 ones, quantity unchanged, and 27 can be taken out, leaving 25. Do this with four different numbers, untying at least once each time, and have children say the renaming aloud: '4 tens and 12 ones is the same as 52.' Lesson 2, representational then abstract: draw the tens as lines and ones as dots, cross out as they remove, then record each column beside the drawing as they do it. Change 'borrow' to 'regroup' or 'rename', because nothing is borrowed and nothing is returned - this single word choice removes a false analogy. Finish by comparing 52 − 27 = 25 with the erroneous 35 and asking the class which answer must be wrong and why (estimation: 52 − 27 is about 50 − 30 = 20).
3. Design the assessment she should use next, and explain how it distinguishes real understanding from a memorised procedure.
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Assess three things separately: regrouping itself, the written algorithm, and problem selection. For regrouping, use items where the ones digit of the subtrahend is larger in an obvious way and the numbers are friendly: 62 − 27, 45 − 18, 70 − 34, and one zero case like 50 − 26, which is the hardest because there is nothing in the tens to give away without chaining. Include at least two items that need no regrouping (58 − 24, 47 − 15): a child who regroups everything has learned a procedure, not a rule, and that shows up only if such items are present. Ask each child to explain one worked example aloud - 'why did you change 62 to 5 tens and 12 ones?' - because verbal justification is the fastest way to separate understanding from imitation. For word problems, give the same numbers in a take-away story and a comparison story ('52 children, 27 are girls; how many are boys?') and see whether the operation is chosen from the situation. Record results in three columns - regroups correctly, algorithm correct, chooses operation - so the next teaching group is decided by evidence rather than by total marks.