The fraction rule that survives re-teaching
In a Class VII diagnostic in a Cuttack school with 40 students, the teacher finds that 32 of 40 write 1/2 + 1/3 = 2/5, and 26 of 40 compute 3/4 of 20 as 17. The same class does well on routine one-step problems: 34 of 40 correctly find the cost of 5 pens when 3 cost ₹45. But only 9 of 40 solve 'If 12 workers build a wall in 8 days, how long will 6 workers take?' - most answer 4 days - and 31 of 40 say that two successive discounts of 20% and then 10% equal a single 30% discount. The textbook chapter is over; the teacher has two weeks before the term test and pressure to move on to mensuration. She is considering re-teaching fractions by repeating the rules and giving thirty practice sums.
1. What single conceptual issue links the 1/2 + 1/3 = 2/5 error and the 3/4 of 20 = 17 error? Which representation would you use to confront it, and why will re-stating the rule not work?
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The shared misconception. Both errors treat the numerator and the denominator as separate whole numbers to be operated on independently: add the tops (1 + 1 = 2), add the bottoms (2 + 3 = 5); take 3/4 of 20 as 20 − 3 = 17. The fraction is not treated as a number or a relationship, only as two written digits. This is a whole-number schema applied to a new domain - and it is coherent to the child, which is why it survives practice.
Representation to confront it. Use a linear model (a strip or measuring tape split into twelfths) and a set model (20 counters shared into four equal groups). The child must see that 1/2 + 1/3 lies between the two fractions, and that three-quarters of 20 must be much closer to 15 than to 17. Estimation before computation makes the wrong answer feel wrong.
Why telling the rule fails. A new rule is stored alongside the old one as a competing verbal fact; on an unfamiliar-looking problem the older, better-practised schema wins. Conflict must be created first: present both answers, ask which must be larger, and settle it on the strip. Thirty more sums with the correct rule will only add a second rule, not replace the first.
2. Are the inverse-proportion error and the successive-discount error the same problem as the fraction error? Justify your answer, and say what follows for the intervention.
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Related in part, but not identical.
- Workers and days: most students answered 4 days, assuming 'fewer workers means fewer days' - a same-direction, direct-rule heuristic. The root is a failure to model the situation (fewer workers means each does more days' work), not the fraction schema.
- Successive discounts: 20% then 10% is not 30% because the second discount applies to a smaller base. The error is additive reasoning about a multiplicative relationship - the same 'operate directly on the numbers' habit seen in the fraction errors.
So the discount error shares the root of the fraction errors; the workers error is a related but distinct failure, a direct-proportion heuristic plus no model of the situation.
What follows. A single intervention on fractions will not fix the workers problem - that needs unit reasoning (worker-days) and a table of values. The discount needs percentage-of-a-changed-base, ideally with a ₹500 shirt worked out both ways. Plan three strands, not one.
3. Design a three-lesson intervention and explain how you would assess whether the misconception is repaired rather than merely masked.
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Lessons.
- Lesson 1 - Conflict and estimation. Pairs receive 1/2 and 1/3 strips, predict the sum, then fold and check. Predictions are recorded; the wrong answer must be argued with, not erased.
- Lesson 2 - Fraction as number and as operator. Number-line placement of 1/2, 1/3, 5/6; set models for 'of' problems; students write the trap answer themselves and explain why a classmate might choose it.
- Lesson 3 - Transfer. Mixed problems in unfamiliar contexts, including one where adding is appropriate (matching units), plus a worker-days table for inverse proportion and a two-step discount worked on a changed base.
Assessing repair, not performance.
- Use fresh items with new contexts and non-standard wording.
- Require an estimate first and an explanation: 'why is 2/5 impossible here?'
- Include a 'find the mistake' item containing the old error, with a written rebuttal.
- Re-test the same concept after two weeks; a correct answer on a practised item may only be pattern-matching. If students still choose 2/5 on a novel item, the schema has not changed, whatever the exercise book shows.