Eighty-Seven Per Cent Can Multiply; Thirty-One Per Cent Can Use It
You teach mathematics to Class VI in a government school in Kurnool. A unit test on fractions produced a split result: 87 per cent of the class could compute 3/4 x 2/3 correctly, but only 31 per cent could solve 'A recipe needs 3/4 cup of rice. If you make 2/3 of the recipe, how much rice do you need?' Roughly a third of the class wrote 3/4 + 2/3; several others multiplied correctly but then wrote the answer as 6/6 and concluded 'one cup'. When you asked one girl why she multiplied, she said, 'Because the last chapter was multiplication.' The textbook's word problems are all of the 'find the product' type. You have three periods before the next unit test, and the district resource person is visiting in the second week.
1. Diagnose the two errors - the addition response and the '6/6, so one cup' answer - as different kinds of failure. What does each tell you about what the students have and have not learned?
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They are different failures needing different remedies. The addition response is a comprehension failure at the level of the situation: the student has read the numbers and ignored the relation the words describe. Saying 'because the last chapter was multiplication' is diagnostic - the student is selecting an operation by topic rather than by meaning, which is exactly what a textbook full of 'find the product' items trains. Multiplication was available; it was simply not called for by the story. The '6/6, so one cup' answer is a different matter: the operation was chosen correctly, so situational understanding was present, but the product was then simplified without attending to what it represents. Six-sixths of a cup is one cup, but the question asked how much rice is needed when the recipe is reduced, and an answer larger than the original quantity is impossible. That student needs a habit of checking the result against the situation, not more practice in multiplying. Treating both as 'fraction errors' and re-teaching multiplication would fix neither.
2. Design the three periods. For each, state the activity and what you want students to be able to say by the end.
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Period 1 - meaning before symbols. Give pairs four stories with the same two numbers (3/4 and 2/3) but different relations: of the recipe, more than, shared among, compared with. No numerical answer is required, only a drawing and a sentence such as 'this one gets smaller, so it is a part of a part'. By the end every student should be able to say that multiplying by a fraction less than one makes the quantity smaller, and show it on a bar model.
Period 2 - representation, then algorithm. Students shade 3/4 of a rectangle, then 2/3 of the shaded part, and see that the overlap is 6/12, not 6/6. The algorithm is derived from the picture rather than announced, and the class writes the rule in its own words. By the end they should be able to explain why the denominators multiply.
Period 3 - checking and transfer. Return to the 'one cup' answer and ask the class to judge it against the original bar; follow with problems in which one plausible answer is wrong because it is larger than the starting quantity. By the end students should routinely ask 'should this be smaller or larger than what I started with?' before writing the answer.
3. How would you satisfy the district resource person's likely expectation of visible, measurable outcomes without returning to drill?
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The resource person will look for evidence that objectives were attained, and that can be shown without drill. Three artefacts: a pre-test and post-test on the same five word problems, with item-level change visible - the addition response should disappear even where the final answer is still wrong, which is a real gain; the students' own bar models, kept in the notebook; and a two-minute interview with three students, including one who originally answered 'one cup', in which they explain why their answer changed. If the visit falls in the second week, the pre-test and the period-2 models are already available. The point to make is that the measurable outcome here is not speed on 3/4 x 2/3 but the proportion of students who, given an unseen story, choose the operation from the situation and reject an impossible magnitude. That is measurable, with a before and an after, which drill on its own cannot show.