Word Problems That Will Not Transfer
An upper-primary mathematics teacher in a Kolkata municipal school has 40 students in Class VII. In a recent unit on the unitary method, 33 of 40 students scored above 75% on a 20-item practice set that used the same structure as the solved examples. On a parallel set of five word problems - same operation, but set in a vegetable market, a bus timetable and a tailoring shop, with decimals in two items - only 9 students scored above 40%, and 12 left the page blank. The teacher's usual method is: explain the rule, solve three examples on the board, assign 15 similar sums for homework. The school timetable gives four 40-minute mathematics periods per week, and the departmental syllabus for the term has two more chapters to cover. The teacher concludes that the students 'lack intelligence' and asks for a slower section.
1. Is the teacher's conclusion justified? Analyse the gap using the concepts of near and far transfer, and explain two likely causes of the failure.
Read the worked answer
The conclusion is not justified. The same students who scored above 75% on the practice set are clearly able to carry out the operation; the failure appears when the context, language and decimal placement change. This is a transfer problem, not evidence of low intelligence.
Near transfer involves applying a skill to a task very similar to the one practised; the 20-item set is near transfer because the structure and wording match the worked examples. Far transfer involves applying the same underlying relationship in a new setting; the market, timetable and tailoring problems are far transfer because the surface features differ. Thorndike's theory of identical elements would predict that transfer occurs only when the learner can identify elements common to both situations. If students have learned to match keywords and imitate steps, they have not abstracted the relation between rate, unit and total. The decimal items add a second demand, and the reading load of word problems may also block access.
A second cause is the worked-example method used without comparison or articulation: students copy a procedure rather than represent the problem. The teacher should test this by asking students to draw or explain one problem before concluding anything about ability.
2. Design a 20-minute classroom activity that would improve transfer, using either Thorndike's or the Gestalt view of transfer. State the steps and what you would look for.
Read the worked answer
A Thorndike-based activity would teach the common elements across contexts. Prepare six cards: three pairs that share the same mathematical structure but have different stories, for example a market rate problem, a bus fare problem and a tailoring cost problem. In the first five minutes, pairs sort the cards into 'same mathematics' groups and write the common relation - rate multiplied by quantity equals total. In the next five minutes, each pair solves one card and annotates which numbers are the rate, the unit and the total. In the next five minutes, two pairs compare their annotations and explain why a different story did not change the operation. In the final five minutes, each student writes one new context for the same relation.
I would look for students naming the relation rather than a keyword, transferring the labels correctly when decimals appear, and producing a sensible new context. A Gestalt alternative would present one complete market problem, ask students to draw it and restructure it, and delay the rule until they see the relation. Either way, the teacher must compare examples rather than simply add more practice.
3. How would you assess transfer differently from the teacher's practice-set test?
Read the worked answer
The practice-set test measures near transfer and rewards imitation, so it cannot show whether students understand the relation. A transfer assessment should use unfamiliar contexts, mix the target structure with superficially similar but different structures, and ask students to represent the problem before solving it. For example, give four items: two require the unitary method in new settings, one is a near-miss that needs a different operation, and one asks students to write a problem for a given equation. Include a delayed retest after a week to see whether the learning persists.
Score more than the final answer: award marks for identifying the rate, unit and total, for a correct diagram or table, and for explaining why an operation was chosen. Use a short rubric and analyse errors by type - reading, representation, operation choice or calculation. Oral probing of two or three students who left the page blank will distinguish a language or confidence barrier from a conceptual one. The purpose is diagnostic, not to rank students or justify a slower section.