Redesigning a Lesson on Direct Proportion After a Failed Exit Ticket
A student-teacher on teaching practice is observed teaching direct proportion to a Form 2 mathematics class of 52 students. The lesson begins with the formula y = kx, followed by one worked example from the textbook: 'If 3 pens cost 600 francs, find the cost of 7 pens.' The student-teacher then assigns 9 minutes of group work on five similar exercises. Only 7 of 52 students solve the exit ticket correctly: 'A recipe for 4 people uses 300 g of rice. How much rice for 10 people?' Several students add 6 to 300 and write 306 g. The observer says the fix is 'more examples'. The mentor says 'concept before formula'. The cooperating teacher says the syllabus must be covered and the class should move on.
1. Diagnose the likely misconception and explain why the observer's advice ('more examples') is insufficient.
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The exit-ticket error, adding 6 to 300, is a classic additive response where multiplicative reasoning is required. Students have noticed the difference between 10 and 4 and applied that difference to the rice quantity. This is not carelessness; it shows that the proportional structure has not been distinguished from an additive structure.
The lesson sequence made this likely. It began with the formula y = kx before students had compared situations that are proportional with situations that are not. One worked example showed a procedure, not the invariant relationship: for every additional person, the rice increases by a constant multiple, not by a constant amount. The five group exercises were all the same type, so students could imitate a routine without deciding whether proportion was appropriate. The formula then becomes a label attached to a procedure rather than a model of a relationship.
The observer's 'more examples' is insufficient because more of the same exercises may strengthen the additive error if students continue to compare differences. What is needed is a contrasting case, multiple representations, and a prediction step before calculation. The mentor's 'concept before formula' is closer, but it must be made concrete: tables, double number lines, graphs, and a non-example.
2. Design a 12-minute teaching segment that addresses the misconception and fits within the same lesson. Include what the teacher says or asks and how understanding is checked.
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0–3 min: Contrast two recipes. Display two situations. A: 4 people use 300 g; 10 people use how much? B: 4 people use 300 g; a second recipe for 10 people uses 600 g because it is a different dish. Ask: 'In which situation can we scale the first recipe? Why?' Students discuss in pairs. The teacher draws out 'same recipe, same rice per person' versus 'different recipe'.
3–7 min: Build a table and a double number line. Students complete: people 4, 8, 2, 10; rice 300, ___, ___, ___. The teacher asks: 'What do we multiply 4 by to get 10? What must we do to 300?' Then show the unit-rate route: 300 ÷ 4 = 75 g per person; 75 × 10 = 750 g. Do not start with y = kx; let k emerge as the amount per person.
7–10 min: Predict before calculating. Give a non-example: 'A 4-person recipe uses 300 g. For 10 people, a cook adds 6 g because there are 6 more people.' Ask students to explain the error using the table. This makes the additive misconception visible and correctable.
10–12 min: Exit check. 'A 5-person recipe uses 400 g. For 15 people, how much?' Students write the answer and one sentence explaining why they multiplied rather than added. The teacher scans for the additive error and addresses it in the next lesson.
3. How should the student-teacher respond to the three conflicting pieces of feedback without simply pleasing the observer?
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The student-teacher should treat the feedback as evidence to be interpreted, not as three orders to obey.
To the observer: agree that examples matter, but explain that the examples must be contrasting, not merely more numerous. Offer the exit-ticket evidence: seven correct responses and a recurring additive error show that repetition of the same worked-example pattern is unlikely to fix the concept. Propose to keep the syllabus pace but change the example sequence.
To the mentor: accept the principle 'concept before formula', but make it operational. The student-teacher can show the 12-minute segment with a table, a double number line, and a non-example before introducing y = kx. This respects the mentor's point and gives a concrete alternative to the failed lesson.
To the cooperating teacher: reassure him on coverage by showing that the redesigned segment teaches the same syllabus objective and adds only a short diagnostic check. The remaining exercises can be completed for homework, so the class does not fall behind.
The professional judgement is to use student evidence as the arbiter. The student-teacher should not simply add more examples to satisfy the observer, nor abandon the formula to satisfy the mentor, nor rush on to satisfy the cooperating teacher. The aim is to teach the same objective with a better sequence and to bring the three adults into a shared conversation about the exit-ticket evidence.