Comparing completed approaches can develop flexibility without asking the pupil to invent several strategies at once. Use two valid representations, or one valid and one carefully chosen flawed approach. Discuss how the problem was understood, represented, planned and checked.

The comparison is about the whole solution, not merely which calculation was fastest.

Use comparison after basic competence

Pupils need enough knowledge to understand the approaches being compared. An incorrect example should be introduced only when they can identify the underlying method without learning the error as a model.

Begin with clear, concise solutions rather than dense pages of work.

Compare two valid approaches

Problem: There are 8 bags with 6 apples in each bag. Five apples are eaten. How many remain?

Approach A:

8 × 6 = 48
48 − 5 = 43

Approach B:

7 × 6 = 42
42 + 1 = 43

Both are valid. Ask:

• How did each approach represent the problem?

• Why does Approach B use 7 full bags and one extra apple?

• Which is easier to explain?

• Which is more dependable for this pupil?

• How was the answer checked?

Do not imply the less obvious method is automatically better.

Compare a valid and flawed approach

A pupil sees:

8 × 6 = 48
48 + 5 = 53

Ask where the reasoning stopped matching the story. The calculation itself is accurate, but the operation does not fit the fact that apples were eaten.

Focus on the decision, not simply marking the answer wrong.

Compare representations

Two pupils may use:

• a bar model;

• equal groups;

• a number sentence;

• a table;

• a number line.

Ask what each model makes visible and whether every quantity is labelled. A representation should fit the structure rather than personal preference alone.

Use practical prompts

• “What did this person notice first?”

• “Where is the unknown shown?”

• “Why was this operation chosen?”

• “Which condition has been used?”

• “What is strong about each approach?”

• “Where does the flawed solution stop making sense?”

• “Which would you choose for a similar problem?”

Follow with a related problem

After discussion, give a new problem with the same underlying structure but different surface details. Ask the pupil to select and justify an approach independently.

This checks whether the comparison has improved transferable decision-making rather than only the completed example.

Common mistakes to avoid

• Comparing methods before pupils understand either one.

• Introducing incorrect examples too early.

• Focusing only on speed.

• Praising novelty over clarity and dependability.

• Comparing arithmetic procedures while ignoring representation and checking.

• Discussing examples without a related independent problem.

• Treating a different valid approach as wrong.

Evidence and further reading

The Education Endowment Foundation discusses mistakes and explanations in worked examples and cautions against using flawed examples before pupils are ready. Its guidance on mathematical problem-solving covers comparing strategies and moving towards independence.