Comparing complete solutions can help pupils examine how a problem was represented, which assumptions were made, how calculations were sequenced and whether the final result was checked. This develops more than calculation choice.

Use two valid approaches, or a valid solution alongside a plausible error. Follow the comparison with a related problem that the pupil solves independently.

Compare after methods are understood

Pupils need enough knowledge to analyse the examples. Introducing a flawed solution too early may reinforce the error rather than expose it.

Keep the solutions concise and ensure the mathematics is accessible before asking for evaluation.

Compare two valid approaches

To increase £240 by 15%, one solution might:

• calculate the increase and add it to the original: 15% of £240 is £36, and £240 + £36 = £276;

• use a multiplier in one calculation: £240 × 1.15 = £276.

Ask:

• How is the original value represented?

• Which approach makes the percentage relationship clearest?

• Which is more efficient for these values?

• Which is easier to check?

• Under what conditions would each approach be suitable?

Do not imply the shortest solution is always best. Dependability and clarity matter.

Analyse a plausible error

For a density problem, a flawed solution may:

• use the correct formula;

• substitute mass in grams and volume in cubic metres;

• calculate accurately;

• produce an invalid result because the units are inconsistent.

For a block with mass 540 g and volume 0.0002 m³, 540 ÷ 0.0002 gives 2,700,000. Written as kg/m³, that is a thousand times too large, because the mass should first be converted to 0.54 kg. The correct density is 2,700 kg/m³.

Ask where the solution stopped matching the problem. The issue is not the arithmetic but the representation and units.

Incorrect examples should use realistic errors and be clearly resolved before pupils move on.

Compare the whole problem-solving process

Consider:

• interpretation of the question;

• choice of representation;

• assumptions;

• step order;

• method selection;

• handling of units and constraints;

• final checking;

• communication of the answer.

Two solutions with the same arithmetic may differ significantly in how well they address the actual problem.

Use structured prompts

• “What did each person identify as the unknown?”

• “Which representation exposed the structure?”

• “Which condition was used here?”

• “Why is this intermediate result needed?”

• “Where is the answer interpreted?”

• “How could the solution be checked?”

• “Which approach would you choose, and why?”

Require mathematical justification rather than preference.

Follow with a related unfamiliar problem

Change the surface context while preserving the underlying structure. Ask the pupil to select, apply and review an approach.

This checks whether the comparison supports transfer rather than only commentary on finished work.

Common mistakes to avoid

• Comparing solutions before pupils understand the mathematics.

• Introducing flawed examples too early.

• Focusing only on speed or number of steps.

• Treating an unusual method as automatically superior.

• Ignoring assumptions, units and constraints.

• Discussing examples without independent follow-up.

• Leaving an error unresolved.

Evidence and further reading

The Education Endowment Foundation discusses mistakes and explanations in worked examples and using worked examples for mathematical problem-solving. Both emphasise explanation, comparison and careful timing of flawed examples.