Worked Examples in Maths: From Teacher Modelling to Independent Practice

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When a learner faces an unfamiliar maths problem, "try harder" is rarely a complete teaching strategy. A worked example shows a solution and makes the decisions behind it visible. It can provide a starting point before students solve related problems themselves.

The Institute of Education Sciences recommends combining worked solutions with problem-solving exercises. The classroom sequence below is a practical illustration of that principle, not a fixed script for every topic.

Model the decision, not only the operation

Consider 3x + 5 = 20. Write the goal first: find the value of x while preserving equality. Subtract 5 from both sides to obtain 3x = 15, then divide both sides by 3 to obtain x = 5. Substitute 5 into the original equation to check the result.

Explain why subtracting is appropriate and why it applies to both sides. If the teacher writes only a sequence of symbols, students may imitate the layout without understanding the relationship.

Use questions that reveal the structure

Pause at selected steps: "What would happen if we subtracted 5 on only one side?" or "Why do we check using the original equation?" Allow thinking time before answers. One student completing your sentence does not establish that everyone is ready.

Keep attention on the current idea. Avoid an example with difficult fractions, unfamiliar notation and a long story if the immediate goal is understanding inverse operations.

Move to a completion problem

Give a related task, such as 4x + 6 = 22, with the first step shown: 4x = 16. Ask students to complete it, explain the missing step and check x = 4 in the original equation. A completion problem provides support while requiring a meaningful decision.

If students struggle, locate the source. Do they understand equality? Do they know the operation? Is arithmetic distracting from the algebra? The response should fit the difficulty rather than automatically providing more of the same worksheet.

Remove support gradually

  1. Show and discuss a complete solution.
  2. Ask students to explain a selected step.
  3. Provide a partly completed example.
  4. Offer a similar problem without the steps.
  5. Later, vary the structure or context and check whether the method still makes sense.

For instance, moving from 3x + 5 = 20 to 5 + 3x = 20 tests whether students recognise the structure rather than rely on the visual position of a term. Introduce a larger change only when prerequisite understanding is secure.

Let mistakes become material for explanation

Present an invented solution with one clear error. Ask students to identify it and explain a correction. Keep the discussion respectful and do not display a named student's error without appropriate consent. Avoid so many incorrect examples that learners lose track of the correct process.

Check independence

A completed page is not enough if students copied the adjacent example. Use a short independent task and ask for reasoning. Return to an important skill in a later lesson. If support is still needed, that is useful planning information rather than a permanent label.

Worked examples are a tool, not the whole curriculum. Students also need application, discussion and opportunities to select methods. For a brief lesson-end check, see ten exit-ticket questions.

Further reading

IES: Organizing Instruction and Study to Improve Student Learning

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