Worked examples: how to design and fade support

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What should a worked example reveal?

Start with one procedure and one intended decision. In a Year 7 algebra lesson, the decision is how to isolate an unknown while preserving equality. Before the example, check whether learners can divide a whole number and understand the equals sign. Otherwise, an error later could come from either prerequisite.

Write the reason beside the operation it explains. A final answer tells learners where the solution ends. It does not show why the route is valid. A page of calculations without explanations leaves that route for the learner to infer.

Here is an authored classroom illustration. The equations and prompts are examples for planning, rather than materials tested in the cited studies.

Solve 3x + 5 = 20.

1. Subtract 5 from both sides: 3x = 15. Applying the same operation to both sides preserves equality.
2. Divide both sides by 3: x = 5. This leaves one x on the left.
3. Substitute 5 into the original equation: 3 × 5 + 5 = 20. The proposed value satisfies the equation.

Ask: “Why do we subtract 5 from both sides?” Listen for the equality relationship, rather than the phrase “move the five”. If learners use that shortcut, ask them to show the operations behind it.

Keep the explanation close to its step. An unrelated story, a second method and extra terminology can all wait if they obscure the decision being taught. Once the procedure is secure, comparing valid methods becomes a different task.

For advice on demonstrating a process aloud, use our guide to modelling learning. Here the focus is the example itself and the work you ask learners to supply.

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Complete example, completion task or independent problem?

These formats allocate different parts of the solution to the learner. They are choices within a lesson, rather than three compulsory phases with fixed timings.

Format	What you provide	What learners supply
Complete example	The problem, solution and reasons	An explanation of a decision
Completion task	The problem and selected solution steps	A missing step and its reason
Independent problem	A fresh problem without solution steps	The method, solution and check

Keep the first completion task close enough to the example for the intended procedure to remain visible. For instance, after 3x + 5 = 20, give 4x + 3 = 19. Supply the first operation and ask learners to finish:

“Subtract 3 from both sides: 4x = 16. Write the next operation and explain why it preserves equality. Then check your value in the original equation.”

The expected continuation is division of both sides by 4, giving x = 4. Dividing both sides by the same non-zero number preserves equality and isolates x. The check is 4 × 4 + 3 = 19. A learner who writes 4 without the operation has supplied the result, but has not yet shown the requested reasoning.

Next, offer 5x + 2 = 27 without solution steps. Ask learners to solve it, explain their first operation and check the result. Here x = 5, because subtracting 2 gives 5x = 25. Substitution gives 5 × 5 + 2 = 27.

Mark the supplied steps clearly. Otherwise, a finished page can disguise how much of the solution the learner produced. When reviewing books, distinguish the teacher's model from the learner's own attempt.

Key takeaways

• Put the reason beside the operation it explains.
• Choose a missing step that reveals the intended decision.
• Use an independent attempt to decide what support comes next.

In this article

• What should a worked example reveal?
• Complete example, completion task or independent problem?
• How do I choose which steps to remove?
• How can I tell whether learners are copying?
• When should guidance change?
• What does the research establish?
• The questions teachers raise
• Questions teachers ask
• References
• Keep a one-page study note
• Plan tomorrow's sequence

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How do I choose which steps to remove?

Choose the omission before you make the worksheet. Ask: “What decision will this gap let me observe?” Removing an arbitrary line can make a task harder without making the evidence more useful.

In the completion task above, the first operation is supplied. The learner decides how to go from 4x = 16 to x = 4. That is appropriate if division is the target. It tells you less about whether the learner can choose the first operation.

If preserving equality is the target, change the gap. Show 4x + 3 = 19, leave the first transformation blank, and supply the later line x = 4. Ask the learner to write the operation that produces 4x = 16 and explain why it is valid. The supplied answer means this is still a completion task, rather than an independent solution.

One studied approach is backward fading: remove later solution steps before earlier ones as learners move from examples towards problems. Renkl and colleagues (2002) examined fading in a school electricity study and university laboratory studies. Their findings support a carefully bounded case for fading, especially on near-transfer tasks. They do not establish that every teacher must always remove the final line first.

Use the studied approach as an option, then judge the gap against your lesson's target. If the final line is only an arithmetic answer, removing it may check calculation rather than the decision you intended. You can ask for the reason as well as the missing line.

Give feedback on that decision. If a learner divides only one side, return to the equality relationship with a short complete example. If both sides are divided correctly but the division is wrong, address the calculation. The same incorrect answer can need different support.

Remove another step when the attempt justifies it. Restore a smaller piece of guidance when it does not. Neither move requires changing the learner's permanent group or labelling them as a novice.

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How can I tell whether learners are copying?

A correct copied solution is a record of what was shown. It is not evidence that the learner could construct the solution. Use a response that requires a decision the example has not already supplied.

After the complete example, cover the explanations and ask: “What makes the first operation valid?” Then give the completion task. Finally, use the fresh problem without visible steps. Each response answers a different question.

Response	What it helps you check	What remains uncertain
Explains a supplied step	Understanding of that decision	Choosing the step independently
Completes a missing step	Acting with partial support	Constructing the whole solution
Solves a fresh problem	Performance without supplied steps	Retention and broader transfer

A fluent explanation can coexist with a weak independent attempt. Equally, a brief spoken answer may reflect difficulty expressing the reasoning. Ask the learner to show the operation as well as describe it, rather than treating eloquence as your success criterion.

You can also use an incorrect transformation as a diagnostic prompt. Write 4x + 3 = 19 → 4x = 19 and ask what operation was applied to each side. The intended response is that removing 3 only on the left changes the equality. This is an authored check, not a claim that error examples always improve learning.

Keep an independent problem for a later lesson too. Immediate success tells you what the learner can do now. It cannot, by itself, settle what they retain after a delay.

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When should guidance change?

Relevant knowledge is specific to the task. A learner can be confident with one-step equations and still need help with a new equation structure. “Novice” describes that relationship to the content, rather than a fixed ability category.

Kalyuga and colleagues (2001) studied changing instructional effectiveness as mechanical trade apprentices acquired expertise. The relevant point is that guidance useful at one stage may become less useful as knowledge grows. This is part of the research behind the expertise reversal effect. It is not a timetable for withdrawing help from every Year 7 class.

In the algebra lesson, offer a brief independent attempt before deciding who needs the complete explanation again. A learner who solves the equation and explains both operations has different needs from someone who changes only one side. Let the observed response guide the next task.

Avoid removing every support at once. If you want to check choosing an operation, keep the vocabulary and layout familiar. Introducing unfamiliar notation, awkward numbers and a new instruction format together makes the cause of an error harder to identify.

For the wider account of working memory and instructional demands, see our cognitive load theory guide. For planning a lesson's changes in responsibility, use I Do, We Do, You Do. A worked example can sit within those approaches without turning them into a fixed fading schedule.

For a wider account of selecting classroom approaches, see our pedagogy guide.

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What does the research establish?

Renkl, Atkinson, Maier and Staley (2002) investigated the transition from example study to problem solving. Their article includes a German ninth-grade electricity quasi-experiment and university laboratory studies. The school study compared existing classes, and its teacher was also an author. That limits how confidently we can isolate fading as the cause of the result.

Across the three studies, fading produced better near-transfer performance than alternating complete examples with problems. Near transfer means applying the learned procedure to a similar problem. Far-transfer findings were mixed, so they do not justify a general promise about substantially different problems or subjects.

Changing 3x + 5 = 20 to 5x + 2 = 27 keeps the same equation structure. Success is useful evidence of applying that procedure. It does not demonstrate that learners can handle every equation, choose a method in an unfamiliar context or use the idea in extended writing.

Kalyuga, Chandler, Tuovinen and Sweller (2001) provides a second lesson about prior knowledge and guidance. Examples helped inexperienced mechanical trade apprentices, while problem solving became more useful with greater domain experience. This finding concerns knowledge within a domain, rather than confidence or a permanent ability label.

Use the research to frame a decision: make the procedure visible, reduce supplied steps deliberately and check the learner's response. Treat the classroom equations here as illustrations of that decision. They are not a tested programme or a guaranteed route to higher attainment.

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The questions teachers raise

These posts raise practical questions about guidance and task design. They are opinions rather than research findings.

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Keeping independence as the goal

“The point of worked examples is not to create permanent dependence.”

@ninja_maths · 1 April 2026

Check a fresh problem without supplied steps before deciding what support to remove.

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Checking the alignment of examples and practice

“The examples don’t align, not enough practice problems, too much spiral in the homework.”

@teachbk · 4 October 2026

This post concerns a scheme the author uses. Keep your example and first completion task on the same procedure, then check the independent problem yourself.

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Questions teachers ask

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Can I leave out the first step?

Yes, when that step is the decision you need to check. Make the remaining support explicit. A supplied later answer changes what you can infer from the completed task.

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Do confident learners need the full model?

Check their relevant knowledge with an attempt. Confidence alone is insufficient. If the procedure and reasoning are secure, use a task that requires more independent choices.

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Does changing the numbers show transfer?

It shows whether the learner can apply the procedure to that related problem. It provides limited evidence about substantially different tasks. Name what changed before deciding what the response proves.

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How many examples are enough?

There is no fixed count established by the sources used here. Look for accurate decisions across related tasks. If success depends on visible steps, keep or restore the needed guidance.

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References

• Renkl, A., Atkinson, R. K., Maier, U. H. and Staley, R. (2002). From Example Study to Problem Solving: Smooth Transitions Help Learning. The Journal of Experimental Education, 70(4), 293–315.
• Kalyuga, S., Chandler, P., Tuovinen, J. and Sweller, J. (2001). When Problem Solving Is Superior to Studying Worked Examples. Journal of Educational Psychology, 93(3), 579–588.

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Keep a one-page study note

Use this A4 note to revisit the three task formats, check the authored equations and retain the research limits. Teacher answers are visible. A text version follows the image.

Download the A4 study note (PDF)

Read the full study note

Study note · teaching practice

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Worked examples: reveal a decision, then fade support

A complete example supplies the problem, solution steps and result. A completion task supplies selected steps. An independent problem supplies no solution steps.

Authored equation sequence

1Complete

Complete: 3x + 5 = 20. Subtract 5 from both sides to get 3x = 15. Divide both sides by 3 to get x = 5. Check: 3 × 5 + 5 = 20.

2Completion

Completion: 4x + 3 = 19. First supplied step: subtract 3 from both sides, giving 4x = 16. Ask: which operation comes next, and why does it preserve equality? Teacher check: divide both sides by 4, so x = 4. Check: 4 × 4 + 3 = 19.

3Independent

Independent: 5x + 2 = 27. Ask learners to choose the operations, explain the first and check the result. Teacher check: subtract 2, then divide by 5, so x = 5. Check: 5 × 5 + 2 = 27.

Decide what the gap checks

Choose the missing step to expose the decision you want to see. A copied answer does not show an independently chosen method. Restore a needed step when an attempt reveals a gap. A similar equation checks a related procedure, not far transfer.

Evidence and limits

Renkl and colleagues (2002): fading improved near-transfer performance compared with example-problem pairs across a school electricity study and two university probability studies. Far-transfer results were mixed. The school study used existing classes and an author-teacher.

Kalyuga and colleagues (2001): inexperienced mechanical trade apprentices benefited from worked examples. With greater domain experience, problem solving became more useful. No fixed readiness threshold follows from this abstract.

These equations are authored planning illustrations, not the research tasks or a tested programme. Choose guidance from relevant knowledge, rather than a permanent ability label.

References

Renkl, A., Atkinson, R. K., Maier, U. H. and Staley, R. (2002). From Example Study to Problem Solving: Smooth Transitions Help Learning. The Journal of Experimental Education, 70(4), 293-315. DOI 10.1080/00220970209599510.

Kalyuga, S., Chandler, P., Tuovinen, J. and Sweller, J. (2001). When Problem Solving Is Superior to Studying Worked Examples. Journal of Educational Psychology, 93(3), 579-588. DOI 10.1037/0022-0663.93.3.579.

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Plan tomorrow's sequence

Choose a procedure from your existing scheme. Write one complete example with a reason beside each operation. Make a related completion task with a gap that reveals your target decision. Prepare a fresh problem without supplied steps and a short route back to support.

Before the lesson, check that the numbers work and the omitted step has a clear purpose. During the lesson, record which operation learners can choose and justify. Next lesson, remove one solution step only after checking the reasoning it will require.