Fading a worked example means removing selected support so the learner takes over a particular decision. A useful completion example keeps enough context to make that decision meaningful. Deleting random numbers may test guessing or copying rather than the reasoning you meant to teach.
This teacher-design exercise follows one small two-step calculation. Its assessed product is the support sequence you create: what the model explains, what the learner must supply, and what evidence would justify removing or restoring help. All situations and responses are fictional.
Name the reasoning you want the learner to take over
Target: Find the number of cards remaining after some cards from equal-sized packs are displayed, and explain why the displayed cards are subtracted from the starting total.
Two numerical operations are needed: multiply packs by cards per pack, then subtract displayed cards. The selected reasoning step is the second operation and its meaning. Before fading it, check whether the learner can identify the starting total and distinguish it from the number displayed. If the multiplication itself is insecure, provide appropriate support instead of hiding both operations at once.
For this task, arithmetic can be checked separately from the operation choice. A learner who writes the correct subtraction but makes a calculation slip needs different support from one who adds displayed cards to the stock.
The complete model: explain every decision
Problem M: Five packs each contain six cards. Eight cards are taken from these packs and placed on a display. How many cards remain in the packs?
| Step | Worked response | Reason made visible |
|---|---|---|
| Find the starting total | 5 × 6 = 30 cards | Five equal packs each contribute six cards. |
| Find what remains | 30 − 8 = 22 cards | The eight displayed cards came from the thirty, so remove them from that total. |
| Check the relation | 22 + 8 = 30 | Remaining cards and displayed cards together account for the starting total. |
Say what each number counts. Thirty is cards, not packs. Eight is already part of the original thirty; it is not an additional delivery. Ask the learner to explain that relationship while the model is available. Reading the final answer aloud is not the same as explaining it.
The completion example: remove the selected support
Problem C: Four packs each contain seven cards. Nine cards are taken from these packs for a display. How many remain in the packs?
Support still supplied: “The starting total is 4 × 7 = 28 cards because there are four equal packs of seven.”
Learner’s completion task: Write the next calculation, give its answer with the unit, and explain why that operation matches taking nine cards from the packs. Then show a calculation that checks the remaining and displayed cards account for the starting total.
Solved completion: 28 − 9 = 19 cards remain. Subtract because the nine displayed cards are taken from the original twenty-eight. The check is 19 + 9 = 28. These are teacher review answers; allow the learner to complete the task before displaying them.
The removal is deliberate: the second operation and its explanation are no longer supplied. The starting-total calculation stays visible. Asking for the check provides a way to inspect the relationship without filling the target step for the learner.
The independent near-transfer task
Problem I: Six packs each contain eight cards. Thirteen cards are taken from those packs for a display. How many remain? Show your method and explain the relationship between the starting, displayed and remaining amounts.
No calculation or operation is supplied. Solution: 6 × 8 = 48 cards initially; 48 − 13 = 35 cards remain. Subtraction removes the thirteen from the original stock. The check is 35 + 13 = 48.
This is near transfer because the numbers change while the pack-and-display structure remains familiar. Success is useful evidence for this structure. It does not establish that the learner can choose methods in every unfamiliar word problem.
| Version | Supplied support | Learner takes responsibility for |
|---|---|---|
| Model M | Both calculations, explanations and check | Explaining how the displayed cards relate to the total |
| Completion C | Starting-total calculation and reason | Remaining calculation, reason and check |
| Independent I | Situation and task only | Both calculations, meanings and check |
Worked planning decision 1: choose the omission
A teacher proposes erasing the 7 from “4 × 7 = 28” while leaving “28 − 9 = 19” visible. That completion might practise finding a multiplication factor, but it does not require choosing subtraction for the remaining cards.
Better design: Keep the first calculation and remove the entire second calculation and reason. Retain the story so the learner has the information needed to reconstruct them. The omission should make the target decision necessary, not conceal unrelated data.
Worked planning decision 2: respond to a completion error
A fictional learner writes “28 + 9 = 37” and explains, “The display has nine.” The arithmetic is correct, but the answer treats cards already in the packs as an extra supply.
Restore a meaning prompt: “Were those nine cards already included in the twenty-eight, or did new cards arrive?” Ask the learner to account for the starting cards. A revised explanation might be, “They were already included, so the packs have fewer: 28 − 9 = 19.”
Record that the revision followed a prompt. Do not remove more support simply because the answer is now correct. If the learner still cannot explain the relationship, revisit the complete model or use a clear verbal partition: starting cards consist of displayed cards plus cards left in packs.
Worked planning decision 3: judge readiness from evidence
Two fictional learners both write 19 for C. One explains the subtraction and checks 19 + 9 = 28. The other says, “I saw that answer on the board,” and cannot identify the original stock.
Decision: Offer I without supplied calculations to the first learner. For the second, conceal the solved completion and return to the relationship prompt with another small example. The difference is evidence about the targeted reasoning, not speed, neatness or a fixed ability group.
A learner who explains the subtraction but calculates 28 − 9 as 21 may need arithmetic support while retaining ownership of the operation choice. Restore only what the evidence suggests is missing. A fixed number of minutes or pages completed does not decide when support should disappear.
Three solved teacher-design tasks
Practice 1: preserve the reasoning demand
Task: Three packs hold nine cards each; five are displayed. You want the learner to choose the second operation. What should the partial example include?
Model: Supply 3 × 9 = 27 with its pack explanation, then ask for the remaining calculation and reason. The solution is 27 − 5 = 22, checked by 22 + 5 = 27. Do not leave the minus sign in place if selecting subtraction is the target.
Practice 2: diagnose what needs restoring
Task: For I, a learner writes 6 × 8 = 48 and 48 − 13 = 36, explaining correctly that the displayed cards came from the total. What support fits?
Model: Preserve the sound operation choice and help check subtraction: 48 − 10 = 38, then 38 − 3 = 35. The proposed check 36 + 13 = 49 exposes the error. A full reteaching of equal packs is not justified by this evidence.
Practice 3: evaluate an answer-only completion
Task: A partial example shows every operation and leaves only the final number blank. Does it test operation selection?
Model: No. It can test calculation, but the operation is supplied. To inspect selection, remove the relevant operation and request a contextual reason. Keep task data available so the learner is solving rather than guessing missing facts.
Questions about fading worked examples
Must support always disappear in three stages?
No. These three versions make the design visible. Add, retain or restore support when the learner’s response calls for it.
Should I hide all earlier models?
During an independent check, make clear which help is available. During learning, revisiting a model can be useful. Record the conditions rather than treating both performances as equivalent.
Can the learner explain orally?
Yes, when mathematical reasoning is the target rather than written expression. Record the reasoning accurately and note any additional prompts.
Is finishing quickly a reason to skip the completion?
Check the relevant explanation. Speed alone does not show whether a learner chose the operation or copied a familiar pattern.
What if multiplication facts slow the task?
Supply or support the starting calculation when the target is interpreting what remains. Revisit multiplication separately if it is also an intended learning goal.
Does restoring support mean the sequence failed?
No. It can be a useful response to evidence. Check whether the support addresses the specific obstacle and then observe what the learner can do.
Is a completion example the same as an easier problem?
Not necessarily. It changes which steps are supplied. The situation and reasoning can stay comparable while responsibility for selected decisions changes.
Further reading
Recommendation 2 in the Institute of Education Sciences practice guide on organising instruction supports alternating worked solutions with problem solving. The specific omissions and card examples here are an original instructional design, not a prescribed sequence.
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