Secondary Mathematics Tuition | How to Use Worked Examples in Maths: Example Fading and Independent Problem Solving

Secondary Mathematics Tuition · Worked examples, example fading and independent problem solving

Adrian is following every line.

The tutor solves a simultaneous-equation problem. Adrian nods when the equations are aligned. He nods when one equation is subtracted from the other. He nods when the remaining variable is solved. He even spots the arithmetic before the tutor writes it.

Then the next question appears.

The coefficients are different. One variable is cheaper to eliminate by addition rather than subtraction. Nothing is labelled. The tutor says nothing.

Adrian stares at the page.

Aisha has the opposite problem. She dislikes looking at worked examples because she thinks they are “too easy”. She skips straight to blank questions, spends ten minutes testing weak routes, and sometimes reaches the correct answer after a great deal of unnecessary search.

Ryan can copy a complete solution perfectly. His notebook looks excellent. Yet if one line is removed, his explanation collapses into “that is how the example did it”.

All three learners are interacting with worked examples. Only one question matters: is the example transferring mathematical control to the learner?

This worldwide guide is written for secondary learners across school systems. Its job is to show how worked examples should be designed, studied, explained, completed, faded, varied, revisited and finally removed so that a learner can move from guided solutions to independent mathematics.

The ownership boundary is deliberate. How Worked Examples & Example Fading Work in Secondary 3 Mathematics remains the stage-specific Secondary 3 owner. How SEC Mathematics Support Fading Works remains the SEC support-fading owner across G1, G2 and G3. Predict the Next Step Before Reading the Worked Solution remains the narrow owner for that specific study technique. How to Practise Maths Effectively remains the wider deliberate-practice owner. This page owns the deep, world-facing Secondary Mathematics worked-example casebook: how examples become independent capability.

Adrian, Aisha and Ryan are fictional recurring learners. Their scenes are explanatory examples, not reported student cases or fixed ability labels.

Choose a route: if you can follow examples but cannot start alone, begin with following versus solving. If you copy model answers, go to worked example versus model answer. If you are a tutor deciding what to show, use what a good example exposes. If you want to know when support should shrink, go to fade by evidence. Later sections build example–problem pairs, partial solutions, backward fading, self-explanation, error examples, route comparison, mixed selection, retrieval, delayed transfer and an independence checkpoint.

The central principle is simple:

A worked example should carry only the parts of the route the learner cannot yet carry. As capability grows, the example should shrink.

1. Following a solution is not the same as generating one

Worked solutions are unusually good at creating a feeling of understanding.

Every completed line supplies context for the next line. Once the solver has chosen the representation, the reader no longer needs to choose it. Once the solver has selected elimination, the reader no longer needs to decide between elimination and substitution. Once the solver has written the first equation, the reader no longer needs to decide what relationship to form.

This means a learner can correctly understand each visible step while still being unable to generate the invisible decision that produced it.

Adrian’s simultaneous-equation example illustrates this.

He can check the tutor’s algebra. He can explain why the subtraction is valid after the tutor has chosen it. But the next question removes the tutor’s route selection. The task changes from “understand the displayed step” to “generate a useful step from the current mathematical state”.

Those are different capabilities.

Use a practical distinction:

Recognition: “This step makes sense when I see it.”

Completion: “I can supply a missing step when the route is mostly visible.”

Reconstruction: “I can rebuild the route after the example is hidden.”

Selection: “I can choose the route when no method cue is supplied.”

Transfer: “I can use the structure when the surface changes.”

Independence: “I can start, execute, check and recover without the example carrying hidden decisions.”

A learner may be strong at one level and weak at the next.

Aisha skips examples because she wants immediate independence. That can also be inefficient. If the mathematical structure is genuinely new, unguided search may consume attention without revealing the key relationship. She may test several routes that fail for predictable reasons an example could have made visible in thirty seconds.

Worked examples therefore solve a real problem: they make expert choices inspectable.

The risk appears only when inspection is confused with ownership.

Ryan’s copying is the clearest case. He writes every line of a factorisation example into his notebook. If asked why the expression was rewritten in that form, he points to the previous page. The paper contains the solution; Ryan does not yet contain the route.

A good worked-example system should therefore ask, at every stage:

Which part of this solution is still being generated by the page rather than by the learner?

That question determines what should be removed next.

If the learner cannot explain the relationship, keep explanatory support.

If the learner can explain but cannot complete a missing line, fade only a small step.

If the learner can complete the route but cannot start, remove more of the setup.

If the learner can start on a familiar surface but fails when wording changes, vary the representation.

If the learner succeeds only when the topic label is visible, mix the problem with competitors.

The worked example is not successful because it is clear. It is successful when clarity eventually survives the example’s disappearance.

2. A worked example is not merely a model answer

A model answer is usually designed to show what a complete correct response could look like. That is useful for checking completeness and communication.

A worked example has a different teaching job.

It should make the decision architecture visible.

Consider a reverse-percentage problem:

An amount is 144 after a 20% decrease. Find the original amount.

A compact model answer might write:

144 ÷ 0.8 = 180.

That is correct. It is also pedagogically thin for a learner who repeatedly reverses percentage changes incorrectly.

A worked example should expose the relationship:

“The final amount is 80% of the original because 20% has been removed.”

“Let original amount = x.”

“Then 0.8x = 144.”

“So x = 180.”

“Check: 20% of 180 is 36, and 180 − 36 = 144.”

The extra value is not simply more words. The example reveals which quantity is 100%, how the multiplier arises and how the answer can be challenged.

Now consider geometry.

A model answer may jump from a diagram to a trigonometric equation.

A worked example should show:

which angle is the reference angle;

which side is opposite it;

which side is adjacent;

why the triangle is right-angled;

which ratio connects the known and unknown quantities;

and how the final answer is checked against the diagram.

The mathematics that matters most is often the line that experts omit because it feels obvious to them.

That is why worked examples must unpack expert compression.

Aisha can solve many equations. When a tutor writes “let x be the original amount”, she sees it as unnecessary. But for a novice learner, defining x may be the decision that turns a story into mathematics.

Ryan sees a factorisation solution:

x² − 7x + 12 = 0

(x − 3)(x − 4) = 0

x = 3 or x = 4.

The worked example should not only show the factorisation. It should expose why factorisation is an attractive route here: integer factors of 12 whose sum is −7 are visible, and the zero-product structure then makes the roots immediate.

Later, a nonfactorable quadratic should appear so the learner does not turn the example into a command: “quadratic means factorise”.

A strong worked example therefore answers four questions:

Cue: What feature in the problem suggests this route?

Move: What mathematical action is being taken?

Reason: Why is the action valid and useful?

Check: How can the result be challenged independently?

These four labels are compact enough for real teaching and broad enough to travel across algebra, geometry, statistics, probability and modelling.

A model answer can remain concise. The worked example exists to make hidden decisions learnable.

3. A good worked example exposes the decisions that determine the route

Not every line deserves equal attention.

If a worked example explains every arithmetic operation in equal detail, the important structural choices can disappear inside commentary.

The learner needs to see the decisions that change the state of the problem.

These commonly include:

Target decisions. What is actually being asked for?

Representation decisions. Should the situation become an equation, diagram, graph, table, tree, coordinate system or ratio?

Method decisions. Which relationship or theorem is authorised?

Form decisions. Should the expression be expanded, factorised, rearranged, substituted or left exact?

Condition decisions. What restrictions, units, domains or assumptions must remain true?

Checkpoint decisions. Where is the working fragile enough to justify a quick check?

Completion decisions. What final form is required?

Take the system:

3x + 2y = 16

x + 2y = 8

A worked example could simply subtract and finish. A better example pauses before subtraction:

“The y coefficients are already equal. Subtracting the second equation from the first removes y without creating larger numbers.”

The decision is now visible.

Change the system to:

2x + y = 11

x − y = 1.

Now addition eliminates y. The learner should notice that the method family is the same but the local move changes with coefficient structure.

This is how examples teach route families rather than photographs.

In geometry, a decision point may be recognising that an unknown height can be created as an intermediate target before an area can be found.

In probability, the decision point may be recognising that the state changes without replacement before any multiplication occurs.

In statistics, the decision point may be recognising that unequal group sizes make a simple average of means invalid.

In a linear model, the decision point may be distinguishing a fixed intercept from a variable rate.

In a proof, the decision point may be recognising that a definition supplies the bridge between the givens and the target.

Adrian’s tutor begins marking decision points with a small dot in the margin. Not every line. Only the places where another reasonable route could have been chosen or where a condition matters.

After the solution is complete, Adrian explains only those marked points.

This reduces unnecessary narration and concentrates attention on transferable structure.

Aisha is asked to compare two valid routes to the same algebra problem. One route is shorter; the other is easier to check. The example now teaches judgment, not obedience.

Ryan is given a worked solution with one invalid line. His task is to locate the first place where validity breaks. The example now teaches diagnosis, not imitation.

The question “What happened next?” is therefore weaker than:

“What changed about the mathematical state, and why was that change chosen?”

That is the level at which worked examples become portable.

4. Self-explanation turns a visible route into a learner-owned relationship

A learner can read a worked example carefully and still remain passive.

Self-explanation interrupts passive recognition by requiring the learner to generate the reason behind a step.

The goal is not to narrate every symbol.

Ask for explanation at the structural points.

Why was this representation chosen?

What rule makes this line equivalent to the previous line?

What condition permits this theorem?

Why is this route cheaper than the alternative?

What would make the route fail?

How could the answer be checked?

Aisha studies the reverse-percentage example. Instead of copying “0.8x = 144”, she must explain why 0.8 multiplies the original rather than the final amount.

Adrian studies an algebraic fraction. He explains that cancellation applies to common factors, not terms separated by addition. This explanation becomes especially valuable when a near non-example appears later.

Ryan studies a probability tree. He explains why the second denominator changes after a draw without replacement.

Self-explanation can also compare alternatives.

“Why is subtraction attractive here?”

“Could substitution also work?”

“Which route would create larger intermediate numbers?”

“Which route has an easier independent check?”

This prevents the learner from treating the author’s route as the only legal route.

Use concise explanation prompts.

Because…

This step preserves…

This method is valid because…

This feature tells me…

I would check by…

For some learners, written annotation is useful. For others, oral explanation is faster.

The objective is evidence of relationship knowledge, not polished prose.

Self-explanation is also a gate before fading.

If Ryan cannot explain why the example uses a common denominator, removing the next line may produce guessing rather than productive reconstruction.

If Aisha can explain every decision and predict the next move before it is shown, the full example may already be too supportive.

This suggests a rule:

Do not fade merely because the learner has seen enough examples. Fade when the learner can explain what the current support is carrying.

The example first makes the route visible. Self-explanation makes the route discussable. Fading then tests whether the learner can generate it.

The sequence matters.

5. Fade by evidence, not by a fixed number of examples

There is no universal rule that every learner needs three full examples, then two partial examples, then independence.

The correct amount of support depends on prior knowledge, topic complexity, representation familiarity and the particular decision the learner is trying to inherit.

Use evidence.

Signal 1: prediction. Can the learner predict the next important move before it appears?

Signal 2: explanation. Can the learner state why the move is valid?

Signal 3: completion. Can the learner fill a missing step accurately?

Signal 4: reconstruction. Can the learner rebuild the route after the example is hidden?

Signal 5: variation. Can the learner handle changed numbers, orientation, wording or unknown?

Signal 6: selection. Can the learner recognise the method without a chapter label?

Signal 7: checking. Can the learner initiate a meaningful check without being told?

When several signals are strong, remove support.

When performance collapses, identify which support was carrying the route.

Suppose Adrian can complete missing algebra steps but cannot decide how to start a new question. Removing arithmetic support has succeeded; method-selection support remains.

The next fade should remove the first strategic line, not more middle arithmetic.

Suppose Aisha can start and solve but does not check. The example should stop showing the checking prompt so she must generate it.

Suppose Ryan can solve a familiar diagram but fails when it rotates. The next change should target representation, not number complexity.

Fading too quickly creates unproductive search.

Fading too slowly creates support-bound competence.

The useful middle is controlled transfer of responsibility.

Official and research-informed guidance supports the general direction but not a universal classroom timetable. The What Works Clearinghouse algebra practice guide includes a recommendation to use solved problems to engage middle- and high-school students in analysing algebraic reasoning and strategies; that recommendation is rated minimal evidence within the guide, so it should not be inflated into a guarantee. The guide also gives examples of alternating solved and unsolved problems. The Education Endowment Foundation has described fading and alternation as ways to bridge from worked examples toward independence, with pace chosen according to curriculum and learner need. These sources support the architecture while leaving the actual fading decision to mathematical evidence.

That evidence can be summarised in one question:

“If I remove this piece of help, will the learner have to think productively—or merely guess?”

Fade when the answer is productive thinking.

6. Pair worked examples with problems so the learner must act immediately

A worked example should not sit alone for too long.

The learner needs to move from inspection to production while the structure is still available enough to compare.

One useful rhythm is:

worked example → matched problem → worked example → matched problem.

The purpose is not to make the second problem a clone. It should preserve the key structure while changing enough surface detail to require reconstruction.

Suppose Adrian studies:

2x + y = 11

x − y = 1.

The worked example adds the equations because the y coefficients are opposites.

The paired problem becomes:

3x + 2y = 17

x − 2y = 3.

Now y can again be eliminated by addition, but the coefficients and later arithmetic are different.

A stronger second pair changes the local decision:

4x + 3y = 22

x + 3y = 10.

This time subtraction is cheaper.

The route family remains elimination, but the learner must inspect coefficient structure rather than replay one operation.

Aisha studies a reverse-percentage example involving a discount. Her paired problem uses population decrease rather than shopping. The percentage relationship stays the same; the vocabulary changes.

Ryan studies a similarity example with one orientation. His paired problem rotates the figure and asks for a different corresponding side.

These pairings do two things.

First, they test whether the learner extracted structure.

Second, they give immediate evidence about what the example failed to teach.

If Adrian chooses elimination correctly but makes a sign error, route selection transferred while execution did not.

If Aisha uses forward percentage instead of reverse percentage, the relationship was not extracted strongly enough.

If Ryan cannot identify correspondence after rotation, the example was too tied to visual layout.

Use a three-part paired problem.

Same idea, changed numbers. This checks basic reconstruction.

Same idea, changed surface. This checks whether the structure survives superficial variation.

Near neighbour. This checks whether the learner knows when not to use the method.

For Pythagoras, the near neighbour might be a non-right triangle.

For direct proportion, it might be a linear relation with nonzero intercept.

For probability without replacement, it might be a with-replacement case.

For factorisation, it might be a quadratic with no convenient integer factors.

Do not require all three after every example. Use them when the decision boundary matters.

The What Works Clearinghouse algebra guide gives an example of alternating solved and unsolved algebra problems with similar structure. Its purpose is not simply repetition: the unsolved problem requires the learner to engage with the solved route. That is the core value of the pair.

A worked example becomes stronger when the next problem can disagree with the learner’s illusion of understanding.

If the learner can produce the mathematics, move forward.

If not, the failure tells you what the next example should expose more clearly.

7. Backward fading can make the first transition from reading to doing less abrupt

One practical way to fade a multi-step procedure is to remove later steps first.

The beginning of the route remains visible while the learner completes the final part.

Then more of the solution is removed progressively.

Consider solving a linear equation:

4(x − 2) + 3 = 19.

A full worked example might show:

4x − 8 + 3 = 19

4x − 5 = 19

4x = 24

x = 6

Check: 4(6 − 2) + 3 = 19.

A first faded example could supply the expansion and simplification but leave:

4x − 5 = 19

4x = ___

x = ___

Check: ___.

The learner inherits the end of the route.

A later example supplies only:

4(x − 2) + 3 = 19

4x − 8 + 3 = 19

then leaves the remaining route blank.

Later still, the equation is presented with no worked line.

Backward fading can be useful because the learner is not asked to invent the entire start-state immediately. The route remains anchored while more responsibility is transferred.

But do not turn backward fading into a rigid law.

Sometimes the strategic first move is precisely the skill that needs training.

In reverse percentage, the first move—identifying the 100% quantity—may matter more than the final arithmetic.

In geometry, identifying the theorem or hidden triangle may be the central capability.

In probability, defining the event and recognising state change may matter more than the multiplication.

In those cases, fade the strategic beginning sooner.

Adrian is strong at algebraic execution but weak at starting word problems. Backward fading leaves the very thing he needs to learn visible for too long. His tutor instead gives the story, removes the equation setup, and lets Adrian define the unknown and form the relationship. Once he starts correctly, more algebraic support can remain if necessary.

Aisha has the opposite profile. She recognises the route immediately but makes errors in long algebraic chains. Her first fades should remove later execution steps because that is where independent control is weak.

Ryan needs work on checking. The example can remain largely complete while the final verification line is removed first. He must generate the check himself.

This suggests a broader rule:

Fade the part of the route the learner is ready to inherit—not simply the part that appears last on the page.

Backward fading is a useful default for some procedures. Evidence should decide when to depart from it.

8. Fade decisions, not only lines of algebra

A partial worked example can look demanding while still carrying all the important thinking.

Suppose a worksheet says:

“Use Pythagoras to find x.”

The triangle is labelled.

The formula is printed.

The substitution is already started.

Only the arithmetic is blank.

The learner is doing something, but almost every strategic decision has already been made.

This can be appropriate if arithmetic is the target. It is not enough if the goal is independent geometry.

Decision fading removes support at the points that determine the route.

Use several fade dimensions.

Target fade. Stop underlining what must be found. The learner identifies the target.

Representation fade. Remove the provided diagram, table or equation setup. The learner creates it.

Method-label fade. Remove “Use Pythagoras”, “Use elimination” or “Use the quadratic formula”.

Condition fade. Stop reminding the learner which theorem conditions matter. They must inspect them.

Algebra fade. Remove intermediate transformations.

Calculator fade. Remove button-sequence cues and require the learner to choose the exact entry.

Checking fade. Remove the instruction “check your answer” and see whether verification becomes spontaneous.

Interpretation fade. Stop writing the final sentence for the learner. They must connect the mathematical result back to the question.

Adrian’s algebra example is faded by method label first because he needs to learn route choice.

Aisha’s geometry example is faded by diagram annotation because she needs to identify the measured objects herself.

Ryan’s statistics example is faded by interpretation because he can calculate means but often fails to state what the comparison actually shows.

A teacher can therefore keep the numerical complexity nearly constant while increasing independence.

This is often better than making every next question “harder” by adding bigger numbers or more steps.

Difficulty has many dimensions.

A problem can become harder because:

the route is no longer named;

the representation must be built;

the condition is implicit;

the surface changes;

the method competes with neighbours;

the problem returns after delay;

or the learner must check independently.

Control these dimensions separately where possible.

If you remove method labels, change representation and increase algebraic complexity at the same time, failure becomes difficult to interpret.

Better fading isolates one transfer of responsibility at a time.

The learner should gradually inherit the invisible work experts perform before and around the visible calculation.

9. Use a hint ladder so help can shrink without becoming all-or-nothing

Worked examples are one form of support. Hints are another.

When a learner is stuck on a faded example, the tutor faces a choice: show the missing step or let the learner continue struggling.

A hint ladder creates more options.

Start with support that removes the least thinking.

Level 1 — orientation: “What is the target?”

Level 2 — state: “What do you know for certain?”

Level 3 — representation: “Would a diagram, table or equation help?”

Level 4 — relationship: “What connects these two quantities?”

Level 5 — method family: “Could elimination help?”

Level 6 — procedural cue: “Try making the y coefficients opposites.”

Level 7 — worked step: the tutor supplies the next mathematical line.

Level 8 — full reload: the tutor reconstructs the whole route if the prerequisite is missing.

The aim is to use the smallest hint that restores useful mathematical movement.

Suppose Adrian is stuck on a word problem.

The tutor first asks, “What are you trying to find?”

If Adrian can then define x and proceed, the weakness was orientation.

If not, the tutor asks, “What quantities change together?”

If that restores the route, the problem may be representation.

If only “form a linear equation” restarts him, method recognition remains externally supported.

The type of hint needed is diagnostic evidence.

Aisha is solving a geometry problem. She has drawn the figure but does not see how to continue. The tutor asks, “What relationships become available because this angle is 90 degrees?” That is more useful than saying “Use Pythagoras” because it preserves some selection work.

Ryan is doing probability. He multiplies two unchanged fractions in a without-replacement problem. Instead of correcting the denominator directly, the tutor asks, “What changed after the first draw?” This targets the state model.

After a hint succeeds, the next problem should use less support.

Do not treat the hint as a permanent feature of the method.

A support ledger can be simple:

Independent.

Orientation prompt.

Representation prompt.

Method cue.

Procedural cue.

Worked step.

Full reload.

Over several sessions, meaningful progress may appear as decreasing support even before raw accuracy changes dramatically.

This makes worked-example fading observable rather than intuitive.

10. Restore support when the wrong difficulty appears—then fade again

Fading is not a one-way march.

A learner may solve one topic independently and need stronger support when a new representation, method or prerequisite enters.

Restoring support is not failure.

The important question is whether the support is restored precisely and temporarily.

Suppose Aisha has become independent with linear equations. A new problem embeds the equation inside a geometry context. She understands the geometry but cannot form the equation from the diagram.

Restoring a full algebra example would be too much.

The tutor might instead supply one representation cue: “Label the unknown length x and write the perimeter relationship.”

Once the equation is formed, Aisha can proceed independently.

The support targeted the new interface.

Adrian handles routine percentage problems independently but fails on a reverse-percentage context. The tutor restores one worked relationship—final = multiplier × original—then immediately removes it on a changed problem.

Ryan is strong on mean and median but meets a combined-mean problem with unequal group sizes. The tutor shows one fully worked example because weighting group totals is genuinely new. A faded example follows.

Restore support when:

a prerequisite is missing;

a new representation adds genuine complexity;

a method has a new condition;

a long chain overwhelms currently stable components;

or repeated guessing shows the learner lacks a viable model.

Do not restore support merely because the learner hesitates.

Productive hesitation is part of independence.

Allow enough time for retrieval and route selection.

A useful rule is:

restore the smallest support that makes the mathematics visible again, then remove it as soon as evidence permits.

This prevents two opposite teaching errors.

One is ideological independence: “They must figure it out alone.”

The other is permanent rescue: “I will keep showing the route so they do not make mistakes.”

Strong teaching moves between the two.

Help enough to create useful thinking.

Observe what the learner can carry.

Remove one support.

Test what remains.

Restore only what the evidence shows is missing.

Fade again.

That loop—model → transfer → test → restore narrowly → transfer again—is more adaptive than any fixed worksheet sequence.

11. Worked examples in algebra should expose invariants, form choices and error risk

Algebra is especially suited to worked examples because much of the visible work can look mechanical while the real learning lives in hidden decisions.

A learner may copy:

3(x − 4) = 18

3x − 12 = 18

3x = 30

x = 10.

But what should the example teach?

It should teach that equality is preserved because equivalent operations are applied consistently. It should expose the distribution rule. It should show which line is high-risk for sign errors. It should include a substitution check when the technique is new or fragile.

A stronger annotation might read:

Structure: product equals a value.

Decision: expand first because the bracket blocks direct isolation.

Risk: multiply both terms inside the bracket.

Check: substitute x = 10 into the original.

Then fade.

In the next example, remove the expansion line.

Later, remove the prompt to expand.

Later still, mix an equation where expanding is not the cheapest first move.

This matters because algebraic independence is not “always expand”. It is choosing a form that makes the target easier to reach.

Worked examples should therefore include contrasting forms.

One quadratic may be best factorised.

Another may invite completing the square.

A third may be handled efficiently by a formula.

The example should reveal the feature that makes one route attractive.

Adrian compares:

x² − 7x + 12 = 0

with

x² − 6x − 2 = 0.

The first has convenient integer factors. The second does not.

If every prior example has factorised neatly, Adrian may have learned “quadratic means factorise”. A comparison repairs that overgeneralisation.

Algebra examples should also show when transformations change the visible form but preserve the same mathematical object.

For example:

(x² − 9)/(x − 3)

= (x − 3)(x + 3)/(x − 3)

= x + 3, for x ≠ 3.

The denominator restriction should remain visible even after cancellation.

If the example shows only the simplification, the learner may infer that the excluded value no longer matters.

A worked example should therefore preserve:

the original domain;

the reason cancellation is legal;

the difference between cancelling factors and cancelling terms;

and a near non-example.

For simultaneous equations, make route selection visible.

Do not merely show elimination. Explain why a particular variable is cheap to eliminate.

Then give a pair where substitution is attractive.

For formula rearrangement, mark the target variable first and show why each inverse operation reduces its entanglement.

For inequalities, make the order reversal under multiplication or division by a negative explicit, then fade the reminder.

For indices, show the condition under which a rule applies. Do not let pattern matching substitute for exponent structure.

The central algebra rule for worked examples is:

show the invariant, show the form choice, show the high-risk transition, then remove the support that chose those things for the learner.

That produces algebra that can survive a changed equation rather than only a familiar page.

12. Worked examples in geometry should teach the learner how to read the diagram before calculating

Geometry worked examples often become calculation demonstrations.

The solver marks one or two lengths, writes a theorem or ratio, calculates, and reaches the answer.

The learner may copy the arithmetic and miss the more important process: how did the solver know what object was being measured and what relationship the diagram permitted?

A strong geometry example begins before the formula.

Take a right-triangle trigonometry problem.

The example should first identify:

the right angle;

the reference angle;

the hypotenuse opposite the right angle;

the side opposite the reference angle;

the side adjacent to it;

which quantities are known;

which quantity is required.

Only then should the ratio be selected.

Aisha can calculate sine, cosine and tangent easily. Her errors come from misidentifying sides when the triangle rotates.

For her, the worked example should not over-focus on calculator entry.

It should expose relational side identity.

Then the fade should remove side labels before removing algebraic steps.

Ryan has a different problem. He can identify sides but chooses trigonometry even when Pythagoras is cheaper. His examples should compare route selection rather than side naming.

Adrian can choose a theorem but assumes unmarked diagram features are exact. His examples should distinguish what is given, what can be inferred, and what merely looks true.

Geometry examples should therefore annotate evidence.

Given: explicitly stated or marked facts.

Derived: facts that follow from valid relationships.

Visual only: features that appear true in the drawing but are not authorised.

This is especially important in angle, similarity and proof problems.

For similarity, show how correspondence is established.

Do not jump directly to a proportion.

Ask which angle pairs or side relationships justify the correspondence.

Then fade the correspondence labels.

For area, teach the measured object.

Area needs a region.

Perimeter needs a boundary.

Volume needs a three-dimensional measure.

Triangle area needs a perpendicular height relative to the chosen base.

A worked example should expose that object choice before computation.

For composite figures, show why a decomposition is useful.

Then give a second example where a different decomposition is cheaper.

The learner should not memorise one cutting pattern.

For circle geometry, include theorem conditions and near cases.

For coordinate geometry, show how algebra and geometry cross-check each other.

For scale, make dimension explicit: length scale, area scale and volume scale do not change the same way.

When fading geometry examples, change orientation before changing conceptual difficulty.

A rotated figure is a useful test because it removes page-position memory while preserving the mathematics.

Later, remove labels.

Later, remove the method cue.

Later, mix the problem with neighbouring geometry methods.

Finally, return after delay.

A geometry worked example has done its job when the learner can rebuild the relevant structure from evidence even though the next diagram does not look like the original.

13. Worked examples in probability should model the event and state before the fractions

Probability examples are easily reduced to fraction arithmetic.

The learner sees:

3/5 × 2/4 = 3/10.

They may copy the multiplication without understanding why the second fraction changed.

A stronger worked example begins with the event.

Suppose a bag contains three red and two blue counters. Two counters are drawn without replacement. Find the probability both are red.

The example should state:

Event: red on first draw and red on second draw.

Initial state: 3 red out of 5 total.

State after first red: 2 red out of 4 total.

Route: multiply along the “red then red” path.

Check: answer must lie between 0 and 1 and be less than first-draw red probability.

Now the fractions have meaning.

Fade in layers.

First remove the second-state count.

Then remove the event statement.

Then remove the tree diagram.

Then mix a with-replacement problem beside the without-replacement problem.

Then change the wording so “at least one” suggests a complement rather than direct enumeration.

Aisha’s probability examples always begin with “What is the event?” before numbers appear.

Ryan compares two nearly identical problems, one with replacement and one without. He annotates only one difference: whether the state resets.

Adrian sees a flawed worked example where the denominator stays unchanged despite no replacement. His task is to locate the first wrong relationship, not merely correct the arithmetic.

Probability worked examples should also teach route choice.

When is a tree useful?

When is a table useful?

When is systematic listing sufficient?

When is a complement cheaper?

When do mutually exclusive cases add?

When does multiplication require independence or conditional probability?

These are decision points.

A good example reveals them explicitly before fading.

Do not let the learner memorise one tree-diagram layout as “probability”.

Change representation.

Use verbal events.

Use tables.

Use ordered outcomes.

Use complements.

The example should leave behind an event model, not a picture template.

14. Worked examples in statistics should make interpretation part of the solution, not an optional final sentence

Statistics examples often show how to compute a mean, median, range or weighted average and then stop.

That trains calculation but can leave the learner weak at interpretation.

Consider two groups.

Group A has 10 students with mean 12.

Group B has 20 students with mean 18.

A weak worked example may write:

(12 + 18)/2 = 15.

A correct worked example should expose why that is invalid.

The groups have unequal sizes.

Group totals are 10 × 12 = 120 and 20 × 18 = 360.

Combined total = 480 across 30 students.

Combined mean = 16.

The key relationship is not the formula alone. It is that the overall mean depends on the total sum divided by total count.

Then fade the group-total line in the next example.

Later remove the “weighted mean” label.

Later mix with a question where the group sizes are equal and a simple average of means happens to work.

The learner must understand why.

Statistics examples should also expose what a measure cannot tell you.

A mean of 10 does not determine the median.

A median does not determine spread.

Equal means do not imply equal distributions.

A graph can look dramatic because of axis scale without the underlying change being proportionally large.

Ryan is given two worked comparisons of data sets.

One uses only the mean.

The other uses centre and spread.

He compares which conclusion is better supported.

Aisha studies a graph example where the scale is read incorrectly. Her task is to explain why the visual impression conflicts with the numerical change.

Adrian studies an example that computes a percentage change in a statistic and must explain what quantity the percentage is relative to.

Worked examples in statistics should therefore model:

choice of summary;

calculation;

interpretation;

limitations;

and communication.

Then fade each layer separately.

First remove the calculation step.

Then remove the choice of measure.

Then remove the interpretation scaffold.

Finally, present a mixed data question where the learner must decide what evidence matters.

The worked example has succeeded when the learner can choose and interpret a summary rather than merely reproduce a formula.

15. Worked examples in modelling should show where mathematics stops being reality

Mathematical models are powerful because they compress a situation into relationships that can be analysed.

They are also dangerous if a worked example treats the symbolic model as valid everywhere.

Suppose a tank contains 30 litres of water and fills at 5 litres per minute until capacity 80 litres.

A worked example might form:

V = 30 + 5t.

That equation is useful.

It is not valid for all nonnegative t.

At t = 20 it predicts 130 litres, which exceeds the tank capacity.

The model is valid only until the tank reaches 80 litres.

Solve:

30 + 5t = 80

5t = 50

t = 10.

The model domain is 0 ≤ t ≤ 10 under the stated assumptions.

A good worked example should therefore expose:

the variables;

the meaning of parameters;

the relationship;

the assumptions;

the domain;

the event that ends or changes the model;

and how to interpret the result.

Adrian can form equations easily. His modelling weakness is forgetting the physical boundary.

His example therefore highlights capacity and asks him to state where the model stops.

Aisha struggles with forming equations from stories. Her example highlights fixed versus variable components.

Ryan forms the model correctly but leaves the final answer as t = 6 without saying “6 hours”. His fade target is interpretation and unit completion.

Model examples should also compare related structures.

Direct proportion:

y = kx.

Fixed-fee linear model:

y = a + kx.

Capacity-limited linear model:

y = a + kx only within a bounded domain.

These look similar algebraically but make different claims about the situation.

Use worked comparisons to reveal those differences.

Then remove the labels.

Later, ask the learner to critique a model that continues beyond a meaningful boundary.

Later still, provide data and ask which model is plausible.

A modelling worked example has done its job when the learner remembers that solving the equation is not the end. The mathematical result must return to the situation that gave the model meaning.

16. Wrong worked examples can teach diagnosis better than perfect examples teach imitation

Most worked examples show only valid routes.

That is necessary at the beginning. A learner needs to see what correct mathematics looks like.

But a mature example system should also include incorrect or incomplete routes that the learner must diagnose.

This changes the task.

Instead of asking, “Can you follow the solution?” the learner must ask, “Where did validity first break?”

Suppose a worked solution says:

3(x − 4) = 18

3x − 4 = 18

3x = 22

x = 22/3.

The final answer is wrong, but the important diagnostic target is the first invalid line: distribution should have produced 3x − 12.

Adrian is asked to circle the first divergence and explain the governing relationship.

That is more valuable than correcting every later line because every later line is contaminated by the first error.

Now consider probability:

A bag contains 4 red and 3 blue counters. Two reds are drawn without replacement.

A flawed example writes:

4/7 × 4/7 = 16/49.

The first wrong decision is failing to update the state after the first red draw.

The repair is not “change the denominator”. It is “without replacement, the composition of the bag changes”.

In geometry, a flawed example may apply Pythagoras to a triangle with no right-angle evidence.

The arithmetic can be flawless and the method still invalid.

In statistics, a flawed example may average group means without weighting unequal group sizes.

In percentage, a flawed example may take 20% of the final amount when trying to reconstruct the original.

These examples teach something perfect solutions cannot: how to detect invalidity.

Use several wrong-example types.

Conceptual wrong example: the relationship itself is wrong.

Condition wrong example: a correct rule is used outside its valid conditions.

Selection wrong example: a legal method is used where it is inefficient or obscures the target.

Execution wrong example: the route is correct but a local step fails.

Completion wrong example: the mathematics is mostly correct but the final answer is incomplete, mislabelled or outside the requested form.

Aisha studies two solutions to the same problem: one valid, one subtly invalid. She must say not only which is correct, but what evidence distinguishes them.

Ryan studies a solution containing an extraneous root created by squaring. He must identify why checking in the original equation is necessary.

Wrong examples can also be faded.

At first, the tutor says, “There is one error. Find it.”

Later, some examples are correct and some are not.

Later still, the learner must decide whether a check is warranted without being told there is a problem.

The final capability is spontaneous scepticism: the learner knows that a neat solution is not automatically a valid one.

This connects directly to Study Wrong Solutions to Find the First Invalid Step and the global error-analysis system. This article uses wrong examples only as one worked-example design tool; the dedicated owners retain full error-analysis scope.

17. Compare two worked solutions to teach strategy choice, not method obedience

A single worked example can accidentally imply that the displayed route is the route.

Many secondary mathematics problems allow more than one valid method.

Comparing two solutions helps the learner see method choice as a mathematical decision.

Take simultaneous equations:

2x + y = 11

x − y = 1.

Method A uses elimination by adding the equations.

Method B rearranges the second equation to x = y + 1, then substitutes.

Both work.

Which is cheaper?

Elimination is probably shorter because the y coefficients are already opposites.

But the comparison matters more than the verdict.

Ask:

Which route creates fewer lines?

Which creates smaller intermediate numbers?

Which has lower sign risk?

Which is easier to verify?

Which generalises if the coefficients change?

Which would you choose under time pressure?

Adrian begins to understand that “method” is not a chapter label. It is a choice among routes shaped by local structure.

Quadratics are another strong case.

Compare factorisation and the quadratic formula on x² − 7x + 12 = 0.

Factorisation is short and transparent.

The formula still works, but it carries more arithmetic and more opportunities for sign error.

Now change to x² − 6x − 2 = 0.

The comparison shifts. The formula or completing the square may be more natural than searching for integer factors that do not exist.

Worked comparisons teach the trigger conditions for method choice.

Geometry can compare algebraic and geometric routes.

One solution may use coordinates.

Another may use similarity.

One can be shorter, while the other exposes structure more clearly.

Statistics can compare direct total-sum reasoning with formula use.

Probability can compare direct enumeration with complement.

Linear models can compare table reasoning with equation solving.

Aisha is given two correct percentage solutions: one multiplier-based, one unitary. She must say when each is clearer.

Ryan compares two area decompositions for a composite figure. Both reach the same answer, but one uses fewer fragile subtractions.

Do not turn comparison into a hunt for one universally “best” method.

Efficiency depends on the problem, learner and checking options.

A route can be mathematically valid but operationally expensive.

A route can be longer but easier to verify.

A route can expose structure that helps future transfer.

Worked-example comparison should therefore teach a broader question:

“What does this route buy me, and what does it cost?”

That is strategy knowledge.

18. Worked examples should model checking—then fade the checking prompt

Many model solutions end when the answer appears.

That quietly teaches that producing an answer is the final mathematical act.

Worked examples should often include verification while the learner is still building the method.

The check should match the problem.

For a solved equation, substitute the root.

For a factorisation, expand.

For a percentage reversal, run the change forward.

For a geometry length, inspect magnitude and theorem conditions.

For an area or volume, inspect units.

For probability, check bounds and whether event totals behave sensibly.

For a graph, compare algebraic behaviour with shape.

For a model, test a boundary or simple input.

Ryan learns simultaneous equations through an example that ends:

x = 5, y = 8.

Then the example substitutes into both originals:

5 + 8 = 13

3(5) − 8 = 7.

The check is not decorative. It shows how a pair of candidate values can be independently challenged.

Aisha studies a reverse-percentage example. The final step reconstructs the original, applies the stated decrease, and verifies the final amount.

Adrian studies a trigonometry problem. The check is not “repeat the calculator entry”. It asks whether the computed side should be longer or shorter than another side given the angle and triangle structure.

Checking should then fade.

First, the example provides the check fully.

Next, it says “verify independently” without naming the method.

Next, the checking instruction disappears.

The tutor observes whether the learner initiates an appropriate check.

This is important because a permanent “check your answer” reminder can become external control.

The learner should eventually decide:

what deserves checking;

which check is cheapest;

and which check is sufficiently independent from the original route to be useful.

A repetition of the same calculation may reproduce the same error.

A stronger check fails differently.

This is the specialist territory of How to Check Maths Answers. In this article, checking appears only as one component a worked example should model and later hand over.

19. Worked examples should sometimes model recovery, not only perfect first-pass solving

Perfect worked examples can create an unrealistic picture of mathematical problem solving.

The solver always sees the right representation.

The first method always works.

No line is reconsidered.

No route is abandoned.

Real problem solving is less tidy.

A learner needs examples of recovery.

Suppose a geometry problem can be solved by coordinates or similarity.

The worked example begins with a coordinate route.

After several lines, the algebra becomes cumbersome.

The example pauses:

“This route is valid, but the cost is growing. Return to the target. Is there a structural shortcut?”

The solver notices similar triangles and switches.

This models something important: route change is not failure.

It is strategic control.

Adrian needs this because he assumes a chosen method must be finished even after it becomes obviously expensive.

Aisha needs a different recovery example. Her route fails a check, so she returns to the last line she trusts instead of restarting the entire problem.

Ryan needs an example where a calculator result is implausible. The solver checks the entry, finds missing brackets, and corrects the input.

Recovery examples can model:

returning to the last justified line;

changing representation;

switching method;

checking a condition;

repairing a copied value;

re-reading the target;

or reducing a complex problem to a simpler subproblem.

Then fade the recovery prompt.

At first, the example explicitly says “stop and reconsider”.

Later, the learner is given a partially flawed route and must decide whether to continue or switch.

Later still, the learner must monitor their own work in ordinary problems.

This prevents another form of example dependence: dependence on the example not only for the route, but also for knowing when the route has stopped being useful.

The dedicated problem-solving owner How to Solve Math Problems owns recovery as part of the wider problem-solving system. Here, recovery is modelled only as a worked-example design feature.

20. Do not add timing until the example has transferred enough mathematical control

Worked examples are often used in examination preparation.

The temptation is to move quickly from worked solution to timed practice.

That can be premature.

Timing adds a new demand: the learner must preserve selection, execution and checking while attention is compressed by the clock.

If the method is still dependent on examples, timing can hide the learning problem behind speed.

Use a staged sequence.

Stage 1 — full example: understand the structure.

Stage 2 — faded example: inherit selected steps.

Stage 3 — independent familiar problem: solve without support.

Stage 4 — changed surface: transfer.

Stage 5 — mixed selection: choose method among alternatives.

Stage 6 — delayed return: retrieve after time.

Stage 7 — short timed cluster: maintain capability under moderate pace.

Stage 8 — fuller examination conditions: integrate timing, endurance and paper control.

Not every topic needs all eight stages explicitly. Secure components can skip ahead.

Adrian can solve linear equations independently and accurately. He does not need a full worked-example sequence every time. Timing may be appropriate immediately.

Aisha is learning a new trigonometric relationship. Timing would add noise before the relationship is stable.

Ryan understands a probability method but needs a method cue. The priority is removing the cue before adding the clock.

Timed fading can also be gradual.

First, give generous time and observe route selection.

Then shorten only after the learner’s working remains accurate.

Do not reward speed that destroys checking or condition control.

Worked examples should eventually disappear inside examination practice, but their exit should be earned by evidence of independent capability.

The destination is not “can solve quickly when shown the method”.

It is “can identify, execute, verify and recover under the time available”.

21. After fading, mix the method with close alternatives so the learner must recognise it

A learner can become independent inside a topic and still remain dependent on the topic label.

If every worksheet says “Simultaneous Equations”, the learner does not need to decide whether simultaneous equations are relevant.

If every geometry page says “Pythagoras”, theorem selection has already happened.

If every probability page is “Without Replacement”, the event-state decision is supplied.

That is why fading must eventually remove chapter context.

Use purposeful mixtures.

Adrian receives four short problems:

one linear equation;

one ratio split;

one simultaneous-equation problem;

one fixed-fee linear model.

Before calculating, he writes only the first move for each.

The task is low-cost but high-information. It reveals whether he can locate the correct route family.

Aisha receives three geometry problems:

one right triangle with two sides known;

one similar-triangle problem;

one general triangle with no right-angle information.

The examples she studied previously no longer sit beside the questions.

She must decide what the givens permit.

Ryan receives probability problems involving:

with replacement;

without replacement;

at least one success;

exactly one success.

The arithmetic is kept simple so method recognition is the main difficulty.

Mixed selection should be introduced when local execution is sufficiently stable.

If the learner still cannot perform the method when named, mixing can create noise.

Use a progression:

named method → faded steps → uncued familiar problem → close alternative → purposeful mixed set → broader mixed set.

The What Works Clearinghouse algebra guide also recommends, with moderate evidence in one recommendation, intentionally choosing among alternative algebraic strategies. That supports the idea that method comparison and selection are meaningful instructional targets rather than afterthoughts.

Do not over-randomise.

A mixed set is most useful when the competing methods are mathematically close enough that the learner must discriminate.

Examples:

direct proportion versus general linear relation;

area versus perimeter;

Pythagoras versus trigonometry versus similarity;

ordinary percentage versus reverse percentage;

factorisation versus another quadratic method;

mean versus weighted mean;

replacement versus non-replacement probability.

The learner should be able to answer:

What feature triggered this method?

What feature ruled out the neighbour?

What would change the route?

Once those answers become independent, the example has transferred method-selection control.

22. Hide the example and return later: delayed reconstruction tests whether the route has become memory rather than dependence

Immediate independence is useful evidence.

Delayed independence is stronger evidence.

A learner can solve a paired problem minutes after studying an example because the route remains active in short-term context.

Time removes some of that support.

Use delayed reconstruction.

After a lesson, close the example.

Later that day or several days later, present a fresh problem with the same mathematical structure.

Do not ask for verbatim reproduction of the old page.

Ask whether the learner can regenerate:

the target;

the representation;

the key condition;

the first justified move;

enough of the execution;

and an appropriate check.

Adrian studies elimination on Monday. On Thursday, he sees a word problem requiring two equations but no chapter label.

If he defines variables, forms the system and chooses elimination independently, the worked example has travelled.

Aisha studies reverse percentage on Tuesday. A week later, the relationship appears in population decrease rather than a sale.

If she identifies the original as 100%, the example has become a relationship memory rather than a shopping template.

Ryan studies similarity. Two weeks later, a rotated diagram returns inside a mixed geometry set.

If he establishes correspondence without prompting, the example has survived both time and surface change.

This is where How to Remember Maths becomes the specialist owner. That article controls retrieval and spacing as a full durable-memory system. Here, delayed return is used only as an exit test for worked-example dependence.

Do not use delayed failure as proof that the learner “never understood”.

Inspect the support required to restore the route.

If one broad prompt restores everything, memory may be close to available.

If a full example is required, the original example may not have transferred enough structure.

If the relationship is recalled but the representation fails, the issue is not pure memory.

Delayed reconstruction is valuable because it separates “I could do it while the example was still nearby” from “I can rebuild it when I need it”.

23. Digital worked examples are useful when they force prediction and completion—not when they become answer theatres

Video, interactive platforms, tablets and AI systems can make worked examples easier to access.

They can also make passivity easier.

A learner watches a solution video.

The teacher writes every line.

The video is paused only to copy.

The learner feels familiar with the route but never generates one.

Digital worked examples need active interruption.

Use pause points before important decisions.

Ask the learner to predict the representation.

Ask for the next move.

Ask why the step is legal.

Ask for an alternative route.

Ask for a check before the video reveals it.

Then hide the example and provide a fresh problem.

Interactive systems can also fade automatically.

At first, show all steps.

Then remove one decision.

Then remove several steps.

Then remove the method label.

Then mix with alternative problems.

But automation should respond to mathematical evidence.

An app may interpret “correct answer” as readiness to fade even when the learner guessed or copied.

Use explanation or intermediate-state checks.

General AI systems can generate worked examples quickly. That is useful for variation, but the mathematics must be checked.

AI should also avoid generating a long stream of nearly identical solutions that encourage template dependence.

A stronger AI workflow is:

generate one clean example;

ask the learner to explain it;

generate a partially faded variant;

generate a near transfer;

generate a close non-example;

then remove the example entirely.

Adrian uses a digital tool that initially shows the next algebraic step only after he predicts it.

Aisha uses a geometry tool that removes labels gradually.

Ryan uses an AI system to generate two alternative solutions and compare them.

Technology is most useful when it lowers the cost of good example variation and fading.

It is least useful when it merely increases the amount of finished mathematics the learner can watch.

24. Tutors and parents should avoid becoming permanent worked examples

Adults often solve too much because it keeps the session moving.

The learner gets stuck.

The tutor writes the first line.

The learner gets stuck again.

The tutor chooses the formula.

The learner reaches the answer.

From the outside, the problem was solved.

But who carried the route?

A tutor should track the kind of support being supplied.

Did the learner identify the target?

Did the learner choose the representation?

Did the learner select the method?

Did the learner execute?

Did the learner check?

Did the learner recover?

These responsibilities can be transferred separately.

A tutor can say:

“I will show the first example fully. On the next one, you will choose the method. On the third, I will not label the topic. On the fourth, you will also choose the check.”

This makes the fade visible.

Parents can use lower-support questions instead of solving.

What are you trying to find?

What do you know?

Which example from class is this closest to?

What is different from that example?

What first move can you justify?

How could you test the answer?

If those prompts still carry too much, fade them too.

Adrian’s tutor stops starting every word problem for him.

Aisha’s parent stops naming the formula and instead asks which quantities are related.

Ryan’s tutor stops saying “check your answer” every time and observes whether checking becomes self-initiated.

Adults should also recognise when a full worked example is justified.

If a new structure is genuinely unfamiliar, withholding all modelling can waste time.

If a prerequisite is missing, a clean example may be more useful than repeated hints.

The goal is not minimal help.

The goal is transferable help.

The adult should leave less of themselves inside each later solution.

25. When worked examples are not improving independence, diagnose the failure mode

Worked examples can fail in several predictable ways.

Failure 1 — passive reading. The learner watches or copies without predicting or explaining. Remedy: pause before decision points and require generation.

Failure 2 — too many full examples. The tutor keeps modelling after the learner is ready. Remedy: fade sooner and test independence.

Failure 3 — too little explanation. The example is concise but hides expert decisions. Remedy: expose cue, move, reason and check.

Failure 4 — fading only arithmetic. The learner fills blanks but never chooses representation or method. Remedy: fade strategic decisions.

Failure 5 — fading too quickly. The learner guesses because prerequisites or conceptual structure are missing. Remedy: restore targeted support.

Failure 6 — identical paired problems. The learner memorises surface patterns. Remedy: vary numbers, wording, orientation, unknown and representation.

Failure 7 — no close non-examples. The learner overgeneralises the method. Remedy: compare situations where the method does and does not apply.

Failure 8 — examples always end perfectly. The learner never sees diagnosis or recovery. Remedy: include wrong solutions and route changes.

Failure 9 — checks remain teacher-owned. The learner solves but does not verify independently. Remedy: fade checking prompts.

Failure 10 — chapter labels remain visible. The learner executes but cannot select. Remedy: purposeful mixed practice.

Failure 11 — no delayed return. Immediate imitation is mistaken for durability. Remedy: hide the example and retest later.

Failure 12 — digital overload. Learners watch many solutions and produce little mathematics. Remedy: alternate viewing with prediction, completion and fresh problems.

Failure 13 — the example is solving the wrong problem. The visible topic is not the first weak link. Remedy: diagnose prerequisites before adding more examples.

Failure 14 — expertise has outgrown the scaffold. The learner finds examples redundant and disengages. Remedy: accelerate fading and move to comparison, transfer or mixed selection.

Failure 15 — worked examples are being used to fix performance under time. The learner already knows the mathematics untimed. Remedy: route into timed and examination practice instead.

Adrian’s worked-example system fails because every worksheet labels the method. His tutor adds mixed selection.

Aisha’s system fails because examples stay fully annotated. Her tutor removes representation labels.

Ryan’s system fails because he copies ten solutions per chapter. His tutor cuts the number of examples and requires one explanation, one partial completion, one fresh problem and one delayed return.

When examples are not producing independence, do not automatically add more examples.

Ask which support is still doing the learner’s thinking.

Then change that part of the system.

26. Worked-example laboratory: eight cases showing how support should change as the learner changes

The cases below treat a worked example as a temporary teaching instrument. Each case starts with evidence, changes the example, and then changes the support again when new evidence appears. The point is not to prescribe one ladder for every learner. It is to show how example design can respond to the part of the route the learner can and cannot yet carry.

Case 1 — Adrian can finish equations but cannot begin word problems

Adrian is accurate once an equation is on the page. He expands brackets correctly, isolates the unknown and checks by substitution. In word problems, however, he waits for someone to write the equation first.

A conventional fading sequence might remove the last algebraic line, then the next line, and eventually ask Adrian to solve the equation independently. That would train a capability he already has.

Instead, the tutor keeps much of the algebra visible and fades the representation decision.

Worked example:

“A delivery service charges a fixed fee of 9 plus 4 per kilometre. A journey costs 37. Find the distance.”

The example shows:

Let distance = d kilometres.

Cost = fixed fee + rate × distance.

37 = 9 + 4d.

Then the remaining algebra is shown.

The paired problem changes the context to printing. This time the page gives the story and only the final algebra after the equation has been formed. Adrian must define the unknown and write the relationship.

On the next problem, the algebra support also disappears.

Design lesson: fade the missing decision, not the last visible line by habit.

Case 2 — Aisha understands the theorem but the diagram is doing the labelling for her

Aisha solves right-triangle trigonometry accurately when the example labels opposite, adjacent and hypotenuse. A rotated triangle with no side-role labels causes hesitation.

The next example keeps the trigonometric ratio visible but removes the labels.

Aisha must mark the right angle, select the reference angle and label side roles herself.

The following example removes the formula cue as well. Aisha chooses sine, cosine or tangent based on the sides that are known and required.

A later problem is a non-right triangle. She must reject the familiar right-triangle ratio method rather than force it onto the diagram.

Design lesson: representation support can hide dependence even when calculation appears fluent.

Case 3 — Ryan copies a model answer perfectly but cannot explain the first transformation

Ryan’s notebook contains neat algebraic solutions. When asked why a common denominator was introduced in an algebraic fraction, he says, “because that is the next step”.

The tutor does not remove more lines yet. The example first becomes more explanatory.

At each decision point, Ryan answers one prompt:

What obstacle is present now?

What will this transformation expose?

What condition must be preserved?

Only after Ryan can explain the route does the tutor hide the next decision line.

Ryan then reconstructs it before revealing the model.

Design lesson: fading before the relationship is understood can turn completion into guessing.

Case 4 — The example has become too familiar

Adrian repeatedly succeeds on reverse-percentage sale questions. Every worked example contains discounts, prices and the phrase “original price”.

When the same mathematics appears as population decline, Adrian performs a forward percentage calculation.

The issue is not a missing arithmetic skill. The example family has trained a surface cue.

The redesign keeps the relationship visible—final = multiplier × original—but changes context across population, concentration and depreciation. Then the relationship is removed, leaving only the problem.

Finally, reverse percentage is mixed beside ordinary percentage change.

Design lesson: an example should vary the surface before the learner mistakes the surface for the structure.

Case 5 — Aisha is ready to compare methods, not see another full solution

Aisha can solve simultaneous equations by both elimination and substitution. Another full demonstration would add little.

The tutor instead places two worked solutions side by side.

One eliminates a variable immediately because coefficients already match.

The other isolates a variable first and then substitutes.

Aisha compares line count, coefficient growth, sign risk and ease of checking.

Then she receives four systems and chooses a route before calculating.

Design lesson: once execution is secure, examples can teach strategy judgement rather than procedure.

Case 6 — Ryan is independent until a check is needed

Ryan solves quadratics accurately but stops as soon as candidate roots appear. In problems that can create extraneous solutions, this is risky.

The next worked example includes a clear verification step in the original equation.

On the following example, the final line says only: “Which candidate still belongs to the original problem?”

Then the checking instruction disappears.

Ryan’s tutor observes whether he now initiates substitution or domain checking on his own.

Design lesson: the example must eventually fade control functions around the solution, not only the calculation itself.

Case 7 — A wrong example reveals more than another correct one

Aisha has seen several correct probability trees but still repeats an unchanged denominator after a draw without replacement.

Instead of showing another perfect tree, the tutor presents:

5 red, 3 blue; draw two reds without replacement.

Proposed working: 5/8 × 5/8.

Aisha must identify the first wrong decision and explain how the state changed after the first draw.

She then repairs the route: 5/8 × 4/7 = 5/14.

A changed retest follows with different colours and numbers.

Design lesson: when a misconception survives correct examples, critiquing an incorrect example can make the boundary visible.

Case 8 — The example should disappear because the learner now owns the route

Ryan’s linear-model work now survives changed contexts, different representations, mixed sets and delayed returns. He can define variables, form the equation, interpret gradient and intercept, solve, check and explain the domain without prompts.

The old worksheet still places a worked example above every set.

The example is now unnecessary support.

It is removed.

If a later model introduces a genuinely new feature—piecewise behaviour, a capacity boundary or another unfamiliar condition—a worked example can return for that new feature only.

Design lesson: worked examples are not permanent furniture. Their success is partly measured by when they can be removed.

27. Worked-example and fading checkpoint: twenty-four cases that test whether support is being transferred

These original cases are not a standardised assessment and have no validated cut score. They are designed to test worked-example judgement: what the example should expose, what should be faded next, whether a learner is actually independent, and which mathematical relationship should survive when the example disappears.

Task 1 — Which line should be faded?

A learner can solve linear equations once brackets have been expanded, but repeatedly waits for the tutor to expand the first bracket. The worked example is:

4(x − 3) + 2 = 18

4x − 12 + 2 = 18

4x − 10 = 18

4x = 28

x = 7.

Which support should be removed first?

Worked explanation: remove the expansion line or leave it blank for the learner to generate. Later algebra is already independent. Fading the final division would train the wrong component. The learner needs responsibility for distributing 4 across the bracket and preserving both terms.

Task 2 — Explain the decision, not the arithmetic

A worked example solves 3x + 2y = 17 and x − 2y = 3 by adding the equations. What is the most useful self-explanation prompt?

Worked explanation: ask, “Why is addition useful here?” The y coefficients are opposites, so adding eliminates y immediately. Asking only “What is 2y + (−2y)?” checks arithmetic but misses the route decision.

Task 3 — Reverse percentage worked example

An amount is 156 after a 35% increase. A model answer writes 156 ÷ 1.35. What relationship should a teaching example make explicit before the calculation?

Worked explanation: the final amount is 135% of the original, so 1.35 × original = 156. The original is 156/1.35 = 1040/9 ≈ 115.56. The key teaching decision is identifying the 100% base and multiplier, not memorising “divide by 1.35”.

Task 4 — Geometry condition before formula

A worked solution uses a² + b² = c². What should the example show before substitution?

Worked explanation: it should establish that the triangle is right-angled and identify c as the hypotenuse opposite the right angle. The theorem condition is part of the method. A later fade should remove the reminder and require the learner to inspect the diagram independently.

Task 5 — Add or subtract?

For the system 4x + 3y = 22 and x + 3y = 10, which local decision should a worked example expose?

Worked explanation: subtracting the second equation from the first eliminates y because the y coefficients are equal. The example should teach coefficient inspection, not a permanent rule that simultaneous equations are always added or always subtracted.

Task 6 — Combined mean

Group A has 8 values with mean 15. Group B has 12 values with mean 21. A worked example averages 15 and 21 to get 18. What should the learner diagnose?

Worked explanation: the groups have different sizes, so the group means cannot be averaged equally. Total for A = 8 × 15 = 120; total for B = 12 × 21 = 252; combined total = 372 across 20 values; combined mean = 18.6. The worked example should expose the total-sum relationship.

Task 7 — Without replacement

A bag has 6 green and 4 yellow counters. A proposed worked example for two greens without replacement uses 6/10 × 6/10. Where should the critique begin?

Worked explanation: after the first green is removed, the state changes to 5 green out of 9 counters. The correct probability is 6/10 × 5/9 = 1/3. The first wrong decision is treating the second draw as if the first draw changed nothing.

Task 8 — Domain restriction

A worked example simplifies (x² − 16)/(x − 4) to x + 4 and stops. What should be added before the example is considered complete?

Worked explanation: preserve the original restriction x ≠ 4. The cancellation is valid for x values where the original denominator is nonzero. A later faded example should require the learner to carry the restriction without being reminded.

Task 9 — Inequality fading

A learner can solve 2x > 8 independently but forgets what happens when dividing by a negative. Design a partial example for −3x > 12.

Worked explanation: the example can show the decision point and leave the outcome blank: “Divide both sides by −3. What happens to the order relation?” The learner should produce x < −4 and explain that multiplying or dividing an inequality by a negative reverses the inequality.

Task 10 — Direct proportion versus general linear relation

A learner has studied worked examples of y = kx. The next task gives y = 5x + 2. Should the example label this as direct proportion?

Worked explanation: no. The nonzero intercept means it is linear but not directly proportional under the standard school definition. A useful close contrast is y = 5x beside y = 5x + 2, with the learner explaining which condition changes the classification.

Task 11 — Similarity and area

Two similar figures have corresponding length ratio 2:5. A worked example asks for the area ratio. What relationship should it expose before the answer?

Worked explanation: area scales with the square of the linear scale factor, so the area ratio is 4:25. The example should connect dimension to exponent. A later faded problem can ask the learner to retrieve the square relationship without a prompt.

Task 12 — Model boundary

A tank contains 30 litres and fills at 5 litres per minute until capacity 80 litres. A worked example writes V = 30 + 5t. What additional line makes the model mathematically responsible?

Worked explanation: state the valid time domain. The tank fills after (80 − 30)/5 = 10 minutes, so the simple model applies for 0 ≤ t ≤ 10 under the stated assumptions. The equation should not be extended indefinitely beyond physical capacity.

Task 13 — Formula memory or execution problem?

A learner writes the quadratic formula correctly from memory for x² − 5x + 6 = 0 but substitutes b = 5 rather than b = −5. Should the next worked example focus on formula recall?

Worked explanation: not primarily. The formula is available. The failure is coefficient reading or sign control. A better partial example should fade less of the coefficient-identification stage: first write a = 1, b = −5, c = 6, then substitute. Later remove that scaffold once the learner reliably reads coefficients from standard form.

Task 14 — Rotated triangle

A learner solves right-triangle trigonometry only when the diagram has the same orientation as the textbook example. What should the next worked-example change be?

Worked explanation: rotate the triangle while keeping the mathematical demand simple. Remove side-role labels and require the learner to identify the right angle, reference angle, hypotenuse, opposite and adjacent sides. This tests relational understanding rather than page-position memory. Do not add harder algebra at the same time unless that is a separate target.

Task 15 — Quadratic route comparison

Compare x² − 9x + 20 = 0 with x² − 9x + 17 = 0. What should two worked examples teach?

Worked explanation: the first factorises conveniently as (x − 4)(x − 5) = 0, giving roots 4 and 5. The second does not have convenient integer factors with product 17 and sum −9, so a different permitted route is likely better. The teaching point is method selection from structure, not “all quadratics factorise” or “always use the formula”.

Task 16 — First invalid line

A proposed worked solution says:

6 − 3(2 − x) = 12

6 − 6 − 3x = 12

−3x = 12

x = −4.

Where does validity first break?

Worked explanation: at the expansion line. −3(2 − x) = −6 + 3x, not −6 − 3x. The correct equation becomes 6 − 6 + 3x = 12, so 3x = 12 and x = 4. A wrong-example exercise should target that first invalid relationship rather than focusing on the final answer.

Task 17 — Build checking into the example

A worked example factorises x² − 9x + 20 as (x − 4)(x − 5). What check should be modelled before checking is faded?

Worked explanation: expand the factors: x² − 5x − 4x + 20 = x² − 9x + 20. Expansion is a structurally independent check of the factorisation. Later, the prompt “expand to check” can disappear and the learner should choose whether verification is warranted.

Task 18 — Design an example–problem pair

A worked example solves a fixed-fee model C = 12 + 5h. What would make a useful paired problem?

Worked explanation: preserve the fixed-plus-variable structure but change the surface and perhaps the unknown. For example: “A printer charges a setup fee of 8 plus 3 per copy batch. A job costs 29. How many batches?” The learner must form 29 = 8 + 3b and solve b = 7. The pair transfers the relationship rather than copying the original wording.

Task 19 — Method-label dependence

A learner solves every worksheet titled “Pythagoras” accurately but fails a mixed geometry set. What should be faded next?

Worked explanation: the topic label. Give several geometry problems with simple arithmetic and ask which relationship applies before calculation. Include at least one right triangle suited to Pythagoras, one problem better suited to trigonometry or similarity, and one near case without right-angle evidence. The main capability to train is method selection.

Task 20 — Immediate success versus delayed transfer

A learner studies a worked reverse-percentage example and solves a nearly identical problem correctly two minutes later. Is the example ready to disappear permanently?

Worked explanation: not on that evidence alone. Immediate paired success is useful but may rely on a still-active route. Hide the example and return after time with changed wording or context. If the learner reconstructs the 100% base and multiplier independently, the example has transferred more convincingly.

Task 21 — Redesign a passive solution video

A ten-minute mathematics video shows three complete worked examples while the learner copies every line. How can it become an active worked-example sequence?

Worked explanation: pause before each major decision. Ask the learner to predict the representation or next move, state why it should work, and then reveal the line. After the first full example, give a partially faded second example. After the second, give a fresh problem with no solution visible. A later return should test whether the route survives without the video.

Task 22 — Fade the checking prompt

A learner always checks equations correctly when the worksheet says “substitute your answer to verify”, but never checks when the instruction disappears. What is the next fade?

Worked explanation: replace the explicit method with a weaker prompt such as “How could you challenge this answer?” Later remove the prompt entirely. The target is spontaneous selection of an appropriate check, not dependence on an instruction that names the verification method.

Task 23 — Interpret the support level

A learner cannot start a geometry problem. The tutor asks, “What is the target?” and the learner immediately marks the unknown, identifies a right triangle and completes the problem without further help. What does this evidence suggest?

Worked explanation: the mathematical method may be largely available, while orientation to the target is not yet spontaneous. The next problem should remove the orientation prompt and see whether the learner identifies the target independently. A full worked example would supply far more help than the evidence justifies.

Task 24 — Learner designs the independence test

A learner says: “Do not give me another example today. Next week, give me a different-looking problem with no topic heading, and make me choose the method and check it myself.” What does this show?

Worked explanation: the learner understands several properties of a fair exit test: delay, changed surface, removal of topic cues, independent method selection and independent checking. This is evidence of worked-example-system independence. The tutor still checks course alignment and difficulty, but more control can be handed over.

After the checkpoint, do not reduce the result to a single score. Look for the first responsibility that remains external: target identification, representation, method selection, condition knowledge, execution, checking, recovery or transfer. The next example or fade should address that responsibility.

28. Frequently asked questions about worked examples and example fading

Are worked examples good for learning Mathematics?

They can be highly useful when they expose relationships and decisions that a learner cannot yet generate efficiently. Their value falls when the learner merely copies or reads them without explanation, completion or transfer. A worked example should lead to learner production.

How many worked examples should a student see?

There is no universal number. A new or complex structure may need more modelling; an experienced learner may need one comparison or none. Use prediction, explanation, completion and transfer evidence to decide when support should shrink.

What is example fading?

Example fading is the gradual removal of parts of a worked solution so the learner carries more of the route. The removed support can be arithmetic, representation, method labels, condition prompts, checking prompts or other decisions. Fading should target the learner’s next achievable responsibility.

Is backward fading always best?

No. Removing later steps first can make the transition from example to problem smoother for some procedures, but the most useful fade is the one that targets the learner’s actual weak component. If startup decisions are weak, fade the beginning sooner.

What is an example–problem pair?

It is a worked example followed by a related problem that the learner must solve. The problem should preserve the key structure while changing enough detail to require reconstruction. Later pairs should vary the surface and introduce close alternatives.

Should the paired problem be almost identical?

Early pairs can be close when a method is new, but perfect clones risk template matching. Change numbers, order, unknown, wording or representation as the learner becomes more secure. Eventually, mix the method with neighbouring methods.

Should students copy worked examples into their notes?

Copying can be useful if it supports annotation and later reconstruction, but copying itself is weak evidence of learning. A stronger routine is predict, inspect, explain, cover, reconstruct and solve a fresh problem.

What should a student write beside a worked example?

Useful annotations include the cue that suggested the method, the reason a major step is valid, a high-risk transition, a condition that limits the rule, and a possible check. Avoid turning every arithmetic line into commentary.

Should students memorise worked solutions?

They should preserve transferable structure rather than cosmetic sequence. Remember the relationship, conditions, first useful moves and checking logic. If the exact numbers or wording change and the route disappears, the example has been memorised too superficially.

How can I tell whether a student is dependent on worked examples?

Common signs include waiting for the first line, needing the method label, copying a representation without understanding it, solving only when a near-identical example is visible, or being unable to check without prompting. Remove one support and observe what fails.

When should a tutor stop modelling?

Reduce modelling when the learner can predict and explain the current decisions, complete missing steps, solve a near problem and begin to select the method independently. Do not keep showing full examples simply because the lesson feels smoother when the tutor does more.

When should support be restored?

Restore support when the learner lacks a necessary prerequisite, a genuinely new representation or condition has appeared, or fading has produced guessing rather than productive reasoning. Restore the smallest useful support, then fade again.

Is productive struggle the same as leaving a student stuck?

No. Productive struggle occurs when the learner has relevant knowledge and is working among plausible routes. Unproductive struggle occurs when a prerequisite or representation is missing and there is no viable entry point. A hint ladder helps distinguish the two.

Are incorrect worked examples useful?

Yes, after learners have enough knowledge to critique them. Wrong examples can train first-error location, theorem conditions, method boundaries and checking. They should not replace clear correct examples when a structure is first being learned.

Should worked examples show more than one method?

Often, yes. Comparing valid routes can teach strategy choice, efficiency, error risk and checking. The goal is not to declare one universal best method but to understand what each route costs and exposes.

How should worked examples be used with calculators?

Show where exact mathematical structure should be established before calculator entry, how grouping and mode matter, and how the numerical output can be checked. Then fade button-level guidance once the learner can enter expressions accurately.

Can AI generate worked examples?

Yes, but generated mathematics should be checked. Use AI to create variation, partial examples, near transfers and alternative solutions rather than a large stream of complete answers. The learner should still predict and produce the mathematics.

What is the difference between a worked example and a hint?

A worked example displays a substantial route. A hint supplies less information to restart the learner. Both belong to the same support family and should shrink as capability grows. Use the least support that creates useful mathematical movement.

What is the difference between worked-example fading and general practice design?

Worked-example fading specifically controls how responsibility moves from displayed solutions to the learner. General deliberate practice also decides what capability to train, question difficulty, feedback, variation, spacing and transfer. Use the dedicated deliberate-practice guide for the wider system.

When should a worked example return after it has been removed?

When a genuinely new feature appears, when a maintenance check reveals that the structure has become unavailable, or when a recurring misconception shows that the earlier model did not transfer. Restore only the part of the example that addresses the new evidence.

29. Independence gate: the example can leave when the learner can carry the decisions it used to carry

The final question is not whether the learner has completed a certain number of examples.

It is whether the learner can now perform the work those examples were doing.

Use an independence gate built from responsibilities rather than a score.

Target responsibility. Can the learner identify what the question actually asks without someone underlining or restating it?

State responsibility. Can the learner separate what is given, what is unknown and what conditions must remain true?

Representation responsibility. Can the learner choose or construct an equation, graph, table, diagram, tree or other useful representation?

Selection responsibility. Can the learner choose a plausible method without the chapter heading or tutor naming it?

Execution responsibility. Can the learner carry the route with enough algebraic, arithmetic, notational and calculator accuracy?

Condition responsibility. Can the learner remember domain restrictions, theorem conditions, units and model boundaries?

Checking responsibility. Can the learner select a meaningful check without being told exactly how to verify?

Recovery responsibility. If a check fails or a route becomes expensive, can the learner return to the last trusted state and change course deliberately?

Transfer responsibility. Can the learner recognise the structure after numbers, wording, representation or context change?

Memory responsibility. Can the learner rebuild the route after enough time has passed that the worked page is no longer fresh?

These are practical teaching indicators, not a validated assessment scale.

Adrian reaches the independence gate for simultaneous equations when he can:

recognise that two unknowns require two independent relationships;

form the equations from a story;

inspect coefficients;

choose elimination or substitution for a reason;

execute accurately;

check the ordered pair in both original equations;

and repeat the process on a changed problem several days later.

At that point, another full worked example is unlikely to be the best use of his time.

Aisha reaches the independence gate for right-triangle trigonometry when she can:

identify right-angle evidence;

choose a reference angle;

label side roles relationally;

choose the appropriate ratio;

enter the calculation correctly;

judge whether the output makes sense;

and reject the method when a near problem does not satisfy its conditions.

Ryan reaches the independence gate for weighted mean when he can:

recognise that unequal group sizes matter;

reconstruct total sums;

combine totals and counts;

interpret the result;

and distinguish the case from one where equal group sizes make a simple average of means valid.

Notice that independence is not defined as “never makes an error”.

An independent learner can make a mistake and still retain control.

They can inspect their own state.

They can challenge the result.

They can recover.

This matters because perfection is not a realistic exit criterion for scaffolding.

Use several exit tests.

Exit test 1 — cover the example. Can the learner reconstruct the route?

Exit test 2 — change one surface feature. Can the learner preserve the relationship?

Exit test 3 — remove the method label. Can the learner select the route?

Exit test 4 — add a close alternative. Can the learner discriminate?

Exit test 5 — delay the return. Can the learner retrieve enough structure later?

Exit test 6 — remove the checking prompt. Does verification become self-initiated when it matters?

Exit test 7 — introduce a small failure. Can the learner recover without needing the whole worked route restored?

When the learner passes these conditions for a topic, the example should become optional reference rather than active support.

If a later topic introduces a new structural feature, the example can return locally.

This creates a mature support system:

examples appear where they create leverage and disappear where they no longer do.

30. The complete worked-example system: model the invisible decisions, then transfer them one by one

Return to Adrian, Aisha and Ryan.

Adrian originally looked competent while the route was visible. His real weakness appeared when the example disappeared. He needed method selection and representation to be transferred, not more arithmetic demonstrations.

Aisha originally wanted to skip examples entirely. She needed to see that a good example is not a shortcut around thinking. It can expose a relationship efficiently so that later thinking has something solid to work with.

Ryan originally produced beautiful notebooks. He needed to move from copying complete solutions to explaining, predicting, completing, critiquing, reconstructing and finally solving without them.

The complete system is:

MODEL → NOTICE THE DECISION → EXPLAIN WHY IT WORKS → PREDICT THE NEXT MOVE → COMPLETE A MISSING PART → FADE ONE RESPONSIBILITY → SOLVE A PAIRED PROBLEM → CHANGE THE SURFACE → COMPARE ALTERNATIVES → CRITIQUE WRONG ROUTES → REMOVE THE METHOD LABEL → RETRIEVE AFTER DELAY → CHECK AND RECOVER INDEPENDENTLY → RETIRE THE EXAMPLE WHEN THE LEARNER OWNS THE ROUTE.

This is not a compulsory sixteen-step ritual for every topic.

A simple familiar skill may move from one example directly to independent work.

A new multi-step structure may use several stages.

A learner with strong prior knowledge may enter at method comparison.

A learner with a missing prerequisite may need the example to become more explicit before it becomes less explicit.

A strong tutor therefore asks two questions repeatedly:

What useful mathematical decision is still hidden inside the example?

Is the learner ready to carry that decision now?

The first question protects against thin model answers.

The second protects against permanent scaffolding.

The world-facing worked-example owner also stays deliberately connected to neighbouring BTT owners.

Use How to Practise Maths Effectively when the question is broader task design.

Use How to Remember Maths when retrieval and spacing become the main problem after the example has disappeared.

Use How to Solve Math Problems when route choice and recovery across unfamiliar problems are the main goal.

Use How to Stop Repeating Math Mistakes when the learner repeatedly reconstructs the same wrong relationship.

Use How to Check Maths Answers when verification itself needs specialist training.

For Singapore SEC-specific support fading, retain the dedicated SEC Mathematics Support Fading owner. For the Secondary 3 stage-specific worked-example treatment, retain Worked Examples & Example Fading in Secondary 3 Mathematics.

Evidence and scope note: the fictional scenes, worked-example sequences, fade decisions, checkpoint cases, support ladder and design matrices in this guide are original explanatory material. They are not a validated diagnostic instrument and do not guarantee examination outcomes. The What Works Clearinghouse algebra practice guide recommends using solved problems to engage students in analysing algebraic reasoning and strategies; the guide rates that recommendation as minimal evidence. The same guide rates intentionally choosing among alternative algebraic strategies as moderate evidence. Readers should consult the source for exact study populations, evidence ratings and recommendation details. Other teaching decisions in this article are presented as bounded instructional design guidance rather than universal causal claims.

Public evidence references: What Works Clearinghouse — Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students; Education Endowment Foundation — Supporting pupils with worked examples; NCETM — Five Big Ideas in Teaching for Mastery.

A worked example is finished teaching when the learner can do the mathematical work the example used to do for them.

Appendix A — Worked-example design matrix: evidence → what to show → what to fade → next test

This matrix is a routing tool. Begin with evidence from the learner’s work. Choose the narrowest support that makes the missing relationship visible, then remove that support as soon as the learner can carry it.

Evidence: learner cannot explain the basic relationship even with notes open

Show: a complete worked example with explicit meaning, conditions and representation. Fade: nothing strategic yet; first require self-explanation. Next test: a very close example–problem pair where the learner explains the same relationship in their own words.

If the relationship remains unclear, the learner may need conceptual teaching before fading is appropriate.

Evidence: learner can explain the example but cannot supply one missing procedural line

Show: the route up to the unstable transition. Fade: that transition only. Next test: one changed problem requiring the same operation.

Example: the learner understands equation balance but loses a negative during bracket expansion. Leave the equation setup visible and remove the expansion line.

Evidence: learner completes missing steps but cannot start a fresh question

Show: less of the startup. Fade: variable definition, representation or first method decision. Next test: an unlabeled fresh problem with manageable arithmetic.

This is common when execution is strong but representation is tutor-owned.

Evidence: learner starts correctly but waits for method confirmation

Show: no full route; at most a broad question. Fade: method labels and approval. Next test: a close contrast where two methods are plausible.

The learner needs evidence that their own route selection can be trusted and checked.

Evidence: learner executes one method well but applies it everywhere

Show: paired examples with different route triggers. Fade: topic headings. Next test: purposeful mixed practice containing near neighbours.

Examples include factorisation versus other quadratic methods and Pythagoras versus trigonometry or similarity.

Evidence: learner succeeds only when the diagram is labelled

Show: the formula or later procedure if necessary. Fade: diagram labels and side-role annotations. Next test: a rotated diagram with the same mathematical demand.

Keep arithmetic simple so representation reading remains observable.

Evidence: learner succeeds on identical contexts but fails a changed story

Show: the invariant relationship across two contexts. Fade: vocabulary cues. Next test: another context preserving the same structure plus one close non-example.

The aim is to detach the method from surface language.

Evidence: learner remembers the route but omits conditions

Show: condition alongside the method. Fade: condition prompt after examples/non-examples have made the boundary visible. Next test: a near case where the method is not authorised.

A theorem remembered without its condition is incomplete method knowledge.

Evidence: learner is accurate but never checks independently

Show: a worked verification once or twice. Fade: first the check method, then the instruction to check. Next test: a problem with a plausible but incorrect candidate where an independent check has clear value.

Evidence: learner copies examples without explanation

Show: fewer examples, more pause points. Fade: immediate access to the next line. Next test: prediction before reveal, then fresh application.

Reduce quantity if necessary. Ten copied examples can provide less evidence than one explained and reconstructed example.

Evidence: learner can reproduce a complete route almost verbatim

Show: nothing extra. Fade: the entire original example. Next test: change unknown, numbers, representation or route opportunity.

The question is whether structural memory survives when answer-memory is removed.

Evidence: learner can solve independently but fails after a week

Show: a compact reactivation or one representative example if necessary. Fade: quickly; do not restart the full instructional sequence automatically. Next test: a nearer delayed return followed by changed transfer.

If full reloading is repeatedly required, route into the durable-memory system and revisit original learning quality.

Evidence: learner is accurate untimed but weak under timing

Show: examples only if a route is actually missing. Fade: do not reintroduce unnecessary solution support. Next test: short timed clusters or examination practice.

Worked examples should not absorb a performance-under-load problem they no longer own.

Evidence: learner discovers a new representation inside an otherwise secure topic

Show: one example that exposes the interface between old mathematics and new representation. Fade: the representation cue quickly. Next test: another representation change without a method label.

Restore support locally rather than treating the entire topic as new.

Evidence: learner can critique wrong routes but still chooses poor routes personally

Show: two correct alternatives with costs and benefits. Fade: the comparison verdict. Next test: several problems where the learner selects a route before solving and later evaluates the choice.

Diagnosis of others is useful, but independent strategy choice is the target.

Evidence: learner is ready to design the next example themselves

Show: the criteria rather than the solution. Fade: tutor control of example selection. Next test: learner creates a useful full example, faded version, near transfer and close non-example, then justifies why each exists.

This is a high-level form of independence: understanding what kind of support would actually test the current capability.

Appendix B — The five-line worked-example annotation template

A long explanation is not always practical. This compact template captures the parts of a worked example most likely to transfer.

CUE — What feature tells me this route is plausible?

Examples: opposite coefficients; right-angle evidence; final amount is a percentage of an original; unequal group sizes; fixed fee plus variable rate.

MOVE — What mathematical action am I taking?

Examples: eliminate y; form a proportion; factorise; use a complement; reconstruct total sums; introduce an intermediate height.

REASON — Why is the move valid or useful?

Examples: the equations remain equivalent; corresponding lengths scale consistently; complement is cheaper than enumerating several events; factorisation exposes zero-product structure.

RISK — What is the first high-risk transition?

Examples: negative sign distribution; denominator state after no replacement; correspondence order; coefficient sign; unit conversion; premature rounding.

CHECK — How could I challenge the result independently?

Examples: substitution; expansion; reverse percentage forward; unit/bound inspection; alternative representation; boundary test.

At first, all five lines may be supplied.

Then one line is left blank.

Then several are removed.

Eventually, the template disappears and the learner carries these questions mentally when the problem warrants them.

A learner does not need to write all five lines for every routine problem. The template is scaffolding for building judgement. Its success is measured by its eventual invisibility.

Example: reverse percentage

Cue: a final amount is given after a percentage change; original is required.

Move: write final = multiplier × original.

Reason: the final amount represents a stated percentage of the original.

Risk: treating the final amount as 100%.

Check: apply the stated percentage change forward to the reconstructed original.

Example: simultaneous equations

Cue: two unknowns linked by two independent equations.

Move: choose elimination or substitution based on coefficient structure.

Reason: reduce two unknowns to one while preserving the common solution.

Risk: sign errors when adding/subtracting equations or substituting.

Check: substitute the ordered pair into both originals.

Example: geometry theorem

Cue: marked geometric conditions that authorise a theorem.

Move: apply the theorem to the correct objects.

Reason: the given conditions create the stated relationship.

Risk: assuming unmarked visual features or mismatching corresponding parts.

Check: inspect conditions again, magnitude and an alternative geometric relationship where available.

Example: combined mean

Cue: separate group means and group sizes must be combined.

Move: reconstruct each group total, then divide combined total by combined count.

Reason: mean is total sum divided by number of values.

Risk: averaging group means equally when group sizes differ.

Check: combined mean should lie between the two group means, weighted toward the larger group.

Example: probability without replacement

Cue: sequential draws and no replacement.

Move: update the state before the next probability.

Reason: both favourable count and total count can change after a draw.

Risk: repeating the original denominator.

Check: ensure each branch probability matches the state and final probability lies in [0,1].

When to stop using the template

Stop requiring it when the learner selects routes, notices high-risk transitions and checks spontaneously without writing the labels.

Keep it available as a recovery tool for genuinely new or confusing structures.

The template is not the mathematics. It is a temporary lens for seeing the mathematical decisions the learner will later make without it.