The Tutor Handbook Vol No.0150 | Erroneous-Example Gate

The Tutor Handbook · Volume 0150 · Series ID THB-0150

Series route: The Tutor Handbook — Complete Series Index.

The tutor writes a wrong solution on the board.

Alicia sees it immediately. “That line is wrong.”

“Why?”

She points to the sign. Good.

The next learner also spots the error. The third does not. The tutor explains it, everyone nods, and the lesson moves on.

It looks like a useful error-analysis exercise. But several important questions remain. Did the learners recognise the misconception or merely notice an odd-looking line? Did the wrong method become memorable in a way the correction did not? Was the learner knowledgeable enough to compare the alternatives? Did the tutor choose an error that represented a real misunderstanding, or invent a cartoon mistake no learner would plausibly make? And after the discussion, could learners solve a fresh problem without reproducing the faulty route?

Incorrect worked examples can be powerful because they make reasoning visible and create something concrete to diagnose. They can also seed confusion, reward superficial fault-finding or become a performance game in which learners hunt for anything that looks unusual.

The Erroneous-Example Gate is the tutor’s decision about when a deliberately incorrect solution will help learners distinguish valid reasoning from a plausible misconception, and when the learner first needs a correct model, more prior knowledge, a narrower comparison or a different form of practice.

The purpose is not to celebrate mistakes. It is to make the boundary between a tempting wrong model and a defensible model observable.

Quick Read

An erroneous example should contain a plausible instructional error, not random nonsense.
Learners need enough correct knowledge to evaluate the error; otherwise the wrong route can become another candidate to memorise.
Compare incorrect and correct reasoning where possible.
Ask learners to locate the first invalid step, not merely the final wrong answer.
Require a reason: what condition or principle was violated?
Separate conceptual errors from arithmetic slips, notation errors and presentation problems.
Do not expose novices to many competing misconceptions at once.
Correct the error explicitly; do not leave ambiguity unresolved.
Follow with a fresh problem where the learner must avoid the error independently.
Use delayed checks because immediate correction can produce recognition without durable discrimination.
In small groups, protect private first analysis before the most fluent learner announces the fault.
Do not use humiliating real learner work as public “bad examples”.
Constructed or anonymised examples are safer for professional teaching.
The research base is promising but not a licence to replace correct worked examples wholesale with erroneous ones.

1. What This Volume Owns

This volume owns deliberate instructional use of incorrect worked solutions, explanations or examples.

It does not own ordinary error correction after a learner makes a mistake. It does not own the Tutor-Side Check, where the tutor tests whether their own explanation created the difficulty. It does not own the Disconfirmation Check, which tests competing diagnostic hypotheses.

The specific job is whether the tutor should intentionally present an error for analysis, how that error should be designed, and what evidence is needed before the tutor concludes that learners have learned from it.

2. Why Show a Wrong Example at All?

Correct worked examples reduce problem-solving search and make expert procedures visible. That can be especially useful for novices.

An erroneous example serves a different purpose. It externalises a plausible faulty model so the learner can inspect it without having to produce the error first.

misconception discrimination;
error detection;
justification of rules;
comparison between strategies;
awareness of common traps;
metacognitive checking.

The learner can ask, “Where does the reasoning first stop being valid?”

That question can be more powerful than simply hearing “do not do this”.

3. The Wrong Answer Is Not the Teaching Object

A poorly designed error exercise focuses on the final answer.

“Here is an incorrect answer. Find the mistake.”

The learner scans until a number differs.

A better example exposes the reasoning chain. The teaching object is the first invalid inference, assumption or operation.

If a learner can identify where validity is lost, the tutor gains evidence about the governing principle.

The same final wrong answer can arise from very different causes. Error analysis should therefore begin upstream.

4. Plausibility Matters

Random errors are easy to reject and teach little.

Suppose the tutor wants to address sign errors in solving equations. Writing 2+3=9 is wrong but irrelevant. Writing 2x+3=9, then 2x=12, is plausible because it reflects an incorrect inverse operation.

The erroneous route should resemble a misconception or procedural temptation that a learner could realistically entertain.

Plausibility creates a meaningful discrimination problem.

But do not claim a constructed error is a documented common misconception unless evidence supports that claim. Label it as a constructed teaching example.

5. Prior Knowledge Matters

A learner cannot reliably diagnose an error if both the correct and incorrect routes are unfamiliar.

Imagine showing a novice two algebraic derivations, one valid and one subtly invalid, before they understand the governing rule. The learner may choose based on surface appearance.

Incorrect examples therefore often belong after at least one correct model has established a reference.

The tutor can ask: “What rule did the correct example rely on?” “Where does the incorrect example violate that rule?”

Without the reference, error analysis risks becoming guessing.

6. Research on Erroneous Examples

Research has investigated incorrect worked examples in mathematics and other domains for many years. IES-funded projects such as AlgebraByExample and GeometryByExample studied combinations of correct and incorrect examples, explanation prompts and comparison activities.

A 2025 Review of Educational Research meta-analysis synthesising 42 papers and 177 effect sizes reported a statistically significant but weak overall effect of erroneous examples on learning. The analysis also found that the design of error-explanation activities moderated outcomes: self-explanation prompts or instructional explanations produced better learning than leaving errors unexplained.

The magnitude and conditions vary. That supports careful use, not a blanket “wrong examples are better” claim.

7. Correct Examples Still Matter

A tutor should not replace a strong correct model with a wrong one merely because error analysis feels more advanced.

Worked-example research generally supports correct examples as an efficient learning support for novices. Some research syntheses also report that correct worked examples can be highly effective under novice-learning conditions.

That makes pedagogical sense. Before learners can judge a violation, they often need a usable model of valid reasoning.

The gate therefore asks whether the learner needs acquisition or discrimination.

8. Acquisition Before Discrimination

If the learner cannot perform the method when named, begin with a correct model.

If the learner can perform it but repeatedly falls into a plausible trap, an erroneous example may sharpen the boundary.

Acquisition: How do I solve simultaneous equations?
Discrimination: Why is dividing only one term on this side invalid?
Strategic discrimination: This method is valid, but why is it a poor choice here?

These are different jobs.

9. Compare Correct and Incorrect Side by Side

One of the safest designs is contrast.

Show two solutions that are identical until one crucial step. Ask where they first diverge, which step is licensed, what principle decides, and what the incorrect step would imply if it were allowed.

The correct example gives the learner an anchor. The erroneous one makes the boundary visible.

Side-by-side comparison can reduce the risk that the wrong route floats alone in memory.

10. Ask for the First Invalid Step

Learners often point to a later consequence rather than the original error.

A sign mistake on line two causes a strange answer on line five. If the learner corrects line five only, the causal model remains broken.

Train upstream diagnosis: “Everything above this line is valid. What is the first line that no longer follows?” “What operation happened?” “What should have happened instead?”

This builds checking habits that transfer to the learner’s own work.

11. Ask What Rule Was Violated

Error spotting can become visual pattern recognition.

“This looks wrong.”

Push one step deeper: “What makes it wrong?”

The learner might name equality preservation, denominator conditions, evidence sufficiency, subject–verb relation, fair-comparison control, or another relevant principle.

If the learner cannot state the rule, the tutor may need to teach or retrieve it.

12. Separate Conceptual Errors From Slips

An erroneous example can represent conceptual misconception, procedural misunderstanding, arithmetic slip, notation error, misread command, unsupported inference, or a strategically inefficient but valid method.

The response should match the type.

Do not build a grand misconception lesson around one arithmetic slip. Conversely, do not dismiss a repeated invalid assumption as carelessness.

The error example should clarify the mechanism, not flatten every mistake into one category.

13. Constructed Case: Alicia and Equation Balance

This is a constructed case. Alicia knows that the same operation must preserve equality, but under speed she sometimes “moves” terms by changing signs mechanically.

The tutor writes: 3x+5=20; then 3x=20+5; then 3x=25.

Alicia is asked to identify the first invalid step.

She says, “The five should become minus five.”

The tutor asks the deeper question: “Why?”

Alicia replies that subtracting five from both sides preserves equality.

The tutor then writes a correct transformation with the subtraction shown explicitly.

On a fresh equation, Alicia writes the operation on both sides before compressing it into the familiar transposition shortcut.

The erroneous example has repaired the meaning underneath the shortcut.

14. Constructed Case: Beatrice and Evidence Selection

Beatrice answers an inference question with a plausible claim but chooses evidence that is related rather than sufficient.

The tutor constructs two short responses: A uses the claim plus indirect evidence; B uses the same claim plus evidence that directly supports it.

Beatrice must decide which response is defensible and explain what the weaker evidence fails to establish.

The exercise does not present one answer as “silly”. Both are plausible. The distinction is evidential strength.

On a fresh passage, Beatrice selects evidence independently.

The receipt is not that she criticised example A. It is that her next evidence choice improves.

15. Constructed Case: Ciara and Fair Tests

Ciara knows that only one tested variable should differ between comparison conditions, but she often overlooks hidden differences.

The tutor shows a constructed investigation where both light exposure and water amount change between two plants. The conclusion attributes growth difference entirely to light.

Ciara identifies the extra changed variable.

The tutor asks: “What does that do to the conclusion?”

Ciara explains that the result cannot isolate light because water is a competing explanation.

A fresh scenario changes temperature and container size. Ciara must detect the confound without the familiar plant context.

The erroneous example has become a causal-reasoning exercise.

16. Constructed Case: Denise and Differentiation

Denise differentiates x³ correctly but sometimes mishandles composite functions.

The tutor shows a solution that differentiates the outer function but ignores the derivative of the inner function.

Denise recognises that something is missing but initially says, “You forgot the chain rule.”

The tutor asks her to connect the rule to structure: “What feature of the original expression makes the chain rule necessary?”

Denise identifies the function-inside-function relation.

That relation becomes the check she can use later, rather than the slogan “remember chain rule”.

17. Constructed Case: Emily and Study Planning

Emily’s error is not mathematical. Her weekly plan schedules every available hour.

The tutor constructs two plans with the same tasks. One uses every open slot. The other leaves a small recovery margin around uncertain tasks.

Emily initially prefers the full schedule because it appears productive.

The tutor asks what happens if one task takes thirty minutes longer.

The first plan has no recovery path.

The erroneous example teaches a planning principle: maximum allocation is not maximum robustness.

18. Avoid Using Real Learner Errors as Spectacle

A tutor may be tempted to project one learner’s wrong work for the group.

This can create embarrassment, status differences and defensive participation.

If the error is pedagogically useful, reconstruct it anonymously or create a composite version unless the learner has genuinely agreed and the context is safe.

The learning goal does not require identifying who made the mistake.

Professional trust is more important than the entertainment value of “look what someone did”.

19. Private First Analysis in Small Groups

When one learner announces the error immediately, others can borrow the diagnosis.

circle the first invalid line;
write one reason;
choose between two candidate principles.

Then discuss.

The tutor can see whether each learner independently recognised the fault before peer explanation enters.

This preserves the evidence value of the exercise.

20. Do Not Make the Wrong Route More Memorable Than the Correction

Incorrect information can stick, especially when it is vivid or repeated.

Do not spend ten minutes elaborating a misconception and thirty seconds stating the correct rule.

identify the invalid step;
state why it is invalid;
replace it with the valid step;
complete the correct route;
apply the correct relation in a fresh problem.

The lesson should end with the correct model stronger than the erroneous one.

21. Label the Status Clearly

If a deliberately wrong solution is displayed, label it as a solution to analyse rather than silently mixing it with correct notes.

This is especially important if learners photograph boards or review materials later.

A visible heading such as “Constructed incorrect example — diagnose before copying” can prevent the artefact from travelling without context.

The tutor may remove the label only when error detection itself is being assessed under controlled conditions.

22. Use One High-Value Error at a Time

A worked solution containing five unrelated mistakes is noisy.

The learner may find one and miss the target. The tutor cannot tell whether the principle was understood.

For focused teaching, construct one main error while keeping the rest of the solution valid.

Later, advanced learners can analyse multi-error work as an authentic checking task.

Early clarity earns later complexity.

23. Errorful Examples and Cognitive Load

Diagnosing an error requires holding the correct principle in mind, comparing it with the presented reasoning and repairing the route.

That can be cognitively demanding.

short solutions;
one key error;
visible steps;
familiar notation;
immediate explanation;
comparison with a correct example.

As knowledge grows, increase subtlety.

24. Erroneous Examples and Self-Explanation

Error analysis becomes more powerful when the learner explains the violation.

But the Self-Explanation Readiness Gate still applies.

A learner who says “I do not know why it is wrong” may need a correct model, not repeated demands to generate an explanation from nothing.

The tutor can move from recognition to explanation: “Which of these two rules applies?” “Now why?” “Now fix the line.”

25. Erroneous Examples and Interleaving

Once learners understand several methods, erroneous examples can test method selection.

Show a plausible solution using a method whose conditions do not hold. The arithmetic may be flawless. The flaw is the initial choice.

This is more advanced than spotting a calculation error.

It prepares learners to check not just execution, but whether the selected route was legitimate from the start.

26. Erroneous Examples and Example Variation

Do not show only one stereotyped version of a misconception.

If every wrong percentage solution uses the same number arrangement, learners may recognise the pattern visually.

Vary the surface while preserving the faulty principle.

Then include near-correct and fully correct cases.

The learner should diagnose by principle, not appearance.

27. Erroneous Examples and Feedback

correctly located and correctly explained;
correctly located but explanation weak;
later consequence identified, upstream error missed;
false positive: learner marks a valid step as wrong;
error missed;
correct repair but wrong reason.

This richer classification tells the tutor what to do next.

28. False Positives Matter

Error-hunting can teach learners to distrust correct work.

If every example contains a mistake, the learner assumes there must be one and starts changing valid solutions.

Include correct examples in some sets.

Ask, “Is there an error? If yes, locate it. If not, justify why the route is valid.”

Now the learner must decide whether intervention is needed at all.

29. Do Not Reward Suspicion Alone

A learner who says “something feels wrong” has begun checking, but the tutor should not stop there.

Professional reasoning requires a criterion.

Ask for the violated condition, evidence or operation.

Otherwise, error analysis can become style preference: unfamiliar method equals wrong.

This is especially important when learners encounter valid alternative methods.

30. Valid but Nonstandard Is Not an Error

An unfamiliar solution can be valid.

Before presenting an “incorrect example”, the tutor must verify that it truly violates the task, mathematics, language rule or evidence standard.

The Alternative Method Gate remains relevant. Do not teach conformity by labelling every non-model route wrong.

If a method is valid but inefficient, frame the comparison around cost, transparency or examination suitability rather than correctness.

31. Error Examples in Writing

Incorrect examples are not limited to mathematics.

a claim with evidence that does not support it;
a paragraph with an unresolved pronoun;
a sentence whose grammar is valid but meaning is ambiguous;
a conclusion that overstates the evidence.

The learner identifies the first point at which the text stops doing its intended job.

Avoid using deliberately poor prose merely for ridicule. The erroneous text should isolate a real decision.

32. Error Examples in Science

changing two variables but claiming one caused the result;
inferring mechanism from correlation alone;
extending a graph beyond measured range without qualification;
confusing observation with explanation;
reporting precision not supported by measurement.

These can teach evidence discipline.

But keep the science appropriate to the learner’s level and do not overclaim research methodology beyond curriculum needs.

33. Error Examples in Study Strategy

rereading only before a retrieval-heavy exam;
scheduling all difficult tasks at the end;
interpreting immediate familiarity as mastery;
changing strategy after one bad session;
using a tool that performs the target thinking.

The learner diagnoses the decision flaw.

This helps transfer error-analysis habits into self-regulated learning.

34. The Erroneous-Example Readiness Gate

What exact misconception or invalid inference is being contrasted?
Is the error plausible and instructionally important?
Does the learner have enough correct knowledge to judge it?
Should a correct example appear beside it?
Can the error be isolated to one main decision?
Will the learner identify the first invalid step?
What principle should they use to explain the error?
Could a legitimate alternative be mistaken for wrong?
Is the example safely constructed or anonymised?
How will the correct model be made explicit?
What fresh problem will test whether the learner avoids the error independently?
When will a delayed return occur?

35. When Not to Use Erroneous Examples

the learner has no stable correct model yet;
the wrong route is more complicated than the concept itself;
the learner is likely to memorise surface forms without understanding;
the emotional context makes mistakes threatening or humiliating;
a real learner’s work would be exposed unnecessarily;
the tutor is not certain the example is actually wrong;
the task goal is initial fluency and error analysis would distract;
the evidence base is being used as an excuse to make lessons adversarial.

A correct worked example may be the better tool.

36. Parent Communication

Parents may be confused when tuition deliberately shows wrong work.

I am not teaching the wrong method. She already has the correct model. I am using one plausible incorrect solution so she can practise locating the first invalid step and explaining the rule it violates. Then she solves a fresh question correctly. The aim is to make the misconception easier to detect in her own work.

This makes the purpose explicit.

37. Learner Communication

Tell the learner the exercise is diagnostic, not a trap.

This solution contains either one important error or no error. Your job is to decide which, locate the first invalid step if there is one, and tell me the rule that decides.

That wording prevents automatic error-hunting.

After the discussion: “Now close the example. Solve a fresh one without copying either route.”

The fresh performance is the receipt.

38. Research Foundation: IES Projects

IES-funded projects such as AlgebraByExample and GeometryByExample have investigated correct and incorrect worked examples paired with explanation prompts and comparison activities.

These projects reflect serious research interest in whether analysing errors can support mathematical learning and conceptual understanding.

Their existence should not be simplified into a claim that erroneous examples always outperform conventional instruction. Designs, learners and outcomes differ.

39. Research Foundation: 2025 Meta-Analysis

A 2025 Review of Educational Research meta-analysis by Alemdag, Eichelmann and Narciss synthesised 42 papers and 177 effect sizes comparing erroneous examples with correct examples or problem solving.

The reported overall effect was statistically significant but weak, with g=.136. Error-explanation activities mattered: self-explanation prompts or instructional explanations enhanced learning from erroneous examples more than conditions without error explanations.

For tutors, the implication is sensible: merely showing something wrong is not the active ingredient. Learners need structured comparison, explanation and correction.

40. Research Foundation: Correct Worked Examples

The broader worked-example literature supports correct examples as a strong instructional approach, especially for novice learners.

That provides the baseline against which erroneous examples should be used. Their value is not that wrong is superior to right. Their value is that, under suitable conditions, comparison with a plausible error can sharpen discrimination and checking.

A tutor should preserve the correct model as the reference point.

41. Research Limits

Studies vary substantially in age, domain, task, example design, prompts, feedback and outcome measures. Some use mathematics; others use different learning contexts.

No source cited here validates a universal erroneous-example protocol for Singapore three-student tuition. No evidence supports a fixed percentage of lesson time that should be spent on mistakes.

The Handbook proposal is therefore bounded: establish a correct model; choose one plausible consequential error; compare and locate the first invalid step; explain the violated principle; correct explicitly; verify on fresh work; and check again after delay if the misconception matters.

42. The Independence Direction

Eventually the learner should generate an internal erroneous example without the tutor.

They ask: “What is the tempting wrong move here?” “What condition would make my method invalid?” “Where would a sign mistake first change the logic?” “What evidence would be insufficient for this claim?” “What shortcut am I most likely to use incorrectly?”

This is advanced checking.

The learner is no longer afraid of mistakes, nor fascinated by them. They use possible errors as counterfactual tests of their own reasoning.

43. One More Boundary: Productive Error Versus Error Exposure

There is an important difference between allowing a learner to make an error during genuine problem solving and deliberately presenting an erroneous worked example.

Productive struggle begins with the learner attempting the task. An erroneous example begins with a designed artefact whose flaw is already present. The tutor should not blur the two.

If the learner can attempt the problem safely, their own error may provide richer evidence because it reveals the route they actually chose. If repeated attempts would waste time, reinforce an invalid procedure or create unnecessary frustration, a constructed erroneous example can externalise the misconception more efficiently.

Choose the form that best preserves learning and evidence. The point is not to maximise exposure to mistakes. It is to make the relevant reasoning boundary visible at the lowest useful learner cost.

Evidence and Connected Reading

Final Compression

An incorrect example is useful when the learner has enough correct knowledge to reject it for the right reason.

Choose a plausible error. Isolate the first invalid step. Compare with a correct model. Ask which principle was violated. Correct the route explicitly. Do not humiliate real learners. Do not let error-hunting become suspicion of valid alternatives. Follow immediately with fresh independent performance and return later if the misconception matters.

The goal is not to make learners good at criticising bad solutions on a board.

The goal is to make invalid reasoning easier to detect before it becomes their own final answer.

That is the Erroneous-Example Gate.

That is Tutor Handbook Volume 0150.

Small Group Tutorials

The Tutor Handbook Vol No.0150 | The Erroneous-Example Gate — How a Tutor Uses Deliberately Incorrect Solutions to Teach Error Detection Without Seeding the Misconception or Rewarding Superficial Fault-Finding