HSW-0246
A teacher places two solutions side by side.
One is correct. The other reaches the wrong answer through a plausible mistake.
The ordinary instinct is to hide the wrong one. Students should see the right method, copy the right method and practise the right method. Why risk contaminating memory with an error?
There is a serious reason for that caution. Wrong examples can confuse novices, become memorable for the wrong reason, or create extra processing without improving understanding. But there is also a serious reason to use them: a carefully chosen error can make a boundary visible that a correct solution leaves implicit.
A 2025 meta-analysis by Ecenaz Alemdag, Anja Eichelmann and Susanne Narciss synthesised 42 papers and 177 effect sizes comparing erroneous examples with correct examples or problem solving. The overall advantage of erroneous examples was statistically significant but weak, with a reported Hedges’ g of .136. Crucially, the design of the error-explanation activity moderated the effect: self-explanation prompts or instructional explanations were more helpful than presenting errors without explanation support.
A separate 2025 systematic review of 40 studies reached a compatible conclusion: learning from erroneous or contrasting erroneous examples can work, but outcomes depend on how errors are highlighted, how learners are prompted to process them, and what prior knowledge and cognitive resources learners bring to the task.
HSW-0246 therefore has a narrower claim than “mistakes are good for learning.” Its subject is erroneous examples: intentionally showing a learner somebody else’s plausible wrong solution so that the learner can identify the fault, explain why it fails, contrast it with a correct principle and then perform independently.
The direct answer
A wrong solution becomes useful study material when it helps the learner discriminate between a tempting but invalid rule and the rule that actually governs the task. Merely exposing students to errors is not enough.
The mechanism is not “error magic.” The learner must do cognitive work around the error: locate it, explain it, reconstruct the correct relation, compare alternatives and later solve a fresh problem without the annotated example present.
If that processing is absent, an erroneous example can become decoration, confusion or a second answer to remember.
An erroneous example is not the same as making your own mistake
This distinction matters because several learning-from-error literatures are easy to collapse.
- Own accidental error: the learner attempts a task, gets it wrong, then receives correction or feedback.
- Deliberate error: the learner intentionally generates a plausible wrong answer before correcting it.
- Erroneous example: the learner studies a prepared solution containing an error made by an example student or fictional solver.
- Contrasting erroneous example: the wrong solution is deliberately compared with a correct example or another solution route.
These activities share error content, but they do not create identical cognitive jobs. HSW-0227, How Studying Works | Deliberate Errors, owns the question of intentionally producing your own wrong response before correction. HSW-0246 owns a different job: learning from a wrong solution that is already present in instructional material.
Why a correct example can hide the decision point
Imagine a learner studying algebraic expansion:
Correct: 3(x + 4) = 3x + 12.
A learner can copy the visible transformation without understanding which term the multiplier applies to. The route looks smooth because the example contains no competing move.
Now add a plausible erroneous example:
Incorrect: 3(x + 4) = 3x + 4.
The contrast exposes a decision: does the factor outside the bracket multiply the first term only, or every term inside? The wrong line is useful because it turns an implicit structural relation into an explicit discrimination problem.
But if the teacher simply labels the second solution “wrong” and moves on, the learner may not reconstruct the principle. The instructional value comes from interrogating the error, not from displaying it.
What the meta-analysis says—and what it does not say
Alemdag and colleagues used robust variance estimation to synthesise experimental studies comparing erroneous examples with correct examples or problem-solving conditions. Across 42 papers and 177 effect sizes, the overall effect was positive but small.
That is important for two reasons.
First, it argues against the simplistic claim that wrong worked examples are a dramatic universal upgrade over conventional instruction. The average advantage was modest and the studies varied in domain, learner population, comparison condition and design.
Second, the moderation analysis points toward a more useful question: what does the learner have to do with the error? Error-explanation activities mattered. Self-explanation prompts and instructional explanations were associated with better learning from erroneous examples than conditions without those supports.
This does not prove one exact prompt is universally optimal. It does tell us that “show mistakes” is too crude an instructional prescription.
The error must be plausible enough to diagnose
A ridiculous mistake teaches little. If a worked example says 2 + 2 = 97, the learner has almost nothing to infer about the mechanism of misunderstanding.
A productive erroneous example usually represents a mistake a reasonable learner might make: applying a familiar rule in the wrong condition, dropping a negative sign, confusing correlation with causation, answering the question they expected rather than the one asked, treating evidence as explanation, or transferring a method across a boundary where it no longer applies.
The point is not realism for its own sake. Plausible errors reveal competing rules. They let the learner ask: “What model would make this answer seem sensible?” That question is often more diagnostic than “Where is the wrong number?”
Illustrative case: the science answer that contains the right facts in the wrong relationship
The following case is illustrative, not a result from the reviewed experiments.
A student is asked why increasing temperature can increase the rate of a chemical reaction. A prepared erroneous answer says:
Particles move faster, so every collision causes a reaction.
The answer contains a correct fact—average kinetic energy increases—but an invalid causal step. Not every collision is effective. The learner’s job is not merely to replace the sentence with the textbook wording. It is to identify the exact break in the explanation and repair the causal chain.
A useful prompt sequence might be:
- Which statement is supported?
- Which statement goes beyond the evidence or model?
- What condition is missing?
- Rewrite the explanation so the mechanism is correct.
- Test the repaired rule on a new reaction-rate question.
The wrong answer becomes a map of the conceptual boundary.
Why explanations matter more than red crosses
A red cross communicates outcome. It does not necessarily communicate cause.
Suppose a student sees a wrong geometry proof with one invalid step. If the correction simply supplies the right theorem, the student can memorise the replacement without learning how to detect the misuse. An explanation asks a deeper question: what condition for this theorem is missing here?
Error explanation can therefore create what is sometimes called negative knowledge: knowledge about what does not work, under which conditions, and why. That negative knowledge is valuable when future tasks contain tempting alternatives.
But the phrase should not be romanticised. The learner still needs a correct positive model. Knowing that a route fails is not the same as knowing what to do instead.
Prior knowledge changes the risk
The 2025 systematic review on conditions for effective learning from erroneous examples highlights prior knowledge and cognitive capabilities among the factors that can influence effectiveness.
This is intuitive. An expert can inspect a wrong proof and immediately see the violated condition. A novice may not yet possess the rule needed to diagnose the error. Asking the novice to “find what is wrong” can therefore create unproductive search.
For lower prior knowledge, the error may need stronger scaffolding: highlight the relevant line, provide the governing principle, ask a constrained comparison, or pair the erroneous solution with a correct one. As knowledge grows, support can fade and the learner can diagnose increasingly subtle errors independently.
Contrasting examples can turn correction into discrimination
A wrong example is often most informative when placed beside a correct solution that differs at the decisive point.
The learner can then compare:
- What is identical in both solutions?
- Where do they first diverge?
- Which rule justifies one branch?
- What tempting cue made the wrong branch plausible?
- What future cue should trigger a check?
This architecture changes studying from “memorise the correct path” to “learn the decision boundary between two paths.”
A five-stage erroneous-example protocol
1. Commit before seeing the diagnosis
Ask the learner to mark the first suspicious step or state whether the solution is valid. This prevents passive agreement with the later explanation.
2. Locate the first causal or procedural break
Later wrong steps may merely inherit the first error. Find the earliest point where the reasoning stops being licensed.
3. Name the violated condition
“Sign mistake” is often too shallow. Ask what rule should have controlled the sign and under what condition.
4. Repair, do not merely label
Reconstruct the solution from the point of failure so the learner leaves with a valid route.
5. Remove the example and test a fresh case
The final test is whether the learner can avoid the same trap when the original wrong solution is gone.
Do not turn a mistake catalogue into revision
There is a tempting failure mode: collect dozens of “common mistakes” and read them before an exam. That can feel sophisticated because the learner is studying errors, but it may simply create recognition without discrimination.
A better error catalogue contains the wrong move, the reason it is tempting, the violated condition, the repair and a fresh item that tests whether the learner can now detect the boundary. Fewer errors processed deeply can be more useful than a long list of warnings.
When correct worked examples should remain the default
Erroneous examples should not replace correct examples wholesale. Novices often need a clean model of successful performance. If the underlying procedure is not yet known, a wrong route may add unnecessary alternatives before the correct route is stable.
Use erroneous examples when there is a real discrimination problem: a recurring misconception, a seductive shortcut, a boundary condition students repeatedly miss, or a transfer situation where knowing why the alternative fails matters.
The meta-analytic effect being small is a useful warning against turning the method into a universal doctrine.
Delayed and independent performance check
Immediately after an error explanation, give one fresh item with the same underlying boundary but different surface features. Then wait and return with another item later. Finally, add an item where the criticised method is actually appropriate.
That last case matters. If the learner now avoids a rule everywhere because it appeared in an erroneous example, they learned a prohibition rather than a condition.
A strong result is conditional knowledge: “Use this method when these conditions hold; reject it when this particular condition fails.”
For parents: ask the child to repair the mistake
When reviewing a marked paper, it is easy to say, “You made this mistake again.” That labels the history but does not necessarily teach the boundary.
A better sequence is: “Show me the first place the reasoning went wrong. What rule should have operated? Why did the wrong move look reasonable? Now repair it. Can you do a different question without looking?”
The aim is not to make mistakes emotionally dramatic. It is to convert them into discriminations the child can use later.
For tutors and teachers: curate errors, do not merely collect them
A high-quality erroneous example has a teaching job. It represents a plausible misconception, contains enough correct structure to make the error diagnostic, and supports a comparison that reveals the governing principle.
Do not publicly attribute real student errors when that would embarrass the learner. Rewrite or anonymise them. The pedagogical object is the reasoning pattern, not the identity of the person who made it.
Also track whether the class can explain the error before increasing subtlety. An erroneous example that only the teacher can diagnose is not yet functioning as learner practice.
Misconceptions
“Mistakes are always good for learning.” No. The average meta-analytic advantage of erroneous examples was small, and design conditions mattered.
“Just show the common mistakes.” Exposure is not the mechanism. Error explanation, comparison and correction are central candidates for making the material useful.
“Wrong examples replace worked examples.” They serve different jobs. Correct examples model successful performance; erroneous examples can sharpen discrimination around tempting failures.
“If a learner can spot the error, the concept is mastered.” Recognition with the wrong solution visible is weaker evidence than independent delayed performance.
“The more errors we show, the better.” More alternatives can increase load and confusion. Select errors for instructional value.
Evidence and limits
The main synthesis is Alemdag, Eichelmann and Narciss, “A Framework for Learning From Erroneous Examples and Meta-Analysis of Empirical Research”, first published online 20 November 2025 in Review of Educational Research. It synthesised 42 papers and 177 effect sizes and reported a statistically significant but weak overall effect of erroneous examples, g = .136. Error-explanation design significantly moderated outcomes.
A complementary systematic review, “Conditions for Effective Learning from Erroneous Examples”, synthesised 40 studies and emphasised that effects depend on how errors are explained or highlighted, prompt design, prior knowledge and cognitive capabilities. Contrasting erroneous examples were often promising, but the literature is heterogeneous.
These syntheses cover multiple domains and comparison conditions. They should not be used to claim that a particular erroneous-example format will improve every learner, every subject or every outcome. Nor do they imply that students should practise producing wrong answers. The instructional object here is a prepared example whose error is made available for diagnosis and repair.
The return
A wrong solution is not valuable because it is wrong.
It becomes valuable when the error exposes a decision that the learner must learn to make.
Locate the first break. Name the violated condition. Explain why the wrong move was tempting. Repair the solution. Then remove the example and ask the learner to perform under new conditions.
If the learner can now distinguish where the method works from where it fails, the erroneous example has done something a clean correct solution sometimes cannot: it has made the boundary visible.
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