Worked examples first: why studying solved problems helps beginners

Should you study a solved problem before trying one yourself? If you are learning an unfamiliar method, the research gives good reasons to do so. Studying a complete solution can help beginners learn how a problem works. The next question matters just as much: when should the help give way to independent practice?

The worked-example effect, associated with John Sweller and colleagues, describes how studying solutions can help learners more than solving conventional problems under the conditions studied. In practice, this means choosing support that fits the learner's knowledge and changing it as that knowledge grows.

What a worked example provides

A worked example gives the full solution to a problem for careful study. The learner studies the steps as well as the final answer.

Sweller, Jeroen van Merriënboer and Fred Paas make this distinction in their 2019 review in Educational Psychology Review. When choosing study material, look for a solution that gives you a method to examine. The research concerns this kind of instruction, beyond simply providing the correct answer.

Why searching can crowd out learning

Cognitive load theory distinguishes working memory, which processes new information with limits on both capacity and duration, from long-term memory. Teaching has to respect those limits, without assuming that every learner can hold the same fixed number of things in mind.

In his 1988 paper in Cognitive Science, Sweller described domain-specific schemas as a major distinction between experts and novices. A schema is an organised knowledge structure that helps someone recognise a kind of problem and the moves associated with it. It gives a learner something reusable, beyond the answer to one question.

Sweller proposed that conventional means-ends search can use processing capacity that might otherwise support learning these structures. Finding an answer can leave a learner without a method to use next time. A full solution offers beginners a way to study the method with less demand to discover it unaided. This explains how particular demands of problem solving can hinder learning.

What the evidence supports

Sweller and Graham Cooper's 1985 algebra paper reported five experiments, with the second to fifth examining worked examples. Reported benefits included faster subsequent solving and fewer mathematical errors. The test problems had the same structure as the initial problems, so the findings apply to that setting rather than unrelated tasks or school subjects.

A broader mathematics meta-analysis by Christina Areizaga Barbieri and colleagues, published in Educational Psychology Review in 2023, included 43 articles and 55 studies, covering elementary through postsecondary settings. It found a medium average effect on mathematics performance, reported as g = 0.48. That is an effect-size statistic, not a 48 per cent increase in marks.

The evidence included experimental and quasi-experimental studies. The average finding supports worked examples in mathematics, while allowing for differences between learners and study designs. It gives no target for examination results or ideal number of examples to study.

Keep examples close to practice

Example study and solving problems belong together. Robert Atkinson and colleagues' 2000 review in Review of Educational Research recommends placing examples close to matched practice problems. The learner studies a solution, then has a chance to use the method.

The same review discusses encouraging learners to explain examples to themselves, through direct training or the design of the material. The aim is to help learners think actively about a solution. The findings on prompts that ask them to explain steps are mixed.

In two experiments reported in 2003, Atkinson, Alexander Renkl and Mary Margaret Merrill combined fading with prompts to identify the principle behind each solution step. That combination produced medium-to-large effects on near and far transfer, without additional time on task. Yet the 2023 mathematics meta-analysis found smaller effects in studies using self-explanation prompts than in those without them.

These findings address different questions. A particular combination can help, while the value of added prompts varies. Comparisons across studies leave open why effects differ; they cannot show that explaining a solution is inherently harmful. The meta-analysis also found larger effects for correct-only examples than for incorrect-only or mixed examples, without establishing that every activity involving errors is ineffective.

Let the learner take over

Fading provides a transition from studying complete solutions to solving independently. In their 2003 review, Renkl and Atkinson proposed gradually increasing the problem-solving demands within example study. Support is reduced as learners take responsibility for more of the work.

One sequence they reviewed began with a full solution, then omitted the last step, then the last two steps, and eventually all three. The learner completed the missing parts. This illustrates backward fading; the three steps and their order are examples rather than rules for every topic.

Renkl, Atkinson and Cornelia Grosse's 2004 experiments add detail. In the first experiment, the position of the omitted steps did not affect learning outcomes; learners learned most about the principles whose steps they completed. The second linked fading to fewer unproductive learning events, without finding an effect on productive events including self-explanations.

Transfer has limits, too. Atkinson, Renkl and Merrill noted that earlier fading research reliably helped near-transfer tasks, but not far-transfer tasks. Success with substantially different problems remains a separate challenge.

Why experience changes the advice

Instruction that helps a beginner can lose its benefit, or become counterproductive, as relevant expertise increases. Slava Kalyuga and colleagues described this expertise reversal effect in their 2003 review in Educational Psychologist. What matters here is how well the learner knows the task, rather than their age.

Their explanation is that guidance may duplicate knowledge the learner already has. Trying to reconcile the supplied information with existing schemas can consume working-memory resources, and unnecessary information may be hard to ignore. Experienced learners can still need explanations. Adjust guidance to what they know as learning progresses.

What to do with the next problem

For an unfamiliar method, start with a complete worked solution and study its steps. Keep a matched practice problem close by. Use partially completed solutions as a way to take on more of the solving, with independent work as the destination.

For parents and teachers, this means choosing support according to the learner's knowledge and adjusting it as that knowledge grows. For learners, it means treating a solved problem as material to study, then moving towards doing the work yourself. The research supports this progression, but sets no fixed timetable, universal prompt or guaranteed result.