Worked Examples: Learn Problem Solving, Then Fade the Help

When you're new to a type of problem, the usual advice is "just do lots of practice problems." But if you don't yet know how to solve them, practice can turn into guessing, getting stuck and copying the answer from the back of the book. There's a better first step. A worked example is a problem with every step of the solution shown. Studying a few carefully, then gradually doing more of the work yourself, is one of the best-researched ways to learn problem solving in math and science.

What a worked example is

A worked example shows a problem, the steps to solve it, and the final answer. Textbooks and lecture slides are full of them. The difference is in how you use them:

• Skimming an example on the way to the exercises gives you little.
• Studying it means following each step, knowing why it was taken, and being able to reproduce the method afterwards.

The idea is simple: when you're a beginner, solving a problem from scratch uses up so much effort on searching for a method that there's little left over for learning the method itself. A worked example shows you the method directly, so your effort goes into understanding it.

What the research says

• The original studies. In a series of experiments on algebra, Sweller and Cooper (1985) found that students who studied worked examples took much less time to get through them than students who solved the same problems conventionally, and then solved similar problems faster and with fewer errors. The benefit was specific, though: it showed up on problems with the same structure as the examples, not on different kinds of problem.
• Across many studies. A meta-analysis of 55 studies on math found a medium-sized benefit of worked examples overall, whether they were used when first learning a skill or for later practice. Examples showing correct solutions helped more than examples showing mistakes, alone or mixed with correct ones (Barbieri et al., 2023).
• Good examples share some features. A review of the worked-examples research by Atkinson and colleagues (2000) recommends several examples for each type of problem, with varied formats, and placing practice problems right next to the examples they match.

Fading: from examples to solving on your own

Studying examples forever won't make you a problem solver. At some point you have to solve problems alone, as you will in the exam. The research suggests getting there gradually.

Renkl and colleagues (2002) tested a fading approach. Learners started with a complete worked example. The next example had one step missing for them to fill in, the one after that had two, and so on until they were solving whole problems on their own. Across a classroom study and two lab experiments, fading helped learning, at least on problems similar to the ones practiced. Removing steps from the end of the solution first (backward fading) worked better than removing them from the start.

A simple fading sequence looks like this:

1. Study a fully worked example.
2. Complete a similar example with the last step missing.
3. Complete another with the last two steps missing.
4. Solve a similar problem from scratch.
5. Check your solution against a worked answer.

When to stop using worked examples

Worked examples help most when you're a beginner. Kalyuga and colleagues (2003) reviewed studies showing that techniques that help inexperienced learners can lose their benefit, and even get in the way, once learners are more experienced. They called this the expertise reversal effect.

So the rule is: examples first, then fade, then practice. Once you can reliably solve a type of problem, studying more examples of it is mostly wasted time. Move on to solving problems, and mix different problem types together; the interleaving guide explains why.

How to study a worked example

Cover the solution and predict each step. Before you read the next line, try to say what it will be. Then check. This makes you think rather than just follow along.

Say why each step was taken. "They used the chain rule here because the function is nested." Explaining steps to yourself is a useful habit, as the guide on asking "why?" explains, and students who learn most from examples tend to do it (Chi et al., 1989). The evidence on adding prompts to worked examples is mixed, though: the meta-analysis above found that, in the studies it reviewed, examples with self-explanation prompts did worse than examples without them. Keep your explanations short and focused on the steps you don't yet understand.

Study two or three examples of each problem type. One example shows you one problem. Several, with different numbers and wording, show you the method.

Pair each example with a similar problem. Study an example, then immediately solve a matching problem. Then fade.

Watch for the type, not just the steps. Notice what about the problem tells you which method to use. That's what you'll need when the problem isn't labeled.

Common mistakes

• Skimming examples. Following along feels like understanding. Covering and predicting shows whether it is.
• Jumping to hard problems too early. Struggling for an hour without a method teaches you little.
• Never fading. Studying examples alone won't prepare you for an exam with no examples.
• Staying with examples too long. Once you can solve a problem type, practice it, don't study it.

Doing this in Atheneum

• Keep examples and practice together. Upload your lecture slides, problem sheets and solutions to a notebook for the subject, so examples and matching problems are in one place. Write your own faded versions in the notebook's Notes tab, in Markdown.
• Ask Athena for a matching problem. After you've studied an example from your notes, ask Athena for a similar problem with different numbers, and to hold back the solution until you've tried. You can also ask her to show only the first steps, so you finish the rest. Check her working: AI tutors can make mistakes in calculations, and the guide on studying with an AI tutor explains how to stay safe.
• Test yourself once you're ready. Quizzes in Review check whether you can choose the right method without the example in front of you. Start at beginner difficulty and move up to intermediate and advanced as your scores rise; every quiz can be retaken.
• Use a study guide for extra practice. A notebook's study guide includes examples, common mistakes and practice questions built from your material.

A quick start

1. Pick one type of problem you find hard.
2. Study two worked examples, covering each step and predicting it before you read it.
3. Complete a similar problem with the last step left for you, then one with the last two steps.
4. Solve one from scratch, check it, and if you got it right, move on to mixed practice.

Watch, then help, then do it alone. That's how most skills are learned, and problem solving is no different.

Read next: how to study for a math exam, on putting examples, mixed practice and timed papers together.

Sources

• Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89. https://doi.org/10.1207/s1532690xci0201_3
• Barbieri, C. A., Miller-Cotto, D., Clerjuste, S. N., & Chawla, K. (2023). A meta-analysis of the worked examples effect on mathematics performance. Educational Psychology Review, 35, 11. https://doi.org/10.1007/s10648-023-09745-1
• Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning from examples: Instructional principles from the worked examples research. Review of Educational Research, 70(2), 181–214. https://doi.org/10.3102/00346543070002181
• Renkl, A., Atkinson, R. K., Maier, U. H., & Staley, R. (2002). From example study to problem solving: Smooth transitions help learning. The Journal of Experimental Education, 70(4), 293–315. https://doi.org/10.1080/00220970209599510
• Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The expertise reversal effect. Educational Psychologist, 38(1), 23–31. https://doi.org/10.1207/S15326985EP3801_4
• Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13(2), 145–182. https://doi.org/10.1207/s15516709cog1302_1