Worked examples: how to use them in math, and what the research says

What it is

A worked example is a problem with its full solution, step by step, that students study rather than solve. It's the written version of what a teacher does at the board when modeling: here's the problem, here's each step, here's why.

The research usually pairs examples with practice. A common pattern is the example-problem pair: study a worked example, then solve a similar problem yourself, then study another example, then solve another. The IES practice guide on organizing instruction recommends exactly this alternation.

Variations teachers use:

• Faded examples: the first example is fully worked, the next has the last step missing, the next has two steps missing, until students solve whole problems.
• Incorrect examples: a solution with a common mistake that students must find and fix.
• Compare two examples: two solutions to the same problem, or two problems that look similar but need different methods, side by side.

Worked examples are a core move of explicit instruction, where they come in the "I do" stage, and they pair naturally with interleaved practice once students can solve each type.

Why it works

A student solving a new kind of problem from scratch has to hold the problem, the goal, possible methods and their own progress in mind at once. Studying a worked solution frees them to focus on the steps and why they work. Paul Kirschner, John Sweller and Richard Clark argued that this kind of strong guidance works better for novices than leaving them to search for a method, and that the advantage shrinks once learners know more.

Christina Barbieri and colleagues' 2023 meta-analysis is the first large synthesis for math. From thousands of abstracts, they kept 55 high-quality experimental and quasi-experimental studies. Worked examples had a medium average effect on math performance, from elementary grades to college, and helped whether used when first learning a skill or during later practice.

The details matter. In that meta-analysis, correct examples alone worked better than incorrect examples or a mix, and adding prompts asking students to explain the examples reduced the effect rather than increasing it. Julie Booth and colleagues' randomized experiments in Algebra I classes found something that sits alongside this: when students explained worked examples during guided practice, their conceptual understanding improved compared with practice alone, and incorrect examples looked especially useful for understanding concepts. Correct examples seem safest for building skill; well-chosen incorrect examples can help understanding once students know the basics.

How to start, by grade

Examples

• Example-problem pairs on a worksheetGrades 3–8

  A two-column sheet. Left: "Find 3/4 of 12. Step 1: 12 ÷ 4 = 3 (one fourth of 12). Step 2: 3 × 3 = 9 (three fourths). Answer: 9." Right: "Find 2/3 of 15." Four rows like this, with the problems on the right changing a little each time. Students study the left before solving the right.

• Faded examples for two-step equationsGrades 6–8

  Example 1 is fully solved: 3x + 5 = 20, subtract 5, divide by 3, x = 5. Example 2 shows the first step, and students do the second. Example 3 shows only the problem. By example 4, students are solving alone. The fade makes the move from watching to doing gradual.

• Find the mistakeGrades 5–9

  "Sam solved 2(x + 3) = 14 like this: 2x + 3 = 14, 2x = 11, x = 5.5. What went wrong?" Students find the error (the 3 wasn't multiplied by 2), explain it, and solve correctly. Use it after students can solve the problem type, not before.

• Compare two that look alikeGrades 4–8

  Two worked examples side by side: the area of a triangle and the area of a parallelogram with the same base and height. Students write what's the same and what's different, and why the triangle's answer is half. Then they solve a mixed set.

Common mistakes

Too many steps shown at once.

A worked example with every detail crammed in can overwhelm. Show clear steps, one idea per step, with the reason in a few words.

Examples without practice.

Studying examples alone isn't enough; alternate them with problems students solve. The IES guide recommends interleaving worked solutions with problem solving.

Never fading.

Examples help novices most. As students improve, remove steps and then the examples, or they'll rely on copying the pattern.

Incorrect examples too early.

In the meta-analysis, correct examples alone worked best for performance. Use incorrect examples once students know the right method, to sharpen understanding.

Asking for an explanation of every example.

Barbieri and colleagues found self-explanation prompts reduced the effect of worked examples on performance. Ask students to explain selectively, such as on one key example or one error, rather than every time.

Time and cost

Time: a little more planning at first to write good examples; no extra class time, since examples replace some of the problems students would otherwise struggle through.

Money: nothing.

Training: low. Most teachers already model problems; the change is writing examples down, pairing them with practice and fading them deliberately.

Debates and open questions

Should students explain the examples? This is where the research isn't settled. Barbieri and colleagues found self-explanation prompts reduced the benefit on performance, while Booth and colleagues found explaining worked examples during guided practice improved conceptual understanding in algebra. Different outcomes (performance versus conceptual understanding) and different setups may explain the difference.

Correct or incorrect examples? Correct examples alone did best for performance in the meta-analysis; incorrect examples looked useful for conceptual understanding in Booth's experiments. A sensible order is correct examples first, then targeted incorrect ones.

Do worked examples stop students thinking? Critics worry that showing the answer removes the productive struggle. The research suggests struggle without a method is least useful for novices, and that examples followed by practice give students something to struggle productively with.

Tools that do it

Free, on this site, nothing to sign up for. Class lists stay in your browser.

• Math worksheet generator: make practice problems to pair with your worked examples, and answer keys that show each step.
• Base-ten blocks: model a worked example with blocks on the projector before students solve one themselves.
• Fraction bars: work fraction examples visually, step by step.
• Math worksheets by topic and grade: problem sets to alternate with your examples.

Questions teachers ask

What is the worked example effect?

The finding that novices learn better from studying worked solutions, alternated with solving problems, than from solving the same problems unaided. Barbieri and colleagues' meta-analysis found a medium effect on math performance.

Can students make their own worked examples?

Yes, once they can solve the problems. Writing a worked example for a classmate, with a reason for each step, is a good test of understanding. Check their examples before they're shared, so nobody studies a wrong one by mistake.

How many worked examples should I show?

Enough that students can solve a similar problem: often two or three for a new skill, alternating with problems. Fewer for students who already know related skills.

Are worked examples just for math?

The meta-analysis on this card is about math, and that's where the classroom evidence is strongest. The same idea (model, then practice, then fade) is used in writing and science, but we haven't added studies on those to this card.

What about students who copy the example without understanding?

Change the problems slightly from the example, fade the steps, and ask occasional why questions. Mixing problem types later, as in interleaved practice, also stops students copying one pattern.