Learning from worked examples the right way

TL;DR:

• Reading a solution feels like understanding, but only redoing it shows whether you can.

• Explain the reason for each step to yourself, then cover the solution and redo it.

• Finish with a similar problem with different numbers, so you learn the method and not the answer.

You open the solution to a chemistry problem, read it line by line, and nod. Every step makes sense. Ten minutes later you try a similar problem and have no idea where to begin. The solution was clear because someone else had already made all the decisions. You watched the decisions, and never made one.

A worked example is one of the best things in a textbook, but only if you use it as a workout and not as a story. Here is how.

Why a solved problem is easy to read and hard to learn from

A finished solution shows you the path but hides the choices: why this equation, why that unit, why this step came before the other. Those choices are the skill. When you only read, you absorb the surface and skip the thinking, and the feeling of following along is mistaken for the ability to do it yourself.

What you need

• A problem with a full worked solution, from your textbook or teacher.

• A sheet of paper and a piece of card or paper to cover the solution.

• A second, similar problem with a different set of numbers, ideally with an answer to check against.

The method

1. Read the problem, then try the first step yourself: Before looking, write what you would do first. Even a wrong guess makes the real first step stand out.

2. Explain each step to yourself: For each line, say why it is there: what rule it uses, what it gets you closer to. Write a short reason in the margin.

3. Cover and redo: Hide the solution and redo the whole problem from the question alone. Uncover and compare only when you are stuck.

4. Mark what you could not reproduce: Circle the steps you missed. They are the ones to study, not the whole example.

5. Do a similar problem: Change the numbers or the substance and solve it without the example in view. Then check.

Worked example: a solved stoichiometry problem

Here is the problem. How many grams of water form when 4.0 g of hydrogen gas burns completely in oxygen? The balanced equation is 2H₂ + O₂ → 2H₂O. Use molar masses of about 2.0 g/mol for H₂ and 18 g/mol for H₂O.

1. Grams of H₂ to moles: 4.0 g ÷ 2.0 g/mol = 2.0 mol. (Reason: the equation talks in moles, not grams.)

2. Moles of H₂ to moles of H₂O using the coefficients: the ratio is 2 to 2, so 2.0 mol H₂ → 2.0 mol H₂O. (Reason: the coefficients are the mole ratio.)

3. Moles of H₂O to grams: 2.0 mol × 18 g/mol = 36 g. (Reason: the question asks for grams.)

Notice that the margin reasons carry the learning. Anyone can copy the arithmetic. The skill is knowing the path is grams, then moles, then moles of the other substance, then grams.

Check. Mass should be conserved. The oxygen needed is 1.0 mol (half the 2.0 mol of hydrogen, from the 2 to 1 ratio), and 1.0 mol × 32 g/mol = 32 g. Then 4.0 g + 32 g = 36 g, which equals the water produced.

Cover and redo, then the similar problem

Now cover the solution and redo it from the problem. Suppose you forgot the middle step and went straight from grams of H₂ to grams of H₂O. That is the circled step. Practise it once more.

Then the similar problem: How many grams of water form from 6.0 g of hydrogen? 6.0 ÷ 2.0 = 3.0 mol H₂, so 3.0 mol H₂O, and 3.0 × 18 = 54 g. Check with conservation: oxygen is 1.5 mol, 1.5 × 32 = 48 g, and 6.0 + 48 = 54 g. It matches.

Using worked examples at different stages

Early in a topic, studying examples is more efficient than struggling alone with problems you have no method for. As you improve, shift towards solving first and checking the example after. The point is to hand the work back to yourself gradually.

Why self-explanation is the part that counts

The margin reasons are the whole trick. When you write why a step is there, you have to connect it to a rule you know, and that connection is what you will reach for on a new problem. If you cannot write the reason, you have found a gap, which is exactly what you wanted to find while the solution is still in front of you.

A good test is to read your margin notes without the arithmetic. If they alone describe the method (convert to moles, use the ratio, convert back), you understand the example. If they read like "then multiply", go back and ask what the multiplication is for.

Mistakes to avoid

• Reading the solution while nodding and calling it studying.

• Copying the steps without writing the reason for each.

• Skipping the similar problem, so you never learn if the method transfers.

• Memorizing the numbers of the example instead of the path.

• Uncovering the solution at the first sign of trouble. Give yourself a minute.

Your plan for this week

1. Choose two worked examples from this week's homework topic.

2. Explain every step in the margin, cover and redo each.

3. Do one similar problem for each, with no example in view.

4. Next week, redo the circled steps once more from memory.

Frequently asked questions

Is it better to try the problem first or read the example first?

If you have no method at all, read the example first. If you have some idea, try first and use the example to repair it.

How many examples should I study per topic?

Two or three varied ones is plenty. More examples without more self-testing mostly adds reading time.