You Understand the Example but Can’t Solve the Next Problem. Here’s Why.

The short answer: if you understand the examples but can’t solve the problems, the reason is that these are two different skills. Following a solution only asks you to recognise that each step is correct. Solving a problem asks you to recall the method and decide which method it needs. Worked examples and grouped homework sets train the first and skip the second. Cover the solution, predict each step and practice with mixed problems to train the part the exam tests.

It is one of the most common complaints in any math or science course. The lecture made sense. The worked example made sense. The textbook solution made sense. Then you sit down with a blank problem and nothing comes. It feels like you understood and then forgot. What actually happened is more interesting, and much easier to fix.

Following is not the same as solving

When you read a worked solution, every step is handed to you. Your only job is to check that it follows from the one before. That is recognition, and recognition is easy. It is why a multiple choice question feels easier than a blank one.

Solving a problem from scratch asks for two harder things. You have to recall the method without a prompt. Before that you have to decide which method the problem needs. A worked example does both of those for you, which is why it feels so smooth. It is also why it leaves you stuck.

This is the same illusion that makes rereading feel productive. In a 2006 study by Henry Roediger and Jeffrey Karpicke, students who reread a passage felt more confident than students who tested themselves. A week later the students who tested themselves remembered far more.

Why homework sets hide the hardest part

Look at a typical problem set at the end of a textbook section. Every problem uses the method from that section. You never have to decide which method to use, because the chapter title already told you.

Exams do not work that way. Problems come mixed together and choosing the method is half the question. Doug Rohrer and Kelli Taylor showed how much this matters in a 2007 experiment. College students learned to find the volume of four unusual solids. One group practiced the problems grouped by type. The other practiced them shuffled together. During practice the grouped students did better, getting 89% right compared with 60%. On a test a week later the result flipped. The shuffled group scored 63%. The grouped group scored 20%.

Mixed practice felt harder and tripled the test score a week later.

Researchers call this interleaving. It works because every shuffled problem forces the decision that grouped practice lets you skip.

How strong students use worked examples

Worked examples are not the enemy. For a beginner they are one of the fastest ways to learn a new method. What matters is what you do while you read them.

In a classic 1989 study Michelene Chi and her colleagues watched students learn physics from worked examples. The strongest students kept explaining the steps to themselves. They asked why each step was allowed and linked it to the principle behind it. The weaker students read the steps and moved on. Later the weaker students leaned heavily on the examples while solving new problems. The self-explainers needed them far less.

A method that fixes it: cover, predict, check

1. Cover the solution with a sheet of paper or your hand.
2. Read the problem and decide which method it needs before you look. Say why.
3. Predict the first step and write it down.
4. Reveal only that step and compare. If yours is different, work out why before you go on.
5. Repeat for each step until the end.
6. Solve a fresh problem of the same type with the example closed.

The first few times this will feel slow. That is the point. Every prediction is a small act of recall, and every mismatch shows you exactly what you did not understand. Researchers have found that gradually removing steps from worked examples until students solve problems alone, an approach they call fading, helps learners make the jump from following to solving.

Try it with the chain rule

Here is the method on a derivative that trips up a lot of calculus students.

Find the derivative of sin(x²).

Decide the method first. The function is sine of something, and that something is not just x. A function inside a function means the chain rule.

Predict step one. Name the outside function and the inside function. The outside is sin(u). The inside is u = x².

Predict step two. Differentiate the outside and leave the inside alone. The derivative of sin(u) is cos(u), which gives cos(x²).

Predict step three. Multiply by the derivative of the inside. The derivative of x² is 2x.

Answer: 2x cos(x²)

The most common mistake here is writing cos(2x). That happens when you differentiate the inside and put the result back inside the cosine. If you made it, you now know exactly which step to practice.

Now try these two with the answers covered. They are deliberately different, so you have to choose the method each time.

1. Find the derivative of e^{3x + 1}.

2. Find the derivative of x² sin(x).

The answers are at the end of the article.

Using AI without falling into the same trap

An AI tutor can make this problem worse or better. A full AI solution is just another worked example, and reading it gives you the same false sense of understanding. In a 2025 study published in PNAS, students who practiced math with a plain ChatGPT-style assistant did better on practice problems and 17% worse on the exam. Students who got hints instead of answers held their ground.

Use AI for the parts of the method that are hard to do alone:

“I think this needs the chain rule because it’s a function inside a function. Is that the right call? Don’t solve it.”

“Here is my working. Tell me the first line that’s wrong but don’t fix it.”

“Give me five derivative problems that mix the chain rule, the product rule and the quotient rule. Don’t label which is which.”

Check any arithmetic an AI gives you. Language models still make calculation mistakes, especially in long problems with many steps. For more on this, read what the research says about studying with AI.

The night before the exam

Make a mixed set of ten problems from different sections of the course, the kind you would expect on the test. Do them without notes and with a timer. Mark every problem where you could not decide which method to use. Those are what to review tonight. Focus less on the ones you got slightly wrong and more on the ones where you did not know where to start.

Answers to the practice problems

1. The derivative of e^{3x + 1} is 3e^{3x + 1}. It uses the chain rule because 3x + 1 sits inside the exponential.

2. The derivative of x² sin(x) is 2x sin(x) + x² cos(x). It uses the product rule because two functions are multiplied together. If you reached for the chain rule on this one, you have just found the decision that mixed practice trains.

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