18 MINDS · How people actually learn
Show one whole solution.
You will learn why beginners learn a new procedure faster from one complete example than from the rule alone. It takes 4 minutes.
Here's a rule. Use it.
Is this number valid? Follow the rule exactly.
- Starting from the rightmost digit and moving left, double every second digit;
- if doubling gives a number greater than 9, subtract 9;
- add up all the digits;
- the number is valid if the total is a multiple of 10.
79927398713
Try it before you scroll. Most people get stuck on one of these:
- Does the last digit count?
- Every second from which end?
- Add 16, or 1 + 6?
The rule is complete and correct. It just isn't clear yet, because you've never seen it done.
Watch the clip
The clip shows the same rule, then one complete example, start to finish. Then it shows the rule again.
The name for it
THE NAME FOR IT
The worked-example effect
John Sweller and Graham Cooper · 1985
Beginners learn a new procedure faster by studying a complete solution than by trying to apply the rule alone.
One whole solution
Here's the same number, worked from start to finish. Read every step.
Step 1. Start at the right. Skip the last digit. Mark every second digit going left.
| Digit | 7 | 9 | 9 | 2 | 7 | 3 | 9 | 8 | 7 | 1 | 3 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marked | • | • | • | • | • |
Step 2. Double the marked digits.
| Digit | 7 | 9 | 9 | 2 | 7 | 3 | 9 | 8 | 7 | 1 | 3 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marked | • | • | • | • | • | ||||||
| After doubling | 7 | 18 | 9 | 4 | 7 | 6 | 9 | 16 | 7 | 2 | 3 |
Step 3. If a doubled digit is over 9, subtract 9. Then add everything up.
| Digit | 7 | 9 | 9 | 2 | 7 | 3 | 9 | 8 | 7 | 1 | 3 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marked | • | • | • | • | • | ||||||
| After doubling | 7 | 18 | 9 | 4 | 7 | 6 | 9 | 16 | 7 | 2 | 3 |
| Over 9? Subtract 9 | 7 | 9 | 9 | 4 | 7 | 6 | 9 | 7 | 7 | 2 | 3 |
Add them up: 7 + 9 + 9 + 4 + 7 + 6 + 9 + 7 + 7 + 2 + 3 = 70. 70 ends in 0. Valid.
Read the rule again
- Starting from the rightmost digit and moving left, double every second digit;
- if doubling gives a number greater than 9, subtract 9;
- add up all the digits;
- the number is valid if the total is a multiple of 10.
79927398713
It's the same rule, but now each line points to something you watched happen. "Every second digit" means the marked ones. "Greater than 9" means 16 and 18. The questions that stopped you before have answers.
This is the Luhn check, used on payment card numbers.
Now you try one
One digit changed: the last one. Is this number valid? Work it the same way, then check.
79927398710
Show the worked answer
| Digit | 7 | 9 | 9 | 2 | 7 | 3 | 9 | 8 | 7 | 1 | 0 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Marked | • | • | • | • | • | ||||||
| After doubling | 7 | 18 | 9 | 4 | 7 | 6 | 9 | 16 | 7 | 2 | 0 |
| Over 9? Subtract 9 | 7 | 9 | 9 | 4 | 7 | 6 | 9 | 7 | 7 | 2 | 0 |
Add them up: 7 + 9 + 9 + 4 + 7 + 6 + 9 + 7 + 7 + 2 + 0 = 67. 67 doesn't end in 0. Not valid.
Changing one digit broke the total. That's why card numbers end in a check digit: most typos make the number fail.
TAKEAWAY
Show one whole solution.
Then the rule makes sense.
Teach a procedure this way
- Show one complete, correct example first, with every step visible.
- Then state the rule, and point to where each part of it happened in the example.
- Give a second problem with some steps done for them. Then one with none done.
- Stop giving examples once people can do it. Experts learn more from solving problems than from reading worked ones (this is called the expertise reversal effect).
Ask your agent
Teach me the procedure below with the worked-example method. 1. Pick one realistic input and solve it completely, showing every step and every intermediate value. 2. Then state the general rule, and for each part of the rule, point to the step in the example where it happened. 3. Give me a second input with the first half done, and ask me to finish it. Show the answer only after. 4. Give me a third input to do on my own. Procedure: [paste here]
What good output looks like: a full example with every number shown, the rule mapped back onto it, then practice that fades the help.
Sources
- John Sweller and Graham A. Cooper, "The use of worked examples as a substitute for problem solving in learning algebra," Cognition and Instruction, 1985.
- On fading and the expertise reversal effect: Alexander Renkl and Robert Atkinson's work on faded worked examples, and Slava Kalyuga's research on expertise reversal (2003 onward).
- The Luhn check was patented by Hans Peter Luhn of IBM in 1960. The number on this page is the standard test example, not a real card.