The last step of problem solving
Mathematician George Pólya’s final step of problem solving is "look back". After finding an answer, strong problem solvers pause to reflect. Reflection builds deeper understanding and makes the next problem easier.
Reflection questions
| Question | Purpose |
|---|---|
| Does the answer make sense? | catch errors |
| Can I verify it another way? | confirm correctness |
| Is there a quicker method? | improve efficiency |
| What was the key idea? | remember the insight |
| Where else could I use this? | generalize |
| What slowed me down? | metacognition |
Many roads to one answer
Problem: Find the sum 1 + 2 + 3 + … + 100.
| Strategy | Idea |
|---|---|
| brute force | add one by one (slow, error-prone) |
| pairing (Gauss) | 50 pairs of 101 = 5,050 |
| formula | n(n + 1)/2 = 5,050 |
| geometry | two staircases form a 100 × 101 rectangle; half is 5,050 |
Each alternative strategy reveals a different connection.
Example: solving a quadratic
x² − 6x + 8 = 0
| Method | Work |
|---|---|
| factoring | (x − 2)(x − 4) = 0 → 2, 4 |
| completing the square | (x − 3)² = 1 → 3 ± 1 |
| quadratic formula | (6 ± √4)/2 → 2, 4 |
| graphing | x-intercepts at 2 and 4 |
Factoring is fastest here, but the formula works when factoring doesn’t.
Example: percent problem
"A $80 item is 25% off. Sale price?"
- Method 1: 25% of 80 = 20; 80 − 20 = $60
- Method 2: 75% of 80 = $60 (one step)
Reflecting shows that Method 2 is more efficient, and that idea (multiply by 1 − rate) generalizes to all percent decreases.
Generalizing
After solving "the sum of the first 10 odd numbers is 100", ask: is the sum of the first n odd numbers always n²? Testing and proving it turns one answer into a general rule.
Learning from mistakes
Errors are valuable data. Ask:
- Where exactly did it go wrong?
- Was it a slip (arithmetic) or a misunderstanding (concept)?
- How will I catch it next time?
Keeping an error log of common mistakes helps.
Metacognition
Metacognition means thinking about your thinking: noticing which strategies you reach for, when you get stuck, and what helps you get unstuck. Students who reflect regularly tend to become more flexible problem solvers.
Worked examples
Example 1: You found 15% of 60 = 9. Verify another way. 10% = 6, 5% = 3 → 9 ✓.
Example 2: You solved 3x + 5 = 20 → x = 5. Verify. 3(5) + 5 = 20 ✓.
Example 3: Generalize: the area of a triangle with base 8 and height 5 is 20. → A = ½bh for any triangle.
Common mistakes
- Stopping at the first answer without checking.
- Sticking with one method for every problem.
- Ignoring mistakes instead of learning from them.
Try it yourself
- Find two ways to calculate 25 × 16.
- Solve x² = 2x two ways (and avoid losing a root).
- What reflection question helps you generalize?
- Name one benefit of comparing strategies.
Answers: 1) e.g. 25 × 16 = 100 × 4 = 400; or 25 × 10 + 25 × 6 = 400 2) factoring: x(x − 2) = 0 → 0, 2; graphing y = x² and y = 2x → intersect at 0 and 2 (dividing by x would lose x = 0) 3) "Where else could I use this?" 4) e.g. finding a more efficient method
Quick recap
- Reflection is the final step: check, compare and learn.
- Try an alternative strategy to verify and improve efficiency.
- Generalize results and use metacognition to grow as a problem solver.