George Polya’s “How To Solve It” Summary
George Pólya’s “How to Solve It” is a classic guide to problem-solving, originally published in 1945. The book presents a systematic four-step framework for tackling problems, particularly mathematical ones, though the principles can be applied to virtually any type of problem.
The Four-Step Problem-Solving Framework
1. Understand the Problem
- Clearly identify what you’re being asked to do
- Restate the problem in your own words
- Identify the unknown, the data, and the conditions
- Create visual representations or diagrams where helpful
- Ask questions like: “What am I trying to find or show?” and “Do I understand all the terms used?”
2. Devise a Plan
- Search for patterns and draw on similar problems you’ve solved before
- Break down complex problems into manageable parts
- Consider working backward from the goal
- Look for ways to simplify the problem
- Apply heuristic strategies such as:
- Analogy: Finding and solving a similar problem
- Decomposition: Breaking the problem into parts
- Generalization or Specialization: Making the problem broader or narrower
- Working Backwards: Starting from the solution and working toward the given
- Auxiliary Elements: Adding helpful constructions or notation
3. Carry Out the Plan
- Execute your strategy systematically
- Verify each step as you proceed
- Document your attempts and be prepared to start over if needed
- Ensure each step is clearly understood and can be proven correct
4. Look Back and Reflect
- Evaluate the reasonableness of your solution
- Verify your results
- Consider alternative approaches
- Extract lessons learned for future problem-solving
- Analyze how the solution could be improved or simplified
The book emphasizes that problem-solving is a skill that can be learned and taught. Pólya suggests that if you cannot solve a problem directly, try solving a related, more accessible problem first. The approach is designed not just for students but also for teachers to help guide students through the problem-solving process.
Examples of Problems Being Solved
Research Project: Writing a Thesis
Understand: Define the research question and scope.
Devise a Plan: Outline methodology, identify sources, create a timeline.
Execute: Conduct research, analyze data, write drafts.
Look Back: Revise based on feedback, consider implications for future research.
Computer Programming: Debugging Code
- Understand: Identify the symptoms and reproduce the error.
- Devise a Plan: Isolate the problematic section, check for common errors.
- Execute: Make targeted changes and test them.
- Look Back: Document the solution, review for similar issues elsewhere.
Geometry Problem: Finding the Area of an Irregular Shape
- Understand: The problem asks for the area of an irregular polygon.
- Devise a Plan: Divide the shape into familiar geometric figures (triangles, rectangles).
- Execute: Calculate the area of each component shape and sum them.
- Look Back: Verify calculations and consider if there was a more elegant approach.