Pólya’s Problem-Solving Method

1. Understand the Problem

• What is the unknown? What are the data? What is the condition?
• Is it possible to satisfy the condition?
• Draw a figure. Introduce suitable notation.
• Separate the parts of the condition.

For research: What exactly am I trying to explain or predict? What are my constraints?

2. Devise a Plan

• Have you seen this problem before? In a slightly different form?
• Do you know a related problem? A useful theorem?
• Could you solve part of the problem?
• Could you derive something useful from the data?

For research: What methods have worked for similar questions? What simplifications make this tractable?

3. Carry Out the Plan

• Check each step. Can you see clearly that it is correct?
• Can you prove that it is correct?

For research: Execute the method systematically. Document what works and what doesn’t.

4. Look Back

• Can you check the result?
• Can you derive the result differently?
• Can you use the result or method for some other problem?

For research: Does the solution make sense? What did I learn about the method? Where else might this apply?

Against Trial and Error

Random attempts waste time. Pólya’s method provides structure: understand first, then plan, then execute.

Metacognitive Awareness

The method forces you to think about your thinking. “Looking back” is where learning happens—not just solving the problem, but understanding the solution.

Transferable Structure

Works for math problems, research questions, implementation challenges, debugging. The phases adapt to different domains.

Problem Formulation

“Understand the problem” prevents solving the wrong question. Forces clarity about what success looks like.

Method Selection

“Devise a plan” by analogy to solved problems. What worked in related domains?

Systematic Execution

“Carry out the plan” with checking at each step. Prevents compounding errors.

Reflection

“Look back” generates insight. What made this hard? What made it tractable? What’s generalizable?