George Pólya’s four principles
(https://en.wikipedia.org/wiki/How_to_Solve_It )
1. Understand the problem e.g.,
- Restate the problem using your own diagram and words
- Is there enough information to enable you to find a solution?
2. Devise a plan e.g.,
- Look for a pattern
- Solve a simpler problem
3. Carry out the plan e.g.,
- Persist with the plan
- If it continues not to work discard it and choose another.
4. Look back e.g.,
- Could you have solved it in a different way?
- Could you use the method for other problems?
What other approaches could you use in Devise a Plan?
Here are a few:
- Guess and check
- Eliminate possibilities
- Use symmetry
- Consider special cases
- Use direct reasoning
- Solve an equation
- Find a counterexample
- Estimate expected results
- Look for a pattern
- Draw a picture
- .Solve a simpler problem
- Use a model
- Work backwards
- Use a formula
- Use the pigeonhole principle
- Use induction
- Find an invariant
- Create a table
- Create a graph
- Use calculus
What might we value in addition to getting a solution?
Here are a few:
- Collaborative work in problem solving
- Sustained work on problems and resilience
- Using computer software to solve problems
- Planning on how a problem might be solved
- Assessment, improvement and completion of another’s method
- Appropriate confidence in answers
- Comparison of methods, why one method rather than another
- Ability to explain a problem and its solution to another
- The creation of notation to help solve a problem
- Understanding the application of mathematics