George Pólya in his 1957 book How to Solve It suggests the following steps when solving a problem:
- First, you have to understand the problem
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?
If this technique fails, Pólya advises: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” (1957:114)
First principle: Understand the problem
Understand the problem is often neglected as being obvious – try these questions:
- What are you asked to find or show?
- Can you restate the problem in your own words? (write it down)
- Can you think of a picture or a diagram that might help you understand the problem?
- What is the unknown? What are the data?
- Is there enough information to enable you to find a solution?
Second principle: Devise a plan
There are many ways to solve problems including:
- Can you restate the problem?
- Make an orderly list of all the components of the problem.
- Eliminate possible solutions.
- Look for a pattern. Find the connection between the data and the unknown.
- Can you draw a picture of the problem or describe as a model?
- Can you find a problem analogous to your problem and solve that?
- Have you seen it before? Or have you seen the same problem in a slightly different form? Could you use its method?
- Can you start with the goal and work backwards to something you already know?
- Be creative. Use your head!
Develop a plan for solving the problem.
Third principle: Carry out the plan
This step is usually easier than devising the plan. All you need is patience, given that you have the necessary skills. Persist with the plan that you have chosen. Carry out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
If it continues not to work go back a step and develop another plan of attack.
Fourth principle: Review/extend
Much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn’t. Doing this will enable you to predict what strategy to use to solve future problems.
In summary: Think, Plan, Do, Review.