Teaching University students to carry out critical and independent science research is challenging, and they need to learn to flex new muscles and approaches in their brain, that are not always well stretched at the school stage. I have found the summary of George Polyas lessons that I reproduce below on a number of websites (e.g. here) and I do not know the original source, but its great – have a read:
In 1945 George Polya published a book How To Solve It, which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this book he identifies four basic principles of problem solving.
Polya’s First Principle: Understand the Problem
This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don’t understand it fully, or even in part. Polya taught teachers to ask students questions such as:
- Do you understand all the words used in stating the problem?
- What are you asked to find or show?
- Can you restate the problem in your own words?
- Can you think of a picture or diagram that might help you understand the problem?
- Is there enough information to enable you to find a solution?
Polya’s Second Principle: Devise a Plan
Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
- Guess and check
- Look for a pattern
- Make an orderly list
- Draw a picture
- Eliminate the possibilities
- Solve a simpler problem
- Use symmetry
- Use a model
- Consider special cases
- Work backwards
- Use direct reasoning
- Use a formula
- Solve an equation
- Be ingenious
Polya’s Third Principle: Carry Out the Plan
This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work, discard it and choose another. Don’t be misled, this is how things are done, even by professionals.
Polya’s Fourth Principle: Look Back
Polya mentions that 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.
These principles and more details about strategies of carrying them out are summarized in this document:Polya’s Problem Solving Techniques
George Polya (1887–1985) was one of the most influential mathematicians of the twentieth century. His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career. Even after his retirement from Stanford University in 1953, he continued to lead an active mathematical life. He taught his final course, on combinatorics, at the age of ninety.