In 1945, Hungarian mathematician George Pólya wrote an extremely successful book called How to Solve It, which sold over one million copies and has been translated into 17 languages. In this book, Pólya identifies the four basic principles of problem solving.
1. Understand the Problem
Firstly, you must understand the problem you are tackling. Although this may seem like common sense, it is often overlooked; the amount of times I’ve been staring at a problem for hours because I haven’t fully understood what it is asking! Consider asking the following questions:
- What is the unknown? What are the data? What is the condition?
- What are you asked to find or show?
- Can you restate the problem in your own words?
- Do you understand all the words used in stating the problem?
For some areas in maths, such as mechanics, drawing a labelled figure can often be extremely helpful to visualise what is going on.
2. Devise a Plan
There are many different suitable ways to solve a problem, as mentioned by Pólya. However, it is important to choose an appropriate strategy, which is a skill learnt by solving many problems. Some strategies include:
- Guess and check
- Use symmetry
- Consider special cases
- Solve an equation or use a formula
- Look for a pattern
- Solve a simpler problem
- Work backwards
- Eliminate possibilities
When choosing a strategy, it may help to consider whether you have solved a related problem already or whether you can think of a familiar problem that has the same or similar unknown.
3. Carry out the Plan
Once you have devised the plan, this step is simple if you already have the necessary skills, often only required care and patience -try to avoid silly mistakes!
Persist with the plan that you have chosen, only discarding it after multiple failed attempts. Then, return to step 2.
4. Look back
Pólya highlights how a lot can be gained from looking back and reflecting on the work you have done, asking yourself what worked and what didn’t. This way you can implement strategies that were successful for future problems that are similar.
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