Learning how to solve problems in mathematics is simply to know what to look for.
Mathematics problems often require established procedures. To become a problem solver, one must know What, When and How to apply them. To identify procedures, you have to be familiar with the different problem situations. You must also be good in gathering information, extracting strategies and use them. But exercise is must for problem solving. Problem solving needs practice!! The more you practice, the better you become. Sometimes it may happen that you knew the formula for a problem, but as you haven't tried it earlier -- you failed to solve it by a minor margin. On the same topic, George Polya published "How to Solve It" in 1957. Many of his ideas that worked then, do still continue to work for us. The central ideas of Polya's books were to identify the clues and to learn when & how to use them into solution strategies.
By experience, one may work on her problem solving approach in four essential steps.
Essential Step One
Identifying the clues
- First of all read the problem carefully understanding each word precisely and underline the clue words in it. As a student I know that this also requires practice. A new problem-solver, in his/her early days, should work on practicing on identifying clues from a problem. Some when it is said that Problems have their solutions in understanding the problems.
- Ask yourself if you've seen a problem similar to this one?
◊ If yes, what is similar about it? What did you do then?
◊ If no, this problem is almost new to you then after using clues, confirm:
What facts are you given here and What you have to find out.
Essential Step Two
The Game Plan
- Define your game plan, and ask again that "Have you seen a problem like this before?".
- Identify what you did (or what have you to do)?
- Define your strategies to solve this problem.
- Try out your strategies, using formulae, simplifying, using sketches, guess and check, look for a pattern, graphing etc..
- If your strategy doesn't work, it may lead you to a new strategy that does work. You can find a new strategy if you know all the concepts related to the topic on which the problem is covered.
Essential Step Three
Solving the Problem
Use your skills of 'strategy' & 'tactics' to solve the problem. Never go over or out of the focus of the problem. This may cause time waste and errors. This step needs specialization and involves a lot of practice.
Essential Step Four
Reflect upon the problem
This one is very critical step and many ignore this while finishing the problem. You should be careful about what you have written. Nobody is perfect, you know it. But still you don't reflect upon your work and then at the problem, you are not going to be great problem solver. This step involves care and it should be adopted as a habit. Because all is well, when end is well. You should look over the solution you arrived at just. Again look at the problem you had.
- Does it look probable?
- Did you answer the question exactly? And are you sure of the answer? If yes , then how much?
- Did you answer in the language of the problem?
- Did you derive the answer in the specified units?
I hope this helps the readers-in-need. If you want to explore the better inside you, explore the following ten external articles:
- Career Advice By Terrence Tao
- John Baez’s page on career advice.
- Fan Chung’s advice for graduate students.
- Lance Fortnow’s “Graduate Student Guide.
- Oded Goldreich’s “On our duties as scientists“.
- Gian-Carlo Rota’s “Ten lessons I wish I had been taught”.
- J. Michael Steele’s “Advice for Graduate Students in Statistics.”
- Ian Stewart’s “Letters to a Young Mathematician“.
- Ravi Vakil’s “For potential students“.
- The Princeton Companion to Mathematics‘ section on advice to younger mathematicians