Learning how to solve problems in mathematics is simply to know what to look for.
Math problems often require established procedures and one must know What & When to apply them. To identify procedures, you have to be familiar with the different problem situations, and be able to collect the appropriate information, identify a strategy or strategies and use the strategy/strategies appropriately. But exercise is must for problem solving. It needs practice!! The more you practice, the better you get. The great mathematical wizard G Polya wrote a book titled How to Solve It in 1957. Many of the ideas that worked then, do still continue to work for us. Given below are the four essential steps of problem solving based on the central ideas of Polya.
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 so, what is similar about it? What did you do then?
If 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 here?
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 iff 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.
Essential Step Four:
Reflect upon the problem
This is very critical step and many students leave this while solving the problems.
• 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?
At last, applying these steps, also need practice and hard work. If you have any other techniques of problem solving then drop those into the comment box . This will help other readers.
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- 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.”
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- Ravi Vakil’s “For potential students“.
- The Princeton Companion to Mathematics‘ section on advice to younger mathematicians