A great discovery solves a great problem but there is a
grain of discovery in the solution of any problem. Your problem may be
modest; but if it chanllenges your curiosity and brings into play your
inventive facilities, and if you solve it by your own means, you may experience
the tension and enjoy the triumph of discovery.
Step 1 Understand the problem
---George Polya, How to solve it.
In trying the problems in tutorial sheet, some students may encounter
some difficulties such as `I don't understand the questions' or
`I understand the question, but I don't know what or how to do?'
or `I have done/proved it this way, but how do I know whether it
is correct or wrong?' or `OK, I am sure I have done the problem correctly,
but this problem seems to have another way of doing it.' We hope that the
following problem-solving process will be helpful to you when you are doing
problems in mathematics. This process is natural to many of us who have
done much in mathematics. Usually, you hear people saying, `I know how
to do this, by experience'. This is quite true.
Problems in mathematics tutorial are selected with some aims: some are
routine questions for familiarization with the skills, some are conceptual
problems, some are open-ended questions
Do you know the meaning of terms or results appear in the problem?
What are you trying to find or do?
What information do you obtain from the problem?
What are the unknowns?
What information, if any, is missing or not needed?
Step 2 Devise a Plan
Some useful strategies are
Look for a pattern or patterns.
Examine a simpler or special case of the problem to gain insight into the
solution of the original problem.
Examine some related problems and determine if the same technique can be
Make a table (if possible).
Draw a diagram (if possible).
Form an equation when appropriate.
Guess and check.
Work backward. (However, be very careful as sometimes it may not work.)
Identify a subgoal.
Recall any results or theorems that seems useful.
Step 3 Carry Out the plan
Implement the strategy/strategies in Step 2 and perform any necessary actions
Check each step of the plan as you proceed. This may be intuitive checking
or a formal proof of each step.
Keep an accurate record of your work. (even if it is a failure step, it
enables you to see what the underlying difficulties or hurdles are. You
may add some comment on why it fails.)
Step 4 Look Back
Check the results in the original problem.
Interpret the solution in terms of the original problem. Does your
answer or solution or proof make sense? Is it reasonable?
Determine whether there is another method of finding the solution.
If possible, determine other related or more general problems for which
the techniques will work.
Explain your work to someone. This may help you to discover any unclear
parts or parts that need further explanation/justifiication.