Important Activities to Complete After Successfully Solving a Problem

After successfully solving a problem, there is still much you can learn form your problem to become better at problem solving, and to also discover other mathematical truths for similar new problems by doing the following:

Review your method of solution.

1.  Check each step to your method of solution afterwards to verify that each step is correct.

2.  See the solution in parts and as a whole to organize the ideas you used in your mind.

3.  Try to see the solution to your problem intuitively in order to strengthen your intuition for improving your defeasible reasoning skills.  This then helps when attempting to solve future problems that have similarities to the one just solved.

Search for applications of the solved problem.

1.  Look for possible implications and special cases of the solution to the problem.

2.  Consider what application the solution to the problem has.

3.  Consider how different specifications of the problem could be applied.

Consider how the solution of your problem can improve your problem-solving skills elsewhere.

1.  Consider how your method of solution could be applied to solve other problems.

2.  Put any learned techniques, methods, or trick in your “toolbox” list of tricks.

Vary the problem you solved, and vary the strategy you used to solve the problem.  

Try to solve those varied problems and try following other plans to solve the original problem.  Do this because your method of solution likely has elements of it that would be key elements to solving similar problems, and you could discover useful mathematical principles.  Problems come in clusters of similarity that are usually solved quite similarly. [7]

1.  Try solving your solved problem more than one way to learn new methods and tricks that may help solve similar problems.  You may discover new solutions to similar useful problems.

2.  Consider possibly generalizing your solved problem and if possible, try finding its solution.

3.  Vary the problem to another similar problem that you may be able to use the same method of solution to.

4.  Look for a closely related problem in analogy, specification, or generalization.  Try the simpler related problems first.

Look back at all you did during the problem-solving process to analyze what worked and what did not work and why.  Do this in order to recognize where you wasted time, what you did to be less efficient, and how you can be more efficient and effective the next time you solve problems.