The Mathematical Perspective
Greeting a Problem
- Learn to welcome a problem. Give it room to roam in your head.
- Make a careful formulation of the problem. It may have arrived improperly attired.
- Identify the assumptions: the ones it brought, the ones you add.
- Decide what type of answer it has requested:
- A number - how accurate?
- A function, vector, matrix, or other mathematical object?
- An algorithm - how efficient?
- A proof - in math, arguments must be conclusive, not merely persuasive.
Solving a Problem
- Learn to recognize common problem types. Does your problem have relatives?
- Reformulate it in a different setting. Everyone likes a change of scenery.
- Ask more questions. Make conversation.
- Get up close. Narrow your perspective or simplify the problem.
- Step back. Widen your perspective or generalize the problem.
- Divide the problem into several shorter ones.
- Experiment - don't be afraid to be wrong. Most problems are playful.
- Take a break. Most problems are patient.
- Come back. It won't solve itself.
Presenting a Solution
- Identify your audience.
- Organize the steps and make that organization obvious.
- For longer solutions:
- Introduce the problem. Be polite.
- Provide context. Not everyone has met your problem.
- Indicate the end of the solution. Bid farewell.
fall 2007 course description
paper and talk guidelines
summary of proof techniques
truth table forms
back to logic & proof
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