Strategies in investigating problems

1. Strategies in
Investigating Problems

2. Introduction
• Solving a problem is not unlike
climbing a mountain.
• A good math problem, one that is
interesting and worth solving, will not
solve itself.
• For beginners especially, strategy is
very important.
• Begin with psychological strategies
that apply to almost all problems.
• The solution to every problem
involves two parts: the investigation
and argument.

3. Psychologial Strategies
A. Mental
Toughness

4. • The moral of the story, is that good
problem solver doesn't give up.
• Most beginners give up to soon,
because they lack the mental
toughness attributes of confidence
and concentration.
• Math problems are easier to deal
with.
• Start with easy problems, to warm up,
but then work on harder and harder

5. Example

6. Solution

9. B. Creativity
• Most mathematicians are "Platonist"
• The good problem solver, then, is highly open
and receptive to ideas.
• The elusive receptiveness to new ideas is
what we call creativity.

10. Example 1

11. Solution

12. • Learn to shamelessly appropriate new
ideas and make them your own.
• One way to heighten your
receptiveness to new ideas is to stay
"loose", to cultivate a sort of mental
peripheral vision.
• You need to relax your vision and get
ideas from the periphery.

13. Example

14. Solution

15. Theme
• Dont let self-imposed,
unecessary restrictions limit
your thinking.
• Break or at least bend a few
rules.
• Toughen up, loosen up, and

16. Strategies for Getting Started

20. Example

24. Methods of Arguments
• Arguments should be rigorous
and clear
• The more complicated the
argument the harder it is to
decide if it is logically correct.
• Three distinct styles of argument:
straightforward deduction,
argument by contradiction, and
mathematical induction.

35. Example

41. Other Important Strategies

43. Strategies for Investigating Problems
1. Psychological Strategies
A. Mental Toughness
B. Creativity
2. Strategies for Getting Started
A. The first step: Orientation
B. I'm oriented. Now what?

44. 3. Methods of Arguments
A. Common Abbreviations and
Stylic Conventions.
B. Deduction and Symbolic Logic
C. Argumented by Contradiction
D. Mathematical Induction
4. Other Importatant Strategies
A. Draw a Picture!
B. Pictures dont help? Recast the
problem in other
ways!
C. Change your point of view.