| 数学专业英语－Polya’s Craft of Discovery
George Polya has a scientific career extending more than seven decades. Abrilliant mathematician who has made fundamental contributions in many fields. Polya has also been a brilliant teacher, a teacher’s teacher and an expositor. Polya believes that there is a craft of discovery. He believes that the ability to discover and the ability to invent can be enchanced by skillful teaching which alerts the student to the principles of discovery and which gives him an opportunity to practise these principles.
In a series of remarkable books of great richness, the first of which was published in 1945. Polya has crystallized these principles of discovery and invention out of his vast experience, and has shared them with us both in precept and in example.These books are a treasure-trove of strategy, know-how, rules of thumb, good advice, anecdote, mathematical history, together with problem after problem at all levels and all of unusual mathematical interest. Polya places a global plan for “How to Solve It” in the endpapers of his book of that name:
HOW TO SOLVE IT
First: You have to understand the problem.
Second: Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
Third: Carry out your plan.
Fourth: Examine the solution obtained.
These precepts are then broken down to “molecular” level on the opposite endpaper. There, individual strategies are suggested which might be called into play at appropriate momentsm, such as:
If you cannot solve the proposed problem, look around for an appropriate related problem.
Work backwards
Work forwards
Narrow the condition
Widen the condition
Seek a counter example
Guess and test
Divide and conquer
Change the conceptual mode
Each of these heuristic principles is amplified by numerous appropriate examples.
Subsequent investigators have carried Polya’s ideas forward in a number of ways. A.H.Schoenfeld has made an interesting tabulation of the most frequently used heuristic principles in college-level mathematics. We have appended it here.
Frequently Used Heuristics
Analysis
1) Draw a diagram if at all possible
2) Examine special cases:
a) Choose special values to exemplify the problem and get a “feel” for it.
b) Examine limiting cases to explore the range of possibilities
c) Set any integer parameters equal to 1,2,3,…,in sequence, and look for an inductive pattern.
3) Try to simplify the problem by
a) exploiting symmetry, or
b) “Without Loss of Generality” arguments (including scaling)
Exploration
1) Consider essentially equivalent problems:
a) Replacing conditions by equivalent ones.
b) Re-combining the elements of the problem in different ways.
c) Introduce auxiliary elements.
d) Re-formulate the problem by
I) change of perspective or notation
II) considering argument by contradiction or contrapositive
III) assuming you have a solution , and determining its properties
2) Consider slightly modified problems:
a) Choose subgoals (obtain partial fulfillment of the conditions)
b) Relax a condition and then try to re-impose it .
c) Decompose the domain of the problem and work on it case by case .
3) Consider broadly modified problems:
a) Construct an analogous problem with fewer variables .
b) Hold all but one variable fixed to determine that variable’s impact .
c) Try to exploit any related problems which have similar
I) form
II) “givens”
III) conclusions
Remember: when dealing with easier related problems , you should try to exploit both the RESULT and the METHOD OF SOLUTION on the given problem .
Verifying your solution
1) Does your solution pass these specific tests:
a) Does it use all the pertinent data?
b) Does it conform to reasonable estimates or predictions?
c) Does it withstand tests of symmetry, dimension analysis , or scaling?
2) Does it pass these general tests?
a) Can it be obtained differently?
b) Can it be sudstantiated by special cases?
c) Can it be reduced to known results?
d) Can it be used to generate something you know?
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