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Pólya lays a big emphasis on the teachers' behavior. A teacher should support students with devising their own plan with a question method that goes from the most general questions to more particular questions, with the goal that the last step to having a plan is made by the student.
Polya begins Volume I with a discussion on induction, not mathematical induction, but as a way of guessing new results.He shows how the chance observations of a few results of the form 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, etc., may prompt a sharp mind to formulate the conjecture that every even number greater than 4 can be represented as the sum of two odd prime numbers.
We were interested in the same kind of questions, in the same topics; but one of us knew more about one topic and the other more about some other topic. It was a fine collaboration. The book Aufgaben und Lehrsatze aus der Analysis , the result of our cooperation, is my best work and also the best work of Gábor Szegő.
Polya’s intention is to teach students the art of guessing new results in mathematics for which he marshals such notions as induction and analogy as possible sources for plausible reasoning. The first volume of the book is devoted to an extensive discussion of these ideas with several examples drawn from various field of mathematics.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
The Polya enumeration theorem translates the recursive structure of rooted ternary trees into a functional equation for the generating function F(t) of rooted ternary trees by number of nodes. This is achieved by "coloring" the three children with rooted ternary trees, weighted by node number, so that the color generating function is given by f ...
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George Pólya (/ ˈ p oʊ l j ə /; Hungarian: Pólya György, pronounced [ˈpoːjɒ ˈɟørɟ]; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician.He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University.