G P'olya. Mathematical discovery. John Wiley & Sons, Inc, 1965. Two volumes. 104  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1: A circle system Concerning a circle system, the following result is known. Theorem 1. If some three circles in a circle system are unit circles, then the remaining circle is also a unit circle. This theorem was first discovered by Roger Johnson in 1916, see [6] A proof is given in Polya [5] (Chapter 10) to show how a useful idea occurs to us in a process of problem solving. See also Davis and Hersh [3] Chapter 6. A sphere system consisting of all unit spheres is called a unit spheresystem. The circle system in Figure 1 is a unit sphere system (unit circlesystem) Theorem 1 ....

G. Polya, Mathematical Discovery, New York (1981) Wiley.

....to come try to disclose the general pattern that lies behind the present concrete situation. 9. Do not give away your whole secret at once let the students guess before you tell it let them find out for themselves as much as feasible. 10. Suggest it; do not force it down their throats. (Polya 1962) Dr. Charles Vonder Embse, Central Michigan University, Mt Pleasant, Michigan, USA states that Technology is the segue to change in the mathematics classroom . These changes might involve the adoption of some new action verbs for mathematics. The Verbs of Mathematics: M use A nalyze T hink H ....

Polya, G. (1962). Mathematical discovery. New Jersey: Princeton University Press.

....Section 3.8 discusses the tools that transform the raw computing power of the hardware into a convenient tool for the user. The material in this section addresses first the concepts of problem and problem solving following the mathematical methodology for problem solving as introduced by Polya [73, 74]. Other readings on this topics can be found in [72, 90, 103] The methodology for problem solving using a computer is then presented as a sequence of transformations undergone by a problem model to the solution algorithm, and further to an executable program. The system software tools that are ....

G. Polya. Mathematical Discovery, volume 1. John Wiley and Sons, Inc., 1962.

....the conjecture to fail. We obtain the new conjecture (IMPLIES (NOT (OCCUR Y X) FUTURE RESEARCH 94 (EQUAL (SUBST X Y (SUBST X Y Z) SUBST X Y Z) which can be proved correct. The sequence of hypothesizing conjectures, modifying and verifying them is an essential part of theory development. [31, 38, 39] Constructing correct conjectures is difficult in general. It is largely because a human s knowledge that is related to the problem may be incorrect or incomplete and he might often miss the special but critical cases in building a conjecture. Using examples for checking and correcting conjectures ....

Polya, G. Mathematical Discovery. Wiley, New York, 1962. Vol. 1 and 2.

.... a proof is not precisely defined; it is a result of imitation, i.e. of generations passing a culture of methods on to the next, and correction (making this culture change) For the fundamental methods of proof in mathematics, more details and interesting expositions on the subject see e.g. Goo51, Pol65, Kol76, EC89] 64 This approach has received a great deal of attention mainly after the work of Hoare[Hoa69] which presented proof rules in a syntax directed manner. The origins of the approach can be traced back to Turing[Tur49] and the main influence on Hoare s work is attributed to ....

G. Polya. Mathematical discovery. John Wiley & Sons, 1965. Two volumes.

....it is inadequate to explain many common observations about mathematical proofs. These observations are so common that they have, as far as we are aware, escaped the attention of psychologists, although they have been noticed by mathematicians reflecting on the processes of theorem proving, e.g. Polya 65] We list them below and hope that other mathematicians will immediately recognise their validity. ffl Mathematicians distinguish between understanding each step of a proof and understanding the whole proof. It is possible to understand at one level without understanding at the other either ....

....was implemented as the limit heuristic and used to guide proofs of such theorems. ffl [Wos McCune 88] describes the common structure in attempts to find fixed points combinators. This common structure was implemented as the kernel method and used to guide the search for such fixed points. ffl [Polya 65] describes the common structure in ruler and compass constructions. This common structure was implemented by [Funt 73] and used to guide the search for such constructions. 7 One approach to capturing such common structure is to build a derived rule of inference out of a sequence of rules of ....

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G. Polya. Mathematical discovery. John Wiley & Sons, Inc, 1965. Two volumes.

....with the second major problem that of flattening the highly convoluted mesh that conforms to the original surface once it has been converted into a topologically defect free piecewise flat surface in 3D space. There is no way to flatten a curved 3D surface without metric and areal distortion [11]; however, there is one type of geometric information that, at least theoretically, can be preserved under flattening, namely conformal (angular) information encoded in the surface. Before reviewing the pertinent mathematics and describing our method for generating quasi conformal flat mappings, ....

Polya, G.: Mathematical Discovery, vol. 2. John Wiley & Sons, New York, 1968

....with the second major problem that of attening the highly convoluted mesh that conforms to the original surface once it has been converted into a topologically defect free piecewise at surface in 3D space. There is no way to atten a curved 3D surface without metric and areal distortion [11]; however, there is one type of geometric information that, at least theoretically, can be preserved under attening, namely conformal (angular) information encoded in the surface. Our starting point is a piecewise at triangulated surface in 3D space that is topologically a 2D disk. The 3D space ....

Polya, G.: Mathematical Discovery, vol. 2. John Wiley & Sons, New York, 1968

....is learned by making connections with physical objects in the physical world. By reflecting upon their actions, they learned to generalize from the specifics of one problem to other problems. Mrs. Clark helped students build knowledge upon what they already knew and what made sense to them. Polya (1981) quoted Lichtenberg in saying, What you have been obligated to discover by yourself leaves a path in your mind which you can use again when the need arises (p. 103) Mathematics textbooks should not contain page after page of procedural computations but should contain significant problems ....

Polya, G. (1981). Mathematical discovery. New York: John Wiley & Sons.

....series. In mapping the standard form method between domains, we merely needed to change the rules used. As I ll show in the next section, the mapping between domains can be more complicated than this. 5 Mapping between Analogical Domains Analogy plays an important role in mathematical creativity [Pol65]. Mapping methods onto analogical domains therefore seems an interesting exercise. Unlike much analogical reasoning [Gen89, Kli71] this involves the mapping not of objectlevel terms but of meta level methods. Consider, for example, the telescope method for summing series. In this method, one term ....

G. Polya. Mathematical discovery. John Wiley & Sons, Inc, 1965. Two volumes.

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G P'olya. Mathematical discovery. John Wiley & Sons, Inc, 1965. Two volumes. 104

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G. Polya, Mathematical Discovery, John Wiley and Sons, New York, 1962.

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G. Polya (Combined Edition, 1981) Mathematical Discovery, John Wiley & Sons.

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