george polya father of problem solving

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George pólya.

... diligence and good behaviour.

I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.

I was greatly influenced by Fejér , as were all Hungarian mathematicians of my generation, and, in fact, once or twice in small matters I collaborated with Fejér . In one or two papers of his I have remarks and he made remarks in one or two papers of mine, but it was not really a deep influence.

On Christmas 1913 I travelled by train from Zürich to Frankfurt and at that time I had a verbal exchange - about my basket that had fallen down - with a young man who sat across from me in the train compartment. I was in an overexcited state of mind and I provoked him. When he did not respond to my provocation, I boxed his ear. Later on it turned out that the young man was the son of a certain Geheimrat; he was a student, of all things, in Göttingen. After some misunderstandings I was told to leave by the Senate of the University.

I was... deeply influenced by Hurwitz . In fact I went to Zürich in order to be near Hurwitz and we were in close touch for about six years, from my arrival in Zürich in 1914 to his passing in ... 1919 . And we have one joint paper, but that is not the whole extent. I was very much impressed by him and edited his works. I was also impressed by his manuscripts.

I came very late to mathematics. ... as I came to mathematics and learned something of it, I thought: Well it is so, I see, the proof seems to be conclusive, but how can people find such results? My difficulty in understanding mathematics: How was it discovered?

... a mathematical masterpiece that assured their reputations.

Pólya was arguably the most influential mathematician of the 20 th century. His basic research contributions span complex analysis, mathematical physics, probability theory , geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career.

For mathematics education and the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Pólya.

The aim of heuristic is to study the methods and rules of discovery and invention .... Heuristic, as an adjective, means 'serving to discover'. ... its purpose is to discover the solution of the present problem. ... What is good education? Systematically giving opportunity to the student to discover things by himself.

If you can't solve a problem, then there is an easier problem you can solve: find it.

Mathematics in the primary schools has a good and narrow aim and that is pretty clear in the primary schools. ... However, we have a higher aim. We wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good school of thinking. But what is thinking? The thinking that you can learn in mathematics is, for instance, to handle abstractions. Mathematics is about numbers. Numbers are an abstraction. When we solve a practical problem, then from this practical problem we must first make an abstract problem. ... But I think there is one point which is even more important. Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems.

Teaching is not a science; it is an art. If teaching were a science there would be a best way of teaching and everyone would have to teach like that. Since teaching is not a science, there is great latitude and much possibility for personal differences. ... let me tell you what my idea of teaching is. Perhaps the first point, which is widely accepted, is that teaching must be active, or rather active learning. ... the main point in mathematics teaching is to develop the tactics of problem solving.

... a remarkable theorem in a remarkable paper, and a landmark in the history of combinatorial analysis.

The whole work displays the taste of the authors for the concrete and explicit result, for elegance and ingenious methods.

With no hesitation, George Pólya is my personal hero as a mathematician. ... [ he ] is not only a distinguished gentleman but a most kind and gentle man: his ebullient enthusiasm, the twinkle in his eye, his tremendous curiosity, his generosity with his time, his spry energetic walk, his warm genuine friendliness, his welcoming visitors into his home and showing them his pictures of great mathematicians he has known - these are all components of his happy personality. As a mathematician, his depth, speed, brilliance, versatility, power and universality are all inspiring. Would that there were a way of teaching and learning these traits.

References ( show )

• G L Alexanderson, The Polya picture album ( Basel, 1987) .
• G L Alexanderson, The random walks of George Pólya ( Washington, DC, 2000) .
• H Taylor and L Taylor, George Pólya : Master of Discovery ( Palo Alto, CA, 1993) .
• D J Albers and G L Alexanderson ( eds. ) , Mathematical People: Profiles and Interviews ( Boston, 1985) , 245 - 254 .
• G L Alexanderson and L H Lange, Obituary: George Pólya, Bull. London Math. Soc. 19 (6) (1987) , 559 - 608 .
• G L Alexanderson and J Pedersen, George Pólya : his life and work ( Hungarian ) , Mat. Lapok 33 (4) (1982 / 86) , 225 - 233 .
• R P Boas, Selected topics from Pólya's work in complex analysis, Math. Mag. 60 (5) (1987) , 271 - 274 .
• R P Boas, Pólya's work in analysis, Bull. London Math. Soc. 19 (6) (1987) , 576 - 583 .
• H Cartan, La vie et l'oeuvre de George Pólya, C. R. Acad. Sci. Sér. Gén. Vie Sci. 3 (6) (1986) , 619 - 620 .
• K L Chung, Pólya's work in probability, Bull. London Math. Soc. 19 (6) (1987) , 570 - 576 .
• F Harary, Homage to George Pólya, J. Graph. Theory 1 (4) (1977) , 289 - 290 .
• P Hilton and J Pedersen, The Euler characteristic and Pólya's dream, Amer. Math. Monthly 103 (2) (1996) , 121 - 131 .
• J-P Kahane, The grand figure of George Pólya ( Czech ) , Pokroky Mat. Fyz. Astronom. 35 (4) (1990) , 177 - 191 .
• J Kilpatrick, George Pólya's influence on mathematics education, Math. Mag. 60 (5) (1987) , 299 - 300 .
• D H Lehmer, Comments on number theory, Bull. London Math. Soc. 19 (6) (1987) , 584 - 585 .
• A Pfluger, George Pólya, J. Graph Theory 1 (4) (1977) , 291 - 294 .
• R C Read, Pólya's theorem and its progeny, Math. Mag. 60 (5) (1987) , 275 - 282 .
• R C Read, Pólya's enumeration theorem, Bull. London Math. Soc. 19 (6) (1987) , 588 - 590 .
• P C Rosenbloom, Studying under Pólya and Szegö at Stanford, in A century of mathematics in America II ( Providence, RI, 1989) , 279 - 281 .
• D Schattschneider, The Pólya-Escher connection, Math. Mag. 60 (5) (1987) , 293 - 298 .
• D Schattschneider, Pólya's geometry, Bull. London Math. Soc. 19 (6) (1987) , 585 - 588 .
• M M Schiffer, George Pólya (1887 - 1985) , Math. Mag. 60 (5) (1987) , 268 - 270 .
• M M Schiffer, Pólya's contributions in mathematical physics, Bull. London Math. Soc. 19 (6) (1987) , 591 - 594 .
• A H Schoenfeld, Pólya, problem solving, and education, Math. Mag. 60 (5) (1987) , 283 - 291 .
• A H Schoenfeld, George Pólya and mathematicvs education, Bull. London Math. Soc. 19 (6) (1987) , 594 - 596 .
• Y S Tseng, On Pólya's Mathematical discovery ( Chinese ) , J. Math. Res. Exposition 3 (1) (1983) , 213 - 216 .
• Y S Tseng, Correction: 'On Pólya's Mathematical discovery' ( Chinese ) , J. Math. Res. Exposition 3 (2) (1983) , 22 .
• A A Wieschenberg, A conversation with George Pólya, Math. Mag. 60 (5) (1987) , 265 - 268 .
• I M Yaglom, George Pólya ( on the 100 th anniversary of his birth ) ( Russian ) , Mat. v Shkole (3) (1988) , 67 - 70 .

Additional Resources ( show )

Other pages about George Pólya:

• Pólya on Fejér
• Pólya and Szegö's Problems and Theorems in Analysis
• Hardy's reference for Pólya at ETH
• Some of Pólya's favourite quotes
• Preface to Pólya's How to solve it
• Heinz Klaus Strick biography

Other websites about George Pólya:

• Australia Mathematics Trust
• Mathematical Genealogy Project
• MathSciNet Author profile
• zbMATH entry

Honours ( show )

Honours awarded to George Pólya

• LMS Honorary Member 1956
• Popular biographies list Number 44

Cross-references ( show )

• Societies: Canadian Mathematical Society
• Societies: Society for Industrial and Applied Mathematics
• Societies: Zurich Scientific Research Society
• Other: 1936 ICM - Oslo
• Other: 2009 Most popular biographies
• Other: Earliest Known Uses of Some of the Words of Mathematics (C)
• Other: Earliest Known Uses of Some of the Words of Mathematics (E)
• Other: Earliest Known Uses of Some of the Words of Mathematics (M)
• Other: Earliest Known Uses of Some of the Words of Mathematics (P)
• Other: Earliest Uses of Symbols of Number Theory
• Other: London Learned Societies
• Other: Most popular biographies – 2024
• Other: Popular biographies 2018

George Pólya & problem solving ... An appreciation

• General / Article
• Published: 06 May 2014
• Volume 19 , pages 310–322, ( 2014 )

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• Shailesh A. Shirali 1  

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George Pólya belonged to a very rare breed: he was a front-rank mathematician who maintained an extremely deep interest in mathematics education all through his life and contributed significantly to that field. Over a period of several decades he returned over and over again to the question of how the culture of problem solving could be nurtured among students, and how mathematics could be experienced ‘live’. He wrote many books now regarded as masterpieces: Problems and Theorems in Analysis (with Gábor Szegö), How to Solve It, Mathematical Discovery , among others. This article is a tribute to Pólya and a celebration of his work.

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Historical Events in the Background of Hilbert’s Seventh Paris Problem

Arnol’d’s 1974 Problems

Commentary on Part II of Mathematical Challenges For All: Making Mathematics Difficult? What Could Make a Mathematical Challenge Challenging?

Suggested reading.

http://www-history.mcs.st-and.ac.uk/Biographies/Polya.html

T Gowers, The Two Cultures of Mathematics , https://www.dpmms.cam.ac.uk/~wtg102cultures.pdf

http://en.wikipedia.org/wiki/George_Polya

T Passmore, Polya’s legacy: fully forgotten or getting a new perspective in theory and practice , http://eprints.usq.edu.au/3625/1/Passmore.pdf

G Pólya, Mathematics and Plausible Reasoning , Princetron University Press, Vols 1&2, 1954.

G Pólya, Mathematical Discovery , Vols 1&2, 1965.

G Pólya, How To Solve It , Princeton University Press, 1973.

Google Scholar  

G Pólya, Teaching us a Lesson (MAA), http://vimeo.com/48768091 (video recording of an actual lecture by Polya).

http://www.math.utah.edu/~pa/math/polya.html

Geoffrey Howson, Review of Mathematical Discovery, The Mathematical Gazette , Vol. 66, No. 436, pp.162–163, June 1982.

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Shailesh Shirali is Director of Sahyadri School (KFI), Pune, and also Head of the Community Mathematics Centre in Rishi Valley School (AP). He has been in the field of mathematics education for three decades, and has been closely involved with the Math Olympiad movement in India. He is the author of many mathematics books addressed to high school students, and serves as an editor for Resonance and for At Right Angles . He is engaged in many outreach projects in teacher education.

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Shirali, S.A. George Pólya & problem solving ... An appreciation. Reson 19 , 310–322 (2014). https://doi.org/10.1007/s12045-014-0037-7

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George Polya of Stanford, 97; Mathematician and Educator

• Sept. 9, 1985

George Polya, a prominent figure in the world of mathematics, an educator and an author, died Saturday after a long illness. He was 97 years old.

Dr. Polya, a retired Stanford professor, was known to the public for his book, ''How To Solve It,'' which sold one million copies, and for his efforts in the wake of Sputnik in 1957 to teach math teachers how to teach math.

He was regarded as the father of the modern emphasis in math education on problem solving.

A leading research mathematician of his time, Dr. Polya made seminal contributions to probability, combinatorial theory and conflict analysis. His work on random walk and his famous enumeration theorem have been widely applied.

Born in Budapest in 1887, he first studied law, then literature and science, and thought, ''I am not good enough for physics and too good for philosophy; mathematics is in between.''

After getting his doctorate in 1912 from the University of Budapest, he taught in Switzerland until 1940 when he and his Swiss wife, Stella, emigrated to the United States.

Dr. Polya wrote 200 research papers and 11 books and monographs. His most important work, ''Problems and Theorems in Analysis,'' written along with the Hungarian mathematician Gabor Szego, is a classic still in wide use.

George Pólya

1 george pólya.

American mathematician, Born: György Pólya in Budapest, Hungary in 1887, ( d. 1985 in Palo Alto, USA)

“ His first job was to tutor the young son, Gregor, of a Hungarian baron. Gregor struggled due to his lack of problem solving skills. ” Thus, according to Long ( [ 1 ] ), Polya insisted that the skill of “ solving problems was not an inborn quality but, something that could be taught ”.

In 1940, George Polya and his wife, Stella, (the only daughter of Swiss Dr. Weber, in Zurich) moved to the United States because of their justified fear of Nazism in Germany ( [ 1 ] ).

Understand the Problem

Devise a Plan on how to approach the Problem; such a plan may include one or several of the following:

Make a first guess to begin with, and then verify the answer

Solve a simpler problem

Consider special cases that are much easier to solve

Look for a pattern

Draw a picture

Use a model

Use direct reasoning but double-check your results

Eliminate possibilities

Carry out the Plan, as modified by partial solutions

If plan doesn’t work, make an improved plan but do not give up

Last-but-not-least, look back and examine critically your solution(s):

Does the solution make sense? Does it check out in particular cases?

Make sure there are no gaps and no steps missing.

He published also a two-volume book, “ Mathematics and Plausible Reasoning ” in 1954, and Mathematical Discovery in 1962.

• 1 Long, C. T., & DeTemple, D. W., Mathematical reasoning for elementary teachers . (1996). Reading MA: Addison-Wesley
• 2 Reimer, L., & Reimer, W. Mathematicians are people too . (Volume 2). (1995) Dale Seymour Publications
• 3 Polya, G. How to solve it . (1957) Garden City, NY: Doubleday and Co., Inc.
• 4 A. Motter,, http://www.math.wichita.edu/history/men/polya.html “A Biography of George Polya”

Beginning Algebra Tutorial 15

• Use Polya's four step process to solve word problems involving numbers, rectangles, supplementary angles, and complementary angles.

Whether you like it or not, whether you are going to be a mother, father, teacher, computer programmer, scientist, researcher, business owner, coach, mathematician, manager, doctor, lawyer, banker (the list can go on and on).  Some people think that you either can do it or you can't.  Contrary to that belief, it can be a learned trade.  Even the best athletes and musicians had some coaching along the way and lots of practice.  That's what it also takes to be good at problem solving.

George Polya , known as the father of modern problem solving, did extensive studies and wrote numerous mathematical papers and three books about problem solving.  I'm going to show you his method of problem solving to help step you through these problems.

If you follow these steps, it will help you become more successful in the world of problem solving.

Polya created his famous four-step process for problem solving, which is used all over to aid people in problem solving:

Step 1: Understand the problem.  

Step 2:   Devise a plan (translate).  

Step 3:   Carry out the plan (solve).  

Step 4:   Look back (check and interpret).  

Just read and translate it left to right to set up your equation .

Since we are looking for a number, we will let 

x = a number

*Get all the x terms on one side

*Inv. of sub. 2 is add 2  

FINAL ANSWER: 

We are looking for two numbers, and since we can write the one number in terms of another number, we will let

x = another number 

one number is 3 less than another number:

x - 3 = one number

*Inv. of sub 3 is add 3

*Inv. of mult. 2 is div. 2  

Another number is 87.

Perimeter of a rectangle = 2(length) + 2(width)

We are looking for the length and width of the rectangle.  Since length can be written in terms of width, we will let

length is 1 inch more than 3 times the width:

1 + 3 w = length

*Inv. of add. 2 is sub. 2

*Inv. of mult. by 8 is div. by 8  

FINAL ANSWER:

Length is 10 inches.

Complimentary angles sum up to be 90 degrees.

We are already given in the figure that

x = 1 angle

5 x = other angle

*Inv. of mult. by 6 is div. by 6

The two angles are 30 degrees and 150 degrees.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem .  At the link you will find the answer as well as any steps that went into finding that answer.

  Practice Problems 1a - 1c: Solve the word problem.  

(answer/discussion to 1c)

http://www.purplemath.com/modules/ageprobs.htm This webpage goes through examples of age problems, which are like the numeric problems found on this page.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

Last revised on July 26, 2011 by Kim Seward. All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.

Guide to the George Pólya papers SC0337

Information about Access

Immediate Source of Acquisition note

Biographical/Historical Sketch

Description of the Collection

Ownership & Copyright

Subjects and Indexing Terms

Papers Accession ARCH-1986-036

Draft: Chapter One: Polanyi Biography Summer 1979

Manuscript: Problems and Solutions

Course Outlines: Math 110, Math 129: How to Solve the Problem

Draft: How To Solve It

Early Manuscript of Mathematical Discovery . The Aims; Part I: Patterns; Chapter 1: Patterns of 2 Loci; Chapter 2: The Cartesian Pattern; Chapter 3: Recursion; Chapter 6: Widening the Scope; Solutions

Pages 2-4, 7, 11 of Typed Manuscript, Essay.

Part 2 of Manuscript. Patterns of Plausible Inference; Chapter 12: Some Conspicuous Patterns.

Manuscript. Preface, Hints to the Reader, Bibliography, Contents, Outline of Chapters.

Manuscript. Chapter 1. Induction and Analogy.

Manuscript. Chapter 2: Generalization, Specialization, Analogy.

Manuscript. Chapter 3. Induction in Solid Geometry.

Manuscript. Chapter 4. Induction in Theory of Numbers.

Manuscript. Chapter 5. Misc. Examples of Induction.

Manuscript. Chapter 6. A More General Statement.

Manuscript. Chapter 7. Mathematical Induction.

Manuscript. Chapter 8. Minima and Maxima.

Manuscript. Chapter 9. Physical Mathematics.

Manuscript. Chapter 10. The Isoperimetric Problem.

Manuscript. Chapter 11. Kinds of Plausible Reasons.

Manuscript. Chapter 12. Further Patterns and First Links.

Manuscript. Chapter 14. Chance, the Ever-present Rival Conjecture.

Manuscript. Chapter 15. The Calculus of Probability and the Logic of Plausible Reasoning.

Manuscript. Chapter 16. Plausible Reasoning in Invention and Instruction.

Mathematical Discovery Manuscript; Supplement to Bibliography; Miscellaneous Sections (Typed); Contents: Chapters 7-15; Chapter 7; Chapter 8; Chapter 9; 3 Pages of Chapter 10. Chapter 11; Chapter 12 (4th Copy).

Mathematical Discovery Manuscript. Chapter 13; Pamphlet "On Learning, Teaching, and Learning Teaching"; Chapter 14; Chapter 15; Chapter 1. Mathematical Language; Chapter 2. Beginnings of Integral Calculus.

Third Essay: Some Notions of Logic; I: The Notions and the Notations; III: Logical Links between the Proposed Problem and Its Auxiliary Problems.

Mathematical Discovery in the Classroom.

Course Descriptions for Math 110, Physical Sciences 115; Course Notes for Math 129; Part 1: Some Problems on Maxima and Minia in Elementary Geometry.

Handwritten Notes: A Simple Method of Approximation Applicable to Some Problems.

Crystallography.

Equations (1 page)

Pages 116-193 of Typed Manuscript.

Designons par le Perimetre et par v le Rayon Conforme Exterieur d'Une Courbe. La Premiere des Deux Inequalites (2 pages)

An Isoperimetric Inequality for Convex Curves (3 pages). Analyse Mathematique (6 pages.

Term Paper Assignments.

Aufgaben Und Lehrsatze Aus der Analysis. 3 Excerpts. Pages 123-131, 156-159, 378-379. Techniques for Planning the Small Experimental Project.

A Generalization of the Theorem on the Capacity of a Condenser.

Exam Book. Equations, Theorems, Proofs (30 pages). Approximations to the Area of the Ellipsoid (30 pages).

Areas of Ellipsoid (11 pages).

Memoires Presentes par Divers Savants a L'Academie des Sciences (1 page). Page 29-C of Tables. 4 Figures.

Equations/Tables (8 pages)

Tables (30 pages)

Equations (8 pages); Torsional Rigidity Increases by Symmetrization (1 page); Equations (3 pages); De Saint-Venant (2 pages).

Problems (29 pages)

Manuscript of Solutions to Vol. I of Mathematics and Plausible Reasoning . Chapter 1 Solutions.

Chapter 2 Solutions

Chapter 3 Solutions

Chapter 4 Solutions

Chapter 5 Solutions

Chapter 6 Solutions

Chapter 7 Solutions

Chapter 8 Solutions

Chapter 9 Solutions

Chapter 10 Solutions

Chapter 11 Solutions

Chapter 12 Solutions

Chapter 13 Solutions

Chapter 14 Solutions

Chapter 15 Solutions

Chapter 16 Solutions

Problem Solving Seminar: 1963; term papers, mid-terms

Math 129: How to Solve the Problem, course outlines, term papers, mid-term examinations, final examinations

Manuscript: Circle, Spheres, Symmetricals and Some Classical Physical Problems

Manuscript: Preface to Mathematical Discovery

Reprint: On Functions of Sequences of Independent Chance Vectors with Applications to the Problems of the "Random Walk" in k Dimensions, by Blackwell, D. and Girshick, M.A.; miscellaneous notes

Mathematical notes: On the Zeros of Successive Derivatives

Draft: Complex Variables , by Pólya, George and Latta, G.E.

Project with Peter? Desordre

Draft: Complex Variables , By Pólya, George and Latta, G.E.

Remarks on Certain Aspects of Some Processes, Resembling to and Differing from the Poisson-Process, by Binet, F.E.

Final Exam for Math 187-b

Class Roll Books; Class Lists; NSF Summer Institute for High School Teachers of Mathematics 1972; Group Photo 1969

NSF Summer Institute for High School Teachers of Mathematics 1966, 1967. Class Lists and Addresses

Math 111-S Final Summer 1965; Math 187-C Final July 29, 1965; Shell Fellows 1967 List; Math 111-S Final - Parts 1,2, Summer 1967; The Mathematics Curriculum

Seminar on Solving Math Problems 1966: Notes Jan-May 1966; Dept. of Mathematics Graduate Students Office Directory Winter 1966; Memo from Webb to Freshman Seminar Instructors

Competitive Examination in Math: Solutions and Comments 1951-1956, 1958, 1964-1965; Meeting of Math Association of America, Aug. 20, 1956; Meeting of National Council of Teachers of Mathematics, Aug. 21 1956: Minutes/Speeches

Letters to Winners and Alternates of Stanford-Sylvania Competitive Exam in Math; Announcement on Competitive Exam March 1963, Instructions

1954, 1956-1958, 1961, 1963 Exams

Los Angeles Mathematics Newsletter April 1956

Math Contest Sponsored by Metropolitan New York Section of the Math Association of America 1950-1952

Committee on Contests and Awards Releases 1951

Correspondence Pólya-Boyer 1951

The National Status of Math Contests in Secondary Schools

National Council of Teachers of Math

Henry L. Alder Letter (Co-Chairman of Committee on Contests)

Report on the Math Contests held in Northern California

Information on the Contest

Letters from Henry L. Alder

Registration for 1956 Annual Math Contest

Rules of the 1956 Annual Contest

Henry L. Alder Letter to Chairmen of Dept. of Math

Letter from Pólya to Prof. W. H. Fagerstrom, Jan. 23, 1952

Letter from Pólya to Prof.John C. Schoonmaker, Nov. 23, 1948

Letter from John C. Schoonmaker to Pólya, Oct. 29, 1948

Pólya to John C. Schoonmaker, Nov. 8, 1948

John C. Schoonmaker to Pólya, March 25, 1949

Reply by Pólya's Secretary

John C. Schoonmaker to Pólya, Nov. 12, 1948

Pi Mu Epsilon Contest and Solutions, March 13, 1948

Emory P. Starke to Pólya, May 29, 1953

Emory P. Starke to Gabor Szego, April 23, 1953

Math Prize Test, New Brunswick, NJ, April 25, 1953. State Math Day Program

Emory P. Starke to Gabor Szego, June 30, 1952

Gabor Szego to Emory P. Starke, July 7, 1952

Students' Assignments

Math 188-A Final, Summer 1966

Shell Merit Fellows List, 1965

NSF Summer Institute for High School Teachers of Math: 1965 List

Academic Year Institute for High School Teachers of Chemistry and Math 1960-61

Instructor's End Quarter Grade Report for Math 111-S, Summer 1966

Shell Merit Fellows Summer 1966 Directory

Final for Math 111-S, Aug. 11, 1966

Mid-term and End-Quarter Report for Math 188-A, April, 1965

Pólya's Notes/Grades for Math 187-C (NSF) and Math 111-S (Shell), Summer 1965

Final for Math 187-C, July 29, 1965

The Mathematical Log, September 1958

High School Math Contest Student Answerbook

Letters to Pólya and Szego, April 25, 1966

Letter to Pólya from Murray S. Klamkin, April 22, 1966

Letter to M. Gail Adams from F.R. Watson, Dec. 21, 1965

Letter from Adams to Watson, Dec. 9, 1965

Letter from Watson to Adams, Nov. 23, 1965

Letter from Virginia S. Boles to Watson, March 30, 1961

Letter from Watson to Boles, Aug. 24, 1960

Letter from Woolum to Boles, Oct. 19, 1960

Letter from Boles to Woolum, Oct. 6, 1960

Watson to Math. Department, April 9, 1960

Boles to Watson, July 29, 1960

Boles to Watson, Nov. 9, 1960

Problems and Solutions for 1960 Stanford-Sylvania Competitive Exam in Math by Pólya

Stanford Competitive Exam in Math, 1946-1964; Article on Exam, 1946

Exams, 1946-1964

Mathematikai Lapoh. Budapest, 1956

Letter to Pólya from James H. McKay, Feb. 14, 1966

Pamphlet and Typed Speeches on the Stanford University Math Examination, 1946 (2 copies)

Pamphlet "A Look at Mathematical Competitions", March 1959

Exams, 1947-1948, 1950-1953

Letter from Edwin Eagle to Pólya, April 5, 1954

Letter from Eagle to Szego, March 20, 1954

Letter from Szego to Eagle, April 9, 1954

1954 Exam Solutions, Written and Typed

Letter from Boles to Eagle, April 14, 1954

Letter from Boles to J.D. Wilson, July 5, 1956

1956 Exam Solution, Hand-written

1956 Results

1957 Exam and Solutions, Remarks, Results (2 Copies)

9-Math, 10-Math, 11-Math by Pólya

1956 Exam and Solutions

1949 Exam - Problems and Results

"On the Mathematics Curriculum of the High School"

"Analyse Mathematique", Institut de France, 3 pgs.

The California Mathematics Council Bulletin, Fall, 1961, Oct. 1960

"Ten Commandments for Teachers"

"On the Ratio of Consecutive Eigenvalues"

The Mathematical Log, Jan. 1960

Nieuw Archief Voor Wiskunde

"Bounds for Eigenvalues of Pólya's 'Plane-Covering Domains' by Filling a Top of a Cylinder" by Hersch

"Pólya's Contributions to Chemical Enumerations" by Harary, Palmer and Robinson

"An Extension of Pólya's Theorem on Power Series With Integer Coefficients" by Robinson

"Hadamard Products of Schlicht Functions and the Pólya-Schoenberg Conjecture" by Ruscheweyh and Sheil-Small

"Pólya's Enumeration Formula by Example" by Tucker

Letter from Andrew F. Acker to Pólya, Feb. 21, 1977, and Enclosures

Letter from Sami Beraha to Pólya, Aug. 18, 1961, and Enclosures

Letters from Albert Edrei to Pólya, March 29, 1978, April 16, 1978, April 26, 1978, and Enclosures

Letter from Doris Schattschneider to Pólya, Sept. 13, 1977 Plus Photograph of Escher's Writing

Letter from W. Wm. Funkenbusch to Pólya, May 29, 1979 and Paper

Letter from Frank Harary to Pólya, March 16, 1978 with "From Euler's Partitions to Cayley's Rooted Trees" by Harary; Letter from Barnabas B. Hughes to Pólya, July 21, 1977 with "A Study in Medieval Problem Solving" (2 Copies)

Letter from Donald E. Knuth to Pólya, Aug. 16, 1976; Various Calculations; Letter from Jean (?) to Pólya, Aug. 10, 1976

Letter from Thomas James Higgins to Pólya, Nov. 14, 1948

Calculations

"Experimental Determination of the Capacitance of Heavy-Current Busses Comprised of Solid or Tubular Rectangular Conductors" by Teasdale and Higgins

Letter from Higgins to Pólya, Aug. 20, 1947

Letter from Higgins to Pólya, May 21, 1947 and Enclosure.

Pólya's Reply, June 24, 1947

Postcard from Higgins to Pólya, June 30, 1947

Letter from Higgins to Pólya, June 28, 1947

Letters from D.H. Lehmer to Pólya, April 27, 1943, May 10, 1943 and Enclosures on "Inequalities for the Area of the Ellipsoid"

Postcard DHL to Pólya, February 1943

Letter DHL to Pólya, Feb. 1, 1943

Letter DHL to Pólya, Fe. 9, 1943

Letter from Lehmer to Pólya, April 20, 1943

Letter from Lehmer to Pólya, May 26, 1943 and Enclosures

Letter from Lehmer to Pólya, Dec. 23, 1943

Letter from Lehmer to Pólya, July 11, 1943 and Enclosures

Letter from Lehmer to Pólya, July 18, 1943 and Enclosure

Letter from Lehmer to Pólya, July 20, 1943

"A Simpler Theory of the Mathematical Infinite" by Arthur Marystone with Letter to Pólya

Letter from Donald A. McQuarrie to Pólya, March 31, 1965

Letters from Carl Prather to Pólya, June 2, 1978, March 28, 1978, March 3, 1978

"A Note on an Extension of the Asymptotic Formula for the Bell Numbers" by Prather

Draft of Pólya Letter to Prather, July 4, 1978

Letter from Sister Madeleine Rose to Pólya, Undated and Enclosed Preliminary Draft of Dissertation Proposal

Information and Instructions on Experiment

"The Use of Heuristic Methods for Problem Solving in Algebra" (Revised Proposal by Rose)

3 Problems and Solutions

Letters from I.J. Schoenberg to Pólya - undated (incomplete), May 30, 1970 and Enclosure

Schoenberg Flow Chart on Problem Solving Model, Math 191-A Discussion; Letter from Schoenberg to Pólya, Oct. 25, 1977; "Can Heuristics be Taught?" by Schoenberg; Excerpt from Boole: Collected Works, Vol. 2

Letter from Jack Williamson to Pólya, March 15, 1977 and Enclosures; Letter from John J. Wavrik to Pólya, Jan. 27, 1978 and Enclosures

Letter from A.W. Heuberg to Pólya, Nov. 28, 1962; Letter from Alexander Wittenberg to Moise, Nov. 27, 1962; Letter from Philip H. Abelson to Wittenberg, Nov. 21, 1962; "Qui Aurait Pu Penser A Cela?" by Wittenberg

"Der Mathematische und NaturWissenschaftliche Unterricht"

Identification and Development of the Mathematically Talented -- The Hungarian Experience , by Agnes Arrai Wieschenberg, PhD. Dissertation submitted to the School of Arts and Sciences, Columbia University, 1984

Addenda, 1986-036 ARCH-1986-036

Polanyi Biography - Draft of chapter 1 1979

Math problems and notes

Mathematics course descriptions

Book and essay drafts

Class finals and term papers 1961-1963

Class exams and term papers

Manuscript for "Circle, Sphere Symmetrization and Some Classical Physical Problems"

Manuscript preface to Mathematical Discovery

On the zeros of successive derivatives. An example

Complex Variables: Notes for Mathematics 106, 107

Project with Peter?

Complex Variables: Notes for Mathematics 106, 107 1963 August

Papers on animal genetics

Mathematics 187b Final examination

NSF Summer Institute for High School Teahcers of Mathematics 1969

High school examination 1950 Spring

Math 111s final exam 1965 Summer

Seminar on solving mathematical problems notes 1966 January-May

Stanford competitive examination in mathematics 1951-1956

High school competition forms and form letters 1961-1963

Other high school contests

NSF - Shell summers

Stanford-Sylvania correspndence

Stanford-Sylvania samples

Sample of items - Stanford Mathematics Competition 1946-1964

Miscellaneous inquiries

High school mathematics materials

Publications about Polya by various authors

Andrew Acker preprints 1977 February

Correspondence and work from Sami Beraha, Albert Edrei, Doris Shattschneider, W. Wm. Funkenbusch

Correspondence and work from Frank Harary, Barnabas B. Hughes

Calculations and correspondence from Donald Knuth

Correspondence - Thomas J. Higgins

Correspondence and calculations - D. H. Lehmer

A Simple Theory of the Mathematical Infinite by A. Marystone 1957

Carl Prather

Sister Madeleine Rose

Letter from I. J. Schoenberg

Schoenfeld flow chart

Correspondence and work from Jack Williamson, John J. Wavrik

Alexander Wittenberg

Identification and Development of the Mathematically Talented - The Hungarian Experience by Agnes Arvai Wieschenberg 1984

Addenda, 1986-114

"George Polya's Outstanding Thought and Contributions" by Xue Di-qun

Addenda, 1987-034 ARCH-1987-034

Partition of a Set into Structures Classes, research notes 1970-1971

List of Papers

Lectures 1969

Integer Valued Entire Function

Lausanne: Methodologia on Heuristigue, Strategic Tactique (manuscript)

Manner of Correcting

Essays: On the Zeros of Successive Derivatives; Guessing and Proving

Manuscript: Suggestion on the Teaching of College Mathmatics

Partitioning of Sets, Cycles in Permutations

Methodology on Heuristics, Strategy or Tactics

On Plausible Reasoning

Reprints on Papers

Draft: Preface, Chapter VII, Cauchy's Internal Formula and Application (photocopy)

Some Mathmaticians I have known, on the isoperimetric theorem...

Polya, in memorium

Correspondence: Miscellaneous Publication, Papers given (drafts and originals) 1968-1969

Lectures: Riemann Hypothesis, Hadamard's Problem, Guessing and Proving 1970-1971

Photographs: Summer Institution 1957-1967

Complex Variables, 1 1963

Complex Variables, 2 1963

With Bowden: Chapter 1 Mathmatical Language, Chapter 2: Beginnings of the Integral Calculus

On the Number of Certain Lattice Polygons, Reprint 1969

The other isoperimetric quotients for polygons and polyhedra

Term Paper Topics, Exercises (Class Unknown)

Stanford University Competitive Examination in Mathmatics by Polya and Kilpatrick

Lectures, Correspondence, re: topics and travel 1970-1971

Varied Papers by Polya

Kadesch, R.R. to Polya 1985

Correspondence and Proofs

Includes Articles by Polya 1964

German title, by Hartkopf, typed manuscript 1960

New Math, Correspondence, Newspaper Articles 1962

Math and Education 1962

On an Irreducibility Theorem

On the Number of Real Roots of Polynominals

Programmed Correspondence Course in Geometry, Introduction

A Programmed Correspondence Course in Geometry for Teachers, 1

A Programmed Correspondence Course in Geometry for Teachers, 2

A Programmed Correspondence Course in Geometry for Teachers, 3

Cognitive Mechanisms

Leibniz mint, paper and correspondence

Work Citing Polya?

Sequential Sampling

Surreal Numbers, by D.K. Ervin

Heauristics Applied to Duplication of the Cube, paper and correspondence

Manuscripts, Papers by Polya

Inequalities and the Principles of Non Sufficient Reason, by Polya

Great and Small Examples of Problem Solving, by Polya 1953

Mathmatical Discovery, by Polya Vol I, Vol II 1962

The Determinant of the Adjacency Matrix of a Graph

Seminar in Problem Solving Mathmatics, Lecture Notes 1875, 1957

Problems and Theorems in Analysis, Volume I (manuscript)

Manuscript Materials, Polya

Papers, by Various Authors

Inspirational Materials for Polya's Symmetry Book

Polya's List of His Papers

Mathmatical Discovery 1980

Polya Personal: Poetry, Drawings, Dreams

Papers, written by different authors

Societe Mathmatique Suisse and others

Typed Paper, in french

Polya's Reprint

Tripos 1925

Polya's Reprints

Manuscript Re: Problem Solving, 1 1940-1941

Manuscript Re: Problem Solving, 2 1940-1941

Manuscript re: Problem Solving, 3 1940-1941

Polya: Personal Papers: Polya, John B. (brother) and others 1957-1962

Exam: How to Solve It 1960s

Graduate Special on Math 1955

Stanford Competitive Examinaion in Mathmatics 1946-1965

Reviews of Polya's Publications 1964

Let Us Teach Guessing, Polya

Problems-Heuristic, To: Little Discoveries

Research Unidentified

Manuscript...Stanford Math 110

Stanford Math 110,111, 129

Review of Polya's Work

Mathematics and mechanics journals and papers

Research reports and papers

Addenda, 1987-112 ARCH-1987-112

Articles, Papers

Alfred Haar

Einar Hille

Hugo Hadwizer

Ludwig Bieberbach

Adolf Hurwitz

Jacques Hadamard

Edmund Landau

Paul Bernays

Harald Buhr

Emile Borel

Franz Brentano

Constantin Czratheodory

Lipot Fejer

Mihaly Fekete

Bruno di Finetti

Friedrich Ludwig Gottlob Frege

Godfrey Harold Hardy

Kurt Hensel

Arthur Hirsch

R. Jentzsch

Louis Kollros

Lindwart, E.

John Edensor Littlewood

H. M. McDonald

Gosta Mittlag-Leffler

Paul Montel

Rolf Nevan Iinna

N. E. Nørlund

A. Ostrowski

Albert Pfluzer

Max von Pidoll

Alfred Pringshrim

Frigyes Riesz

Marcel Riesz

Ludwig Schlesinger

Carl Ludwig Siegel

Waclaw Sierpinski

Arnold Sommerfeld

Alfred Stern

S. Straszevicz

Gabor Szego

Heinrich Tietze

Otto Toeplitz

J. M. Whittaker

Eugene Wigner

Grace Chisholm Young

Miscellaneous

Issai Schur

Hilbert Köaigsberg, photo, "FC 2A"

Edmund Landau, photo, "FC 2E"

Hilberts letter on Hurnritz, FC 2B" 89

"FC 3C" letter

H. Weyl and Pólya, Mathematische Zeitschrift "FC 4E" 1972

Copied cover of The Stanford Mathematics Problem Book with Hints and Solutions" by Pólya and J. Kilpatrick"WC 2E"

Copied cover of "Teaching and Learning: A Problem-Solving Focus", labelled "WC 2F"

Copied cover of "Developments in Mathematical Education: Proceedings of the Second International Congress on Mathematical Education", labelled "WC 2B"

Copied cover of "Proceedings of the Fourth International Congress on Mathematical Education", labelled "WC 2C"

Articles, notes, papers re: Isoperimetric Inequalities "FC 7F"

Articles, correspondence, papers re: honorary diploma and membership in the Hungarian Academy of Sciences "HC 2B" 1977

Copied poster of Pólya "HC 2C"

Floor plans

Pólya reprints: 81, 147, 154-171

Pólya reprints: 172-190

Pólya reprints: 191-204

Pólya reprints: 205-220

Pólya reprints: 221-230, 232-233, 235-238, 240, 245

Pólya reprints: Miscellaneous, no assigned number

Two Year College Mathematical Journal , Vol. 14 No. 41 1983 September

Certificate of Fellowship in the American Academy of Arts and Sciences 1974 May 8

Certificate of Membership in the National Academy of Sciences of the United States of America 1976 April 27

Honorary Law Doctorate from the University of Alberta 1961 May 20

Addenda, 1987-137 ARCH-1987-137

Personal papers, immigration

Correspondence, incoming

Vancouver 1958, Belmont 1963, etc. 1958-1963

Correspondence re: Open University and miscellaneous

Correspondence

Correspondence, incoming re: books and articles

Correspondence: Harary, Frank

Correspondence with Springer-Verlag Publishers re: Problems and Theorems in Analysis II

Miscellaneous correspondence / articles

Correspondence / articles in German

Correspondence re: lectures 1974-1976

Correspondence re: lectures 1971

Correspondence re: lectures 1970

Lecture: Galileo Galilei

Lectures (includes correspondence and outlines)

Lectures: Conferences (includes correspondence and outlines)

Lecture Illustrations

Lectures (old)

Series of Lectures: Aspects of Calculus 1955 June 27-August 19

Conference: Great and Small Examples of Problem Solving, University of Colorado 1953

Commonwealth Conference on Mathematics in Schools reprints, Folder 1

Commonwealth Conference on Mathematics in Schools reprints, Folder 2

Math 180S Exams

Littlewood, John; Plancherel, Michel; Van Neumann, John

Weyl, Herman; Weyl, Mrs. Herman

Bio for Publisher

Misellaneous

Journal: L'ensignment Mathematique; Article: Descarte, Euler, Poincaré, Pólya — and Polyhedra

Pólya Reprint: Mathematics and Plausible Reasoning, annotated

Reprints: American Authors re:problem solving

Reprints: European Authors re: logic, creativity, discovery in math

Manuscript in German

Mathematical Notes

Correspondence, manuscripts

Notes for "As Their Students See Them"

Pólya manuscript: Teaching Mathematics with Emphasis on Problem Solving

Manuscript: Some Methods of Approximation Applicable to Problems Depending On A Minimum Principle

Manuscript: Observations on the Eigenvalues of a Vibrating Membrance; Pólya and Peter Szegö

Manuscript: Circle, Sphere, Symmetrization and Some Classical Physical Problems

Chapter 1: Mathematical Language; Chapter 2: Beginnings of Integral Calculus

Manuscript: Problems in Elementary Calculus, Analytic Geometry, Fundamental Concepts

Manuscript: The Working of the Mind in Mathematics

Manuscript: Mathematical Discovery in the Classroom; Patterns of Heuristic Reasoning 1963

Mathematical Discovery Vol. II

Notes for Mathematical Discovery

Manuscript: Mathematical Discovery, Vol. II

Manuscript: Mathematical Discovery

Manuscript: Mathematical Discovery: Hypothetico-Dedeuctive Systems

Manuscript: Mathematical Discovery, The Greatest Common Division and The Lattice Points on a Circle

Manuscript: Mathematical Discovery, Preface and Epilogue

Manuscript: Mathematical Discovery, Chapter 7, Chapter 8

Manuscript: Mathematical Discovery, Chapter 9

Manuscript: Mathematical Discovery, Chapter 10, Chapter 11

Manuscript: Mathematical Discovery, Chapter 12, Chapter 13

Manuscript: Mathematical Discovery, Chapter 14

Manuscript: Mathematical Discovery, Chapter 15

Manuscript: Mathematical Discovery, Déchets Chapter 7-9

Manuscript: Mathematical Discovery, Déchets Chapter 10-13

Manuscript, Mathematical Discovery Vol. II, Déchets Chapter 14

Manuscript, Mathematical Discovery Vol. II, Déchets Chapter 15

Mathematical Discovery Vol. II Illustrations

Manuscript notes: How to Solve the Problem (Folder 1)

Manuscript notes: How to Solve the Problem (Folder 2)

Manuscript notes: How to Solve the Problem (Folder 3)

Manuscript draft: How to Solve the Problem (Folder 1)

Manuscript draft: How to Solve the Problem (Folder 2)

Manuscript draft: How to Solve the Problem (Folder 3)

Manuscript notes, draft 2: How to Solve the Problem

Manuscript draft 2: How to Solve the Problem (Folder 1)

Manuscript draft 2: How to Solve the Problem (Folder 2)

Manuscript notes: Euclid Guessed It

Manuscript: Logic of Discovery 1946

Manuscript: Problems (part of How to Solve the Problem)

Correspondence with Publishers re: How to Solve the Problem

How to Solve the Problem, Corrected Galleys

How to Solve the Problem, miscellaneous

How to Solve the Problem, first copy

How to Solve the Problem, preface

How to Solve the Problem, manuscript (Folder 1)

How to Solve the Problem, manuscript (Folder 2)

How to Solve the Problem, manuscript (Folder 3)

How to Solve the Problem, third copy

How to Solve the Problem, homework, final, selected topics

How to Solve It, third copy

How to Solve It, manuscript (Folder 1)

How to Solve It, manuscript (Folder 2)

How to Solve It, third essay

How to Solve It, fourth essay

How to Solve It, fifth essay

How to Solve It, seventh essay

How to Solve It, Essays 5-7

Die Strukturformen der Probleme

Rényi Kató: Dissertation, Budapest, 1957 1957

Math Notebooks 1917

Math Notebooks 1920

Math Notebooks 1921

Math Notebooks 1921-1922, 1927

Math Notebooks 1923

Math Notebooks 1924

Math Notebooks 1926

Math Notebooks 1929

Math Notebooks 1929-1933

Math Notebooks 1933-1935

Math Notebooks 1936

Math Notebooks 1940

Math Notebooks 1941-1944

Math Notebooks 1944-1947

Math Notebooks 1947-1949

Math Notebooks 1949-1952

Math Notebooks 1952-1954

Math Notebooks 1953

Math Notebooks 1954-1957

Math Notebooks 1960

Math Notebooks 1966

Math Notebooks Undated

Letter from Mrs. H. Albert Einstein 1973

Bound volume of dissertations in German

Addenda, 1989-132 Accession ARCH-1989-132

Subseries 1. Biographical Subseries 1

George Pólya - Biographical 1979-1985

Biographical Notes 1975-1979

Biographical - on Zurich and ETH 1914, 1977-1978

Pólya Gyorgy, Stanford Egyetem

Pólya Bibliography circa 1976

Grade Books from Grammar Schools 1893-1896

Report Books After Grammar School 1905-1912

Miscellaneous Correspondence about Teaching Posts and Lectures 1920-1940

Pólya's Ph.D. Students and Collaborators

Dissertations by Pólya Students 1928-1962

Pólya Mentioned (Egocentrics) 1974-1982

Scope and Contents note

Recommendations 1972-1975

Two-Year College Mathematical Journal 1978-1980

Stanford Faculty Reports 1965-1971

John Pólya 1952-1956

J.B. Pólya on Fluoridation 1966

Family 1884

Subseries 2. Correspondence Subseries 2

Varia 1934-1940

Communications 1930-1954

Correspondence 1979-1985

Miscelaneous Correspondence 1981-1985

Personal Correspondence 1961-1966

Correspondence 1971-1983

Miscellaneous Correspondence 1961-1979

Alexander Israel Wittenberg 1963-1965

Alfred Renyi, Kato (Catherine) Renyi, Paul Erdos 1957

Correspondence 1919-1972

Pólya Prize 1971

Permission Requests 1967-1971

Consulting Agreements and Requests 1965-1969

Stanford Library Archives 1973

Stella Pólya 1987-1988

Reviews and Reports of Papers 1967

Subseries 3. Class Materials Subseries 3

Theory of Numbers, Problem Solving seminar (NSF) 1962-1967

Miscellaneous notes - Psyc. 262 1951-1953

Probability, Statistics - Math. 123, 125 1947-1953

Calculus of Probability - Math. 123 1946-1948

Probability and Statistics - Math. 123, 125 1943-1951

Calculus of Variations - Seminar 1952-1953

Group Theory with Applications 1938-1946

Potential Theory 1939

Differential Geometry 1941-1959

Introduction to Elementry and Analytic Number Theory 1935

Introduction to Elementry and Analytic Number Theory 1941-1950

Minimum Principle Approximations 1954

Minimum Principle Approximations 1956-1957

Algebra and Number Theory 1928-1929

Singularities of Power Series 1933

Vectors 1935-1948

Integral Equations 1944

Smith College, Mathematics 35B, 36B 1942

Introduction to Combinatorics - CS150 1977-1978

Mathematics 5 (Berkeley) 1957-1958

Aspects of Calculus 1955

Mathematics 111s 1965

Combinatorial Theory 1966

Introduction to Combinatorics - CS150 1978

Pólya Miscellaneous Classes 1958-196?

E.T. Jaynes 1957

Elementary Mathematics From Higher Point of View - Math 129

How To Solve The Problem - Math 129 1931

Lattice Points - Math Seminar undated

Subseries 4. Manuscripts Subseries 4

Essays On Mathematical Method, Discovery, and Teaching undated

Essays On Mathematical Method, Discovery, and Teaching

Drafts and notes for Essays On Mathematical Method, Discovery, and Teaching ~1950

Mathematics and Plausible Reasoning - Appendix 1955-1965

Mathematics and Plausible Reasoning - Illustrations undated

Mathematical Discovery undated

Mathematical Discovery 1962

Corrections for Mathematical Discovery 1981

Complex Variables 1971-1972

Mathematical Methods in Science undated

Notebook 1, Aufgaben und lehrsatze 1955

Notebook 2, Aufgaben und lehrsatze undated

Aufgaben und Lehrsatze - Addenda undated

Aufgaben und Lehrsatze 1972

IX: Psychologie undated

III: Auffassung, Einfall, Ansatz undated

V: typische Anschlusse von Hilfsaufgaben undated

VI: Schetzung der Aussichten undated

VII: Die Phasen der Untersuchung undated

VII: Die Phasen der Untersuchung 1939-1940

Notes for a book undated

Mathematical Discovery in the Classroom

Notes 1949-1954

Notes for a Book ~1941

Rules and Moves

Essential Texts in German How To Solve It undated

Figures and Drawing

Bound Notebook 1957-1977

Notes from previous bound notebook

Talks 1940-1946

Teaching Mathematics 1950

Notes for Short Papers - Problems 1971-1979

How To Solve It - Symposium 1947-1948

Miscellaneous Manuscripts 1962

Rules of Thumb 1977

Seattle for Oldsters undated

San Diego, June, 1970 1970

Galileo 1976

Slides undated

Talks: Seattle, San Diego 1977-1978

Daylight Arc 1976

Symmetry of Ornaments undated

Slides for a talk undated

Films 1970-1976

Lectures on Travels 1958-1959

Notebook undated

Notebook 1942

Mathematical Education 1968

Unfinished and Planned Writing undated

Fragments of Writings undated

Henryk Lauer undated

Misc. Pólya Manuscripts undated

Lecture Notes 1966-1968

Various Lectures 1967-1968

Summer Institute Lecture 1972-1976

Numerical Integration 1943-1946

Lectures - Plausible Reasoning 1954

Pólya Counting Method 1978

George Pólya Article for Szego's Collected Papers 1981

Calculus of Probabilities (French) undated

Papers by Pólya 1952

Project Hilgard ~1953

Subseries 5. Research Materials Subseries 5

Experimental Tables Undated

"The Heuristic of George Pólya and its Relation to Artificial Intelligence", by Allen Newell, Feb. 1981, draft, 50pp, plus Pólya's letters of thanks to Newell 1981

Miscellaneous Research Notes 1951-1953

Solow, Daniel 1980-1981

Miscellaneous Problems 1966-1969

Miscellaneous Problems Collected from Various Sources 1954-1962

Notebook for Mathematical Discovery 1953-1977

Notebook: Heuristic Logic

Odlyzko - Zeros of Zeta Function 1982

Michael Klass 1972

Rubik's Cube undated

Generalizing Hardy's Theorem undated

Problems 1933-1939

Stochastic Proceses - Feller

Infinite Series and Fourier Series 1947-1948

Pointsets and Real Functions (Punktmengen und reale Funktionen) undated

Positions of Zeros 1929-1937

Schur Darstelungstheorie 1936

Entire Series Satisfying Algebraic Differential Equations 1935

Miscellaneous Notes 1978

Miscellaneous Notes and Problems 1956-1985

Pinboards and Geoboards undated

117A and Ingham 1966-1967

Polyhedra and Other Enumerations 1964-1965

Miscellaneous Notes on Problems 1972-1977

"Inequalities involving integrals of functions and their derivatives," by Donald Benson, 23pp (mimeograph) 1966

"Our Aim is to Find an Approximation to n!" (typed paper, unknown author and date) undated

Miscellaneous Notes undated

"The basic concepts of Pólya's theory of enumeration, with examples from the structural classification of mechanisms," by F. Freudenstein, 26pp (typescript) 1967

"Variational Properties of Steady Fall in Stokes Flow", by H.F. Weinberger, 51pp (typescript) undated

Williamson, S.G. undated

Combination Problems Involving Points and Lines in a Plane undated

"Modern Heuristic in Historical Perspective With Implications for Research Pedagogy," by Elton Carter and Robert Richey, in General Semantics Bulletin (offprint) 1961-1962

Pascal Letters, by Renyi (Manuscript of the Hungarian version of the Letters on Probability by Alfred Renyi) undated

Martin Gardner Problem 1966

"The Problem of Packing a Number of Equal Nonoverlapping Circles on a Sphere," by H.S.M. Coxeter, in Transactions of the New York Academy of Sciences (reprint), with notes by Pólya 1962

"Example Generation," by Edwina Rissland, Computer and Information Science Technical Report 80-14 1980

"On the Art of Problem Solving," by Murray S. Klamkin, Scientific Laboratory, Ford Motor Co., 41pp (publication preprint), plus graph by Pólya 1966

"Transfinite Diameter and Analytic Continuation of Functions of Two Complex Variables", by M. Schiffer and J. Siciak, Tech. Rep. No. 1100, Applied Mathematics and Statistics Lab, Stanford, plus three pages of notes 1961

"On Pólya Frequency Functions IV: The Fundamental Spline Functions and Their Limits", by H.B. Curry and I.J. Schoenberg, in Journal D'Analyse Mathematique 1966

Handbook for Planning an Effective Mathematics Program , California State Department of Education 1982

Radius of Circumscribed Sphere undated

Pal Turan and George Alexits death notices from Hungarian Academy of Sciences 1976, 1978

"Counting of Isomeric Hydrocarbons with Asymptotic Results," by Joel G.W. Rogers, 26pp. (manuscript) 1971

Subseries 6. Reprints and Articles Subseries 6

Bib. #163: Heuristic reasoning and the theory of probability 1941

Bib. #168: Approximations to the area of the ellipsoid 1943

Bib. #172: A minimum problem about the motion of a solid through a fluid 1947

Bib. #174: On patterns of plausible inference 1948

Bib. #195, 196, 199: Two Notes on Minimum Principle Approximations, Techical report No. 29, Stanford, 1953; Estimates for Eigenvalues, Page Proofs, 1954 1953-1954

Bib. #200: More isoperimetric inequalities proved and conjectured 1955

Bib. #202, 203, 206: On the Characteristic Frequencies of a Symmetric Membrane, 15pp, Tech. Rep. No 40, Stanford, 1955 (2 copies); On the Ratio of Consecutive Eigenvalues, 21pp (with L.E. Payne and H.F. Weinberger), Tec. Rep. No 41, Stanford, 1955; The Mathematics Teacher, LII/1, 1959: Mathematics as a Subject for Learning Plausible Reasoning (translation) 1955-1959

Bib. #209: Remarks on De La Vallee-Poussin Means and Convex Conformal Maps of the Circle, 76pp, Tech. Rep. No 70, Stanford (with I.J. Schoenberg), 1957 1957

Bib. #215, 219, 217: On the Eigenvalues of Vibrating Membranes; Two More Inequalities Between Physical and Geometrical Quantities -- Tech. Rep. No 88, Stanford, 1960; Archimedes (magazine), 1960, containing: Die Mathematische Erziehung (p.. 103) and Die Mathematikals Schule der plausiblen Schliessens (page proof, in back of magazine; The Mathematics Teacher (magazine), 1961, containing: The Minimum Fraction of Popular Vote that Can Elect the President of the United States 1960-1961

Bib. #222: Intuitive outline of the solution of a basic combinatorial problem 1963

Bib. #232, 236, 241, 249: Entiers Algebriques ... (1969); Two Incidents (1970); Stanford Competitive Exam (1973); On the Zeros ... (1976) 1969-1976

Bib. #233: Some mathematicians I have known 1969

Bib. #234: On the isoperimetric theorem: History and strategy 1969

Bib. #235: Gaussian binomial coefficients and the enumeration of inversions 1970

Bib. #237: Gaussian Binomial Coefficients (with G.L. Alexanderson) 1971

Bib. #238: Methodology or Heuristics, Strategy or Tactics? 1971

Bib. #239: Eine Erinnerung un Hermann Weyl 1972

Bib. #240: A letter by Professor Pólya 1973

Bib. #242: A story with a moral 1973

Bib. #243: Formation, not only Information 1972

Bib. #244: As I read them 1973

Bib. #245: Partition of a finite set into structured subsets 1975

Bib. #246: Probabilities in Proof-Reading 1976

Bib. #247: Guessing and Proving 1976

Bib. #248: As their students see them 1976

Bib. #250: A note of welcome 1977

Bib. #251: More on Guessing and Proving 1979

Norsk Matematisk Tidsskrift 1950

California Mathematics Council Bulletin 1954-1958

Mathematical Log 1960

Generalization, Specialization, Analogy 1961

Dedication of Stanford University Computation Center 1963

Die Heuristik and Conference Procedings 1964-1966

Sylvania Competitive Examination 1960

Miscellaneous Articles by Pólya 1962

Some Articles by Pólya 1980

Book: The 1953 Jennings Scholar Lectures 1963

Book: Inequalities: Theory of Majorization and Its Applications 1979

Book: Research Papers in Statistics 1966

Book: Do You Teach? Views on College Teaching 1969

Book: A Taste of Science 1975

Pólya's Theory of Counting 1959-1965

Articles about Pólya 1963-1988

Articles and Journals related to Pólya 1966-1980

Articles and Journals related to Pólya undated

"An Elementary Solution of Pólya's Orchard Problems," by Tracy Allen, 16pp (typescript) 1983

"Omzien in bewondering," by N.G. de Bruijn, pamphlet, 18pp, retirement speach in Dutch (references to Pólya) 1984

"II: Drunkard's Walk on an Infinite Lattice", by Peter Doyle, pp. 34-73 (typescript) 1980

"A Method for Technological Predictions", by D.G. Ellson, 20pp (typescript), plus letter from Ellson comparing his paper to "Patterns of Plausible Inference" 1954-1955

Two AI papers by Friedman (on Plausible Inference), typescript, with Pólya letter (presumably unsent) 1980

David Hawkins publications 1964-1966

"The Ancient Tradition of Geometric Problems", Part I, Section 7: Apollonius, by Wilbur Knorr, 65pp (typescript), with note by Pólya, manuscript, crossed out, on back of letter 1981

"Imre Lakatos e la 'filosofia della scoperta,'" by Genner Luigi Linguiti, Lucea 1981

"What Every Secondary School Mathematics should Read" (references Pólya books), by Lowell Leake, Mathematics Teacher 1983

Subseries 7. Miscellaneous Subseries 7

Stanford University Mathematics Examination 1946-1965

Committee on the Teaching of Undergraduate Mathematics 1976-1979

Miscellaneous articles undated

Choice [Combinatorial Analysis] undated

Eckford Cohen 1968

Bruno de Finetti 1971

P. Cartier and D. Hejhal 1979

Ernst Mohr 1962-1983

S.C. Bhatnagar 1984

Colin Thompson 1969-1971

John Brillhart 1981

Jean Pedersen undated

Imre Lakatos 1971-1978

Antoine Ehrhard ~1982

Latta - Complex Variables 1974-1977

Paul Halmos 1972

Feldzamen 1966

Polyhedra - Federico 1975-1976

Ars Expositionis: Euler as Writer and Teacher [typescript of paper by Gerald L. Alexanderson] 1982

Fejer Lipot 1960

Arne Baartz undated

Albert Einstein 1958

Stella Pólya 1977

Shiffman 1966

School Mathematics Project 1967-1969

Miscellaneous Hungarian Newspaper Clippings 1971, 1975

Pólya and Wehl 1918

Hilbert to Hurwitz 1900

Herzberger 1972

Miscellaneous Notes and Pieces of Paper undated

"Identification and Development of the Mathematically Talented - The Hungarian Experience," Agnes Arvai Wieschenberg Dissertation 1983

Alvin White 1974-1975

Miscellaneous Items 1911-1968

Certificate 1976

Honorary Ph.D., Univ. of Waterloo; photo

School Records (Hungarian/European) 1888-1914

Assorted Awards 1920-1971

Professional Associations 1950-1977

Poster about lectures by Pólya 1980

Slides for a Talk

Chessboard Thank You

Honorary Membership, Santa Clara Valley Mathematics Association 1976

Santa Clara Valley Mathematics Association Life Member Plaque 1985

Slides (graphs, diagrams, etc.) undated

Photograph Album 1917-1978

Copy prints of images from Photograph Album

School Papers 1887-1905

Research note cards (mostly in German)

Addenda, 2003-185 Accession ARCH-2003-185

Miscellaneous Mathematical Notes

Miscellaneous Papers

Tributes to George Pólya 1957-1989

Reminisces of Gorge Pólya 1973-1978

Articles on Mathematics and Science (in English, German) 1933-1950

Time Magazine article on post-Kennedy assassination 1963

Articles on Lee and Grace Lorch 1957-1974

Mathematical papers by Pólya (in English, French) 1958-1969

Articles and obituaries of mathematicians (in English, French, German) 1938-1955

Student Examinations (Blue Books) undated

Notes in preparation for Pólya's book with Gábor Szego undated

Mathematical papers (in German) 1950-1952

Speech to the International Congress of Mathematicians, Bologna 1928

Summer Mathematical Conferences 1954-1955

Trip to Zurich 1953-1954

National Lecture Tour 1955

Resumes of Pólya, undated

Correspondence, Trinity College, Cambridge, England 1951

Pólya's speeches

Book Reviews, "Problems and Theorems in Analysis" 1977

Book Reviews, "Mathematical Discovery: On Understanding Learning and Teaching Problem Solving" 1963-1965

Book Reviews, "How to Solve It" 1987

Book Reviews, "The Stanford Mathematics Problem Book" 1974

Book Reviews, "Notes on Introductory Combinatories" 1983

Book Reviews, "Isometric Inequalities in Mathematics" 1951-1952

Book Reviews, "Patterns of Plausible Inference" 1969

Book Reviews, "Mathematics and Plausible Reasoning" 1954-1963

Secretary of Defense William Perry, student of Pólya 1996

Pamphlet "How to Teach Guessing" 1956

Pólya Exhibit and Speech, Stanford 1987

Pólya's notes "On Plausible Reasoning" undated

Letters to Pólya 1941-1995

Papers on Mathematics 1941-1975

Correspondence 1932-1991

Papers on Pólya Family History

Pamphlet, "The Four Color Theorem" 1971

The University of Tasmania 1955

Correspondence with Gerald Alexanderson, Santa Clara University 1972-1997

Summer Institute for Teachers of Collegiate Mathematics at Stanford, General File 1955

Summer Institute for Teachers of Collegiate Mathematics at Stanford, Professor Schoenburg 1955

Summer Institute for Teachers of Collegiate Mathematics at Stanford, Professor Allendoerfer 1955

Summer Institute for Teachers of Collegiate Mathematics at Stanford, Professor Lehmer 1955

Manuscript on Probabilities (in French) undated

Letters from Issai Schur and Carl Ludwig Siegel (in German) 1930-1932

Papers for Pólya's "Groups, Graphs and Chemical Compounds" (in German) 1930

Paper "Zweck und Hillel" (in German) undated

Paper "Das Schliemen" (in German) undated

Papers in German and French 1940

Manuscript of JLWV Jensen (in Danish) 1911

Papers and correspondence on Pólya's Collected Works

List of Dissertations 1914-1960

News clippings 1953-1977

Letters of recommendation 1943-1980

Figures for "Mathematics and Plausible Reasoning" undated

Papers from Institute Henri Poincaré, Paris (in French) undated

Papers on Robert Weber, Stella Pólya's father 1850-1915

Correspondence with Swiss educators on problem solving (in German, French) 1931-1932

Physical problems often involving differential equations (in German) undated

Notes on "finite" Fourier integrals 1945

Postcards from Edmund Landau

Examinations, Stanford 1946-1951

Papers from the Swiss Institute of Technology, Zurich 1924-1940

Mathematical Association of America lectures 1956

Stanford Mathematics Fellowship Program 1957

Two papers by Alexandre Wittenberg (in French) 1957

Paper on "Structure-Counting Functions" 1958

Institute for High School Teachers of Science and Mathematics 1957-1958

Social Aspects of Science 1957

Pamphlet "Synthesis" (in Dutch) undated

Putnam Mathematical Competition 1948-1949

"Code Civil Suisse" (in French) 1907

"Seminar Aufgaben" (in German) 1929

"Aufirahme Porifung" (in German) 1935

Notes (in German) 1937

Daily Log Book (in German) undated

Draft of Pólya's doctoral dissertation (in Hungarian) 1912

"Vom Sternenhimmel" (in German) undated

"Twenty-Five Lessons in Citizenship" 1945

"Federal Textbook on Citizenship" 1943

"Buchbinderarbeiten" (in German) 1928

"Biztositasi Szemle" (in Hungarian) 1955

"Methode vol Laguerre zur Bestimmung des Geschlechts einer ganzen Funkton" (in German) 1914

"Tolstoi" (in German) 1907

"Biztositasi" (in Hungarian) 1956

Unsorted papers

Doctorate of philosophy dipoloma 1912 April

Correspondence, notes, and publications (FRAGILE) 1920-1930

Physical Characteristics and Technical Requirements

Az Orvostudomany Regenye volume (FRAGILE)

Unlabeled notebook

Conditions Governing Use

Manuscript material concerning probability theory pre-1940s

"Not in print" - miscellaneous materials

Work with Paul Bernays

"More on Problem Solving" 1931

Additional assorted materials

Addenda, 2004-075 ARCH-2004-075

Korean translation of Mathematics and Plausible Reasoning

Addenda, 2009-240 ARCH-2009-240

Correspondence (some in photocopy form, some to Gerald Alexanderson), papers by others, biographical articles, source materials on other mathematicians, and a few photographs.

Processing Information

Reprints and other published materials by Pólya and others

Addenda, 2016-095 Accession ARCH-2016-095

Materials from exhibit at Mathematics & Statistics (Math-Stat) Library circa 1985-1987

Addenda, 2024-583 accession ARCH-2024-583

Books, conference proceedings, and journals

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George Polya Mathematical Discovery

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MATHEMATICAL DISCOVERY

On understanding, learning and teaching problem solving

Combined Edition

George Polya

Professor Emeritus of Mathematics

Stanford University

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What is Polya’s method of problem solving?

Nearly 100 years ago, a man named George Polya designed a four-step method to solve all kinds of problems: Understand the problem, make a plan, execute the plan, and look back and reflect. Because the method is simple and generalizes well, it has become a classic method for solving problems.

What are the 4 problem solving methods?

• Rubber duck problem solving.
• Lateral thinking.
• Trial and error.
• The 5 Whys.

What is Polya’s third step in the problem solving process?

Third. Carry out your plan. Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct?

What is the part of Polya’s four step strategy is often overlooked?

Understand the Problem. This part of Polya’s four-step strategy is often overlooked. You must have a clear understanding of the problem. To help you focus on understanding the problem, consider the following questions: • • • • • Can you restate the problem in your own words?

What are the 5 problem-solving methods?

• Step 1: Identify the Problem.
• Step 2: Generate potential solutions.
• Step 3: Choose one solution.
• Step 4: Implement the solution you’ve chosen.
• Step 5: Evaluate results.
• Next Steps.

What is the best problem-solving method Why?

One of the most effective ways to solve any problem is a brainstorming session. The gist of it is to generate as many ideas as you can and in the process, come up with a way to remove a problem.

What are the 7 steps of problem-solving?

• 7 Steps for Effective Problem Solving.
• Step 1: Identifying the Problem.
• Step 2: Defining Goals.
• Step 3: Brainstorming.
• Step 4: Assessing Alternatives.
• Step 5: Choosing the Solution.
• Step 6: Active Execution of the Chosen Solution.
• Step 7: Evaluation.

What are the 3 types of problem-solving?

• Social sensitive thinking.
• Logical thinking.
• Intuitive thinking.
• Practical thinking.

What are the 3 stages of problem-solving?

A few months ago, I produced a video describing this the three stages of the problem-solving cycle: Understand, Strategize, and Implement. That is, we must first understand the problem, then we think of strategies that might help solve the problem, and finally we implement those strategies and see where they lead us.

What are the three problem-solving techniques?

• Trial and Error.
• Difference Reduction.
• Means-End Analysis.
• Working Backwards.

Who is the father of problem-solving method?

George Polya, known as the father of modern problem solving, did extensive studies and wrote numerous mathematical papers and three books about problem solving.

What are the examples of problem-solving strategies?

• Guess (includes guess and check, guess and improve)
• Act It Out (act it out and use equipment)
• Draw (this includes drawing pictures and diagrams)
• Make a List (includes making a table)
• Think (includes using skills you know already)

Which step of Polya’s problem-solving strategy where you can freely state the problems in your own word?

The first step of Polya’s Process is to Understand the Problem. Some ways to tell if you really understand what is being asked is to: State the problem in your own words.

Which method is also known as problem-solving method?

Brainstorming and team problem-solving techniques are both useful tools in this stage of problem solving. Many alternative solutions to the problem should be generated before final evaluation.

What is the 5 step approach?

Step 1: Identify the problem. Step 2: Review the evidence. Step 3: Draw a logic model. Step 4: Monitor your logic model. Step 5: Evaluate the logic model.

What is the problem-solving approach?

A problem-solving approach is a technique people use to better understand the problems they face and to develop optimal solutions. They empower people to devise more innovative solutions by helping them overcome old or binary ways of thinking.

What is another term for problem solving?

synonyms for problem-solving Compare Synonyms. analytical. investigative. inquiring. rational.

How many tools are used for problem solving?

The problem solving tools include three unique categories: problem solving diagrams, problem solving mind maps, and problem solving software solutions. They include: Fishbone diagrams. Flowcharts.

What are the stages of problem solving?

• Step 1: Define the Problem. What is the problem?
• Step 2: Clarify the Problem.
• Step 3: Define the Goals.
• Step 4: Identify Root Cause of the Problem.
• Step 5: Develop Action Plan.
• Step 6: Execute Action Plan.
• Step 7: Evaluate the Results.
• Step 8: Continuously Improve.

How do you teach problem solving?

• Model a useful problem-solving method. Problem solving can be difficult and sometimes tedious.
• Teach within a specific context.
• Help students understand the problem.
• Take enough time.
• Ask questions and make suggestions.
• Link errors to misconceptions.

What are the 4 common barriers to problem-solving?

Some barriers do not prevent us from finding a solution, but do prevent us from finding the most efficient solution. Four of the most common processes and factors are mental set, functional fixedness, unnecessary constraints and irrelevant information.

Why is Polya the father of problem-solving?

Pólya is considered the father of mathematical problem-solving in the 20th century. It was his constant refrain that problem-solving was not some innate special ability but can actually be taught to anyone.

What is George Polya known for?

He was regarded as the father of the modern emphasis in math education on problem solving. A leading research mathematician of his time, Dr. Polya made seminal contributions to probability, combinatorial theory and conflict analysis. His work on random walk and his famous enumeration theorem have been widely applied.

What is the most difficult part of solving a problem?

Contrary to what many people think, the hardest step in problem solving is not coming up with a solution, or even sustaining the gains that are made. It is identifying the problem in the first place.

What are 10 problem-solving strategies?

• Guess and check.
• Make a table or chart.
• Draw a picture or diagram.
• Act out the problem.
• Find a pattern or use a rule.
• Check for relevant or irrelevant information.
• Find smaller parts of a large problem.
• Make an organized list.

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2.1: George Polya's Four Step Problem Solving Process

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Step 1: Understand the Problem

• Do you understand all the words?
• Can you restate the problem in your own words?
• Do you know what is given?
• Do you know what the goal is?
• Is there enough information?
• Is there extraneous information?
• Is this problem similar to another problem you have solved?

Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.)