Pólya, George 401

2

Subdivide the region Rinto small pieces appropri-

ate for examination with polar coordinates. The kth

section is thus a region given by values rand θ

bounded between fixed values: rk1< r< rk2and θk1<θ

< θk2. It appears as the region between a sector of angle

θk2– θk1of radius rk2and the sector of the same angle

but at a smaller radius rk1, and so has area:

where dθk= θk2– θk1and drk= rk2– rk1. The double

integral is thus given as a limit

taken over finer and finer subdivisions of the region R.

But in this limit, the values rk1and rk2both approach

a common value r, and so the term = rappears in

the integrand. (Similar calculations explain why one

must insert an rwhen converting an integral to

CYLIN

-

DRICAL COORDINATES

, and a term ρ2sinϕwhen con-

verting to

SPHERICAL COORDINATES

.)

See also C

ARL

G

USTAV

J

ACOB

J

ACOBI

;

ROSE

.

Pólya, George (1887–1985) Hungarian Education,

Geometry, Number theory Born on December 13,

1887, in Budapest, Hungary, mathematician George

Pólya is remembered not only for his influential work

in the fields of

PROBABILITY

theory, mathematical

physics,

GEOMETRY

,

COMBINATORICS

, and

NUMBER

THEORY

, but also for his landmark work in mathemati-

cal education and problem solving. His famous work

How to Solve It provided heuristic tools for breaking

complicated problems into simpler components and

developing strategies of attack, and furnished practical

insights to finding solutions and understanding the cre-

ative process of doing mathematics. The work was

immensely influential and was translated into 17 lan-

guages, selling over 1 million copies. Pólya is also

noted for coining the name

CENTRAL

-

LIMIT THEOREM

for the famous fundamental result used in

STATISTICS

.

Pólya received a doctorate in mathematics from the

University of Budapest in 1912 after completing a the-

sis on the topic of geometric probability. He was inter-

ested in not only understanding mathematics, but also

the process by which it is discovered. In 1913 he began

work on a textbook in

ANALYSIS

that organized topics

and problems not by their subject, but rather by their

method of solution. This thinking soon led him to the

work on the art of problem solving and the writing of

his famous text How to Solve It. In brief summary,

Pólya developed the following basic four-stage plan to

solving a problem:

1. Understand the problem. (Identify the knowns and

the unknowns. Draw a diagram or a table. Use the

diagram to identify relationships between the vari-

ables or do the same with algebraic formulae.)

2. Devise a plan. (Consider if the problem is similar to a

previously solved example. Look for a pattern. Con-

sider known formulae that may be applicable. Make

guesses and use trial and error. Break the problem

into smaller parts or cases. Work backwards from the

stated solution. Restate the problem.)

3. Carry out the plan. (Check each step. Prove or ver-

ify each step.)

4. Look back, check work, and reflect on the problem.

(Does the solution obtained make sense? Can the

solution be found via a different, and perhaps shorter,

method? Ask “what if” questions. What if the ques-

tion is changed in a certain way? What if some condi-

tions or statements in the problem are omitted?)

By identifying the thinking process behind solving

problems for the first time, Pólya revolutionized the art

of mathematics education. He submitted that the great-

est good in teaching mathematics comes from providing

students the opportunity to discover results for them-

selves and that the development of problem-solving

skills provides the appropriate means for this to occur.

Pólya received many honors throughout his life,

including election to the London Mathematical Society,

r+ r

——

2