History of Polya

8/13/2019 History of Polya

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DEFINITION OF PROBLEM

A problem is decided by purposes. If someone wants money and when he or she

has little money, he or she has a problem. But if someone does not want money, little

money is not a problem. Problem can be defined in many things. There are many

problems that we face in our daily life. Other example is, if your computer is damaged,

you do not have a computer to do assignment. The problem here is you do not have acomputer to do your ob. These are the example of a problem.

In mathematics, problem also defined as a preposition that re!uires solution by

mathematical operation and construction. There are many types of solution we can

found when we are solving a mathematic problem. "or example, what is the number

that should be added so that it gets a total of #$ The way to solve this !uestion is ust

divide the number so that we get %. That&s mean % plus % is #. That&s the answer. The

next solution is we can try the solution of trying an error. 'e can add a number that

below six until we get the answer. This is ust a simple type of example. There are many

more examples that can be defined.

Problem also defined as a !uestion that was raised for consideration or solution.

Problem can be solver in many type of solution depend on the problem. Problem also

defined as a !uestion, matter, situation or person that is perplexing or difficult.


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HISTORY OF POLYA

(eorge Polya

)*++ - *+/0

(eorge Polya was a 1ungarian who immigrated to the 2nited 3tates in *45. 1is

maor contribution is for his wor6 in problem solving.

(rowing up he was very frustrated with the practice of having to regularly

memori7e information. 1e was an excellent problem solver. 8arly on his uncle tried

to convince him to go into the mathematics field but he wanted to study law li6e his

late father had. After a time at law school he became bored with all the legal

technicalities he had to memori7e. 1e tired of that and switched to Biology and the

again switched to 9atin and 9iterature, finally graduating with a degree. :et, he tired

of that !uic6ly and went bac6 to school and too6 math and physics. 1e found he

loved math.

1is first ob was to tutor (regor the young son of a baron. (regor struggled due to

his lac6 of problem solving s6ills. Polya );eimer, */0 spent hours and developed a

method of problem solving that would wor6 for (regor as well as others in the same

situation. Polya )9ong, *#0 maintained that the s6ill of problem was not an inborn

!uality but, something that could be taught.

1e was invited to teach in


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1e later did experiment that he called the random wal6 problem. 3everal years

later he published a paper proving that if the wal6 continued long enough that one

was sure to return to the starting point.

In *45 he and his wife moved to the 2nited 3tates because of their concern for

>a7ism in (ermany )9ong, *#0. 1e taught briefly at Brown 2niversity and then, for

the remainder of his life, at 3tanford 2niversity. 1e !uic6ly became well 6nown for

his research and teachings on problem solving. 1e taught many classes to

elementary and secondary classroom teachers on how to motivate and teach s6ills

to their students in the area of problem solving.

In *4/ he published the boo6 ?1ow to 3olve It& which !uic6ly became his most

pri7ed publication. It sold over one million copies and has been translated into *

languages. In this text he identifies four basic principles .

Polyas four steps of Problem Sol!"#


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$% &"'ersta"' t(e Problem

;ead the problem carefully.

@a6e sure you understand the situation that is given.

@a6e sure you understand what information is provided, and what the

!uestion is as6ing.

"or many problems, drawing a clearly labeled picture is very helpful.

)% De!se a Pla"

"irstly, focus on the obective. 'hat do you need to 6now in order to

answer the !uestion$

Then loo6 at the given information. 1ow can you use that information to

get what you need to 6now to answer the !uestion$

If you do not see a clear logical path leading from the given information

to the solution, ust try something.

9oo6 at the given information and thin6 about what you can find from it,

even if i t is not what the !uestion is as6ing for. Often you wil l f ind

another piece of information that you can then use to answer the

!uestion.


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*% +arry out t(e Pla"

>ow you ust have to solve the e!uat ion)s0 for the un6nown)s0.

;emember to answer the !uestion that the problem as6s.

Try to express mathematically the logical connections between the

given information and the answer you are see6ing

Assign variab le names to the un6nown !uanti ti es.

Translate the sentences and words into @athematics.

,% Loo- Ba.-

Thin6 about your answer.

=oes your answer come out in the correct units$

Is it reasonable$ If you made a mista6e somewhere, chances are your

answer will not ust be a little bit off, but will be completely ridiculous


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$$ /ays of Polyas Mo'el Problem Sol!"#

$0 1&ESS AND +HE+2

'hen using this strategy, you are encouraged to ma6e a reasonable guess and

chec6 the guess. ;epeating this process can allow you to arrive at a correct

answer that has been chec6ed. 2sing this strategy does not always yield a

correct solution immediately but provides information that can be used to better

understand the problem and may suggest the use of another strategy.

To use the guess and chec6 strategy, follow these steps

a0 @a6e a guess at the answer.

b0 hec6 your guess. =oes it satisfy the problem$

c0 2se the information obtained in chec6ing to help you ma6e a new guess.

d0 ontinue the procedure until you get the correct answer.

)0 MA2E AN ORDERLY LIST

@a6ing an orderly list is a way to organi7e data presented in a problem.This problemCsolving strategy allows the problem solver to discover

relationships and patterns among data.

*0 MA2E A DRA/IN1

@a6ing a diagram to solve problems can help you understand and manipulate

data. @a6e a diagram strategy is especially useful with problems that involve

mapping, geometry and graphing.


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,0 FIND A PATTERN

A pattern may be in various forms such as numerical, visual or aural. By

identifying a pattern in a problem, you can predict what will happen later in the

se!uence.

30 MA2E A TABLE

@a6ing a table is another way to organi7e the data. This allows the problem

solver to discover relationships and patterns among data.

40 &SE A 5ARIABLE

This problemCsolving strategy is similar to developing a formula or an algebraic

e!uation. In each of these cases the solution re!uires finding values that meet

the conditions of the problem.

60 +ONSIDER A SPE+IAL +ASE

'hen considering a special case one will choose for example the first few

values in a se!uence or a specific value in a formula and then try to ma6e a

generali7ation about the problem

70 SOL5E AN EASIER SIMILAR PROBLEM

This strategy involves changing the given problem into one that can be easier to

solve and, by solving this secondary problem, you can gain insight needed to

solve the original problem.


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80 /OR2IN1 BA+2/ARDS

The strategy of wor6ing bac6wards involves starting with the end results and

reversing the steps you need to get those results, in order to figure out the

answer to the problem.

$90 ELIMINATE POSSIBILITIES

:ou can use the problemCsolving strategy of eliminating possibilities to solve

problems that may have several possible answers from which to choose. :ou

are not loo6ing for the correct answers, rather you are loo6ing for several

incorrect answers. As each incorrect answer is eliminated, you get closer to the

correct answer. The problemCsolving strategy of eliminating possibilities is also

useful when you solve logic problems.

**0 PI1EONHOLE PRIN+IPLE

If >D* pigeons are put into > holes, then at least one hole would have more

than one pigeon.

The pigeonhole principle relies on filling existing spaces )pigeonholes, boxes,

and the li6e0 with items )pigeons, coins, and so on0 to the point where all spaces

are ust one item short of being full. At this point, no matter where the next

items are placed, all spaces are full, and one space has more than one item.

PROBLEM AND PROBLEM SOL5IN1


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Problem

*. a !uestion proposed for solution or consideration

E. a !uestion, matter, situation, or person that is perplexing or difficult

%. A proposition re!uiring solution by mathematical operations, constructions.

4. presenting a problem of human conduct or social relationships aproblemnovel

Problem sol!"#

1. "orms part of thin6ing.

E. onsidered the most complex of all intellectualfunctions, problem solving has

been defined as higherCorder cognitiveprocess that re!uires the modulation and

control of more routine or fundamental s6ills.

RO&TINE PROBLEM
http://en.wikipedia.org/wiki/Thoughthttp://en.wikipedia.org/wiki/Intelligencehttp://en.wikipedia.org/wiki/Cognitivehttp://en.wikipedia.org/wiki/Thoughthttp://en.wikipedia.org/wiki/Intelligencehttp://en.wikipedia.org/wiki/Cognitive


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;outine problem solving involves using at least one of the four arithmetic operations

andFor ratio to solve problems that are practical in nature. ;outine problem solving

concerns to a large degree the 6ind of problem solving that serves a socially useful

function that has immediate and future payoff. hildren typically do routine problem

solving as early as age / or #. They combine and separate things such as toys in the

course of their normal activities. Adults are regularly called upon to do simple and

complex routine problem solving. 1ere is an example.

8xample of routine problem

A sales promotion in a store advertises a jacket regularly priced at $125.98 but

now selling for 2! off t"e regular price. #"e store also waives t"e ta. %ou "ave

$1 in your pocket &or $1 left in your c"arge account'. (o you "ave enoug"

money to buy t"e jacket)

NON:RO&TINE


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>onCroutine problem solving serves a different purpose than routine problem solving.

>onCroutine problem solving concerns that only indirectly. >onCroutine problem solving

is mostly concerned with developing students& mathematical reasoning power and

fostering the understanding that mathematics is a creative 8ndeavour. "rom the point of

view of students, nonCroutine problem solving can be challenging and interesting.

3tatic

3tatic nonCroutine problems have a fixed 6nown goal and fixed 6nown elements

which are used to resolve the problem. 3olving a igsaw pu77le is an example of

a static nonCroutine problem.

Active

Active nonCroutine problem solving may have a fixed goal with changing

elementsG a changing goal or alternative goals with fixed elementsG or changing

or alternative goals with changing

elements

The following is an example of a problem that nonCroutine problem

*onsider w"at "appens w"en +5 is multiplied by ,1. #"e result is 1,+5. -otice

t"at all four digits of t"e two multipliers reappear in t"e product of 1,+5 &but t"ey

are rearranged'. ne could call numbers suc" as +5 and ,1 as pairs of stubborn

numbers because t"eir digits reappear in t"e product w"en t"e two numbers are

multiplied toget"er. /ind as many pairs of 20digit stubborn numbers as you can.

#"ere are pairs in all &not including +5 ,1'.


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;&ESTION $

1&ESS AND +HE+2


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$0 Four .olours

'rite the numbers from * to 4 on four sets of different coloured chips. @a6e a 4 by 4

board and colour it, using the pattern above, with the same colours as the chips.

Put the chips on the board so that each row and column contains all four numbers. :ou

have to put each chip on a s!uare of the same colour. The first four chips have to be

placed as shown.

3tep * 2nderstand the !uestion

The !uestion want we put the chips on the board. 8ach chips must be put on the same

colour of it. 8ach rows and column on the board must have contains all four numbersstarting from * until 4.

3tep E =evising a plan.

'e can use ma6e guess and chec6,so we can see which number will suitable to put on

the board. If our first guess does not wor6, try put different numbers on the board.

3tep % arry the plan.


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F!rst .(e.-

, ) * $

$ $ $ ,

) , ) *

$ * , )

hec6 this guess

It can&t accept because in line two there are one number that are same.

Se.o"' .(e.-

) $ * ,

$ ) , $

* , ) *

, * $ )

hec6 this guess

It can accept because all the row have different numbers and satisfy the !uestion.

3tep 4 9oo6ing bac6.

If we loo6 the pattern from the diagram we can see the numbers are in the up row is

same li6e at the down row on the board. But on at the down row it is inversed from up

row.

;&ESTION )


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Hohn as6s his father for the homewor6 that teacher given. Hohn does not 6now how to

solve this problem. Teacher as6 to place the number of *,E,%,4, and / in these pattern

so that the sum across )hori7ontally0 and down )vertically0 are the same.

Place the number in the box and ma6e sure the sum across and down are the same.

3tep * understanding the problem

Place number *, E, %, 4, and / in the box but the sum across and down number should

be same.

3tep E ma6e a plan

The strategy that we selected is by (uess and hec6.

3tep % devising a plan


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'e start with number * at the middle.

*5 +

# +

Put number E at the middle

** + *5

#


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Put number % at the middle

*E *5

# +

Put number 4 at the middle

*E *5

*E

Put number / at the middle


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*E *5 **

+ *5

If we put number * at the middle we got combination

If we put number % at the middle we got combination


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If we put number / at the middle we got combination

>umber E and 4 cannot be at the middle because it is even number. Odd number such

as *,% and / it flexible being at the middle.

3tep 4 loo6ing bac6

C Actually to chec6 possible solutions, you don&t have to add the number in the middle -

you ust need to chec6 the sum of the two outsideJ numbers.

C E cannot be in the middle, neither can 4.

;&ESTION *


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*0T(e

one hat in each compartment in such a way that every row and column has four

different coloured hats and four different arithmetic signs.

3he has started to put the hats in the crate. The picture shows where she has put

four of the hats. an you finish pac6ing the crate for her$

3tep * 2nderstand the !uestion

The !uestion want we put the hats in the compartment. 8ach hats must be put indifferent colour and different arithmetic sign. 3o in the row and column have no same

colour and same signs.

3tep E =evising a plan.


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'e can use ma6e guess and chec6,so we can see which hats will suitable to put in the

compartment that all row and column have different colur and signs. If our first guess

does not wor6, try put different hats in the compartment.

3tep % arry out the plan.

F!rst .(e.-

? @ C

: @ ?

@ : ?

@ :

hec6 guess

It can&t accept because there have same colour and signs hats in a rows and in a

columns

Se.o"' .(e.-

? C @

: @ ?

@ : ?

? @ :

hec6 this guess

This guess we can accept because all the column and row have different colour and

signs of hats. It also satisfy the !uestion.


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hec6 the answer and ma6e sure in each rows and columns have no same colour and

arimethic signs of hats.


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;&ESTION $ C NON RO&TINE PROBLEM 0

3how how to draw four line segments through the nine dots shown below without lifting

your pencil from the paper.

3tep * understand the problem

- 'e need to connect all the nine dots without lifting your paper.

- 'e cannot ma6e a double line in a same place.

3tep E devising a plan

The strategy we selected by trial and error.

>early everyone who attempts this problem becomes frustrated by assuming that the

line segments must lie within the confines of the % by % array. But by removing this

unnecessary restriction, it opens the door to the solution shown below.


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3tep % carry out the plan

3tep 4 loo6ing bac6

3ometime we have to change our point of view.

This is very hard for most people. But we want the students to consider other

possibilities. >ot ust for this problem, but for any problem in which they get stuc6

First trial

Second trial


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;&ESTION )

)0+ats .o


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3tep % arry the plan.

F!rst .(e.-

+AT +O/ PI1

PI1 +O/ +O/

+AT +AT PI1

hec6 this guess

It can&t accept because there have two same animals in a row and in a column

Se.o"' .(e.-

+O/ +AT PI1

PI1 +O/ +AT

+AT PI1 +O/

hec6 this guess

This guess we can accept because all the column and row have different animals. It

also satisfy the !uestion.


If we can see the pattern in this !uestion all the animal will moves form one pen to the

beside pen. 3o no animal will in same in row or column.


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REFLECTION

Alhamdulillah, finally i managed to finish this basic math coursewor6 very well on

time as i have been given ust two wee6s to finished it. =uring the process to finish the

course wor6, i have learnt many new things that i can use when i are teaching. I have

learnt on how to solve problem using four step of Polya&s model. By using Polya model&s

step, it is easier to solve and chec6 the answer again.

I also 6now how to solve problem using many strategy in Polya, model such as

wor6ing bac6wards, find a pattern, using the table , using variables and so on. There

are many 6ind of way to solve mathematics problem in Polya&s model especially for nonC

routine !uestion. I find the suitable and different strategy to solve the problem for each

!uestion.

I also leart that >onCroutine problem is which the person have a problems but we

don&t have a procedure or don&t have a s6ills to solve the problems. 'e must try to

solving with multi s6ills. >ow, I also 6now that it have a lot of comparison between

routine and nonCroutine problems such as routine problems most basic simple type of

problemCsolving in mathematics however nonCroutine problems needs a set ofsystematic activities with logical planning and some else.

At first, i thought it is difficult to do this tas6 but after i had been explained by my

@athematic&s lecturer, I had found that it is easy. I can finish this tas6 at the time

because all of us give a good cooperation each other. Besides that, i also find the

addition information in reference boo6 in library and internet. Actually, i have learnt

about Polya&s model before I am giving this tas6. 3o i can understand and i hope that i

will get good feedbac6 from the lecturer.

PREPARED BY NORLYANA BINTI SA2AR&DI C 899$)7:9*:33)) 0


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BIBLIOGRAPHY

3at"ematics #etbook %ear ,. 'an :usof 'an >gah, 9ee (i6 9ean dan

;abiyah "a6ir @ohd. =ewan Bahasa dan Pusta6a.

3at"ematics #etbook %ear 5. @ohammad Lhairuddin bin :ahya, @ai7ita binti

Puteh dan 3anthi Periasamy. =ewan Bahasa dan Pusta6a.

3at"ematics %ear #eac"er4s uidebook. 9ai Lim 9eong, ;a7ali bin @ohd Ali,

@ohammad Lhairudin bin :ahya, @ai7ita binti Puteh dan 3anthi Periasamy. =ewan

Bahasa dan Pusta6a.

httpFFwww.math.wichita.eduFhistoryFmenFpolya.html E februari E55

httpFFwww.math.utah.eduFMalfeldFmathFpolya.html E februari E55

httpFFfaculty.salisbury.eduFMdccathcartF@AT1E%5FPolya.html
http://www.math.wichita.edu/history/men/polya.html%2027%20februari%202009http://www.math.utah.edu/~alfeld/math/polya.html%2027%20februari%202009http://www.math.wichita.edu/history/men/polya.html%2027%20februari%202009http://www.math.utah.edu/~alfeld/math/polya.html%2027%20februari%202009


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RECOGNITION PAGE

'e recogni7e this wor6 is the result of our own except for !uotation and a summary of

each of them we describe the source.J

3ignature NNNNNNNNNNN.

NNNNNNNNNNN.

>ame @uhammad 1afi7ie Bin orlyana Binti 3a6arudi

3iti >ur Amalina Binti @ohamed >or

=ate 20thSeptemberE5*5


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TITLE PAGE

P;OB98@ 3O9I>( 3T;AT8(I83

) BA3I @AT18@ATI3 0

MATHEMATI+S DEPARTMENT

INSTIT&TE OF ED&+ATION TEA+HERS

+AMP&S ED&+ATIONAL TE+HNI;&E


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REFLECTION

=uring doing the assignment, the source that I are obtained to get the information

about this topic are li6es from the library, boo6s, and also by searching from the internet

that we are already 6now IT is give us a fast and brief explanation about the some

topics.

Besides that, I also learned to be a more wiser girl in time management aspects

in my daily life. I can manage my time to do many wor6 if I 6now in the step in an

effective time management. "urthermore, during doing the assignment I also found the

difficulties. 8ven though I had faced a some trouble during doing the assignment , but I

could finish all the wor6 successfully. 3ome of the trouble that I got during doing the

assignment were internet interruption during searching the information about the social

ills among the youth nowadays. But the error of the internet was not too often during

searching the information.

Then , the problem that I had faced during the period of tas6 wor6ing was the

difficulties to find the information by using the boo6s, ournal, encyclopedia and article

were too6 a lot of time and there are too much of time we used during searching the

information&s. But I ta6e all that things as an advantages in the step to produce the

!uality product. I had tried to overcome all the trouble during do the wor6 efficiently and

try to do all the all the best for the tas6 gave.

Besides that, I can learn many things about the shape that are very important in

our everyday life because in our daily things, mostly all the things that we are using

involved the shapes and that&s why we have to study and 6now about the concept of

problem solving that we can applied them in our daily routine. 9ast but not least, I

really want to stressed again that all of us should try to learn and try to find a solution to

solve this serious matter in our daily life especially in order to save our next generation,

the youth.PREPARED BY SITI N&R AMALINA BINTI MOHAMED NOR C 8994$7:9*:3**9 0


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+OMPARISON BET/EEN RO&TINE AND NON RO&TINE PROBEM

RO&TINE NON:RO&TINE

1ave many ways to solve it Try and error

;outine problem solving involves

using at least one of the four

arithmetic operations

>onC routine problem solving

stresses the use of heuristics and

often re!uires little

to no use of algorithms

;outine problem solve problems

that are practical in nature and

solving concerns to a large degree

the 6ind of problem solving

>on C routine problem solving

concerns that only indirectly.

There are no types of routine

problem

There are two types of nonCroutine

problem solving situations, static

and active

History of Polya

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