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PROBLEM SOLVING IN MATHEMATICS BY POLYA



A.    Biografy of George Pólya


“Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only byimitation and practice. . . . if you wish to learn swimming you have to go in the water, and if you wish tobecome a problem solver you have to solve problems.- Mathematical Discovery”

The undisputed father of mathematical problem solving is George Pólya (December 13, 1887 – September 7, 1985), one of the giants of classical analysis in the 20th century. 
Pólya was born in Budapest, Hungary and died in Palo Alto, California, almost 98 years later. Both of his parents were born Jewish but converted to Catholicism. His father was born Jakab Pollák, a surname suggesting Polish origins. Jakab changed his name to the more Hungarian , believing this would help him obtain his goal of a university position. He was a talented solicitor, but because he often accepted cases without fees, he was not a financial success. George, who was originally called György, attended Dániel Berzsenyi Gymnasium, where he earned a fine academic reputation, but did not shine in mathematics. Initially he resisted the career that fate had in store for him, because as he later recalled his mathematics instructors who should have been his models were “despicable teachers.”
Even at an early age George had great skill for analyzing and solving problems. His uncle encouraged him to pursue a mathematical career but Pólya wanted to become a lawyer like his father. He entered the University of Budapest, became bored with all the legal technicalities he was required to memorize. After reading Charles Darwin’s The Descent of Man, Pólya briefly took up the study of  biology, but when his brother insisted there was no money to be made in the subject, George shifted to languages and literature. Next he turned to philosophy but to better understand it he had to learn mathematics and he was hooked. He was awarded a PhD in mathematics from the University of Budapest (1912) for an essentially unsupervised thesis in geometric probability. He spent the following year in Göttingen.
Pólya’s first job was tutoring the son of a baron. His pupil struggled with mathematics because he lacked problem-solving skills. To deal with this Pólya began developing his method of problem solving, which he hoped would not only work for his student but for others facing a similar challenge. He was convinced that problem solving was not some special ability that some are born with and others not, but rather was a practical skill that could be taught to anyone, and if students are to have a chance of understanding mathematics, it must be learned. In 1914 he was invited to teach at Zurich and while in Switzerland he made two major discoveries. One was Stella Weber, with whom he spent 67 years of married life. The second discovery came to him as he took walks in a local park. It led to what he called the “random walk problem.” Some years later he published a paper proving that if one continued a walk on a grid long enough, one was certain to return to the starting point. In 1921 he investigated what he called “street networks,” which are now referred to as “lattices.”
In 1924 Pólya was the first International Rockefeller Fellow, spending a year in England, where he worked with G.H. Hardy and John Littlewood. Nine years later, he was once again a Rockefeller Fellow, spending the year at Princeton University. Although his main mathematical interest was in real and complex analysis, he also made contributions to probability, combinatorics, algebra, number theory, voting systems and astronomy. Other mathematicians’ elaborations on his major contributions have become the foundations of several important branches of mathematics. Independently, Pólya and Hilbert conjectured that the zeros of the Riemann zeta function correspond to the eigenvalues of a self-adjoint Hermitian operator. His main contribution to combinatorics is the enumeration theorem. He collaborated with Hardy and Littlewood on the first systematic study of inequalities.
In 1940 George and Stella moved to the United States because of their concerns about Hitler and the Nazis in Germany. He taught at Brown University for two years, and then spent a short time at Smith College before finally moving to Stanford in California, in 1942, where he stayed for the rest of his life. He retired in 1954 but continued to teach until 1978. Pólya was a masterful storyteller, a man of rare wit, insight, enthusiasm, and tremendous curiosity. He was a genuinely friendly individual who enjoyed entertaining visitors by showing them pictures of famous mathematicians he had known, and recalling delightful and amusing instances in their lives. While Pólya was correct in believing that he could teach others his skill for problem solving, it is a shame that his other strengths can’t be taught as easily. In “George  (1887-1985),” Mathematics Magazine, December 1987 M.M. Schiffer stated: “The driving force in his research was the search for beauty and the joy of discovery.”

Books of Polya:
*      His books on the subject How To Solve It (1945)
which has been translated into 21 languages, has sold more than a million copies over the years. with Pólya’s How To Solve It, they have already developed their own strategies for learning mathematics. There seems to be an extra principle that should be added to Pólya’s – one preceding number four. That is, pretending you are interested in finding the solution of a problem until you get to the point that you are. Teaching mathematics is difficult for reasons that appear to be unique to the subject. One of these is that there are several languages integral to the learning process with which teachers and pupils alike must be conversant.
*      Two-volume set Mathematics and Plausible Reasoning (1954),
*      Mathematical Discovery (1962) are classics.
His Ten Commandments for Teachers, found in his book Mathematical Discovery (1981) are as follows:
1. Be interested in your subject.
2. Know your subject.
3. Know about the ways of learning: the best way to learn anything is to discover it by yourself.
4. Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place.
5.  Give them not only information, but “know-how,” attitudes of mind, the habit of methodical work.
6.  Let them learn guessing.
7.  Let them learn proving.
8.  Look out for such features of the problem at hand as may be useful in solving the problems to come – try to disclose the general pattern that lies behind the present concrete situation.
9.  Do not give away your whole secret at once – let the students guess before you tell it – let them find out for themselves as much as feasible.
10.  Suggest it; do not force it down their throats.”

A.    What Is Problem Solving?
Problem solving and reasoning are now more prominent in school mathematics than ever before. As we have stated previously, we believe that reasoning skills will be developed through a continuous emphasis on problem solving. Students are confronted by problems both in school and in their daily activities. For example, in the classroom, problems are represented by the techer or from the textbook.
A problem is a situation, quantitive or otherwise, that confronts an individual or group of individual, that requires resolution, and for which the individual sees no apparent or obvious means or path to obtaining a solition. Using this idea of the problem, problem solving emerges as a process. In fact,
It (problem solving) is the means by which an individual uses previously acquired knowledge, skills, and understanding to satisfy the demands of an unfamilar situation.
Problem Solving also means engaging in a task for which the solution method is not known in advance.  In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understandings.  Solving problems is not only a goal of learning mathematics but also a major means of doing so.  Students should have frequent opportunities to formulate, grapple with, and solve complex problems that require a significant amount of effort and then be encouraged to reflect on their thinking.
By learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom.  In everyday life and in the workplace, being a good problem solver can lead to great advantages.  Problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program.  Problem solving in mathematics should involve all five content areas:  Number and Operations, Algebra, Geometry, Measurement, and Data Analysis & Probability.
Trying to find the solution, we may repeatedly change our point of view, or way of looking at the problem. We have to shift our psition again and again. Our conception of the problem is likely to be rather incomplete when we start the wprk, our out look is different when we have made some progress, it is again different when we have almost abtained the solution.





B.     The Differences of Problem and Task
Not everyone agrees on what exactly is a problem. There may not be a perfect definition of “problem,” but it is possible to make a distinction between a problem and a task. With a task, one knows what to do and how to do it; with a problem, neither what to do nor how to do it may be known, at least initially. One of the first steps in problem solving is to carefully describe the problem one wishes to solve, and this may not be easy. Whereas in classrooms, problems are presented to students to solve, in the real world, problems are not ordinarily assigned by someone to be solved, but are discovered to need solving. Even some of those successful in “solving” problems, may not understand problems. Many have been taught to mimic a procedure, and as good imitators they get correct answers, but may have little awareness of what they have done or of the value of their labor.
If students merely imitate procedures, no plan is needed, and this becomes obvious when they are asked to work on a new, unfamiliar problem. They do not think about the problem or a plan of attack. Instead they ripple through their Rolodex of memorized techniques, hoping to find a trick that works. For them, carrying out a plan consists of mindlessly imitating the technique they have learned. After they get an answer, they don’t look back, not even to see if their result makes any sense in the context of the problem. As the primary goal of good students is to understand things, it should be the goal of good teachers to help others understand. This cannot be accomplished through memorization or imitation. By the time some people become acquainted
There is the mathematics itself, the language of mathematics, the language of logic, and a meta-language, which teachers use to instruct students in the language of mathematics so that mathematics itself may be understood. For instance, meta-language is used to provide students with an intuitive understanding of mathematical concepts before they are formalized or abstracted. By examining meta-languages used by teachers it is possible to determine likely communication failures that prevent willing instructors from helping willing students realize their mathematical potential in problem solving
“A mathematics teacher is a midwife to ideas.” – George Pólya
C.    PROBLEM SOLVING STRATEGIES BY POLYA

In order to group conveniently the question and suggestions of our list, we shall distinguish four phases of the work. Polya  identifies four principles that form the basis for any serious attempt at problemsolving:

SEE, PLAN, DO, CHECK
1.  Understand the problem (SEE)
2.  Devise a plan (PLAN)
3.  Carry out the plan (DO)
4.  Look back (CHECK)

     Each  of these phase has its importance. It may happen that a student hits upon an axcptionally bright idea and jumping all preparation bluits out with the solution. Such lucky idea, of course, are most desirable, but something very undesirable and unfortunate may result if the student leaves out any of the four phase whithout having a good idea. The worst may happen if the students embarks upon computations or constructions whitout having seen the main connection , or having made a sort of plan. Many mistakes can be avoided if, carrying out his plan, the students checks each step. Some of the best effects may be lost if the students fail to reexamine and to reconsider the completed solution. For each phase will explain below :

1.  Understand the problem
"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. The students should understand the problem. But he should not only understand it, he should also desire its solution. If the student is lacking in understanding or interest, it is not always his fault; the problem should be well chosen, not too difficult and not too easy, natural and interesting, and some time should be allowed for natural and interesting presentation. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:
·         What are you asked to find out or show?
·         Can you draw a picture or diagram to help you understand the problem?
·         Can you restate the problem in your own words?
·         Can you work out some numerical examples that would help make the problem more clear?
There are the way can you use to Understand the problem :
·         Carrefully read the problem
·         Decide what you are trying to do
·         Identify the important data
The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.
Then we chould subdivided Understanding the Problem into two stages : “Getting acquainted” and “Working for better understanding.”
Example :
Let us illustrate some of the points examplained in the foregoing section. We take the following simple problem :
“Find the diagonal of rectangular paraleleliped of which the length, the width, and the height are known.”
In order to discuss this problem profitably, the students must be familiar with theorema pythagoras, and with some of its applications in plane geometry, but they may have very little systematics knowledge in solid geometry. The dialogue between the teacher and the students may start as follows :
     “What is the unknown?”
     “the length of the diagonal of a parallelepiped.”
     “What are the data?”
     “The length, the width, and the height of the parallelepiped.”
     “Introduce suitable notation. Which letter shpuld denote the unknown?”
     “x”
     “which letters would you choose for the length, the width, and the height?”
     “a,b,c.”
     “What is the condition, linking a,b,c and x?”
     “x is the diagonal of the parallelepiped of which a,b,c are the length, the width, and the height
     “Is it a reasonable problem? I mean, is the condition sufficient to determine the unknown?”
     “Yes, it is. If we know a,b,c we know the paralleliped. If the parallelepiped is determined, the diagonal is determined.”

2.  Devise a plan
In order to be able to see the student’s position, the teacher should think of his own experience, of his difficulties and successes in solving problem. We know, of course, that it is hard to have a good idea if we have little knoledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge. Mere remembering is not enough for a good idea, but we cannot have any good idea without recollecting some pertinent facts; materials alone are not enough for constructing a house but we cannot construct a house without  collecting the necessery materials. The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems, or formerly proved theorems. Thus, it is often appropriate to start the work with the question: Do you know a related problem?
Pólyamentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
§  Gather together all available information
§  Consider some possible actions
§  Look for a pattern
§  Draw a sketch
§  Make an organised list
§  Simplify the problem
§  Guess and check
§  Make a tabke
§  Write a number sentence
§  Act out the problem
§  Identify a sub-task
§  Check the validity of given information

Example (continuous the example in number 1)
The dialogue between the teacher and the students may start as follows :
“do you know a related problem?”
                .......
    “look at the unknown! Do you know a problem having the same unknown?”
    “well, what is the unknown?”
    “the diagonal of a parallelepiped.”
    “do you know any problem with the same unknown?”
    .......
    “you see, the diagonal is a segment, the segment of a straight line. Did you never solve a problem whose unknown was the length of a line?”
    “of course, we have solved such problem. For instance, to find a side of a right triangle.”
    “good! Here is a problem related to yours and solved before. Could you use it?”
    ..............
“you were lucky enough to remember a problem which is related to your presents one and which you solved before. Would you like to use it? Could you introduce some auxiliary element in order to make its use possible?”
    .............
    “look here, the problem you remembered is about a triangle. Have you any triangle in your figure?”
    ...........
    “you cannot find yet the diagonal; but you said that you could find the side of a triangle. Now, what will you do?”
    “could you find the diagonal, if it were side of a triangle?”
    “I think that was a good idea to draw that triangle.”
    “the unknown is the hyphotenuse of the triangle; we can calculate it by the theorem of pythagoras.”
    “you can, if both legs are known; bu are they?”
    “one leg is given, it is c. And the other, I think, is not difficult to find. Yes, the other leg is the hypothenuse of another right triangle.”
    “very good! Now I see that you have a plan.”

3.  Carry out the plan
This step is usually easier than devising the plan In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals
   Carrying out the plan is usually easier than devising the plan
   Be patient – most problems are not solved quickly nor on the first attempt
   If a plan does not work immediately, be persistent
   Do not let yourself get discouraged
   If one strategy isn’t working, try a different one

Example ( continuous the previous example)
The teacher has no  reason to interrupt the students if he carries out these details correctly except, possibly, to warn him that he sould check each step. Thus, the teacher may ask :
“can you see clearly that the triangle with sides x,y,c is a right triangle?”
“but can you prove that this triangle is a righ triangle?”

4.  Look back (reflect)
Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem
   Does your answer make sense?  Did you answer all of the questions?
   What did you learn by doing this?
   Could you have done this problem another way – maybe even an easier way?
Example( continuous the previous example)
The students finally obtained the solution of the diagona is
The teacher can ask several question about the result which the students may readily answer with “yes” or “no”
“did you use all the data? Do all the data a,b,c appear in your formula for the diagonal?”
“length, width, and height play the same role in our question; our problem is symmetric with respect to a,b,c. Does it remain unchanged when a,b,c are intercanged?”
“our problem is a problem of solid geometry; to find the diagonal of a parallelepiped with given dimensions a,b,c. Is the result of our ‘solid’ problem analogous to the result of ‘plane’ problem?”
“if you put c=0 in your formula, parallelepiped becomes a parallelogram. Do you obtained the correct formula for the diagonal of the rectangular parallelogram?”

A.    APPLYING OF POLYA’S PROBLEM SOLVING IN MATHEMATICS

1.        GUESS AND CHECK

Copy the figure below and place the digits 1, 2, 3, 4, and 5  in these circles so that the sums across (horizontally) and down (vertically) are the same.  Is there more than one solution?






SOLUTION :
Emphasize Polya’s four principles- especially on the first several examples, so that procedure becomes part of what the students knows.
1.      Understand the problem.
Have the students discuss it among themselves in their groups of 3, 4 or 5.
2.       Devise a plan.
Since we are emphasizing Guess and Check, that will be our plan.
3.      Carry out the plan.
It is best if you let the students generate the solutions.  The teacher should just walk around the room and be the cheerleader, the encourager, thefacilitator.  If one solution is found, ask that the students try to find other(s).

Possible solutions:

2    3    2
1 3 5  2 1 5  1 5 4
4    4    3
Things to discuss (it is best if the students tell you these things)

·         Actually to check possible solutions, you don’t have to add the number in the middle – you just need to check the sum of the two “outside” numbers.
·         2 cannot be in the middle, neither can 4.  Ask the students do discuss why.
4.      Look back.
Is there a better way?  Are there other solutions? Point out that “Guess and Check” is also referred to as “Trial and Error”.   However, I prefer to call this “Trial and Success”, I mean, don’t you want to keep trying until you get it right?

Below is an exercise to assign for the next day, which is also included in the STUDENTPROBLEMS.
1.      Put the numbers 2, 3, 4, 5, and 6  in the circles to make the sum across and the sum down equal to 12.  Are other solutions possible?  List at least two, if possible

SOLUTION:  One possibility            2
Other solutions possible.         3 4 5
Have students suggest those.               6

1.      MAKE AN ORGANIZED LIST


Three darts hit this dart board and each scores a 1, 5, or 10.The total score is the sum of the scores for the three darts.There could be three 1’s, two 1’s and 5, one 5 and two 10’s,And so on.  How many different possible total scores could aperson get with three darts?

SOLUTION:
1st.  Understand the problem.
Gee, I hope so.  ☺  But let students talk about it just to make sure.
2nd.  Devise a plan.  Again, it would be what we are studying:  Make an organized or
orderly list.  Emphasize that it should be organized.  If students just start throwing out
any combinations, they are either going to list the same one twice or miss some
possibilities altogether.
3rd.  Carry out the plan.
There are 10 different possible scores.

4th.  Look back.   Point out the there are other ways to “order” the possibilities
List the 4-digit numbers that can be written using each of  1, 3, 5, and 7 once and only once.  Which strategy did you use?
SOLUTION:
1357  1735  3517  5137  5713  7315
1375  1753  3571  5173  5731  7351
1537  3157  3715        5317  7135  7513
1573  3175  3751        5371  7153  7531

24 possible 4-digit numbers.

1.      MAKE A TABLE
Pedar Soint has a special package for large groups to attend their amusement park:  a flat fee of $20 and $6 per person.  If a club has $100 to spend on admission, what is the most number of people who can attend?



SOLUTION:
1st.  Understand the problem.
Students may need to discuss this a little before attempting to tackle the problem.
2nd.  Devise a plan.
Make a table.  But develop what should be in the table with the students.  Let them
assist how you make this table.
3rd.  Carry out the plan.
Answer:  At most, 13 people can attend for $100 and they will have $2 l         eft over.
4th.  Look back. 
Is there another way this could be done?  Yes, guess and check (whichis part of what we did).  The difference is that we tried to do this in an orderly fashion  not just guess randomly.  We tried to “surround” the solution.

Assign the following problem.
Stacey had 32 coins in a jar.  Some of the coins were nickels, the others were dimes.  The total value of the coins was $2.80.  Find out how many of each coin there were inthe jar.  What problem solving strategy did you use?

SOLUTION:   8 nickels, 24 dimes

References :

Polya, George. 1957. How To Solve It : A New Aspect of Mathematical Method. United state of America : Princeton University Press, Princeton, New Jersey.
Polya, George. 1887. Mathematical Discovery. Canada : John Wiley & Sons.
Billstein, Rick. And friends. 2010. A problem Solving Approach to mathematics for elementary School Teachers. United state of America : Pearson














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