Jhaymie P. Dizon June 24, 2011

BEED 3-1 Problem Solving

1. George Pólya (1887-1985)

George Pólya was born in Budapest on 13 December 1887. His father Jakab (who died in 1897) had been born Jakab Pollák, of Jewish parents, and with a surname which suggested Polish origin. It is likely that ancestors had emigrated from Poland to Hungary, where a lesser degree of anti-Semitism existed. However Jakab converted to Catholicism believing that this would help him advancing in a career and changed his name to the more Hungarian Pólya. George’s mother had also been of Jewish background with similar history. Her paternal grandfather, Max Deutsch, had in fact converted to Presbyterianism and worshipped with Greek Orthodox Romanians.

George’s father Jakab had been a solicitor with a great mind, but one who was prepared to pursue a case in which he believed with no fees. He was not financially successful despite the time he lived in being considered a golden age for Hungary.

As a student George attended a state run high school with a good academic reputation. He was physically strong and participated in various sports. His school had a strong emphasis on learning from memory, a technique which he found tedious at the time but later found useful. He was not particularly interested in mathematics in the younger years. Whereas he knew about the Eötvös Competition and apparently wrote it he also apparently failed to hand in his paper.

He graduated from Marko Street Gymnasium in 1905,ranking among the top four students and earning a scholarship to the University of Budapest, which he entered in 1905. He commenced studying law, emulating his father, but found this study boring and changed to language and literature. He had become particularly interested in Latin and Hungarian, where he had had good teachers. He also began studying physics, mathematics and philosophy. His development was greatly influenced by the legendary mathematician Lipót Fejér, a man also of wit and humour, who also taught Riesz, Szegö and Erdös. Fejér had discovered his theorem on the arithmetic mean of Fourier Series at the age of 20.

Pólya soon concentrated his studies on mathematics and in 1910 finished his doctorate studies, except for his dissertation. He took a year in Vienna and returned to Budapest in 1911-12 to give his doctoral dissertation and met Gábor Szegö, seven years younger, who was to become one of his major collaborators.

In the fall of 1912 he went to Göttingen for postdoctoral study and met David Hilbert, Richard Courant, Felix Klein and Hermann Weyl. In 1913 he was offered a position in Frankfurt, but was discouraged from staying in Germany and turned the job down after

being told he was a “bloody Jew” by a ruffian on a train and went on to the University of Paris for further postdoctoral work.

In 1914 he took up a position at the Eidgenössische Technische Hochschule (ETH) in Zurich, an institution which boasted the names of the physicists Röentgen and Einstein (1900) among its graduates. This position was arranged by the mathematician Adolf Hurwitz (1859 to 1919), who had studied at various times under Kronecker, Klein and Weierstrass and was the other great influence on Pólya. The ETH was next door, and closely associated with, the University of Zurich and Polya had joint teaching rights with the University.

In 1914 Pólya was called up by Hungary to fight in the war, but by this time he had adopted Russell’s pacifism and refused to go. The fear that he might be arrested for being unpatriotic meant that he did not return to his native country until after World War II. In Zurich he met his future wife, Stella Weber. They married in 1918 and were still together 67 years later when Pólya died. They had no children.

Inspired by walks in the woods near Zurich, Pólya in 1912 published one of his major results, the solution of the random walk problem. In this problem one walks in an infinite rectangular grid system, at each node having an equal probability of walking to each of the adjoining nodes on his next leg. Pólya was able to show that in the two dimensional case it was almost certain (but with probability 1) that one would eventually return to the original position, but one would almost never (with probabilty 0) return to the origin in the case of three or more dimensions.

Pólya was interested in chemical structure, which led him in 1924 to publishing the classification of seventeen plane-symmetry groups, a result which was later to inspire the Dutch artist M.C. Escher.

In 1924 he spent a year in England, working with G.H. Hardy and J.E. Littlewood at Oxford and Cambridge. This collaboration led to publication in 1934 of the bookInequalities, which included a new proof by Pólya of the AM-GM inequality based on the Maclaurin expansion of the exponential function.

In 1925 Pólya, with Szegö, published arguably one of his most influential books,Aufgaben und Lehrsätze aus der Analysis, volumes 19 and 20 of the series Die Grundlehren der Mathematischen Wissenschaften published by J. Springer, Berlin. A whole generation - the generation of Erdös, Szekeres and their circle, and later, learned their mathematics not so much from the lectures they attended but by trying to solve the problems of this book one after another and debating their solutions with each other. Problem-solving as a method of teaching and learning may never have been practiced on such a scale, and with such success, before (or since).

One of Pólya’s most famous results, the Pólya Enumeration Theorem, was published in 1937. This also arose from his interest in chemical structure and looking at possible configurations of the benzene ring and other figures with 6 vertices. Generalising a

theorem by Burnside in Group Theory, Pólya showed how one can determine the number of different assignments of atoms, or colours, to vertices as sides of geometrical figures.

A special case of this is the Necklace Theorem, which shows how many necklaces of n beads can be constructed with k colours available, assuming there is an infinite supply of each colour.

In the case where n is prime and necklaces are regarded as unchanged by rotation, the number of configurations is k+(k^n-k)/n. Thus if there are 5 beads and 3 colours the number of necklaces is 3+(3^5-3)/5=3+240/5=3+48=51.

In the case where n is composite there is also a formula but it is a little more complicated. However more details of this and another version of the theorem can be found here.

In 1940 the Pólyas became increasingly concerned, with George’s Jewish background, of the possibility of a German invasion of Switzerland, and decided to leave for the United States.

He was offered a research position by his old collaborator, Gábor Szegö, now at Stanford, but he did not initially accept it, going instead to Brown University. In 1942 he did move to Stanford, however, where he stayed until his retirement from teaching in 1953. After 1953 he stayed at Stanford, living at Palo Alto until his death, as a Professor Emeritus.

In 1945, Pólya published one of his most famous books, How to Solve It. Then in 1951 he published, with Gábor Szegö, Isoperimetric Inequalities in Mathematical Physics.

Gábor Szegö had been a winner of the Eötvös Competition in 1912 and Pólya saw the value in competitions. In 1946 Pólya and Szegö founded the Stanford University Competitive Examination in Mathematics. In its first year 322 students from 60 schools in California entered. The competition grew to having typically 1200 students from 150 schools in 3 western states. However the competition was terminated in 1965 when Stanford shifted its emphasis to postgraduate study. Pólya however continued his activity in this area by publishing problem material in books and journals.

Pólya was particularly interested in the high school curriculum and was concerned about the new maths curriculum. He eventually saw the curriculum change back to basics and was not happy with the way this happened either.

In 1954 he published the two volume book Mathematics and Plausible Reasoningand in 1962 and 1965 a further two volume set entitled Mathematical Discovery.

From his retirement in 1953 Pólya took an active interest in improving the standard of teaching and took steps to establish, with NSF funding, eight-week Summer Institutes

for mathematics teachers, first at the college level (1953-1960), then later for teachers of high school and eventually moving the Institutes to Switzerland.

George Pólya died on 7 September 1985.

A.

DATE EventsDecember 13, 1887. George Pólya was born

1897 His father Jakab died1905 He graduated from Marko Street

Gymnasium1905 he entered in University of Budapest1910 He finished his doctorate studies, except

for his dissertation1911-12 He returned to Budapest to give his

doctoral dissertation and met Gábor Szegö, seven years younger, who was to become one of his major collaborators.

1912 He went to Göttingen for postdoctoral study and met David Hilbert, Richard Courant, Felix Klein and Hermann Weyl.

1913 He was offered a position in Frankfurt, but was discouraged from staying in Germany and turned the job down after being told he was a “bloody Jew” by a ruffian on a train and went on to the University of Paris for further postdoctoral work.

1914 He took up a position at the Eidgenössische Technische Hochschule (ETH) in Zurich, an institution which boasted the names of the physicists Röentgen and Einstein (1900) among its graduates

1914 Pólya was called up by Hungary to fight in the war

1918 He met his future wife, Stella Weber and they got married.

1912 published one of his major results, the solution of the random walk problem

1924 publishing the classification of seventeen plane-symmetry groups, a result which was later to inspire the Dutch artist M.C. Escher.

1924 he spent a year in England, working with G.H. Hardy and J.E. Littlewood at Oxford

and Cambridge.1934 publication of the bookInequalities, which

included a new proof by Pólya of the AM-GM inequality based on the Maclaurin expansion of the exponential function

1925 Pólya, with Szegö, published arguably one of his most influential books,Aufgaben und Lehrsätze aus der Analysis, volumes 19 and 20 of the series Die Grundlehren der Mathematischen Wissenschaften published by J. Springer, Berlin.

1937 Pólya Enumeration Theorem was published

1940 Pólyas became increasingly concerned, with George’s Jewish background, of the possibility of a German invasion of Switzerland, and decided to leave for the United States.

1942 He did move to Stanford1953 He stayed until his retirement from

teachingAfter 1953 He stayed at Stanford, living at Palo Alto

until his death, as a Professor Emeritus.1945 Pólya published one of his most famous

books, How to Solve It.1951 He published, with Gábor

Szegö, Isoperimetric Inequalities in Mathematical Physics.

1946 Pólya and Szegö founded the Stanford University Competitive Examination in Mathematics

1965 the competition was terminated1954 He published the two volume

book Mathematics1962 He published Plausible Reasoning1965 a further two volume set

entitled Mathematical Discovery.1953 took an active interest in improving the

standard of teaching and took steps to establish, with NSF funding, eight-week Summer Institutes for mathematics teachers

1953-1960 for teachers of high school and eventually moving the Institutes to Switzerland.

September 7, 1985. George Pólya died

B.

George Polya was a Hungarian who immigrated to the United States in 1940. His major contribution is for his work in problem solving.Growing up he was very frustrated with the practice of having to regularly memorize information. He was an excellent problem solver. Early on his uncle tried to convince him to go into the mathematics field but he wanted to study law like his late father had. After a time at law school he became bored with all the legal technicalities he had to memorize. He tired of that and switched to Biology and the again switched to Latin and Literature, finally graduating with a degree. Yet, he tired of that quickly and went back to school and took math and physics. He found he loved math.His first job was to tutor Gregor the young son of a baron. Gregor struggled due to his lack of problem solving skills. Polya (Reimer, 1995) spent hours and developed a method of problem solving that would work for Gregor as well as others in the same situation. Polya (Long, 1996) maintained that the skill of problem was not an inborn quality but, something that could be taught.He was invited to teach in Zurich, Switzerland. There he worked with a Dr. Weber. One day he met the doctor s daughter Stella he began to court her and eventually married �her. They spent 67 years together. While in Switzerland he loved to take afternoon walks in the local garden. One day he met a young couple also walking and chose another path. He continued to do this yet he met the same couple six more times as he strolled in the garden. He mentioned to his wife how could it be possible to meet them� so many times when he randomly chose different paths through the garden .�He later did experiments that he called the random walk problem. Several years later he published a paper proving that if the walk continued long enough that one was sure to return to the starting point.In 1940 he and his wife moved to the United States because of their concern for Nazism in Germany (Long, 1996). He taught briefly at Brown University and then, for the remainder of his life, at Stanford University. He quickly became well known for his research and teachings on problem solving. He taught many classes to elementary and secondary classroom teachers on how to motivate and teach skills to their students in the area of problem solving.In 1945 he published the book How to Solve It which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this text he identifies four basic principles .Polya s First Principle: Understand the Problem�This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don t understand it fully,� or even in part. Polya taught teachers to ask students questions such as: 

Do you understand all the words used in stating the problem?

What are you asked to find or show? Can you restate the problem in your own words? Can you think of a picture or a diagram that might help you understand the problem? Is there enough information to enable you to find a solution?

Polya s Second Principle: Devise a plan�Polya mentions (1957) that it 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: 

Guess and checkMake and orderly listEliminate possibilitiesUse symmetryConsider special casesUse direct reasoningSolve an equation

Look for a patternDraw a pictureSolve a simpler problemUse a modelWork backwardUse a formulaBe ingenious

Polya�s third Principle: Carry out the planThis step is usually easier than devising the plan. In general (1957), all you need is care and patience, given that you have the necessary skills. Persistent 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. Polya�s Fourth Principle: Look backPolya mentions (1957) 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.George Polya went on to publish a two-volume set, Mathematics and Plausible Reasoning (1954) and Mathematical Discovery (1962). These texts form the basis for the current thinking in mathematics education and are as timely and important today as when they were written. Polya has become known as the father of problem solving.

C.His friends will always remember George Pólya's enthusiasm, his warmth and

humor,and his ready wit. George and Stella Pólya welcomed visitors to their home in College Terrace with pleasure, and George loved to recount anecdotes from his many contacts with the great mathematicians of the world

SETTING -- The time and location in which a story takes place is called the setting.  For some stories the setting is very important, while for others it is not.  There are several aspects of a story's setting to consider when examining how setting contributes to a story (some, or all, may be present in a story):  

a)  place - geographical location.  Where is the action of the story taking place? b)  time - When is the story taking place? (historical period, time of day, year, etc) 

c)  weather conditions - Is it rainy, sunny, stormy, etc? d)  social conditions - What is the daily life of the characters like? Does the story contain local colour (writing that focuses on the speech, dress, mannerisms, customs, etc. of a particular place)? e)  mood or atmosphere - What feeling is created at the beginning of the story?  Is it bright and cheerful or dark and frightening?

PLOT -- The plot is how the author arranges events to develop his basic idea;  It is the sequence of events in a story or play.  The plot is a planned, logical series of events having a beginning, middle, and end.  The short story usually has one plot so it can be read in one sitting.  There are five essential parts of plot:  

a)  Introduction - The beginning of the story where the characters and the setting is revealed.

b)  Rising Action - This is where the events in the story become complicated and the conflict in the story is revealed (events between the introduction and climax).

c)  Climax - This is the highest point of interest and the turning point of the story.  The reader wonders what will happen next; will the conflict be resolved or not?

d)  Falling action - The events and complications begin to resolve themselves.  The reader knows what has happened next and if the conflict was resolved or not (events between climax and denouement).

e)  Denouement - This is the final outcome or untangling of events in the story.

It is helpful to consider climax as a three-fold phenomenon:  1)  the main character receives new information  2)  accepts this information (realizes it but does not necessarily agree with it) 3)  acts on this information (makes a choice that will determine whether or not he/she gains his objective). 

CONFLICT--   Conflict is essential to plot.  Without conflict there is no plot.  It is the opposition of forces which ties one incident to another and makes the plot move.  Conflict is not merely limited to open arguments, rather it is any form of opposition that faces the main character. Within a short story there may be only one central struggle, or there may be one dominant struggle with many minor ones.

There are two   types   of conflict : 1)  External - A struggle with a force outside one's self.

2)  Internal - A struggle within one's self; a person must make some decision, overcome pain, quiet their temper, resist an urge, etc.

There are four   kinds   of conflict : 1)  Man vs. Man (physical) - The leading character struggles with his physical strength against other men, forces of nature, or animals.

2)  Man vs. Circumstances (classical) - The leading character struggles against fate, or the circumstances of life facing him/her.

3)  Man vs. Society (social) - The leading character struggles against ideas, practices, or customs of other people.

4)  Man vs. Himself/Herself (psychological) -  The leading character struggles with himself/herself; with his/her own soul, ideas of right or wrong, physical limitations, choices, etc. 

CHARACTER -- There are two meanings for the word character: 1)  The person in a work of fiction. 2)  The characteristics of a person.

Persons in a work of fiction - Antagonist and Protagonist Short stories use few characters.  One character is clearly central to the story with all major events having some importance to this character - he/she is the PROTAGONIST.  The opposer of the main character is called the ANTAGONIST.

The Characteristics of a Person - In order for a story to seem real to the reader its characters must seem real.  Characterization is the information the author gives the reader about the characters themselves.  The author may reveal a character in several ways: a)  his/her physical appearance b)  what he/she says, thinks, feels and dreams c)  what he/she does or does not do d)  what others say about him/her and how others react to him/her

Characters are convincing if they are:  consistent, motivated, and life-like (resemble real people)

Characters are... 1.  Individual - round, many sided and complex personalities. 2.  Developing - dynamic,  many sided personalities that change, for better or worse, by the end of the story. 3.  Static - Stereotype, have one or two characteristics that never change and are emphasized e.g. brilliant detective, drunk, scrooge, cruel stepmother, etc. 

POINT OF VIEW

Point of view, or p.o.v., is defined as the angle from which the story is told.

1.  Innocent Eye - The story is told through the eyes of a child (his/her judgment being different from that of an adult) .

2.  Stream of Consciousness - The story is told so that the reader feels as if they are inside the head of one character and knows all their thoughts and reactions.

3.  First Person - The story is told  by the protagonist or one of the characters who interacts closely with the protagonist or other characters (using pronouns I, me, we, etc).  The reader sees the story through this person's eyes as he/she experiences it and only knows what he/she knows or feels.

4.  Omniscient- The author can narrate the story using the omniscient point of view.  He can move from character to character, event to event, having free access to the thoughts, feelings and motivations of his characters and he introduces information where and when he chooses.  There are two main types of omniscient point of view:

a)  Omniscient Limited - The author tells the story in third person (using pronouns they, she, he, it, etc).  We know only what the character knows and what the author allows him/her to tell us. We can see the thoughts and feelings of characters if the author chooses to reveal them to us.

b)  Omniscient Objective – The author tells the story in the third person.  It appears as though a camera is following the characters, going anywhere, and recording only what is seen and heard.  There is no comment on the characters or their thoughts. No interpretations are offered.  The reader is placed in the position of spectator without the author there to explain.  The reader has to interpret events on his own. 

THEME -- The theme in a piece of fiction is its controlling idea or its central insight.  It is the author's underlying meaning or main idea that he is trying to convey.  The theme may be the author's thoughts about a topic or view of human nature.  The title of the short story usually points to what the writer is saying and he may use various figures of speech to emphasize his theme, such as: symbol, allusion, simile, metaphor, hyperbole, or irony.  

Some simple examples of common themes from literature, TV, and film are: - things are not always as they appear to be - Love is blind - Believe in yourself - People are afraid of change- Don't judge a book by its cover

D.As a person who became herself--fitting into the mould of society without really

fitting according to its rules. A woman who wasn't afraid to think openly.I want to be remembered for my warm heart, sense of humor and generosity

My mother Lazy but good daugther

My father Kind to everyoneMy bestfriend- Jessica R. Manalang Crazy I a fun way

My friend- Giewelyn N. Calica Love expertMy cousin- Alfredo Santiago Loving Tita

I’m so flattered that they see me like that. My mother said that I’m lazy, yes its true in our house I act like a princess because I’m the only girl and I’m the eldest. But as she said I’m a good daughter and kind to everyone as my father said. Two of close friends said that I’m crazy,yes it also true but in a fun way. Love expert? I don’t think so,maybe sometimes. And lastly he said that im a loving tita, all my nieces and nephew are really close to me because I always spoiled them.

2.

Polya's prescription for solving problems consists of four steps that use the 3 R's of problems solving, Request-Response-Result, and a verification of the result.

1. Understanding the problem. (Recognizing what is asked for.) 2. Devising a plan. (Responding to what is asked for.) 3. Carrying out the plan. (Developing the result of the response.) 4. Looking back. (Checking. What does the result tell me? )

The solutions below show the incorporation of these steps in forming a logically sound, reliable problem solving process. The solution developments are concept-based instead of the prevailing manipulation-based process.

The first 'Plan ' is the 'Plan ' devised for responding to what the problem asks for, the number of apples Mary has, in this case. The information for formulating the 'Plan ' is in the problem statement.

The second 'Plan ' is the plan devised to respond to what the first 'Plan ' asks for. The information for the second 'Plan ' is contained in the problem statement. It is indented one level since such hierarchical organization is universally used in effective communication.

The third 'Plan ' is the plan devised to respond to what the second 'Plan ' asks for. This is indented to the right since it is a sub problem of the previous problem ( 'Plan '). The process is repeated until there are no more requests. The total 'Plan ' for solving the problem consists of the collection of indented 'Plans '.

With the 'Plan ' complete the third step suggested by Polya, 'Carrying out the plan ' can be done. This consists of successive substitutions.

It is to be noted that each of the 'Plans ' is a new problem independent of the previous 'Plan '. Step 1, 'Understanding the problem,' is applied to the 'Plan ' last written to determine what that 'Plan ' asks for.

In the solution above, the third step, 'Carrying out the plan, ' was delayed till the total 'Plan ' was developed. The problem is "solved" once the total 'Plan ' has been developed. The remainder of the steps, 'Carrying out the plan , ' is not problem solving but rather the use of mathematics to generate the final result. It is a mechanical

process. The problem is "solved" once the 'Plan ' is complete. This is quite different than the existing problem solving paradigms. These uniformly focus on 'Carrying out the plan , ' at the expense of devising the 'Plan ' from concepts.

It is characteristic of textbooks and supporting material that only the bottom part of the solution, the left-moving indents here (the manipulation part), is displayed. The problem solving part (the right indents) is not typically shown.

The planning of the solution is the "thinking" part. In general it requires use of knowledge. In simple problems like this one the "knowledge" is contained in the problem statement.

Delaying 'Carrying out the plan ' is not necessary. The display below shows 'Carrying out the plan ' executed as each 'Plan ' is devised.

This arrangement of the solution brings out the Request-Response-Result character of the steps in a problem solution and shows the connection that the solutions presented in these web pages make with Polya's prescription. In the typical problem the unique nested, in parallel/series, structures 'Plan ' developed by this author provide a logically consistent approach to problem solving.

The extension of the four-step process suggested by Polya used in the examples in these web pages applies equally well to more complex problems. The four steps are used recursively, as described above, to generate the solution to a large class of problems. The solution of a physics problem in this manner is shown below.

This solution has the same logical organization as the first one shown. The successive 'Plans ' in this case were devised from knowledge of the subject, resulting in a solution that focuses on concepts as opposed to the usual manipulation-based approach.

Those not possessing the needed knowledge can follow the steps of the solution, being aware of the manner in which using a concept leads to using successive concepts as needed. This is of particular value to those learning the subject.

Textbooks typically display only the bottom part of the solution ('Carrying out the plan '). This is the manipulative part of the solution. It has been observed that students become quite adept at manipulation. Omission of the "thinking" part, 'Devising a plan ' , in examples the students encounter provides no opportunity for the student to develop thinking skills. The 'Devising a plan ' part of the solution depends heavily on understanding concepts. Since coming to an understanding of concepts is a primary goal, the omission of concepts in problem solutions, which are intended to enhance understanding, is particularly distressing.

The solution presentations that students frequently encounter are not the same as the solutions developed by the expert who developed the problem solutions. The expert indeed does the 'Devising a plan ' part of the solution. However, this is frequently done subliminally. The expert may not even be consciously aware of what was done before the first line was written. This effect can be seen quite vividly in a lecture. The lecturer

frequently says the words that build the ' planning ' but the first thing written on the board is the manipulative part ('Carrying out the plan ').

The two examples above are particularly simple logical structures consisting of nested 'Plans '. More involved structures are shown elsewhere in these web pages. In all cases, though, Polya's four steps are identifiable, used in combinations which are nested in a parallel and/or series solution structure.

As a consequence all solutions have the same look and feel. The words change, the symbols change, and the concepts change but the problem solving process remains the same.

The reason that the look and feel is the same for all problem solutions is simply that a problem is a request for some result subject to a set of conditions that must be simultaneously satisfied. The ideas used in solving simultaneous equations are then appropriate. A convenient method which handles many problems is elimination by substitution. It is then a mathematical necessity that the look and feel of all problem solutions will be the same.

There are situations in which difficulty is encountered in developing a solution presentation that has the attributes described above. The cause of this is lack of available tools. This leads to the anything goes approach.

The anything goes approach is unfortunately encouraged by NCTM and is the central theme of contemporary (1999) problem solving instruction. This is particularly distressing when used on students in early grades. If builds destructive problem solving habits which leads to the I can't solve word problems syndrome in later grades where problems are encountered for which the anything goes approach does not work.

The chaotic nature of contemporary K-12 problems solving instruction is wasteful of resources and deprives the student of many opportunities. That this should be the case in view of the extensive understanding of problem solving that exist, as shown in the references of the the previous page, is particularly tragic.