describe a problem situation and how you solved it
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How to Solve It suggests the following steps when solving a mathematical problem:
- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look backon your work.How could it be better?
If this technique fails, PÃ³lya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"
First principle: 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. In order to remedy this oversight, PÃ³lya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:
- 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?
- Do you understand all the words used in stating the problem?
- Do you need to ask a question to get the answer?
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.
Second principle: Devise a plan
PÃ³lya mentions 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:
- Guess and check
- Make an orderly list
- Eliminate possibilities
- Use symmetry
- Consider special cases
- Use direct reasoning
- Solve an equation
- Look for a pattern
- Draw a picture
- Solve a simpler problem
- Use a model
- Work backward
- Use a formula
- Be creative
- Use your head/noggin
Third principle: 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.
Fourth principle: Review/extend
PÃ³lya 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.
The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:
The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.
The book has achieved "classic" status because of its considerable influence (see the next section).
- It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
- Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George PÃ³lya on how to solve problems."
- PÃ³lya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
- Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with PÃ³lya's famous book.
- Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels PÃ³lya's work.
How Do You Solve a Problem Like Maria? was a British television talent series, that was broadcast on Saturday evenings on BBC One between 29 July and 16 September 2006. It documented the search for a new, undiscovered musical theatre performer to play the role of Maria von Trapp in the 2006 Andrew Lloyd Webber and David Ian stage production of The Sound of Music. Presented byGraham Norton and "masterminded" by Andrew Lloyd Webber, the title of the show was derived from a line in the production's song "Maria".
Following a public telephone vote, 23-year-old Connie Fisher was chosen as Maria and performed the role in the West End from November 2006 to February 2008. The series also helped the careers of the other finalists, some of whom later took leading roles in West End shows. Semi-finalist Aoife Mulholland also took up the role of Maria in April 2007 for two shows a week, after Fisher was advised to reduce her performances to six per week.
This was the first programme to allow the public to cast a leading role in a West End show, and it was initially criticised. However, it won International Emmy and Royal Television Society awards and became the first of a series of collaborations between the BBC and Lloyd Webber, including Any Dream Will Do,I'd Do Anything, andOver the Rainbow. The series also led to versions and similar series abroad.
The lead role of Maria von Trapp in the new West End production of The Sound of Music, to be staged byAndrew Lloyd Webber and David Ian, was to be played by American actress Scarlett Johansson. Negotiations fell through, and after a four-year search for an actress to fill the role, it was revealed in November 2005 that Lloyd Webber had approached the BBC to allow the public to cast the role through a Popstars-style talent search, the first time that such a format had been used.
The series was devised by BBC creative head Bea Ballard, and was announced by the BBC in April 2006. It would find an undiscovered talent to perform the role, cast by the public, through a series of live television shows on Saturday nights on BBC One hosted by Graham Norton. The series was named after the first line of the chorus of "Maria", a song from the musical.
To assess and train the potential Marias and judge them during the live shows, an expert panel was chosen. The panel comprised:
- Andrew Lloyd Webberâ€“ musical theatre composer and producer, co-producer of the new stage production
- David Ianâ€“ theatre producer, co-producer of the new stage production
- John Barrowmanâ€“ musical performer, dancer, singer and actor
- Zoe Tylerâ€“ voice coach, singer and performer, vocal coach to the finalists
Lloyd Webber also asked Denise van Outen to participate in the series, but she turned him down, saying that she "felt uncomfortable about being on the panel and giving my criticism". She later became a judge on follow-up series, Any Dream Will Do.
Open auditions were held around the UK in April and May 2006, open to both professionals and amateurs over the age of 17. The top 200 made it through to the London callbacks where they performed for Ian, Barrowman and Tyler to secure one of 50 places at Lloyd Webber's "Maria School", where over four days they would receive vocal and drama training from the expert panel.
Several additional performers were selected over this fifty contestant limit; one being Briony, who had been rejected initially due to nerves hampering her performance, but who returned for a second chance and was allowed in by Ian. A further four, whom the panel had rejected, were contacted by Lloyd Webber himself as he personally believed them to be potential Marias.
During "Maria School", contestants were eliminated to leave twenty, who were then taken to Lloyd Webber's house, where they performed for fifty people from the entertainment business. Ten finalists were then chosen by the panel and taken through to the live studio finals.
The series started on Saturday 29 July 2006, and the first two programmes followed the audition stages of the competition before revealing the final ten at the end of the second programme.
The final ten contestants then competed in the live studio finals held on Saturday nights over six weeks. Each week the contestants sang and performed during the live show, receiving comments from the judges following their performance. The public then got a chance to vote for their favourite Maria, and the two contestants with the fewest votes performed a sing-off in front of Lloyd Webber, who then decided which Maria to keep in the contest. This was repeated with the top ten, the top nine and the top eight. With the top seven and top five, two were voted off in the program, and there were two different sing-offs.
Lloyd Webber had no say in the final casting decision, when in the concluding edition of the series it was left to the public to choose who should play Maria out of the final two contenders, Connie Fisher and From Yahoo Answers
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