The ultimate goal of any problem-solving program is to improve students'
performance at solving problems correctly. The specific goals of problem-solving in
Mathematics are to:
1. Improve pupils' willingness to try problems and improve their perseverance when
2. Improve pupils' self-concepts with respect to the abilities to solve problems.
3. Make pupils aware of the problem-solving strategies.
4. Make pupils aware of the value of approaching problems in a systematic manner.
5. Make pupils aware that many problems can be solved in more than one way.
6. Improve pupils' abilities to select appropriate solution strategies.
7. Improve pupils' abilities to implement solution strategies accurately.
8. Improve pupils' abilities to get more correct answers to problems.
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A problem is a task for which:
1) The person confronting it wants or needs to find a solution.
2) The person has no readily available procedure for finding the solution.
3) The person must make an attempt to find a solution.
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FOUR PHASES IN SOLVING A
In solving any problems, it helps to have a working procedure. You might want to
consider this four-step procedure: Understand, Plan, Try
It, and Look Back.
Understand -- Before
you can solve a problem you must first understand it. Read and re-read the problem
carefully to find all the clues and determine what the question is asking you to find.
What is the unknown?
What are the data?
What is the condition?
Plan -- Once
you understand the question and the clues, it's time to use your previous experience with
similar problems to look for strategies and tools to answer the question.
Do you know a related problem?
Look at the unknown! And try to think of a familiar problem having the same or a similar
Try It -- After
deciding on a plan, you should try it and see what answer you come up with.
Can you see clearly that the step is correct?
But can you also prove that the step is correct?
Look Back -- Once
you've tried it and found an answer, go back to the problem and see if you've really
answered the question. Sometimes it's easy to overlook something. If you missed something
check your plan and try the problem again.
Can you check the result?
Can you check the argument?
Can you derive the result differently?
Can you see it at a glance?
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1. Make a table
2. Make an organised list
3. Look for a pattern
4. Guess and check
5. Draw a picture or graph
6. Work backwards
7. Solve a simpler problem
The list of problem-solving strategies above is by no means exhaustive. You
may like to read up on some other strategies such as
(i) Reading and restating problem.
(iii) Looking in another way
(iv) Making a model
(v) Identifying cases
Note: Different strategies can be used to solve the same problem.
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Goals for Extension
1) Lead pupils to generalisation
2) Introduce or integrate other branches of mathematics
3) Provide opportunities for divergent thinking and making value judgements
Principles for Extending a Set of Problems
At a party I attended recently, I noticed that every person shook hands with each
other person exactly one time. There were 12 people at the party. How many handshakes were
|Principle for Problem
|A. Change the problem context/ setting
(e.g., party to a Ping-Pong tournament).
||A. Twelve students in Ms.Palmer's fifth-grade class
decided to have a Ping-Pong tournament. They decided that each students would play one
game against each other students. How many games were played?
|B. Change the numbers (e.g.,
12 becomes 20 or n).
||B. At a party I attended recently, I noticed that
every,person shook hands with each other person exactly one time. There were 20 people at
the party. How many handshakes were there? That if there were n people at the party?
|C. Change the number of conditions(e.g.,
instead of the single condition that "every person shook hands with each other person
exactly one time," we add the condition "Tim shook hands with everyone
||C. At a party I attended recently I noticed that every
person but the host,Tim, shook hands with each other person exactly one time. Tim shook
hands with everyone twice (once when they arrived, once when they left). There were 12
people at the party. How many handshakes were there?
|D. Reverse given and wanted
information (e.g., in the basic problem you are given the number of people at the party
and you want to find how many handshakes there are; the reverse
is true in the new problem).
||D. At a party I attended recently I noticed that every
person shook hands with each other person exactly one time. If I told you there were 66
handshakes, could you tell me how many people were at the party?
|E. Change some combination of
E. context, numbers' conditions, and given/wanted information (e.g., in problem E both the
context and the numbers have been changed). Note: There are 11 combinations possible!
||E. All 20 students in Ms. Palmer's fifth-grade class
decided to have a Ping-Pong tournament. They decided that each student would play one game
against each other student. How many games were played?
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1. CHARLES, R. I. AND MASON, R. P. AND NOFSINGER, J. M.AND WHITE, C. A.
Problem-Solving Experiences In Mathematics. Addison-Wesley Publishing Company;
2. DOLAN, D. T. AND WILLIAMSON, J.
Teaching Problem-Solving Strategies. Addison-Wesley Publishing Company; 1983.
3.MEYER, CAROL AND SALLEE, TOM.
Make It Simpler. Addison Wesley Publishing Company; 1983.
4. POLYA G.
How To Solve It. Princeton University Press; 1973.5.
CAROLE GREENES, JOHN GREGORY AND DALE SEYMOUR.
Successful problem Solving Techniques. Creative Publications, Inc.
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