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Problem Solving in Mathematics |  |
Content
| Goals | |
| Mathematical problem defined | |
| Four phases in solving a problem | |
| Problem-solving strategies | |
| Problem extension | |
| References |
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 solving problems.
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.
FOUR PHASES IN SOLVING A
PROBLEM
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
unknown?
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?
1. Make a table
8. Reading and restating problem.
9. Brainstorming
10. Looking in another way
11. Making a model
12. Identifying cases
Note: Different strategies can be used to solve the same problem.
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 there?
| Principle for Problem Variation | New 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 twice." | 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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Problem-Solving Experiences In Mathematics. Addison-Wesley Publishing Company;
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Teaching Problem-Solving Strategies. Addison-Wesley Publishing Company; 1983.
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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.