Problem Solving in Mathematics

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Problem Solving Approach in teaching mathematics

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Problem Solving in Mathematics
John Mark T. Lampos Math Division, UPLB

Try this!
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?

Mathematical Problem

A task for which a person:
confronting it needs to find a solution has no readily available procedure for finding the solution
must make an attempt to find solution

Goals of Problem Solving
To improve the willingness to try problems and improve their perseverance when solving problems

The Problem Solving Approach
-focus is on teaching mathematical topics through problem-solving contexts and inquiry-oriented environments -the teacher helping students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics

Why Teach by Problem Solving?
Problem solving is an important component of mathematics education because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics: functional, logical and

aesthetic.

Functional Value
Approaching mathematics through problem solving can create a context which simulates real life
It can help people to adapt to changes and unexpected problems in their careers and other aspects of their lives.
It is a vehicle for students to construct, evaluate and refine their own theories about mathematic s and the theories of others.
NCTM 1989 NCTM 1980 NCTM 1989

Focus of Problem Solving
developing skills and the ability to apply these skills to unfamiliar situations gathering, organizing, interpreting and communicating information

Problem Solving
formulating key questions, analyzing and conceptualizing problems, discovering patterns and similarities, seeking out appropriate data, experimenting, transferring skills and strategies to new situations

developing curiosity, confidence and openmindedness

Implications
Emphasis must be on making the students more responsible for their own learning rather than letting them feel that the algorithms they use are the inventions of some external and unknown 'expert'

A considerable importance must be placed on exploratory activities, observation and discovery, and trial and error

Students must be encouraged to discuss the processes which they are undertaking, in order to improve understanding, gain new insights into the problem and communicate their ideas

4 Phase in Problem Solving GEORGE
POLYA’S HEURISTICS
Understand Plan Try It

Look Back

1. Understand the problem.
Can you draw a picture or diagram to help you understand the problem? Can you work out some numerical examples that would help make the problem more clear?

What are we trying to find out? What is unknown? Can you restate the problem in your own words?

What do we know for sure? What are the data? What is the condition?

Illustration
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 a person get with three darts?

10 5 1

Solution
1. Understand the problem.
The problem is looking for the number of possible scores that the 3 darts can make altogether, which is very clear. But let students talk about it just to make sure that they really understand the problem.

Extending Problems (Principles)
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? A. Twelve students in Ms. Jose's 2nd year class A. Change the decided to have a Pingproblem context/ Pong tournament. They setting decided that each students would play one game (e.g., party to a Ping- against each other students. Pong tournament) How many games were played?

Extending Problems (Principles)
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? 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? What if there were n people at the party?

B. Change the numbers

(e.g., 12 becomes 20 or n)

Illustration
Allen and Jeric each began reading a Harry Potter book today. If Allen reads 8 pages each day and Jeric reads 5 pages each day, what page will Jeric be reading when Allen is reading page 56?

Problem Extension
Mary reads 9 pages a day, Sue reads 10 pages a day, and Molly reads 8 pages a day. What page will Sue and Molly each be reading when Mary is reading page 72? (Sue--Page 80, Molly--Page 64)

Modifying Questions
• To facilitate problem solving • To allow for critical and logical thinking to take place
• To make assessments more authentic

Example
• Find the perimeter of this polygon.

• Most students realize that if the figure has 6 labeled sides, what operation makes sense? • Add them all together, even if they have NO idea what it means to find the perimeter.

Modified Question
Draw a 6-sided figure that has a perimeter of 23 units.
In this alternate item, students will find it difficult to bluff their way through the drawing without knowing anything about perimeter. It also facilitate multiple ways of solving the problem and diverse responses.

Conclusion
• The main reason for learning all about math is to become better problem solvers in all aspects of life.
• It has been suggested that problem-solving techniques should be made available most effectively through making problem solving the focus of the mathematics curriculum. • The challenge for teachers, at all levels, is to develop the process of mathematical thinking alongside the knowledge and to seek opportunities to present even routine mathematics tasks in problem-solving context.

“A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you it by your own means, you may experience the tension and enjoy the triumph of discovery.” -George Polya

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