Sunday, July 12, 2009

MATHEMATICAL PROBLEM SOLVING

What is Problem Solving?

Problem solving is the process by which the unfamiliar situation is resolved. A situation that is a problem to one person may not be a problem to another. For instance; determining the number of people in 3 cars when each car contains 5 people may be a problem to some primary school children. They might solve this problem by placing chips or counters in boxes or by making a drawing to represent each car and each person and then counting to determine the total number of people.

What are the attitudes of teachers in using problem solving in mathematics teaching?

For many people, involvement in problem solving might have been an unpleasant memory because they felt as though they were stumbling blindly through problems without much guidance from the teacher or the textbook.

· Sometimes teachers resist teaching problem-solving skills because of their own frustrating experiences with problem solving.

· On the contrary, some teachers believe that teaching mathematics through problem solving helps students develop mathematical skills and apply those skills to situations they encounter in their daily lives.

CHARACTERISTICS OF GOOD PROBLEM SOLVERS

Cognitive research indicates that students who are successful at solving problems structure mathematical concepts and rules into related chunks that make these problem components easier to recall at appropriate times. Conversely, novice problem solvers tend to memorize mathematical knowledge as unrelated rules, making it far more difficult for them to perceive critical relationships that may play an important role in solving a problem. Such rote learning also makes it more difficult to recognise inconsistencies in a set of procedures.

Good problem solvers demonstrate the ability to identify pertinent information and to ignore irrelevant or superficial problem features. This ability enables the capable problem solver to view a specific problem as one example of a whole class of similar problems, a critical attribute that seems to separate good problem solvers from their less capable peers.

WHAT ARE THE DIFFERENT TYPES OF PROBLEMS?

Souviney (1994), presented mathematical problems under two major categories, namely routine or story problems and nonroutine or process problems.

STORY PROBLEMS OR ROUTINE PROBLEMS

These types of problems are employed on a daily basis to practice reading skills and to foster the development of new mathematical ideas and procedures.

To solve story problems, students first read and evaluate the problem statement and then select an appropriate operation and apply it to the values given. As students gain experience, multi-step exercises are introduced.

NON ROUTINE PROBLEMS OR PROCESS PROBLEMS

Process problems differ from story problems in that they cannot be solved immediately by selecting and applying one or more operations. Process problems can be used daily to motivate new skills and concepts, or as follow-up activities to apply previously learned mathematical ideas and procedures. Solving process problems requires flexible thinking and good organisational skills. Success often relies on an individual’s ability to exploit the underlying mathematical structure of a problem using a general solution strategy.

A FOUR-STEP PROBLEM SOLVING PLAN

Texts on mathematical problem solving and research, refers to George Polya’s (Souviney, 1994) four-step problem solving plan. It consists of four interrelated steps;

1. Understanding the problem

2. Select a strategy

3. Carry out the strategy

4. Evaluate the results

Step 1: Understanding the problem

The first step in solving a problem is to understand the information given in the problem statement and the intended goal.

Some helpful hints at this stage;

* Restate the problem in your own words.

* Use materials or sketches to model the problem situation.

* Make a list of all given facts.

* Make a list of the stated conditions and restrictions.

* List unstated, or implicit, conditions.

* Compare the current problem to problems solved previously.

* Work with a partner or small group to discuss the problem.

In addition, questions carefully posed by the teacher can stimulate student thinking. Questions promote children’s’ critical thinking. Further teachers could encourage novice problem solvers to evaluate the ideas of others and alter their positions in light of new evidence.

Step 2: Select a Strategy

Once students understand the problem, a systematic approach, or solution strategy, can be selected. Some commonly used strategies are;

· Guessing and checking

· Substituting simpler values

· Dividing the problem into sub-tasks

· Making a drawing

· Making a systematic list or a table

· Using a model

· Be creative—act out the problem, create a comic strip, think of ways of teaching the concept to a younger student.

· Pose problems.

Often novice problem solvers move to this step in the plan prematurely, before they have spent enough effort on understanding the problem itself. This frequently results in a failure to identify key structural features of the problem that lead to the selection of an effective strategy. Teachers could play a key role at this stage.

Guess-and-test

This familiar trial and error technique can be used to solve a wide range of problems. The solver makes an educated guess and then tests to see if the answer solves the problem. If it does not, the guess is altered.

Step 3: carry out the strategy

Once the facts, conditions, and goal of a problem are understood, and a solution strategy has been selected, the next step is to apply the chosen strategy persistently. Carrying out a solution requires persistence. The students’ task is to determine whether the chosen strategy generates meaningful clues for unravelling the problem. These clues may take the form of patterns that relate the problem to a previously solved example.

At this stage children could be encourage to:

· Keep accurate records

· Stick to their chosen strategy until some evidence suggests specific changes

· Carefully monitor their thinking during each step in the solution.

Step 4: Evaluate the results

Review provides an opportunity for learners to evaluate and refine their results. Further, it brings the process of solution into sharper focus. After the solution of a problem, children could be led to carry out the following activities.

· Describe the steps taken orally or in writing.

· Discuss the form of the answer

· Compare the problem solution to previously solved problems

· Solve extensions of the problem using the same strategy

WHAT APPROACH COULD YOU USE TO TEACH PROBLEM SOLVING?

The key to effective integration of problem solving strategies into mathematics instruction is careful planning.

* children often make false starts and encounter blind alleys.

* children need to feel comfortable while trying out untested ideas.

PROCEDURES FOR TEACHING PROBLEM SOLVING

Establishing Norms

· Provide an environment that encourages students to take intellectual risks and explore untested alternatives.

· Encourage students to work in pairs or collaboratively in groups. Have them learn to collaborate with each other.

· Set norms to encourage cooperative behaviour.

· Encourage students to learn from members within their own group and have students share across groups and learn from each other.

· Have students recognize that there are multiple solution strategies to solving a problem

· Value all answers as potentially useful.

· Praise persistence and unexpected solutions.

· Evaluate the quality of students’ problem solving efforts

· Encourage students to work on problems at home with their families.

· Establish a systematic schedule for integrating problem solving sessions into each strand of the mathematics curriculum.

· Introduce a systematic problem solving plan.

· Ask appropriate questions during each problem solving session.

Understanding the Problem

· Once the students are more accustomed to working together, have each student in a group read the problem. Discuss what the problem means.

· Identify key words. Identify words that may have multiple meanings.

· Have a student be the facilitator (change roles for each problem), asking questions such as what does the problem mean, are there any words you do not understand, are there any words that are not clear?

Cognitive Modelling

Teach the students to problem solve by understanding the question.

Have them appropriate as you model your strategies for understanding a problem. Here is an example to model problem solving strategies.

What is the probability of getting a head if you flip a coin twice?

1. Read the question aloud. Identify key words. Even point out how the “a head” is critical to this problem.

2. Think aloud—how would I solve this problem? Share your thoughts.

3. Show the students how you would find the list of possibilities. Below is a list

of all possibilities.

HH, HT, TT, TH

4. Discuss various meanings of the terminology: “a head” could be at least 1 head or it could mean only 1 head.

REFERENCE

Souviney, R. J. (1994), Learning To Teach Mathematics, Second Edition, Merill.