WHAT IS A PROBLEM?

PROBLEM SOLVING

1 A problem is a problem because you dont know how to solve itWHAT IS A PROBLEM ???2 WHAT IS A PROBLEM ??? A task for which the person confronting it :wants or needs to find a solutionhas no readily available procedure for finding a solution, andmust make an attempt to find a solution

3 WHAT IS A PROBLEM ??? a problem for one person is not necessarily a problem for another 4Poor Pam has measles. She has one spot on her chin, one spot on each leg, one spot on each arm and one spot on her tummy. How many measles spots does Pam have ? The next morning, Pam wakes up with even more spots! Now she has two on her chin, two on each arm and each leg, and two on her tummy. How many spots does she have now?Examples of problems

5Rosey and Ratu were hunting around in the family car. They each collected together all the marbles that they could find. That night Rosey and Ratu sorted and counted the marbles. They found thatwhen they counted by fours they had three left over;when they counted by fives they had none left over;when they counted by threes they had none left over.Their father knew they had less than 18 marbles. How many marbles had they collected?Examples of problems

6something about the wording ?where to get started ?no obvious strategy What is the right piece of mathematics to use ?How to use them correctly or put them together to come up with a solutionWhat are some possible problems face by school children?

7TYPES OF Mathematical PROBLEMSNON-ROUTINE PROBLEMS

2. ROUTINE PROBLEMS

8ROUTINE PROBLEMSMerely involved an arithmetic operation Presents a question to be answeredGives the facts or numbers to useCan be solved by direct application of previously learned algorithmsThe basic task is to identify the operation appropriate for solving the problem.

9Example of Routine ProblemsWhat is the area of a 100 meter x 1000 meter car lot ?An employee makes RM8.50 per hour. How much will the employee makes in 40 hours?What is the product of 269 x 76 ?Ahmad has 11 marbles and Cheah has 7 marbles. How many more marbles does Ahmad has than Cheah

10NON-ROUTINE PROBLEMunusual problem situationnot aware of any standard procedure for solving itneed to create a procedure

to do so, he or she must collect appropriate informationidentify an efficient strategyuse strategy to solve the problem

11NON-ROUTINE PROBLEMcall for the use of processes far more than those of routine problemsUse of strategies involving some non-algorithmic approachesCan be solved in many distinct ways requiring different thinking processes

12Example of non-routine problems :How much paper of all kinds does your school uses in a fortnight?Approximately how many hairs are there on your head ?

13WHAT IS PROBLEM SOLVING ?Problem solving is the process of applying previously acquired knowledge, skills, and understanding to new and unfamiliar situations.Problem solving is the process used to find an answer to a statement or a question Hamada, R.Y. & Smith

14WHAT IS PROBLEM SOLVING?Find a way where no way is know off handout of difficultyaround an obstacleAttain a desired end, by appropriate means

15WHY A SPECIAL EMPHASIS ON PROBLEM SOLVING?Hiebert,J.CMathematical ideas are the outcomes of problem-solving experience, rather than elements that must be taught before problem solving

16WHY A SPECIAL EMPHASIS ON PROBLEM SOLVING ?lessen the gap between real world and the classroom world which will set a more positive mood Allows interaction between mathematical ideasis an integral part of the larger area of critical thinkingIts a powerful and effective vehicle for learning Mathematics

17WHAT MAKES A GOOD PROBLEM SOLVER ?Have a desire to solve a problemExtremely perseverant when solving problemsShow an ability to skip some of the steps in the solution processNot afraid to guesswho hold conversations with themselves knowing what questions to ask what to do with the answers they receive

18WHAT MAKES A GOOD PROBLEM?The solution to the problem involves the understanding of distinct mathematical concepts or the use of mathematical skills.The solution of the problem leads to a generalization.The problem is open-ended in that it lead to extensions.The problem lends itself to a variety of solutions.The problem should be interesting and challenging to the students.

19ExampleThere are 8 people in a room. Each person shakes hands with each of the other people once and only once. How many handshakes are there?A farmer has some horses and some chickens. He finds that together they have 70 heads and 200 legs. How many horses and how many chickens does he have?

20PROBLEM SOLVING PROCESSGeorge Polya identified four steps in the problem solving

Understanding the problemDevising a planCarrying out the planLooking back

21Problem Solving Process (George Polya)1. Understanding the problemCan you state the problem in your own words?What are you trying to find or do?What information do you obtain from the problem?What are the unknowns?What information, if any, is missing or not needed?

222. Devising a planFind the connection between the data and the unknown.Consider auxiliary problem if an immediate connection can not be foundWhat strategies do you know?Try a strategy that seems as if it will work.

233. Carrying out the planUse the strategy you selected and work the problem.Check each step of the plan as you proceedEnsure that the steps are correct

244.Looking BackReread the questionDid you answer the question asked?Is your answer correct?Does your answer seems reasonable?

25How to establish a positive climate in the classroom for problem solving :Be enthusiastic about the problemHave students bring in problems from their personal experiencesPersonalize problems whenever possible e.g. use students namesRecognize and reinforce willingness and perseverance

26How to establish a positive climate in the classroom for problem solving :Reward risk takersEncourage students to guess answersAccept unusual solutionsPraise students for getting correct solutionsEmphasize the selection and use of problem solving strategiesEmphasize persistence rather than speed

273 stages in teaching problem solvingBeforeDuringAfter

28Read the problem to the class or have a student read the problemdiscuss words or phrases students may not understand.Discussion about the problems Ask questions to help students understand the problem.Ask students which strategies might be helpful for finding a solution. do not evaluate students suggestions. direct students attention to the list of strategies on problem solvingBe sure all the instructions are clearStep 1 : Before

29Step 1 : questions to askWhat are the important ideas here?Can you rephrase the problem in your own words?What is this asking us to find ?What information is given ?What conditions apply ?Anyone want to guess the answer?Anyone seen a problem like this before?

What strategy could we use to get started?

Which one of these ideas should we pursue?

30Observe and question students about their work.Give hints for solving the problems as needed.Require students who obtain a solution to check their work and answer the problem. Give a problem extension to students who complete the original problem much sooner than others.Step 2 : During

31Tell me what you are doing?

Why did you think of that?

Why are you doing this?

What are you going to do with the result once you have it?

Why do you think that that stage is reasonable?

Why is that idea better than that one?

Step 2 : questions to ask

32Youve been trying that idea for 5 minutes. Are you getting anywhere with it?

Do you really understand what the problem is about?

Can you justify that step?

Are you convinced that that bit is correct?

Can you find a counter example?Step 2 : questions to ask33Show and discuss students solutions to the original problemhave students name the strategies used. Relate the problem to previous problems and solve an extension of the original problem.Discuss special features of the original problem, if any.Step 3 : After34Step 3 : Questions to askHave you answered the problem?

Have you considered all the cases?

Have you checked your solution?

Does it look reasonable?

Is there another solution?

35Step 3 : Questions to askCould you explain your answer to the class?

Is there another way to solve the problem?

Could you generalize the problem?

Can you extend the problem to cover different situations?

Can you make up another similar problem?

Ministry of Education Report No. 587, OPEN Plan for Teaching Mathematical Problem Solving, by Holton, Anderson and Thomas (1997)

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Conclusion

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