It usually takes times to develop proficiency in problem solving. Students must be given problems to solve time and again throughout their entire mathematics at school.
Kantowski (1980) suggest that the role of the teacher may differ according to the levels of their students’ problem-solving capabilities. He has outlined the following characteristics of students at each level of problem solving development and the role of the teacher at each level.
s/he demonstrate the use of strategies to solve problems often thinking aloud and taking the students through the four stages of Polya’s problem solving model.
s/he guides them in their problem solving process, suggest alternatives when they come to a dead end. Provides them with problems that are carefully structured so they might fruitfully use the strategy they have observed
s/he provides the with more and more challenging problems to reinforce the usefulness of the strategies and to help them collect a store of related problems for future use
s/he motivates students to pose problems and facilitates the exchange of problems posed by students amongst them. s/he encourages them to view the world mathematically and a source of problems.
According to the teaching model, there are three phases in teaching process. The first phase helps students develop an understanding of a concept or a skill. The second one provides students with opportunities to consolidate the concept or skill. The third one provides opportunities for students to transfer their understanding to new situations. Problem solving involves students transferring their understanding of a concept or skill to familiar situations.
Three conceptions of problem-solving lessons will be discussed. Three lessons are described to illustrate each of the concept.
Itlm teaching problem solving
TEACHING PROBLEM SOLVING
INTRODUCTION“Problem solving must not be treated byteachers as a dish on the buffet table but rathera sauce in every dish that is on the buffet tableserving a feast of mathematical dishes.”
OBJECTIVESTeacher candidates will be able to: Identify the role of the teacher in problemsolving Explain types of problem solving lessons Explain the stages in a problem solvinglesson Assess problem solving lesson
THE ROLE OF THE TEACHER The novice (first level) A beginner (second level) An enthusiast (third level) An expert (fourth level)
THE NOVICE Students have little or no understanding ofwhat problem solving is, of the meaning ofstrategy, or of mathematical structure of theproblem. Most students at this level do notknow where to begin to solve a non-routineproblem. Teacher assumes the role of a model.
A BEGINNER Student understand the meaning of problemsolving, of strategy, and the mathematicalstructure of a problem. They are able tofollow someone’s solution and can oftensuggest strategies to be tried for problemssimilar to those they have seen before.Although they will participate actively ingroup, many feel insecure about independentproblem solving Teacher acts a prosthesis or crutch.
AN ENTHUSIAST Student begin to feel comfortable withproblems. They suggest strategies differentfrom those who have seen used. Theyunderstand and appreciate that problemsmay have multiple solutions or in presentstate of the problem may have “no solution”unless an assumption is made. Teacher becomes problem provider.
AN EXPERT Student are able to select appropriate strategyfor most problems encountered and aresuccessful in finding solutions much of the time.They show an interest in elegance andefficiency of a solution and in finding analternative solutions to the same problem. Theysuggest variation of old problem and areconsistently searching for problems to challengethemselves. Teacher serves a facilitator.
THE TEACHING MODELUnderstandingConsolidatingTransferring
TYPES OF PROBLEM-SOLVING LESSONS Teaching through problem solving; Teaching about problem solving; and Teaching for problem solving
STAGES IN PROBLEM SOLVING LESSONS Modeling; Coaching; Explaining; and Providing.
HOW TO ASSESS PROBLEM SOLVING? Observing and questioning Observation checklists Scoring scales Analytic scoring Focused holistic scoring Problem solving portfolios