As most textbook “problems” are practice task, which can be solved by the direct application of a procedure illustrated in the chapter to which they are appended, they are mere exercices.
PROBLEM SOLVING IN THE MATHEMATICSCLASSROOMITLM Topic 9
INTRODUCTION“The ability to solve problem is the heart ofmathematics… mathematics teaching at alllevels should include opportunities for problemsolving, including the application ofmathematics to everyday situations”Cockcroft Report, 1982
INTRODUCTION“Problem solving should be the central focus ofthe mathematics curriculum. As such it isprimary goal of all mathematics instruction andan integral part of all mathematical activity.Problem Solving is not a distinct topic but aprocess that should permeates the entireprogram and provide a context in whichconcept and skill can be learned”NTCM, 1989
INTRODUCTION Build new mathematical knowledge throughproblem solving; Solve problems that arise in mathematicsand in other contexts; Apply and adapt a variety of appropriatestrategies to solve problems; Monitor and reflect on the process ofmathematical problem solvingNTCM, 2000
WHAT IS A PROBLEM?A problem is a task for which: The person confronting it wants or needs tofind a solution, The person has no readily availableprocedure for finding the solution, The person must make an attempt to find asolution.(Charles & Lester, 1984)
A PROBLEM OR AN EXERCISE?As most textbook “problems” are practicetasks, which can be solved by directapplication of a procedure illustrate in thechapter to which they are appended, they aremere exercises.
A PROBLEM OR AN EXERCISE?Solve the following pair of simultaneousequations using the method of elimination:x + y = 123x – 2y = 16
A PROBLEM OR AN EXERCISE?Cows, Chickens and SnakesOn a farm there are cows, chickens andsnakes. Altogether they have 10 heads and 24legs. How may cows, how many chickens andhow many snakes are there? Explain how youworked it out?
WHAT IS PROBLEM SOLVING?Problem solving is a complex process whichrequires an individual to coordinate previousexperiences, knowledge, understanding andintuition, in order to satisfy the demands of anovel situation.In simple terms it is the mental journey onetakes to arrive at a solution starting with thegiven of a situation.
PROBLEM SOLVING STAGESRead the problemUnderstand the problemThink of a way to solve the problemTranslate the problem into a mathematical model/sentenceDo the mathematical computations, etc.Arrive at a solutionCheck solution for accuracy/reasonableness, etc.
COMMON DIFFICULTIES Inability to read the problem Misinterpretation of the conditions of theproblem Lack of strategy knowledge Inappropriate strategy used Inability to translate the problem Incorrect formulation Computational errors Imperfect mathematical knowledge
POLYA’S FRAMEWORKUnderstandthe problemDevise aplanCarry outthe planLook back
UNDERSTAND THE PROBLEM Look for information given Visualize the information Organize the information Make a table Connect the information
DEVISE A PLAN To make representation Draw a diagram Make a systematic list Use equations To make calculated guess Guess and check Look for a pattern Make suppositions
DEVISE A PLAN To go through the process Act it out Work backwards Before after To change the problem Restate the problem Simplify the problem Solve part of the problem
CARRYING OUT THE PLAN Use mathematical knowledge Use mathematical skills Use logical thinking
LOOK BACK (REFLECTING) Check the solution – is it reasonable? Improve on the method used Seek alternative solutions Extend method to other problems
PROBLEM TO SOLVEOn a farm there are cows and chickens.Altogether they have 41 heads and 100 legs.How many chickens are there?
PROBLEM TO SOLVEPlanet Marston has only two types ofcreatures, triads with 3 legs and pentads with 5legs. Their legs glow in the dark. While onMarston, one nights Balpz counted 34 legs.How many triads and pentads were there onMarston?