Problem Solving in Mathematics Essay
WHY PROBLEM SOLVING? • Problem solving is the most basic of mathematical skills- the reason for studying mathematics • Problem solving is an integral part of the larger area of critical thinking, which is universally accepted goal for all education • Problem solving shows an interaction between mathematical ideas • In the classroom can lessen the gap between real world problem and the classroom worlds and thus set more positive mood in the classroom. WHAT IS PROBLEM? The normal process for solving a problem will initially involve defining the problem we want to solve.
We need to decide what we want achieve and write it down. Often people keep the problem in their head as a vague idea and can so often get lost in what they are trying to solve that no solution seems to fit. Merely writing down the problem forces us to think about what we are actually trying to solve and how much we want to achieve. The first part of the process not only involves writing down the problem to solve, but also checking that we are answering the right problem.
It is a check-step to ensure that we do not answer a side issue or only solve the part of the problem that is most easy to solve.
People often use the most immediate solution to the first problem definition that they find without spending time checking the problem is the right one to answer. • A problem is a task for the person confronting it • Wants or need to find a solution • Has no readily available procedure for finding a solution and • Must make an attempt to find a solution. Charles & Lester (1982) TYPES OF PROBLEMS • ROUTINE PROBLEM • NON- ROUTINE PROBLEM ROUTINE PROBLEM From the curricular point of view, routine problem solving involves using at least one of the four arithmetic operations and/or ratio to solve problems that are practical in nature.
Routine problem solving concerns to a large degree the kind of problem solving that serves a socially useful function that has immediate and future payoff. Children typically do routine problem solving as early as age 5 or 6. They combine and separate things such as toys in the course of their normal activities. Adults are regularly called upon to do simple and complex routine problem solving. The research evidence suggests that good routine problem solvers have a repertoire of automatic symbol-based and context-based responses to problem situations.
They do not rely on manipulating concrete materials, nor on using strategies such as ‘guess and check’ or ‘think backwards’. Rather, they rely on representing what is going on in a problem by selecting from a limited set of mathematical templates or model. • Routine problem are those that merely involved an arithmetic operation with the characteristics • Presents a question to be answered • Gives the facts or numbers to use • Can be solved by direct application of previously learned algorithms and the basic task is to identify the operation appropriate for solving the problem.
NON – ROUTINE PROBLEM Non-routine problem solving serves a different purpose than routine problem solving. While routine problem solving concerns solving problems that are useful for daily living (in the present or in the future), non-routine problem solving concerns that only indirectly. Non-routine problem solving is mostly concerned with developing students’ mathematical reasoning power and fostering the understanding that mathematics is a creative endeavour. Non-routine problem solving can be seen as evoking an ‘I tried this and I tried that, and eureka, I finally figured it out. reaction. That involves a search for heuristics (strategies seeking to discover). There is no convenient model or solution path that is readily available to apply to solving a problem. That is in sharp contrast to routine problem solving where there are readily identifiable models (the meanings of the arithmetic operations and the associated templates) to apply to problem situations. • It occurs when an individual is confronted with an unusual problem situation, and is not aware of a standard procedure for solving it. The individual has to create a procedure.
To do so, he or she must become familiar with the problem situation, collect appropriate information, identify an efficient strategy to solve the problem • Non-routine problems are those that call for the use of processes far more than those of routine problems with the characteristics: • Use of strategies involving some non-algorithmic approaches • Can be solved in many distinct ways requiring different thinking processes. Comparing routine and non-routine problem solving To make clearer the distinction between routine and non-routine problem solving, consider the following two problems.
Problem 1 My mom gave me 35 cents. My father gave me 45 cents. My grandmother gave me 85 cents. How many cents do I have now? |Problem 2 | |[pic] | | |Place the numbers 1 to 9, one in each circle so that the | | | | |sum of the four numbers along any of the three sides of | | | | |the triangle is 20.
There are 9 circles and 9 numbers to | | | | |place in the circles. Each circle must have a different | | | | |number in it. | | | | | | | | | Notice that addition is required for both problems. In problem 1, you need to figure out that you need to add.
Understanding addition as modeling a ‘put together’ action helps you realize that. In problem 2, you are told to add by the word ‘sum’. Understanding addition as modeling a ‘put together’ action does not help you with solving problem 2. Being good at arithmetic might help you a bit, but the matter really concerns a search for strategies to apply to the problem. Guess and check is a useful strategy to begin with. WHAT IS PROBLEM SOLVING? [pic] We should know the basic terminology for Problem Solving. This report proposes seven terms such as Purpose, Situation, Problem, Cause, Solvable Cause, Issue, and Solution. Purpose
Purpose is what we want to do or what we want to be. Purpose is an easy term to understand. But problem solvers frequently forget to confirm Purpose, at the first step of Problem Solving. Without clear purposes, we can not think about problems. Situation Situation is just what a circumstance is. Situation is neither good nor bad. We should recognize situations objectively as much as we can. Usually almost all situations are not problems. But some problem solvers think of all situations as problems. Before we recognize a problem, we should capture situations clearly without recognizing them as problems or non-problems.
Without recognizing situations objectively, Problem Solving is likely to be narrow sighted, because problem solvers recognize problems with their prejudice. Problem Problem is some portions of a situation, which cannot realize purposes. Since problem solvers often neglect the differences of purposes, they cannot capture the true problems. If the purpose is different, the identical situation may be a problem or may not be a problem. Cause Cause is what brings about a problem. Some problem solvers do not distinguish causes from problems. But since problems are some portions of a situation, problems are more general than causes are.
In other words causes are more specific facts, which bring about problems. Without distinguishing causes from problems, Problem Solving can not be specific. Finding specific facts which causes problems is the essential step in Problem Solving. Solvable Cause Solvable cause is some portions of causes. When we solve a problem, we should focus on solvable causes. Finding solvable causes is another essential step in Problem Solving. But problem solvers frequently do not extract solvable causes among causes. If we try to solve unsolvable causes, we waste time. Extracting solvable causes is a useful step to make Problem Solving efficient.
Issue Issue is the opposite expression of a problem. If a problem is that we do not have money, the issue is that we get money. Some problem solvers do not know what Issue is. They may think of “we do not have money” as an issue. At the worst case, they may mix the problems, which should be negative expressions, and the issues, which should be positive expressions. Solution Solution is a specific action to solve a problem, which is equal to a specific action to realize an issue. Some problem solvers do not break down issues into more specific actions. Issues are not solutions. Problem solvers must break down issues into specific action.
Therefore: • Problem solving involving a situation whereby an individual or a group is required to carry out the working solution • Problem solving is the process of applying previously acquired knowledge, skills and understanding to new and unfamiliar situations • Problem solving is the process use to a statement or a question (Hamada, R. y. & Smith). PROBLEM SOLVING PROCESS GEORGE POLYA George Polya identified four steps in the problem solving processes: ? Understanding the problem ? Devising a plan ? Carry out the plan ? Looking back BURTON (1984) He identified four phases in the problem solving processes : ?
Entry ? Attack ? Review ? Extension THE REALITY AND MATHEMATICS EDUCATION (RIME) PROGRAM The Reality and Mathematics Education (RIME) program recommended the following steps for mathematical problem solving: ? Introduce the problem ? Pose the problem ? Allow students to carry out initial investigations ? Encourage students to check their predictions ? Assist students to develop a summary an conclusion to what they have been doing. PROBLEM SOLVING PROCESS (GEORGE POLYA) UNDERSTANDING THE PROBLEM Understanding the problem is the most important step before we can devise the plan for its solution.
The problem cannot be solved until we fully understand what to find. We have to read the problem carefully for several times and try to analyze and understand it clearly. We need to look for clues and information, and then identify quantities and the unknowns. We need to analyze the problem an ask ourselves to answer the following questions: • Can 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 need? DEVISING A PLAN
Think of all possible methods or strategies to solve the problem and then choose the best strategy that fit the problem. Decide what plan is appropriate for the particular problem. Try to relate information to past experience and consider auxiliary or smaller problems, if an intermediate connection cannot be found. Find the connection between the given data or information to the unknowns and choose the best strategy to solve the problem. • Find the connection between the data and the unknown • Consider auxiliary problem if an immediate connection can be found • What strategies do you know? Try a strategy that seems as if it will work CARRY OUT THE PLAN After devise a plan, we are in better position to carry out the selected strategy. Be persistent to overcome all the obstacles and continue the struggle to solve the problem until we reach the dead end. • Use the strategy you selected and work the problem • Check each step of the plan as you proceed • Ensure that the steps are correct LOOKING BACK The final solution to the problem has to be counterchecked whether it’s reasonable or not. Does the solution answer all questions and satisfy all conditions of the problem?
Is there other ways that could give the same answer to the problem? • Reread the question • Did you answer the question asked? • Is your answer correct? • Does your answer seems reasonable? PROBLEM SOLVING STRATEGIES General strategies are procedures which guide a person’s choice of what skills to use or what knowledge to draw upon at each stage while in the course of solving a problem or investigations or verification of a discovery. STRATEGIES • Guess and check • Organize information in a chart, table or graph • Look for pattern • Simulation / acting it out • Drawing a picture / making a diagram Work backwards • Simplify the problem Guess and Check This is probably the simplest and most natural of all problem-solving strategies. Often referred to Trial and Error, it is important for students to realize that an error really isn’t a mistake at all; it help to guide the problem solver to the next attempt at the answer. Unfortunately, beginning with the student’s earliest experiences in a mathematics classroom, solving problems through the application of the guess and check strategy instead of the direct application of a particular algorithm is often discouraged.
As a result, many students are very reluctant to guess at the solution to a problem. Thus, to teach this strategy, it is first necessary to encourage students to make guesses. Only after they are comfortable with making them, can they be taught the essential features of the guess and check strategy: 1. Make an “educated” guess at the solution. 2. Check the guess against the conditions of the problem. 3. Use the information obtained in checking to make a better guess. 4. Continue this procedure until the correct answer is obtained.
Collect and Organize Data • Organizing information in a chart, table or graph is a way of : – presenting information so it can be read quickly and easily – listing information in an orderly way • Students must first learn how to read charts, tables, or graphs for information an dthen learn how to construck them to report information • Reading and constructing graphs are skills that student must have before interpreting, analyzing, and using the information. Suggested sources for these included : make a list : a procedure for constructing an organized list contaning all the possibilities for a given situation is developed – make a table : develop methods for setting a table for a problem and for using antries in a table to solve word problem. It is also shows how tables can be used as an organizer when using guess and check strategy. Look For a Pattern Patterns pervade most of mathematics. Students are introduced to them at the most elementary level in counting processes, and continue to utilize them in addition tables, multiplication tables, and so forth.
While students are exposed to a variety of pattern situations in the early grades, generally there is no application of them as a strategy for problem solving. With proper guidance from the teacher, students progress from simple identification of a pattern and completion of next terms to a final stage in which they will find the term of a sequence given any term number and in the case of a linear sequence, determine a rule to describe it. At this stage, the table (no pun intended) is set for the student to explore the rudiments of algebra.
Because patterns play an integral role in the discovery and application of mathematical concepts, at some point in their mathematics education, students must be taught: 1. To analyze patterns and make generalizations based on their observation 2. To check the generalization against known information 3. To construct a formal proof to verify the generalization Act It Out (SIMULATION) • Teachers in Kindergarten and year one use this approach when developing models for addition and subtraction. Rebus problems, which use pictures to convey words, are used as basis for children physically ’acting out’ a problem. Acting out a problem forces an understanding of the nature of the problem. If someone is capable of acting out the problem, we can almost best be certain that he or she understands it. • Some manipulative such as bottles caps or chips can be used to represent people or things. We can simulate the action with pencil and paper, by making a drawing or table. Draw A Picture • Children’s textbook use drawings and pictures throughout the early development of mathematical concepts, as well as beginning problem-solving exercises. However, teachers must design activities that encourage children to use this skill. A drawing is a paper and pencil simulation of the action described in a problem. Drawing will enable the students to convert a verbal situation into a visual representation. This pictorial representation represents the intermediate stage between the concrete and the abstract. • Emphasis should be placed on neatness, accuracy and proper scaling Making Diagram • A diagram is a picture or sketch taht shows how things or parts are arranged. Working Backward • A strategy to solve problems by starting with the final data or statements and working backwards to get the starting value.
Simplify The Problem Simplifying a problem involves changing its form so that: 1. The problem becomes more understandable. 2. A method of solution is more easily discovered. 3. The solution process is facilitated. Since it is not a method of solving a problem which is complete in itself, it is generally used in conjunction with one or more of the other strategies. It would be impossible to define and enumerate all of the methods of simplifying. However, there are four techniques which are frequently employed and which can be taught successfully. 1.
Reword the problem using different numbers or a more familiar setting to help discover the operation which should be used. This is probably the most commonly used method of simplifying. 2. Separate the problem into distinct subproblems which can be solved individually or in sequence. 3. Solve the problem by cases. Start with a simpler case of the problem and work through successive cases until a general method of solution is discovered. ROUTINE PROBLEMS EXAMPLE 1 : A sales promotion in a store advertises a jacket regularly priced at $125. 98 but You have $100 in your pocket ( or $100 left in your charge account ).
Do you have enough money to buy the jacket ? $125. 98 – $100 = $25. 98 Therefore : I dont have enough money to buy the jacket. EXAMPLE 2 : Elizabeth bought milk for $1. 10, carrots for $0. 69, cottage cheese for $1. 29, and kidney beans for $0. 59. These was no tax. What was the bill for these items? $1. 10 + $0. 69 + $1. 29 + $0. 59 = $3. 67 PROBLEM SOLVING STRATEGIES Guess and Check Handmade Frienship bracelets use 20 cm of thread. Handmade rings use 8 cm of thread. Merry used a total of 184 cm of thread to make 14 items. How many frienship bracelets did she make? UNDERSTANDING THE PROBLEM I.
How much thread is needed for a bracelet? – 20 cm II. How much thread is needed for a ring? – 8 cm III. How much thread did Marry use in all? – 184 cm IV. How much items Marry did? – 14 V. What does the problem ask to find? – the total of bracelets Marry make DEVISING A PLAN I. Multiple the possible value of bracelet and ring to get a total of 184 cm II. Calculate the items of both bracelets and rings if they are 14 items. III. Find the total of items need for Frienship and bracelet IV. Guess and check to get the right total of items of bracelet and ring CARRY OUT THE PLAN Total needed : 184 cm Bracelet |Ring |Bracelet + ring |High / low | |2 x 20 = 40 |2 x 8 =16 |56 |< 184 | |6 x 20 = 120 |5 x 8 = 40 |160 |< 184 | |5 x 20 = 100 |6 x 8 = 48 |148 |< 184 | |8 x 20 = 160 |7 x 8 = 56 |216 |> 184 | |6 x 20 = 120 |8 x 8 = 64 |184 |184=184 |
LOOKING BACK I. Is the total number of bracelets and rings 14? II. Is the total number of thread use 184 cm? ( 6 x 20 ) + ( 8 x 8 ) = 184 6 bracelets + 8 rings = 14 items or ( a x 20 ) + ( 8 x 8 ) = 184 20a + 64 = 184 20a = 184 – 64 a = 120/20 a = 6, 6 bracelets ( 6 x 20 ) + ( b x 8 ) = 184 120 + 8b = 184 8b = 184 – 120 b = 64/8 b = 8, 8 rings Therefore, the sum of bracelets and rings are 14 items. MAKE A TABLE A bus can hold 45 passenger. It starts outn empty and picks up 1 passenger at the first stop, 2 at the second stop, 3 at the third stop and so on. If no one gets off of the bus, at which stop will the bus become full? UNDERSTANDING THE PROBLEM I.
How many passenger the bus can hold? – 45 passengers II. Is there any passenger in the bus before the bus arrived at the first stop? – No one. The bus starts out empty III. How many passsenger at the first, second and third stop picks up by the bus? – starts with 1,2 and 3 passengers IV. Is there any of the passenger gets off along the journey? – no one gets off from the bus DEVISING PLAN I. Draw a table to shows the total of the passenger and the stage of the stop. II. Calculate the total of the passenger until reach the maximum total of 44 passenger. III. Choose the right stop where the total of the passenger at maximum. CARRY OUT THE PROBLEM STOP |PASSENGER | |1 |1 | |2 |2 | |3 |3 | |4 |4 | |5 |5 | |6 |6 | |7 7 | |8 |8 | |9 |9 | |TOTAL |45 | 1 + 2 + 3 + 4 +5 + 6 + 7 + 8 + 9 = 45 The 9th stop = 45 passengers LOOKING BACK • If the bus starts empty, then : 45 – ( 9 + 8 + 7 +6 + 5 + 4 + 3 + 2 +1 ) = 0 Therefore, the stop where the total of the passenger at maximum is at 9th. LOOK FOR PATTERN
Linda plans to save $1 the first week, $2 the second week, $4 the third week, $8 the fourth week, and $16 the fifth week. If Linda can continue this pattern, how much money will she save the twelth week? UNDERSTANDING PROBLEM I. How much Linda planned in the first, second, third, fourth and fifth week? -1, $2, $4, $8 and 16 II. what i need to find? -The money Linda save in twelth week III. Did Linda continue the same pattern? – yes DEVISING PLAN I. Look at the numbers given (total of money Linda save on each week ) to find the pattern. II. The new number depends upon the number before it. III. Guess and check to find the total of money Linda save on the twelfth week. CARRY OUT THE PLAN WEEKS |MONEY | |1 | 2 ( 1 x 2 ) | |2 | 4 ( 2 x 2 ) | |3 | 8 ( 4 x 2 ) | |4 | 16 ( 8 x 2 ) | |5 | 32 (16 x 2 ) | |6 | 64 ( 32 x 2 ) | |7 | 128 ( 64 x 2 ) | |8 | 256 ( 128 x 2 ) | |9 | 512 ( 256 x 2 ) | |10 | 1024 ( 512 x 2 ) | |11 | 2048 (1024 x 2 ) | |12 | 4096 ( 2048 x 2 ) | LOOKING BACK Twelfth week = 4096 4096 / 2 = 2048 2048 / 2 = 1024 1024 / 2 = 512 512 / 2 = 256 256 / 2 = 128 128 / 2 = 64 64 / 2 = 32 32 / 2 = 16 16 / 2 = 8 8 / 2 = 4 4 / 2 = 2 Therefore, the total money on twelfth week is $4096. SIMULATION / WORK OUT A cafeteria always serves the same three main courses and three deserts. Each day, Marny choose a combination of one course and one dessert. How many meals can Marny eat before she repeat a combination? UNDERSTANDING THE PROBLEM i.
How many courses and desserts the cafeteria serves? – 3 courses and 3 desserts ii. What kind of combination per day did Marny chose? – one course and one dessert iii. What Marny need to find? – the total of meals Marny eat before repeat a combination. DEVISING PLAN I. Assume that the three main course are named as A. B, and C as well as the three desserts as a, b, and c. II. Find all combination between the courses and desserts. III. Calculate the total of meals Marny eat before repeat a combination ? CARR OUT THE PLAN Courses = A, B, C Desserts = a, b, c a ——– Aa A b ——– Ab c ——– Ac a ———- Ba
B b ———- Bb c ———- Bc a ———— Ca C b ———— Cb c ———— Cc The total of meals before Marny repeat a combination = 9 meals LOOKING BACK A ———— aA a B ———— aB C ———— aC A ———— bA b B ———— bB C ———— cC A ———— cA c B ———— cB C ———— cC Aa = aA Therefore : aA ———— 1 aB ———— 2 aC ———— 3 bA ———— 4 bB ———— 5 bC ———— 6 cA ———— 7 cB ———— 8 cC ———— 9 The total meals before repeat a combination = 9 meals
WORK BACKWORDS Ani made silk bouquet in 3 days. Ani give half of the silk bouquet to her friend and divided the remaining evenly into three part. If the first part contain 2000 silk bouquet, how many silk bouquets did Ani made in 3 days? UNDERSTANDING PROBLEM I. How long did Ani make silk bouquets? – 3 days II. How many Ani gave to her friend – half III. How about the remaining silk bouquets? – Ani divided them evenly into 3 part IV. How many silk bouquets in each part? – after divided, each part contains 2000 silk bouquets. DIVISING PLAN i. Calculate the total of silk bouquet in three part. ii. Calculate the total of silk bouquets in three days. iii.
Work backwords to check the total of silk bouquets in three days. CARRY OUT PLAN • The total of silk bouquet in the three part is half for the total of silk bouquet in the three days. • Aini divided the remaining silk bouquets evenly into three part (2000 each) • Together, the total of silk bouquet in three days are 6000 (2000 x 3 ) • Aini gives half of the silk bouquet to her friend (6000 + 6000 = 12000) LOOKING BACK • Start with 2000 silk bouquets, • Multiply by 3 parts ( 3 x 2000 = 6000), • Multiply by 2 for the half that Aini gives to her friend (2 x 6000 = 12000). Therefore, the of total silk bouquet in three days is 12000. THE ROLE AND PURPOSE OF PROBLEM SOLVING IN MATHEMATICS CURRICULUM
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 listed at the outset of this article: functional, logical and aesthetic. Let us consider how problem solving is a useful medium for each of these. It has already been pointed out that mathematics is an essential discipline because of its practical role to the individual and society. Through a problem-solving approach, this aspect of mathematics can be developed. Presenting a problem and developing the skills needed to solve that problem is more motivational than teaching the skills without a context.
Such motivation gives problem solving special value as a vehicle for learning new concepts and skills or the reinforcement of skills already acquired (Stanic and Kilpatrick, NCTM, 1989). Approaching mathematics through problem solving can create a context which simulates real life and therefore justifies the mathematics rather than treating it as an end in itself. The National Council of Teachers of Mathematics (NCTM, 1980) recommended that problem solving be the focus of mathematics teaching because, they say, it encompasses skills and functions which are an important part of everyday life. Furthermore it can help people to adapt to changes and unexpected problems in their careers and other aspects of their lives.
More recently the Council endorsed this recommendation (NCTM, 1989) with the statement that problem solving should underlay all aspects of mathematics teaching in order to give students experience of the power of mathematics in the world around them. They see problem solving as a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others. According to Resnick (1987) a problem-solving approach contributes to the practical use of mathematics by helping people to develop the facility to be adaptable when, for instance, technology breaks down. It can thus also help people to transfer into new work environments at this time when most are likely to be faced with several career changes during a working lifetime (NCTM, 1989).
Resnick expressed the belief that ‘school should focus its efforts on preparing people to be good adaptive learners, so that they can perform effectively when situations are unpredictable and task demands change’ (p. 18). Cockcroft (1982) also advocated problem solving as a means of developing mathematical thinking as a tool for daily living, saying that problem-solving ability lies ‘at the heart of mathematics’ (p. 73) because it is the means by which mathematics can be applied to a variety of unfamiliar situations. Problem solving is, however, more than a vehicle for teaching and reinforcing mathematical knowledge and helping to meet everyday challenges. It is also a skill which can enhance logical reasoning.
Individuals can no longer function optimally in society by just knowing the rules to follow to obtain a correct answer. They also need to be able to decide through a process of logical deduction what algorithm, if any, a situation requires, and sometimes need to be able to develop their own rules in a situation where an algorithm cannot be directly applied. For these reasons problem solving can be developed as a valuable skill in itself, a way of thinking (NCTM, 1989), rather than just as the means to an end of finding the correct answer. Many writers have emphasised the importance of problem solving as a means of developing the logical thinking aspect of mathematics. If education fails to contribute to the development of the intelligence, it is obviously incomplete. Yet intelligence is essentially the ability to solve problems: everyday problems, personal problems … ‘(Polya, 1980, p. 1). Modern definitions of intelligence (Gardner, 1985) talk about practical intelligence which enables ‘the individual to resolve genuine problems or difficulties that he or she encounters’ (p. 60) and also encourages the individual to find or create problems ‘thereby laying the groundwork for the acquisition of new knowledge’ (p. 85). As was pointed out earlier, standard mathematics, with the emphasis on the acquisition of knowledge, does not necessarily cater for these needs.
Resnick (1987) described the discrepancies which exist between the algorithmic approaches taught in schools and the ‘invented’ strategies which most people use in the workforce in order to solve practical problems which do not always fit neatly into a taught algorithm. As she says, most people have developed ‘rules of thumb’ for calculating, for example, quantities, discounts or the amount of change they should give, and these rarely involve standard algorithms. Training in problem-solving techniques equips people more readily with the ability to adapt to such situations. A further reason why a problem-solving approach is valuable is as an aesthetic form.
Problem solving allows the student to experience a range of emotions associated with various stages in the solution process. Mathematicians who successfully solve problems say that the experience of having done so contributes to an appreciation for the ‘power and beauty of mathematics’ (NCTM, 1989, p. 77), the “joy of banging your head against a mathematical wall, and then discovering that there might be ways of either going around or over that wall” (Olkin and Schoenfeld, 1994, p. 43). In the past decade it has been suggested that problem-solving techniques can be made available most effectively through making problem solving the focus of the mathematics curriculum.
Although mathematical problems have traditionally been a part of the mathematics curriculum, it has been only comparatively recently that problem solving has come to be regarded as an important medium for teaching and learning mathematics (Stanic and Kilpatrick, 1989). In the past problem solving had a place in the mathematics classroom, but it was usually used in a token way as a starting point to obtain a single correct answer, usually by following a single ‘correct’ procedure One of the aims of teaching through problem solving is to encourage students to refine and build onto their own processes over a period of time as their experiences allow them to discard some ideas and become aware of further possibilities (Carpenter, 1989). As well as developing knowledge, the students are also developing an understanding of when it is appropriate to use particular strategies.
Through using this approach the emphasis is 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’. There is considerable importance placed on exploratory activities, observation and discovery, and trial and error. Students need to develop their own theories, test them, test the theories of others, discard them if they are not consistent, and try something else (NCTM, 1989). Students can become even more involved in problem solving by formulating and solving their own problems, or by rewriting problems in their own words in order to facilitate understanding.
It is of particular importance to note that they are encouraged to discuss the processes which they are undertaking, in order to improve understanding, gain new insights into the problem and communicate their ideas (Thompson, 1985, Stacey and Groves, 1985). CONCLUSION It has been suggested that there are many reasons why a problem-solving approach can contribute significantly to the outcomes of a mathematics education. Not only it is a vehicle for developing logical thinking, it can provide students with context for learning mathematical knowledge, it can enhance transfer of skills to unfamiliar situations and it is an aesthetic form in itself. A problem-solving approach can provide a vehicle for students to construct their own ideas about mathematics and to take responsibility for their own learning.
There is little doubt that the mathematics program can be enhanced by the establishment of an environment in which students are exposed to teaching via problem solving, as opposed to more traditional models of teaching about problem solving. 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 contexts. Furthermore, by finding the way in Solving Problem can encourage the move from specific to general thinking, in other words, encourage the ability in more abstract ways. From the point of view students, growing to adulthood, that ability is becoming important in today’s technological, complex, and demanding world.