PROBLEM SOLVING IN MATHEMATICS
Problem solving in mathematics instruction is a fundamental means of developing mathematical knowledge at any level, including primary school. Problem solving is one of the most important, if not the most important, aspects of doing mathematics. Everyone who learns or uses mathematics will face any kind of mathematical problems to solve. Therefore,developing the skills in problem solving should be part of the objectives in the school mathematics curricula.
Problem solving is one of the ten standards in the 2000 NCTM's Standards. As proposed in the 2000 NCTM's Principles and Standards, the standards in the school mathematics curriculum from prekindergarten through grade 12 consist of contents standards and processes standards. The content standards (the content that students should learn) are:
(1) number and operations, (2) algebra, (3) geometry, (4) measurement, and (5) data
analysis and probability. The process standards (ways of acquiring and using content
knowledge) are: (1) problem solving, (2) reasoning and proof, (3) communication, (4)
connections, and (5) representation.
Furthermore, the NCTM states that problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program. Problem solving in mathematics should involve all the five content areas described in the Standards.
The contexts of the problems can vary from familiar experiences involving students' lives or the school day to applications involving the sciences or the world of work. Good problems will integrate multiple topics and will involve significant mathematics. Not only in the US school mathematics curricula, problem solving has been integral parts of school mathematics at any other countries, including Australia, Asian or even South East Asian countries such as Indonesia, and Singapore. According to Lenchner (2005: 2), the ultimate goal of school mathematics at all times is to develop in our students the ability to solve problems. Lenchner also argues that the ability to solve problems cannot always develop automatically from mastery of computational skills, but it should be taught, and mathematics teachers must make a special effort to do so. Through problem solving, students acquire and apply mathematical concepts and skills, so they experience the power
and usefulness of mathematics, both in mathematical contexts and everyday situations as well. In this way, the mathematics they are learning makes sense to them. Problems can also be used to introduce new concepts and extend previously learned knowledge.
In the NCTM's Principles and Standards it is described that:
Problem solving means engaging in a task for which the solution method is not
known in advance. In order to find a solution, students must draw on their
knowledge, and through this process, they will often develop new mathematical
understandings. Solving problems is not only a goal of learning mathematics
but also a major means of doing so. Students should have frequent
opportunities to formulate, grapple with, and solve complex problems that
require a significant amount of effort and should then be encouraged to reflect
on their thinking. (NCTM, 2000. http://standardstrial.nctm.org/document/
chapter3/prob.htm)
The NCTM's Principles and Standards also recommend that instructional programs from
prekinder-garten through grade 12 should enable all students to:
? build new mathematical knowledge through problem solving,
? solve problems that arise in mathematics and in other contexts,
? apply and adapt a variety of appropriate strategies to solve problems, and
? monitor and reflect the process of mathematical problem solving.
Integrating problem-solving experiences in the school mathematics curricula should also build and develop the innate curiosity of young children. Teachers need to value the thinking and efforts of their students as they develop a wide variety of strategies for tackling problems. In integrating problem solving in the mathematics lessons, the teacher should create an environment in which students' effort to discover strategies for solving a problem is appreciated. Such environment is conducive in promoting learning for all students and supports students with different learning styles (Ng Wee, 2008: 7).
Problem solving is one of the most important aspects of doing mathematics. Everyone who learns or uses mathematics will face any kind of mathematical problems to solve. Therefore, developing the skills in problem solving should be part of the objectives in the school mathematics curricula.
As the emphasis has shifted from teaching problem solving to teaching via problem solving (Lester, Masingila, Mau, Lambdin, dos Santon and Raymond, 1994), many writers have attempted to clarify what is meant by a problem-solving approach to teaching mathematics. The focus is on teaching mathematical topics through problem-solving contexts and enquiry-oriented environments which are characterized by the teacher 'helping students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics: creating, conjecturing, exploring, testing, and verifying' (Lester et al., 1994, p.154). Specific characteristics of a problem-solving approach include:
� interactions between students/students and teacher/students (Van Zoest et al., 1994)
� mathematical dialogue and consensus between students (Van Zoest et al., 1994)
� teachers providing just enough information to establish background/intent of the problem, and students clarifing, interpreting, and attempting to construct one or more solution processes (Cobb et al., 1991)
� teachers accepting right/wrong answers in a non-evaluative way (Cobb et al., 1991)
� teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems (Lester et al., 1994)
� teachers knowing when it is appropriate to intervene, and when to step back and let the pupils make their own way (Lester et al., 1994)
� A further characteristic is that a problem-solving approach can be used to encourage students to make generalisations about rules and concepts, a process which is central to mathematics (Evan and Lappin, 1994).
Schoenfeld (in Olkin and Schoenfeld, 1994, p.43) described the way in which the use of problem solving in his teaching has changed since the 1970s:
My early problem-solving courses focused on problems amenable to solutions by Polya-type heuristics: draw a diagram, examine special cases or analogies, specialize, generalize, and so on. Over the years the courses evolved to the point where they focused less on heuristics per se and more on introducing students to fundamental ideas: the importance of mathematical reasoning and proof..., for example, and of sustained mathematical investigations (where my problems served as starting points for serious explorations, rather than tasks to be completed).
Schoenfeld also suggested that a good problem should be one which can be extended to lead to mathematical explorations and generalisations. He described three characteristics of mathematical thinking:
1. valuing the processes of mathematization and abstraction and having the predilection to apply them
2. developing competence with the tools of the trade and using those tools in the service of the goal of understanding structure - mathematical sense-making (Schoenfeld, 1994, p.60).
As Cobb et al. (1991) suggested, the purpose for engaging in problem solving is not just to solve specific problems, but to 'encourage the interiorization and reorganization of the involved schemes as a result of the activity' (p.187). Not only does this approach develop students' confidence in their own ability to think mathematically (Schifter and Fosnot, 1993), it is a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others (NCTM, 1989). Because it has become so predominant a requirement of teaching, it is important to consider the processes themselves in more detail.
Math problem solving in elementary school requires teaching problem solving strategies. Problem solving exercises should require a variety of strategies for kids to use. Problem solving activities are part of everyday life. Teaching math problem solving strategies provides our students with:
� choices of how to solve for an unknown answer
� improved abilities to choose appropriate strategies
� scaffolding support for improving self-concepts about abilities to problem solve.
The learning process in knowing how to choose a problem solving strategy is most important. There are different methods to solving math problems, and students need to know how to make an appropriate choice. Problem solving exercises are chosen with the goal of either teaching a strategy or practicing a technique to mastery. Daily problem solving is a must. Students need continual practice with problem solving activities for their skills to become fluid. One way to do this is to start with teaching problem solving steps.
4 Steps for Mathematics Problem Solving
1. Understand the Problem
Read and re-read. State the problem in your own words. Decide if there are multiple steps that will need to be taken to arrive at the final answer. Determine what the question is asking you to actually do.
2. Make a Plan
Choose your strategies. Think back to similar problem solving exercises and recall what was successfully used.
3. Try It Out
Go step-by-step with the problem solving strategies chosen. Draw pictures to show each step is correct. Self-talk your way through the problem.
4. Look Back
Always check your work. Did you actually answer the question? Did you use all relevant data? Does your answer make sense? Is there another way to solve the problem or show your answer differently?
These are heuristics, or methods for solving a problem. Students must be taught how to use a variety of methods to solve the same problem. Only then can we see the results of their conceptual understanding of applying mathematics.
1. Make A Pattern
When we tell elementary students to make a pattern, they often do not understand. Essentially, a pattern is a figure that repeats itself, such as a number or word. It is best to use manipulative to study pattern making to provide a concrete understanding. There are two basic types of patterns: repeating and growing. A repeating pattern has an identifiable core that repeats over and over, such as ABACABACABAC. A growing pattern repeats a mathematical process that makes the figure or number grow. Fibonacci numbers are a great example of growing patterns and are found throughout nature.
2. Make A Table
Often a table is used to help organize information found in Making a Pattern. A table helps to quantify a pattern so students can visualize the growing numbers. Students note changes in stages and explain how the change in the table is occurring. This is done by identifying the repeating element. Teacher can use picture books to support teaching how to make a table.
3. Make an Organized List
We make talk about making a list as part of problem solving strategies, we generally refer to a Tree Diagram. This is important because part of efficient problem solving techniques is to not make random guesses. A tree diagram is useful to determine the number of given outcomes (possibilities) from a set situation. You can tell the children that the trunk of the tree is the problem. Each variable is a branch. Little branches grow from larger branches, and to calculate the final answer you only count the final number of little branches.
4. Guess and Check
This is one of the more difficult mathematics problem solving strategies for children to understand, but is often one of the first taught and used. They get the "Guess" part but have difficulty with the "Checking." Often they aren't sure what they are checking for. Teacher can always point out that a good way to check an answer is to re-read it and see if it actually makes sense. Is the given answer actually possible? The guess and check problem solving techniques helps students to think logically, make predictions and use mathematical equations. It all leads to deeper understanding.
5. Draw a Picture
Drawing a picture is a problem solving technique that has students make a visual representation of what the problem is. This really helps solidify concrete thinking. Drawing a picture is the step between the visual and symbolic language of math. Pictures and diagrams are problem solving strategies that many students learn at the earliest stages of math development. This is a good strategy as it is a way to communicate mathematical thinking. With just a bit of encouragement, most students will draw pictures.
The Role of Problem Solving in Teaching Mathematics as a Process
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, 1989, 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 underly 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 emphasized 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). They also speak of the willingness or even desire to engage with a task for a length of time which causes the task to cease being a 'puzzle' and allows it to become a problem. However, although it is this engagement which initially motivates the solver to pursue a problem, it is still necessary for certain techniques to be available for the involvement to continue successfully. Hence more needs to be understood about what these techniques are and how they can best be made available. 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. More recently, however, professional organisations such as the National Council of Teachers of Mathematics (NCTM, 1980 and 1989) have recommended that the mathematics curriculum should be organized around problem solving, focusing on:
(i)developing skills and the ability to apply these skills to unfamiliar situations
(ii) gathering, organising, interpreting and communicating information
(iii)formulating key questions, analyzing and conceptualizing problems, defining problems and goals, discovering patterns and similarities, seeking out appropriate data, experimenting, transferring skills and strategies to new situations
(iv)developing curiosity, confidence and open-mindedness (NCTM, 1980, pp.2-3).
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)
Problem solving in mathematics instruction is a fundamental means of developing mathematical knowledge at any level, including primary school. Problem solving is one of the most important, if not the most important, aspects of doing mathematics. Everyone who learns or uses mathematics will face any kind of mathematical problems to solve. Therefore,developing the skills in problem solving should be part of the objectives in the school mathematics curricula.
Problem solving is one of the ten standards in the 2000 NCTM's Standards. As proposed in the 2000 NCTM's Principles and Standards, the standards in the school mathematics curriculum from prekindergarten through grade 12 consist of contents standards and processes standards. The content standards (the content that students should learn) are:
(1) number and operations, (2) algebra, (3) geometry, (4) measurement, and (5) data
analysis and probability. The process standards (ways of acquiring and using content
knowledge) are: (1) problem solving, (2) reasoning and proof, (3) communication, (4)
connections, and (5) representation.
Furthermore, the NCTM states that problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program. Problem solving in mathematics should involve all the five content areas described in the Standards.
The contexts of the problems can vary from familiar experiences involving students' lives or the school day to applications involving the sciences or the world of work. Good problems will integrate multiple topics and will involve significant mathematics. Not only in the US school mathematics curricula, problem solving has been integral parts of school mathematics at any other countries, including Australia, Asian or even South East Asian countries such as Indonesia, and Singapore. According to Lenchner (2005: 2), the ultimate goal of school mathematics at all times is to develop in our students the ability to solve problems. Lenchner also argues that the ability to solve problems cannot always develop automatically from mastery of computational skills, but it should be taught, and mathematics teachers must make a special effort to do so. Through problem solving, students acquire and apply mathematical concepts and skills, so they experience the power
and usefulness of mathematics, both in mathematical contexts and everyday situations as well. In this way, the mathematics they are learning makes sense to them. Problems can also be used to introduce new concepts and extend previously learned knowledge.
In the NCTM's Principles and Standards it is described that:
Problem solving means engaging in a task for which the solution method is not
known in advance. In order to find a solution, students must draw on their
knowledge, and through this process, they will often develop new mathematical
understandings. Solving problems is not only a goal of learning mathematics
but also a major means of doing so. Students should have frequent
opportunities to formulate, grapple with, and solve complex problems that
require a significant amount of effort and should then be encouraged to reflect
on their thinking. (NCTM, 2000. http://standardstrial.nctm.org/document/
chapter3/prob.htm)
The NCTM's Principles and Standards also recommend that instructional programs from
prekinder-garten through grade 12 should enable all students to:
? build new mathematical knowledge through problem solving,
? solve problems that arise in mathematics and in other contexts,
? apply and adapt a variety of appropriate strategies to solve problems, and
? monitor and reflect the process of mathematical problem solving.
Integrating problem-solving experiences in the school mathematics curricula should also build and develop the innate curiosity of young children. Teachers need to value the thinking and efforts of their students as they develop a wide variety of strategies for tackling problems. In integrating problem solving in the mathematics lessons, the teacher should create an environment in which students' effort to discover strategies for solving a problem is appreciated. Such environment is conducive in promoting learning for all students and supports students with different learning styles (Ng Wee, 2008: 7).
Problem solving is one of the most important aspects of doing mathematics. Everyone who learns or uses mathematics will face any kind of mathematical problems to solve. Therefore, developing the skills in problem solving should be part of the objectives in the school mathematics curricula.
As the emphasis has shifted from teaching problem solving to teaching via problem solving (Lester, Masingila, Mau, Lambdin, dos Santon and Raymond, 1994), many writers have attempted to clarify what is meant by a problem-solving approach to teaching mathematics. The focus is on teaching mathematical topics through problem-solving contexts and enquiry-oriented environments which are characterized by the teacher 'helping students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics: creating, conjecturing, exploring, testing, and verifying' (Lester et al., 1994, p.154). Specific characteristics of a problem-solving approach include:
� interactions between students/students and teacher/students (Van Zoest et al., 1994)
� mathematical dialogue and consensus between students (Van Zoest et al., 1994)
� teachers providing just enough information to establish background/intent of the problem, and students clarifing, interpreting, and attempting to construct one or more solution processes (Cobb et al., 1991)
� teachers accepting right/wrong answers in a non-evaluative way (Cobb et al., 1991)
� teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems (Lester et al., 1994)
� teachers knowing when it is appropriate to intervene, and when to step back and let the pupils make their own way (Lester et al., 1994)
� A further characteristic is that a problem-solving approach can be used to encourage students to make generalisations about rules and concepts, a process which is central to mathematics (Evan and Lappin, 1994).
Schoenfeld (in Olkin and Schoenfeld, 1994, p.43) described the way in which the use of problem solving in his teaching has changed since the 1970s:
My early problem-solving courses focused on problems amenable to solutions by Polya-type heuristics: draw a diagram, examine special cases or analogies, specialize, generalize, and so on. Over the years the courses evolved to the point where they focused less on heuristics per se and more on introducing students to fundamental ideas: the importance of mathematical reasoning and proof..., for example, and of sustained mathematical investigations (where my problems served as starting points for serious explorations, rather than tasks to be completed).
Schoenfeld also suggested that a good problem should be one which can be extended to lead to mathematical explorations and generalisations. He described three characteristics of mathematical thinking:
1. valuing the processes of mathematization and abstraction and having the predilection to apply them
2. developing competence with the tools of the trade and using those tools in the service of the goal of understanding structure - mathematical sense-making (Schoenfeld, 1994, p.60).
As Cobb et al. (1991) suggested, the purpose for engaging in problem solving is not just to solve specific problems, but to 'encourage the interiorization and reorganization of the involved schemes as a result of the activity' (p.187). Not only does this approach develop students' confidence in their own ability to think mathematically (Schifter and Fosnot, 1993), it is a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others (NCTM, 1989). Because it has become so predominant a requirement of teaching, it is important to consider the processes themselves in more detail.
Math problem solving in elementary school requires teaching problem solving strategies. Problem solving exercises should require a variety of strategies for kids to use. Problem solving activities are part of everyday life. Teaching math problem solving strategies provides our students with:
� choices of how to solve for an unknown answer
� improved abilities to choose appropriate strategies
� scaffolding support for improving self-concepts about abilities to problem solve.
The learning process in knowing how to choose a problem solving strategy is most important. There are different methods to solving math problems, and students need to know how to make an appropriate choice. Problem solving exercises are chosen with the goal of either teaching a strategy or practicing a technique to mastery. Daily problem solving is a must. Students need continual practice with problem solving activities for their skills to become fluid. One way to do this is to start with teaching problem solving steps.
4 Steps for Mathematics Problem Solving
1. Understand the Problem
Read and re-read. State the problem in your own words. Decide if there are multiple steps that will need to be taken to arrive at the final answer. Determine what the question is asking you to actually do.
2. Make a Plan
Choose your strategies. Think back to similar problem solving exercises and recall what was successfully used.
3. Try It Out
Go step-by-step with the problem solving strategies chosen. Draw pictures to show each step is correct. Self-talk your way through the problem.
4. Look Back
Always check your work. Did you actually answer the question? Did you use all relevant data? Does your answer make sense? Is there another way to solve the problem or show your answer differently?
These are heuristics, or methods for solving a problem. Students must be taught how to use a variety of methods to solve the same problem. Only then can we see the results of their conceptual understanding of applying mathematics.
1. Make A Pattern
When we tell elementary students to make a pattern, they often do not understand. Essentially, a pattern is a figure that repeats itself, such as a number or word. It is best to use manipulative to study pattern making to provide a concrete understanding. There are two basic types of patterns: repeating and growing. A repeating pattern has an identifiable core that repeats over and over, such as ABACABACABAC. A growing pattern repeats a mathematical process that makes the figure or number grow. Fibonacci numbers are a great example of growing patterns and are found throughout nature.
2. Make A Table
Often a table is used to help organize information found in Making a Pattern. A table helps to quantify a pattern so students can visualize the growing numbers. Students note changes in stages and explain how the change in the table is occurring. This is done by identifying the repeating element. Teacher can use picture books to support teaching how to make a table.
3. Make an Organized List
We make talk about making a list as part of problem solving strategies, we generally refer to a Tree Diagram. This is important because part of efficient problem solving techniques is to not make random guesses. A tree diagram is useful to determine the number of given outcomes (possibilities) from a set situation. You can tell the children that the trunk of the tree is the problem. Each variable is a branch. Little branches grow from larger branches, and to calculate the final answer you only count the final number of little branches.
4. Guess and Check
This is one of the more difficult mathematics problem solving strategies for children to understand, but is often one of the first taught and used. They get the "Guess" part but have difficulty with the "Checking." Often they aren't sure what they are checking for. Teacher can always point out that a good way to check an answer is to re-read it and see if it actually makes sense. Is the given answer actually possible? The guess and check problem solving techniques helps students to think logically, make predictions and use mathematical equations. It all leads to deeper understanding.
5. Draw a Picture
Drawing a picture is a problem solving technique that has students make a visual representation of what the problem is. This really helps solidify concrete thinking. Drawing a picture is the step between the visual and symbolic language of math. Pictures and diagrams are problem solving strategies that many students learn at the earliest stages of math development. This is a good strategy as it is a way to communicate mathematical thinking. With just a bit of encouragement, most students will draw pictures.
The Role of Problem Solving in Teaching Mathematics as a Process
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, 1989, 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 underly 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 emphasized 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). They also speak of the willingness or even desire to engage with a task for a length of time which causes the task to cease being a 'puzzle' and allows it to become a problem. However, although it is this engagement which initially motivates the solver to pursue a problem, it is still necessary for certain techniques to be available for the involvement to continue successfully. Hence more needs to be understood about what these techniques are and how they can best be made available. 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. More recently, however, professional organisations such as the National Council of Teachers of Mathematics (NCTM, 1980 and 1989) have recommended that the mathematics curriculum should be organized around problem solving, focusing on:
(i)developing skills and the ability to apply these skills to unfamiliar situations
(ii) gathering, organising, interpreting and communicating information
(iii)formulating key questions, analyzing and conceptualizing problems, defining problems and goals, discovering patterns and similarities, seeking out appropriate data, experimenting, transferring skills and strategies to new situations
(iv)developing curiosity, confidence and open-mindedness (NCTM, 1980, pp.2-3).
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)