Math Problem Solvingamc8

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Problem Solving In Mathematics Essay Sample

Learning problem solving is never a spectator game. The learners have to be actively involved if any meaningful learning has to take place. Different teachers use different strategies and techniques. In teaching contents, the teacher has no option but to master strategies and skills that will inspire learners to become motivated and actually enjoy learning. (Leamson, R, 2000). Also learners must be taught intensively and extensively the strategies. The learners '… must make what they learn part of themselves.' (Chickening, A.W. & Gamson, Z.F, 1987). The teacher should strive to trigger intrinsic motivation in the learners, as this is likely to make them succeed.

Poor masterly of strategies deny students the power to be flexible as they lack approach options to choose from in solving problems. When learners know a variety of techniques, they do not give up at the first failure, they tend to apply different approaches until they get it correct (persistence). When they get it correct, they get motivated and learn to be flexible and persistent instead of giving up.

Most students at the grade between 7-12 have a dislike for mathematics. Poor masterly of techniques has been held culprit for this. In the attempt to enable the learners between 7-12 grades develop persistence and flexibility in mathematics, a number of problem solving strategies and techniques can be instrumental and they include:

1). Work Backward Strategy

Problem: Cleo got his salary on Wednesday, on Thursday he spent $1.50 at the hotel. On Friday, Ben paid Cleo the $1.00 he owed him. If Cleo now has $2.00, how much is his salary?

Understanding the problem

How much money did Cleo have on Friday? ($2.00)

How much money did Cleo spent in the hotel? ($1.50)

How much money was given on top of his salary? ($ 1.00)

Planning a solution

Had Cleo got Ben's $1.00 on Thursday night? (No)

How much money did Cleo have at the end of Thursday?($2.00-$1.00=$1.00)

How much money did Cleo have before he spent $ 1.50 on Tuesday? (2.50)

Finding the answer

Start with $ 2.00

Subtract $ 1.00

Add $ 1.50

End with $ 2.50

Extension of problem

This strategy can be applied in solving all problems that deal with spending. For instance, John spend ¼ of his gas on day 1, and 2/3 of the remaining on the second day, if the remaining was one liter, what capacity was his gas before use?

2) Make a table strategy

Cleo and Tom began reading a novel the same day. If Cleo reads 5 pages each day and Tom 3 pages each day, what page will Tom be when Cleo will be reading page 20?

Fig.1

Understanding the problem

How many pages does Cleo read each day? (5) Tom? (3). Did they start reading their books on the same day? (Yes)

Planning a solution

How many pages had each read at the end of the day 1? Cleo (5) Tom (3)

Find the number of pages read for the first 3 days. 5, 10, 15

Finding the solution

Fig 1 shows that, Tom will be reading page 12 when Cleo is reading page 20

Problem extension

Cleo digs 10 hectors a day, Tom digs 8 while Ben digs 6, what hector will Tom and Ben be digging when Cleo will be digging his 50^th hector? Using the table the student will be able to work it.

• Bean Toss

Materials: the student will need 10 beans, a cup, pencil and a plain paper.

Purpose Contactriver oak saddlery. : Not many students are abstract thinkers. The beans painted on one side with a color like white and black on the other side to represent positive and negative is a concrete reference for the concept of integers. This will represent a positive and a negative side in an activity such as (+2) – (-1) = +3

Activities and procedure: the students' pair up and decide on which color is negative and which represents positive. Each student tosses the beans and records the outcome, for instance, (+3) + (-2) = +1. As the game continues, the students internalize the rules of working with signed numbers and the rule can be extended to division and multiplication

Extension: they learn the real life application. For example I received $ 6. I owe Tom $ 4. What does my account read? (+6) + (-4) = +2

4) Guess and check technique

This strategy arrives at a verifiable answer through guessing possible answers and checking to see which answer fits the problem. Students should predetermine a likely starting point and work in the right direction to solve the problem. The best student achieves this by eliminating as many numbers as possible with every guess.

Example: Caleb has 40 balls. If he had 10 more white types than the black type, how many of each ball did he have?

Understanding the problem.

There are more 10 white balls than black balls so white ball +10. Black balls –10.Half of 40 = 20 so if there were 20 black the white would be 30 (20+10) hence not correct. So the number of black is less than 20. Half the difference between the 2 total s and subtract from previous guess different is 10 half = 5 hence the black balls are 20-5 = 15.White ball = 25 so that 15+25 = 40. The second guess is correct

Amc Math Problems 8

5). Solve a simpler problem

Some problems are too complex to solve in one step. The learner should divide it into cases and solving each separately.

Example; how many palindromes are there between 0-1000? The student can solve this by starting at how many of the numbers 1-9 are palindromes? All the nine are palindromes. How many of the number 10-99 are palindromes?

11

22

33 ý = 9

.

• From 100-999
• 202 ….909

111 212 …. 919

121 222 …. 929

…. …. …

191 292 …. 999

Working out

9 columns´10 palindromes=90

90 palindromes from 100-999

The answer is (90+9+9)=108

The above techniques and others must be accompanied with creative problem variation, such as changing context/setting. It is only after students have mastered various techniques to solve similar or varied problems that they can develop flexibility and persistence. So the teacher should competently teach the application of various strategies. The learners should know how to select appropriate techniques for each problem and how to justify their solutions using different approaches. When students develop flexibility and persistence, they learn to view the difficulty of complex mathematical investigations as a challenge rather than a bother. When they solve a problem successfully, they experience a feeling of accomplishment. This motivates them to attempt harder problems.

Remember the extension of problems help in generalization of problems and makes the learner to be creative, make value judgment and incorporate other branches of mathematics.

In conclusion, techniques for solving different problems coupled with plenty of examples, motivating exercises that build skills and confidence, visually appealing graphics presented in fun, and highly engaging manner should all be used to help learners develop flexibility and persistence in solving problems.

References

Charles, R.L., Mason, R. P., Nofsinger, J.M, & White, C. A. (1985).

problem-solving experiences in mathematics. Addison: Wesley publishing

company.

Leamson, R. (2000). Algebra in simplest terms. From

www.learner.org/resources/series66htm

as retrieved on Nov 1 2007. 19:43:52. GMT

Polya, G. (1973). How to solve it. Princeton: Princeton university press.

We can write a custom essay

According to Your Specific Requirements

What Is A 'Problem-Solving Approach'?

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 characterised 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).
3. 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.

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 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). 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:

• developing skills and the ability to apply these skills to unfamiliar situations
• gathering, organising, interpreting and communicating information
• 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
• 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).

Conclusion

It has been suggested in this chapter that there are many reasons why a problem-solving approach can contribute significantly to the outcomes of a mathematics education. Not only is it a vehicle for developing logical thinking, it can provide students with a 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.

References

Carpenter, T. P. (1989). 'Teaching as problem solving'. In R.I.Charles and E.A. Silver (Eds), The Teaching and Assessing of Mathematical Problem Solving, (pp.187-202). USA: National Council of Teachers of Mathematics.

Clarke, D. and McDonough, A. (1989). 'The problems of the problem solving classroom', The Australian Mathematics Teacher, 45, 3, 20-24.

Cobb, P., Wood, T. and Yackel, E. (1991). 'A constructivist approach to second grade mathematics'. In von Glaserfield, E. (Ed.), Radical Constructivism in Mathematics Education, pp. 157-176. Dordrecht, The Netherlands: Kluwer Academic Publishers.

Cockcroft, W.H. (Ed.) (1982). Mathematics Counts. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, London: Her Majesty's Stationery Office.

Evan, R. and Lappin, G. (1994). 'Constructing meaningful understanding of mathematics content', in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp. 128-143. Reston, Virginia: NCTM.

Gardner, Howard (1985). Frames of Mind. N.Y: Basic Books.

Lester, F.K.Jr., Masingila, J.O., Mau, S.T., Lambdin, D.V., dos Santon, V.M. and Raymond, A.M. (1994). 'Learning how to teach via problem solving'. in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp. 152-166. Reston, Virginia: NCTM.

National Council of Teachers of Mathematics (NCTM) (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s, Reston, Virginia: NCTM.

National Council of Teachers of Mathematics (NCTM) (1989). Curriculum and Evaluation Standards for School Mathematics, Reston, Virginia: NCTM.

Olkin, I. & Schoenfeld, A. (1994). A discussion of Bruce Reznick's chapter. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 39-51). Hillsdale, NJ: Lawrence Erlbaum Associates.

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Polya, G. (1980). 'On solving mathematical problems in high school'. In S. Krulik (Ed). Problem Solving in School Mathematics, (pp.1-2). Reston, Virginia: NCTM.

Resnick, L. B. (1987). 'Learning in school and out', Educational Researcher, 16, 13-20.

9 columns´10 palindromes=90

90 palindromes from 100-999

The answer is (90+9+9)=108

The above techniques and others must be accompanied with creative problem variation, such as changing context/setting. It is only after students have mastered various techniques to solve similar or varied problems that they can develop flexibility and persistence. So the teacher should competently teach the application of various strategies. The learners should know how to select appropriate techniques for each problem and how to justify their solutions using different approaches. When students develop flexibility and persistence, they learn to view the difficulty of complex mathematical investigations as a challenge rather than a bother. When they solve a problem successfully, they experience a feeling of accomplishment. This motivates them to attempt harder problems.

Remember the extension of problems help in generalization of problems and makes the learner to be creative, make value judgment and incorporate other branches of mathematics.

In conclusion, techniques for solving different problems coupled with plenty of examples, motivating exercises that build skills and confidence, visually appealing graphics presented in fun, and highly engaging manner should all be used to help learners develop flexibility and persistence in solving problems.

References

Charles, R.L., Mason, R. P., Nofsinger, J.M, & White, C. A. (1985).

problem-solving experiences in mathematics. Addison: Wesley publishing

company.

Leamson, R. (2000). Algebra in simplest terms. From

www.learner.org/resources/series66htm

as retrieved on Nov 1 2007. 19:43:52. GMT

Polya, G. (1973). How to solve it. Princeton: Princeton university press.

We can write a custom essay

According to Your Specific Requirements

What Is A 'Problem-Solving Approach'?

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 characterised 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).
3. 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.

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 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). 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:

• developing skills and the ability to apply these skills to unfamiliar situations
• gathering, organising, interpreting and communicating information
• 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
• 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).

Conclusion

It has been suggested in this chapter that there are many reasons why a problem-solving approach can contribute significantly to the outcomes of a mathematics education. Not only is it a vehicle for developing logical thinking, it can provide students with a 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.

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George Harrison, My sweet Lord, Hm, my Lord, Hm, my Lord, I really want to see you, Really want to be with you, Really want to see you Lord, But it takes so long, my Lord, My sweet Lord, Hm, my Lord, Hm, my Lord, I really want to know you, Really want to go with you, Really want to show you Lord, That it won't take long, my Lord (hallelujah), My sweet Lord (hallelujah), Hm, my Lord (hallelujah. George harrison my sweet lord demo.

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Amc 8 Math Problems Pdf

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Art Of Problem Solving Amc

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