## Problem Solving - National Council of Teachers of Mathematics

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 . Problem Solving Strategies - Examples and Worked Solutions of Math Problem Solving Strategies, Verbal Model (or Logical Reasoning), Algebraic Model, Block Model (or Singapore Math), Guess and Check Model and Find a Pattern Model, examples with step by step solutions. Others think of program goals in which the strategies and techniques of problem solving are emphasized. Some discuss mathematics problem solving in the context of a method of teaching, i.e., a problem approach. Indeed, discussions of mathematics problem solving often combine and .

## Problem Solving in Mathematics

The focus is on teaching *problem solving method in mathematics* 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.

Specific characteristics of a problem-solving approach include:. Schoenfeld in Olkin and Schoenfeld,p. 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 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:. 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 *problem solving method in mathematics* how problem solving is a useful medium for each of these, **problem solving method in mathematics**.

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, 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, 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, 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 **problem solving method in mathematics** as a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others.

According to Resnick 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, 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.

Cockcroft 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.

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,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.

Yet intelligence is essentially the ability to solve problems: everyday problems, personal problems Modern definitions of intelligence Gardner, talk about practical intelligence which enables 'the individual to resolve genuine problems or difficulties that he or she encounters' p.

As was pointed out earlier, standard mathematics, with the emphasis on the acquisition of knowledge, does not necessarily cater for these needs. Resnick 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, *problem solving method in mathematics*, for example, **problem solving method in mathematics**, quantities, discounts or the amount of change they should give, *problem solving method in mathematics*, 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,p.

They also speak of the willingness *problem solving method in mathematics* 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. **Problem solving method in mathematics** more needs **problem solving method in mathematics** 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, *problem solving method in mathematics*, 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, 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, *problem solving method in mathematics*, usually by following a single 'correct' procedure.

More recently, however, professional organisations such as the National Council of Teachers of Mathematics NCTM, and have recommended that the mathematics curriculum should be organized around problem solving, focusing on:.

One of the aims *problem solving method in mathematics* 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, 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 **problem solving method in mathematics** 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, 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,Stacey and Groves, 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 *problem solving method in mathematics* 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.

Carpenter, T. Charles and E. Clarke, D. Cobb, P. In von Glaserfield, E. Cockcroft, W. Mathematics Counts. Evan, R. Professional Development for Teachers of Mathematicspp. Lester, F. Olkin, I. A discussion of Bruce Reznick's chapter. Schoenfeld Ed. Mathematical Thinking and Problem Solving. Polya, G. Krulik Ed. Problem Solving in School Mathematics, pp. Romberg, T. Classroom instruction that fosters mathematical thinking and problem solving: connections between theory and practice.

Schifter, D. Reconstructing Mathematics Education. NY: Teachers College Press. Schoenfeld, A. Reflections on doing and teaching mathematics. Stanic, G. Swafford, J. Prospects for School Mathematicspp. Thompson, P. Silver Ed. Hillsdale, N. J: Lawrence Erlbaum. Van Zoest, L. Mathematics Education Research Journal.

By signing up, you agree to receive useful information and to our privacy policy. Shop Math Games. Skip to main content. Search form Search. He described three characteristics of mathematical thinking: valuing the processes of mathematization and abstraction and having the predilection to apply them 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, **problem solving method in mathematics**,p.

As Cobb et al. Not only does this approach develop students' confidence in their own ability to think mathematically Schifter and Fosnot,it is a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others NCTM, 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.

More recently, however, professional organisations such as the National Council of Teachers of Mathematics NCTM, and have recommended that the mathematics curriculum should be organized around problem solving, focusing on: developing skills and the ability to apply these skills to **problem solving method in mathematics** 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,pp.

### METHODS OF TEACHING MATHEMATICS: Module 9: Problem Solving Method

Every year my students can be fantastic at math until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don't know how to complete them. Every year in math, I start off by teaching my students problem-solving skills. Jul 10, · Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation. Others think of program goals in which the strategies and techniques of problem solving are emphasized. Some discuss mathematics problem solving in the context of a method of teaching, i.e., a problem approach. Indeed, discussions of mathematics problem solving often combine and .