| As the emphasis has shifted from teaching | | | | can enhance logical reasoning. Individuals |
| problem solving to teaching via problem | | | | can no longer function optimally in society |
| solving (Lester, Masingila, Mau, Lambdin, dos | | | | by just knowing the rules to follow to obtain |
| Santon and Raymond, 1994), many writers have | | | | a correct answer. They also need to be able |
| attempted to clarify what is meant by a | | | | to decide through a process of logical |
| problem-solving approach to teaching | | | | deduction what algorithm, if any, a situation |
| mathematics. The focus is on teaching | | | | requires, and sometimes need to be able to |
| mathematical topics through problem-solving | | | | develop their own rules in a situation where |
| contexts and enquiry-oriented environments | | | | an algorithm cannot be directly applied. For |
| which are characterised by the teacher | | | | these reasons problem solving can be |
| 'helping students construct a deep | | | | developed as a valuable skill in itself, a |
| understanding of mathematical ideas and | | | | way of thinking (NCTM, 1989), rather than |
| processes by engaging them in doing | | | | just as the means to an end of finding the |
| mathematics: creating, conjecturing, | | | | correct answer. |
| exploring, testing, and verifying' (Lester et | | | | |
| al., 1994, p.154). Specific characteristics | | | | Many writers have emphasised the importance |
| of a problem-solving approach include: | | | | of problem solving as a means of developing |
| | | | the logical thinking aspect of mathematics. |
| interactions between students/students and | | | | 'If education fails to contribute to the |
| teacher/students (Van Zoest et al., 1994) | | | | development of the intelligence, it is |
| | | | obviously incomplete. Yet intelligence is |
| mathematical dialogue and consensus between | | | | essentially the ability to solve problems: |
| students (Van Zoest et al., 1994) | | | | everyday problems, personal problems ... |
| | | | '(Polya, 1980, p.1). Modern definitions of |
| teachers providing just enough information to | | | | intelligence (Gardner, 1985) talk about |
| establish background/intent of the problem, | | | | practical intelligence which enables 'the |
| and students clarifing, interpreting, and | | | | individual to resolve genuine problems or |
| attempting to construct one or more solution | | | | difficulties that he or she encounters' |
| processes (Cobb et al., 1991) | | | | (p.60) and also encourages the individual to |
| | | | find or create problems 'thereby laying the |
| teachers accepting right/wrong answers in a | | | | groundwork for the acquisition of new |
| non-evaluative way (Cobb et al., 1991) | | | | knowledge' (p.85). As was pointed out |
| | | | earlier, standard mathematics, with the |
| teachers guiding, coaching, asking insightful | | | | emphasis on the acquisition of knowledge, |
| questions and sharing in the process of | | | | does not necessarily cater for these needs. |
| solving problems (Lester et al., 1994) | | | | Resnick (1987) described the discrepancies |
| | | | which exist between the algorithmic |
| teachers knowing when it is appropriate to | | | | approaches taught in schools and the |
| intervene, and when to step back and let the | | | | 'invented' strategies which most people use |
| pupils make their own way (Lester et al., | | | | in the workforce in order to solve practical |
| 1994) | | | | problems which do not always fit neatly into |
| | | | a taught algorithm. As she says, most people |
| A further characteristic is that a | | | | have developed 'rules of thumb' for |
| problem-solving approach can be used to | | | | calculating, for example, quantities, |
| encourage students to make generalisations | | | | discounts or the amount of change they should |
| about rules and concepts, a process which is | | | | give, and these rarely involve standard |
| central to mathematics (Evan and Lappin, | | | | algorithms. Training in problem-solving |
| 1994). | | | | techniques equips people more readily with |
| | | | the ability to adapt to such situations. |
| Schoenfeld (in Olkin and Schoenfeld, 1994, | | | | |
| p.43) described the way in which the use of | | | | A further reason why a problem-solving |
| problem solving in his teaching has changed | | | | approach is valuable is as an aesthetic form. |
| since the 1970s: | | | | Problem solving allows the student to |
| | | | experience a range of emotions associated |
| My early problem-solving courses focused on | | | | with various stages in the solution process. |
| problems amenable to solutions by Polya-type | | | | Mathematicians who successfully solve |
| heuristics: draw a diagram, examine special | | | | problems say that the experience of having |
| cases or analogies, specialize, generalize, | | | | done so contributes to an appreciation for |
| and so on. Over the years the courses evolved | | | | the 'power and beauty of mathematics' (NCTM, |
| to the point where they focused less on | | | | 1989, p.77), the "joy of banging your head |
| heuristics per se and more on introducing | | | | against a mathematical wall, and then |
| students to fundamental ideas: the importance | | | | discovering that there might be ways of |
| of mathematical reasoning and proof..., for | | | | either going around or over that wall" (Olkin |
| example, and of sustained mathematical | | | | and Schoenfeld, 1994, p.43). They also speak |
| investigations (where my problems served as | | | | of the willingness or even desire to engage |
| starting points for serious explorations, | | | | with a task for a length of time which causes |
| rather than tasks to be completed). | | | | the task to cease being a 'puzzle' and allows |
| | | | it to become a problem. However, although it |
| Schoenfeld also suggested that a good problem | | | | is this engagement which initially motivates |
| should be one which can be extended to lead | | | | the solver to pursue a problem, it is still |
| to mathematical explorations and | | | | necessary for certain techniques to be |
| generalisations. He described three | | | | available for the involvement to continue |
| characteristics of mathematical thinking: | | | | successfully. Hence more needs to be |
| | | | understood about what these techniques are |
| 1. valuing the processes of mathematization | | | | and how they can best be made available. |
| and abstraction and having the predilection | | | | |
| to apply them | | | | In the past decade it has been suggested that |
| | | | problem-solving techniques can be made |
| 2. developing competence with the tools of | | | | available most effectively through making |
| the trade and using those tools in the | | | | problem solving the focus of the mathematics |
| service of the goal of understanding | | | | curriculum. Although mathematical problems |
| structure - mathematical sense-making | | | | have traditionally been a part of the |
| (Schoenfeld, 1994, p.60). | | | | mathematics curriculum, it has been only |
| | | | comparatively recently that problem solving |
| 3. As Cobb et al. (1991) suggested, the | | | | has come to be regarded as an important |
| purpose for engaging in problem solving is | | | | medium for teaching and learning mathematics |
| not just to solve specific problems, but to | | | | (Stanic and Kilpatrick, 1989). In the past |
| 'encourage the interiorization and | | | | problem solving had a place in the |
| reorganization of the involved schemes as a | | | | mathematics classroom, but it was usually |
| result of the activity' (p.187). Not only | | | | used in a token way as a starting point to |
| does this approach develop students' | | | | obtain a single correct answer, usually by |
| confidence in their own ability to think | | | | following a single 'correct' procedure. More |
| mathematically (Schifter and Fosnot, 1993), | | | | recently, however, professional organisations |
| it is a vehicle for students to construct, | | | | such as the National Council of Teachers of |
| evaluate and refine their own theories about | | | | Mathematics (NCTM, 1980 and 1989) have |
| mathematics and the theories of others (NCTM, | | | | recommended that the mathematics curriculum |
| 1989). Because it has become so predominant a | | | | should be organized around problem solving, |
| requirement of teaching, it is important to | | | | focusing on: |
| consider the processes themselves in more | | | | |
| detail. | | | | (i)developing skills and the ability to apply |
| | | | these skills to unfamiliar situations |
| The Role of Problem Solving in Teaching | | | | |
| Mathematics as a Process | | | | (ii)gathering, organising, interpreting and |
| | | | communicating information |
| Problem solving is an important component of | | | | |
| mathematics education because it is the | | | | (iii)formulating key questions, analyzing and |
| single vehicle which seems to be able to | | | | conceptualizing problems, defining problems |
| achieve at school level all three of the | | | | and goals, discovering patterns and |
| values of mathematics listed at the outset of | | | | similarities, seeking out appropriate data, |
| this article: functional, logical and | | | | experimenting, transferring skills and |
| aesthetic. Let us consider how problem | | | | strategies to new situations |
| solving is a useful medium for each of these. | | | | |
| | | | (iv)developing curiosity, confidence and |
| It has already been pointed out that | | | | open-mindedness (NCTM, 1980, pp.2-3). |
| mathematics is an essential discipline | | | | |
| because of its practical role to the | | | | One of the aims of teaching through problem |
| individual and society. Through a | | | | solving is to encourage students to refine |
| problem-solving approach, this aspect of | | | | and build onto their own processes over a |
| mathematics can be developed. Presenting a | | | | period of time as their experiences allow |
| problem and developing the skills needed to | | | | them to discard some ideas and become aware |
| solve that problem is more motivational than | | | | of further possibilities (Carpenter, 1989). |
| teaching the skills without a context. Such | | | | As well as developing knowledge, the students |
| motivation gives problem solving special | | | | are also developing an understanding of when |
| value as a vehicle for learning new concepts | | | | it is appropriate to use particular |
| and skills or the reinforcement of skills | | | | strategies. Through using this approach the |
| already acquired (Stanic and Kilpatrick, | | | | emphasis is on making the students more |
| 1989, NCTM, 1989). Approaching mathematics | | | | responsible for their own learning rather |
| through problem solving can create a context | | | | than letting them feel that the algorithms |
| which simulates real life and therefore | | | | they use are the inventions of some external |
| justifies the mathematics rather than | | | | and unknown 'expert'. There is considerable |
| treating it as an end in itself. The National | | | | importance placed on exploratory activities, |
| Council of Teachers of Mathematics (NCTM, | | | | observation and discovery, and trial and |
| 1980) recommended that problem solving be the | | | | error. Students need to develop their own |
| focus of mathematics teaching because, they | | | | theories, test them, test the theories of |
| say, it encompasses skills and functions | | | | others, discard them if they are not |
| which are an important part of everyday life. | | | | consistent, and try something else (NCTM, |
| Furthermore it can help people to adapt to | | | | 1989). Students can become even more involved |
| changes and unexpected problems in their | | | | in problem solving by formulating and solving |
| careers and other aspects of their lives. | | | | their own problems, or by rewriting problems |
| More recently the Council endorsed this | | | | in their own words in order to facilitate |
| recommendation (NCTM, 1989) with the | | | | understanding. It is of particular importance |
| statement that problem solving should underly | | | | to note that they are encouraged to discuss |
| all aspects of mathematics teaching in order | | | | the processes which they are undertaking, in |
| to give students experience of the power of | | | | order to improve understanding, gain new |
| mathematics in the world around them. They | | | | insights into the problem and communicate |
| see problem solving as a vehicle for students | | | | their ideas (Thompson, 1985, Stacey and |
| to construct, evaluate and refine their own | | | | Groves, 1985). |
| theories about mathematics and the theories | | | | |
| of others. | | | | Conclusion |
| | | | |
| According to Resnick (1987) a problem-solving | | | | It has been suggested in this chapter that |
| approach contributes to the practical use of | | | | there are many reasons why a problem-solving |
| mathematics by helping people to develop the | | | | approach can contribute significantly to the |
| facility to be adaptable when, for instance, | | | | outcomes of a mathematics education. Not only |
| technology breaks down. It can thus also help | | | | is it a vehicle for developing logical |
| people to transfer into new work environments | | | | thinking, it can provide students with a |
| at this time when most are likely to be faced | | | | context for learning mathematical knowledge, |
| with several career changes during a working | | | | it can enhance transfer of skills to |
| lifetime (NCTM, 1989). Resnick expressed the | | | | unfamiliar situations and it is an aesthetic |
| belief that 'school should focus its efforts | | | | form in itself. A problem-solving approach |
| on preparing people to be good adaptive | | | | can provide a vehicle for students to |
| learners, so that they can perform | | | | construct their own ideas about mathematics |
| effectively when situations are unpredictable | | | | and to take responsibility for their own |
| and task demands change' (p.18). Cockcroft | | | | learning. There is little doubt that the |
| (1982) also advocated problem solving as a | | | | mathematics program can be enhanced by the |
| means of developing mathematical thinking as | | | | establishment of an environment in which |
| a tool for daily living, saying that | | | | students are exposed to teaching via problem |
| problem-solving ability lies 'at the heart of | | | | solving, as opposed to more traditional |
| mathematics' (p.73) because it is the means | | | | models of teaching about problem solving. The |
| by which mathematics can be applied to a | | | | challenge for teachers, at all levels, is to |
| variety of unfamiliar situations. | | | | develop the process of mathematical thinking |
| | | | alongside the knowledge and to seek |
| Problem solving is, however, more than a | | | | opportunities to present even routine |
| vehicle for teaching and reinforcing | | | | mathematics tasks in problem-solving |
| mathematical knowledge and helping to meet | | | | contexts. |
| everyday challenges. It is also a skill which | | | | |