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