Development of problem solving skills

According to the curriculum document, mathematics should be learned through problematic contexts, to develop problem-solving skills in the learners. Helping learners to become better problem solvers is an important aim in algebra teaching (French, 2002, p. 5). While to some people problem solving in mathematics may be thought of as just about getting answers for questions in an exercise at the end of a topic, to mathematics educators, it is much more than

that. For the purposes of this study, we will consider two perspectives of problem solving namely: a) problem-solving as a skill and b) problem solving as a teaching approach. We thus view problem solving not only as a goal of learning algebra but as a means of learning algebra as well (NCTM, 2000, p.52).

i) Problem solving as a goal of algebra learning

Problem solving as defined by Bell (1996, p.167) is a mathematical activity that involves “exploring problems in an open way, extending and developing them in the search for more results and more general ones”. When confronted with a realistic problematic situation, learners bring their mathematical tools to unravel the problem. They discover and organize the mathematics through mathematising horizontally and vertically. Learners, with as little guidance as possible, are given opportunity to “aspire, to climb, and to dive to heights and depths as steep and as deep as they can reach and afford” (Freudenthal, 1991, p. 47) according to demands of each particular mathematical problem situation. In this way learners go through what Freudenthal (1991, p.46) termed guided reinvention. When mathematics is learned in this way, the valuable knowledge and abilities so acquired are there to stay, they will be more easily retained and transferred; and mathematics learning becomes enjoyable as learners discover (reinvent) knowledge and the “aha”(eureka!!) effect is felt.

The experience that learners must go through in problem-solving, demands thoughtful planning and a lot of creativity on the part of the teacher, regarding the nature of problems to be solved. A mathematical problem is one that has no immediate solution. It requires mathematical knowledge and skills to solve. According to this description, the problems that learners engage with should be those that provide for exploration thus investigation, reflective enquiry and discovery come into the picture. They should require from learners, answers to “what if” type of questions, thus offering learners opportunity to make conjectures, make extensions and analyze their own solution procedures. Learners should be able to generalize about the problem situation they are confronted with. They should even be able to make predictions based on data gathered during the solution process.

The problem solving process involves four stages as identified by Polya (1973). These are sequentially arranged as: Understanding the problem, devising a plan or deciding on an approach for tackling a problem, carrying out the plan, and looking back (at the problem, the answer, what has been done to get there and interpreting the solution in terms of the question asked). This means that the problem solver needs to understand the problem clearly; this may involve representing it in different ways, identifying different variables and establishing the existing relationships between variables. The problem solver would then devise and implement a strategy for solving the problem. Having found the solution, s/he then reflects on the solution and checks if it makes sense; that is, interpreting it in terms of the original problem situation. The question is now “how can these problem solving skills be developed in the learners?” This then brings us to the discussion on problem solving as an approach to teaching of algebra.

ii) Problem solving as a means to algebra learning

Problem solving skills cannot be taught to learners directly. They should be acquired through real experience of solving problems. Hiebert, Carpenter, Fennema, Fuson, Human, Olivier &Wearne (1996, p. 12) argue that mathematics instruction should be geared towards students “problematizing the subject”. According to them, the subject matter should be made more problematic, that is

allowing students to wonder why things are, to inquire, to search for solutions and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students (Hiebert et al, 1996, p. 12).

Learners should be given opportunity to grapple with the mathematics and discover the solutions themselves. The result of this is that learners will invent their own knowledge and gain better understandings of the subject matter. The problems posed should offer learners opportunity to acquire the symbolic language and be able to represent information in different ways. They should be able to conjecture about situations, critique and justify their solutions. Thus through logical reasoning, they would be able to make decisions about the problem at hand.

The teaching here is aimed at enhancing problem solving skills in the learners. Use of models is also very vital in problem solving. Learners should be able to analyze the problem confronting them, identify the variables, establish the relationships between the different variables, and tackle the problem. They should be able to use the different representations for situations. A mathematical problem situation may be represented algebraically, be it graphically or in the form of equations. Mathematical models serve as a bridge between the informal contexts, related mathematics and a more formal (abstract) mathematics (Van den Heuvel-Panhuizen, 2000, p. 5). As these forms can be easily manipulated, a solution may be sought using these representations and the results then interpreted in terms of the original problem situation. This would therefore increase learners’ understanding of the problem and help them make better decisions about the situation confronting them.

Indeed, contextualized mathematical problems are ideal for enhancing problem-solving skills. They offer learners opportunity to move from one level of understanding to another, from informal solution procedures to more advanced, formal mathematical understandings. Learners first develop solution strategies (representations) that are closely related to the context and later on, the context can be treated more generally thus acquiring the character of a model that can then be used to solve problems of a similar nature. In this way, the model shifts from being only of a particular situation to being a model for a certain class of situations. Through reflection on activities at the lower level, the learner is able to move to the next level of understanding (van den Heuvel-Panhuizen, 2000, p. 6).

This approach has many implications on the part of the teacher, with regard to their role in teaching through problem solving. These concern mainly the nature and organization of problems, and the classroom environment and culture.

a) Nature and organization of problems

Teaching through problems is very demanding on the part of the teacher. It demands that the teacher is very conversant with the curriculum and knows his/her learners’ mathematical abilities very well. The teacher needs to very resourceful and creative in designing the problems. This needs a lot of time and commitment.

Teaching through problem solving requires that the teacher has the learners’ interests at heart and is determined to see them gain mathematical understandings. The teacher should therefore have a collection of problems from where learners would choose. The problems should cater for a wide range of abilities so that at the end of each session everyone would have been successful. This has motivational value, as learners would always look forward to the next session with all eagerness. The problems should also have multiple paths to solutions, thus allowing learners to use their existing knowledge and ideas to get to the solution. In executing their solution strategies, learners expand on their knowledge and grow in their understanding as they interact with the problem and listen to other students’ ideas as well (Van de Walle, 2004, p. 38).

Problems that learners engage with should take into consideration the mathematical knowledge that learners have acquired but should still challenge their thinking. As has been discussed earlier, they should be those that are appealing to the learners. They should be related to the learners’ experiences and be in situations that learners can imagine and understand. The problems should provoke learners’ cognitive abilities, engage them in mathematical “habits of mind” (Cuoco, Goldenberg, & Mark, 1996, pp. 378-383) and develop their mathematical power. Cuoco et al (1996, pp.3-8) suggests that, learners should be pattern sniffers, experimenters, tinkerers, describers, inventers, visualizers, conjecturers, and guessers. The problems they tackle should be focused on the mathematical concepts that learners are to learn. This means that in teaching algebra, the problems chosen should be based on the current algebraic knowledge and skills that learners have and be aimed at enhancing acquisition of knowledge that is more advanced and skills. Problems that they engage in should require them to use algebraic reasoning to justify their procedures and thus solutions.

Teaching through problem solving also means confronting learners with different problems with varying levels of complexity. This is important, as the problems would be representative of problem situations that learners encounter in real life (Van den Heuvel-Panhuizen, 1996, p.13). Problems that learners encounter in real life are not all solvable; some require a certain

level of maturity while some may just need more time before a solution could be realized. Problems of this nature thus develop in the learners, perseverance towards problem solving. I now turn to another feature of the recommended teaching approach, interaction and guidance.