Problem solving frameworks

In this section, two problem-solving frameworks are described, namely that of Polya and Schoenfeld.

2.3.4.1 Polya’s problem-solving framework

According to Polya (1988:5), there are four phases of problem solving. He outlines the four phases as understanding the problem, devising a plan, carrying out the plan and looking back. Proponents of Polya’s framework follow a step-by-step instruction in teaching.

(a) Understanding the problem

Polya (1988:5) is of the opinion that in order to understand the problem, the verbal statement of the problem should be understood. Learners should be able to repeat the statement using their own words. To make the problem understandable, the teacher should design and pose the problem in such a way that learners are capable of determining what information is being requested (Hatfield et al., 2000:96). Learners should become involved in the problem and be able to state the parts of the problem, namely the unknowns, the data and the conditions. Modes of representation can also be used to enable the teacher to make the problem understandable and interesting to the learners. Modes of representations are described as forms for representing mathematical concepts and principles externally through the use of written or oral language, manipulatives, diagrams, calculators or computers (Artzt et al., 2008:12).

(b) Devising a plan

In this stage, the role of the teacher is to offer ‘unobtrusive help’ to learners. Artzt et al. (2008:1) state that the teacher is supposed to help the learners organise and formalise their

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ideas. Learners ought to be allowed to find strategies of solving the problem with little interference from the teacher. If the teacher interferes too much, the problem becomes his or hers and the reason for the activity is lost. The teacher can ask the learners to identify the unknown in a given equation or expression and to try to think of familiar problems having the same or similar unknowns. During this stage, the responsibility of the teacher shifts from providing information to asking questions and providing resources (Burton, 1989:20).

(c) Carrying out the plan

In this phase, the leaner should be convinced about the correctness of each step. The teacher can emphasise the difference between seeing clearly that the step is correct and proving that the step is correct. Learners should be able to apply various strategies until the problem is solved. According to Hatfield et al. (2000:96), strategies are methods by which the problem can be solved. These strategies are determined by the skill and mathematical tools that the learner has previously mastered. In this phase, learners need to be patient in order to succeed with solving the problem (Burton, 1989:21). Teachers and learners have to realise that the time for problem solving is open-ended, in the sense that the problem can be continued in the next session without feeling that a solution must be reached at the end of each session (Burton, 1989:21).

(d) Looking back

After arriving at the solution, learners should re-examine and reconsider the path that led to the required solution. Learners should be able to check each step and have a good reason to believe that their solution is correct. If the solution is not correct, another strategy should be tried. Hatfield et al. (2000:96) suggest that learners should always write the numerical answer in a full sentence. Writing numerical answers in full sentence forces learners to reflect on their answers as they translate them (Hatfield et al., 2000:96). Buschman (2004:304) supports the view of writing the solution processes into words, as such actions enhance learners’ deep understanding of mathematics. Moreover, learners should be able to notice that the knowledge gained and the procedure used in arriving at the solution could be applied to some other problems. The teacher can also emphasise to the learners that no problem is completely completed, because we can always improve on our understanding of the solution.

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2.3.4.2 Schoenfeld’s problem-solving framework

Schoelfeld’s framework is based on Gestaltism learning theory developed from a cognitive science perspective. According to him, Gestaltists believed in rich mental structures and felt that the object of instruction should be to help learners develop these structures (Schoenfeld, 1987:4). Schoenfeld (1985:12) states that any mathematical problem-solving activity is built on a foundation of basic mathematical knowledge, which is called ‘resources’. Furthermore, Schoenfeld (1985:12) outlines categories such as heuristics, control and belief systems that are required when one is working on problems with mathematical content.

(a) Resources

According to Schoenfeld (1985:12), resources are tools, procedures, facts and skills potentially accessible to the problem solver. Resources imply the mathematical knowledge and procedures that the individual is bringing to bear on a particular problem. The resource stage can also be referred to as the problem solver’s ‘initial search stage’. In this stage, the individual must have an intuition and informal knowledge regarding the problem. The resource stage consists of four classes, namely a set of relevant facts known by the problem solver, algorithmic procedures known by the individual, routine procedures and procedural knowledge about the agreed-upon rules for working in the domain.

(b) Heuristics

Heuristics are strategies, techniques or rules of thumb for successful problem solving, and suggestions that help an individual to understand a problem better and make progress towards its solution. Larson (1983:1) also affirms that heuristics are strategies or tactics of solving problems such as mathematical problems. These strategies can include drawing figures, exploiting related problems, reformulating problems, introducing relevant notations, arguing by contradiction, working forward from available data, working backwards, and testing and verifying procedures.

(c) Control

Control implies how the individuals use the information at their disposal to solve problems. This category focuses on decisions about what to do in a problem and decisions that may ‘make or break’ an attempt to solve the problem. Decisions include making plans, selecting

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goals and sub-goals, monitoring and assessing solutions, and revising or abandoning directions when the assessments indicate that such actions should be taken.

(d) Belief systems

The category of belief systems means one’s mathematical worldview; the perspective with which one approaches mathematics and mathematical tasks. This category includes the belief about the self, the topic, mathematics and the environment. One’s belief about mathematics can affect how one chooses to approach a problem, which techniques will be used or avoided, and how long and how hard one will work on the problem.

From Polya and Schoenfeld’s problem-solving frameworks, it follows that rather than seeking a single correct answer, learners deduce the problem, gather appropriate information, identify possible solutions, evaluate options and present conclusions. Furthermore, they contend that learners become good problem solvers by learning mathematical knowledge through the use of strategies or heuristics. The frameworks seem to be based on constructivist learning approaches that having learners construct their own solutions leads to effective learning experiences. Proponents of problem-based learning, such as Polya and Schoenfeld, believe that when learners develop frameworks for constructing their own procedures, they are integrating their conceptual knowledge with their procedural skill.

The two frameworks of teaching and learning to promote deep mathematical understanding offered the researcher an integrated view of problem solving, although Kilpatrick et al. (2001:116) argue that “no framework captures completely all aspects of expertise, competence, knowledge and facility in mathematics”. However, the researcher associates the two frameworks as one of the elements necessary for anyone to teach and learn mathematics successfully.