Polya (1988:5) and Schoenfeld’s (1985:12) problem-solving frameworks were referred to as the researcher linked data from the teachers’ experiences to discuss the findings. Such problem-solving frameworks include an understanding of the problem (resources available to the problem solver), devising a plan (heuristics for successful problem solving), carrying out the plan (control and belief systems) and finally looking back. The researcher used this framework to synthesise the findings from the empirical part of the study.
Understanding of the problem and resources
Having considered the meaning of understanding, the four selected teachers agreed that understanding plays an important role in mathematics teaching. The teachers indicated that a problem-solving approach underpins deep mathematical understanding. However, from the classroom observation data, there was little evidence of learners been given opportunities to demonstrate and justify their own ideas in order to foster understanding, as illustrated in sections 4.3.1.1 (a) and (b) and 4.3.2.
To enhance understanding, learners should have the required prior knowledge regarding the problem. Even though the four cases stated the importance of establishing learners’ prior knowledge, there was little evidence to show attempts by the four cases to establish learners’ prior knowledge from the data generated through interviews and the classroom observation visit lists. To the researcher, the teacher’s role seems of essence to help the learners develop connections between their current knowledge and new information in order to promote understanding. However, according to the four cases, the time factor remains a challenge for teaching through problem solving, yet crucial for learners in terms of learning with understanding. Teachers tend to focus on covering content rather than on developing learners’ understanding (Artzt et al., 2008:4). As discussed in Section 4.3.3.1, the teachers
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acknowledged that it is sometimes time-consuming to allow learners to make sense of mathematics in their own way.
Devising a plan and heuristics
From extracts of interviews and classroom observation data presented in Section 4.3.1, there was no evidence of teachers providing learners with opportunities to create strategies, suggestions and techniques that help learners understand a problem better and make progress towards its solution. It seems the four teachers continuously told their learners what to do, as depicted in Table 4.2, 4.6 and 4.10 However, from the literature reviewed in Chapter 2, in problem solving the responsibility of the teacher shifts from providing information to asking questions and providing resources (Burton, 1989:20).
From the classroom observation data presented in Section 4.3.1, there is however little evidence from the four cases of shifting from being transmitters of information to being resource providers. The learners were not given sufficient opportunities to make plans and create strategies to solve problems. Yet, it was evidenced in the literature presented in Chapter 2 that learners should be able to apply various strategies until the problem is solved. According to Artzt et al. (2008:1), the teacher is supposed to help learners organise and formalise their ideas. Learners should 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. Driver and Oldham (1986:112) state that teachers’ key role should be to lead learners in their own discovery and understanding of mathematical concepts.
Carrying out the plan, control and belief system
From the classroom observation data there was limited evidence of the learners being convinced of the correctness of each step by teachers B, C and D, except in the case of Teacher A, as explored in Section 4.3.1. Table 4.16 shows how the researcher observed the learners engaging in some discussions to illustrate how they arrive at the solution. However, in general, the four teachers were observed selecting goals and sub-goals as well as providing directions, indicating actions to be taken to solve a particular problem, as discussed in Section 4.3.1. The teachers could emphasise the difference between seeing clearly that the action taken is correct and proving that the action is correct (Polya, 1988:5).
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From the classroom observation data presented in Section 4.3.1, it was evident that the ‘control’ of solving problems seemed to lie entirely with the teacher and not with the learners. As presented in Chapter 2, teaching through problem solving means that learners solve problems in their own ways, use mental tools already available to them and are able to learn important mathematical concepts (Schroeder & Lester, 1989:33). In addition, if learners are able to become involved in what they are doing, then they have reached significantly high levels of mathematics (Brousseau, 1997:28). This is not the case with the four selected teachers, as they were normally dominating the discussions and not the learners, as shown, for example, in tables 4.12, 4.13 and 4.18.
Looking back
From the classroom observation data, there was little evidence of the learners re-examining and reconsidering the ‘path’ they took to arrive at the solution. The learners were not able to check each step taken to arrive at a solution. Instead, after the leaners completed the task, they should share the process used in solving the problem. With the teacher’s guidance, learners should reflect on the problem, their work and the important mathematical ideas that have emerged.
For the four teachers who constituted the case study, it seemed to be ones who reflected on the ‘path’ taken and indicated whether the answer is correct or not as illustrated in tables 4.11, 4.13 and 4.14. However, Manouchehri (2007:299) states that ending learners’ discussions by giving them the correct formulas or answers would likely close the door not only on their mathematical investigations but also on the formation of a learning community in which members willingly explore mathematics and engage in collaborative construction of knowledge. Moreover, Schroeder and Lester (1989:33) state that the facilitation of mathematics using problem solving entails more than simply posing the correct type of problems and then allowing learners to solve them without understanding.
4.6 CONCLUSION
This chapter has covered an analysis and discussion of the findings of the study. The data generated through the interviews, classroom observations and questionnaires were analysed. By doing so, the researcher noted that there is little evidence to indicate that teachers in the RUMEP programme have made a definitive move towards a problem-solving approach. It
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appears that the teachers are still finding their way to implement the teaching and learning of mathematics through problem solving. From the study it was evident that there is a mismatch between what the four teachers in the case study advocate and their classroom practices. The data generated through the face-to-face interview seem to contradict the data in the classroom observation notes. From the interview data, the four selected teachers seem to be in favour of problem solving, but in the actual classroom, their teaching is mostly dominated by ‘telling and showing’. According to Brodie et al. (2001:541), teachers can create a mismatch or gap between what actually happens in the classroom and curriculum demands.
The teachers in this study seem to have the knowledge of the theoretical aspects of a problem-solving approach, but the implementation aspects are still problematic. This may be attributed to some of the factors (challenges) mentioned by the participants in this study. Panaoura (2012:291) states that mathematics teachers often experience some challenges in making use of proper mathematical problem-solving tools. However, it seems the understanding and interpretation of problem solving is not the main obstacle, because the participants in this study seem to share a common understanding of problem solving, as can be seen in their responses in the interviews and the questionnaire. However, for teachers to become invested sufficiently in this process of professional development, they must first come to believe that their current practice is in some way problematic. In essence, it is very difficult for teachers to make a mind shift in their teaching orientation. It is very difficult to move out of their comfort zone.
The next chapter outlines the conclusions and implications as well as the limitations of the study.
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