The questionnaire, the MPSSI, the pre- and post- multiple-choice and word- problem tests were administered to both the experimental and control group at the beginning and at the end of the intervention.
The control group: As indicated in section 1.10.3 the control group was
taught by the current grade 8 mathematics teacher using the traditional teaching approach. With the traditional approach (see section 1.4 and 2.5.1) the teacher represents all source of knowledge and learners are empty vessels to be filled with knowledge. For this study the control group teacher followed the Department of basic education CAPS syllabus and focussed on the teaching of algorithms that could be employed by learners to solve problems. After teaching an algorithm, the teacher worked a few examples with learners. Learners got similar problems as exercises or homework for the day and used the demonstrated algorithms to solve the assigned problems. If learners in the control group did not understand any algorithm rule, the teacher would show the rule again “drilling each piece in sequence”. The researcher observed that learners in the control group expected their teacher to explain all the algorithm rules and did not attempt to solve unfamiliar problems for which no algorithms were provided.
The experimental group: The researcher started the intervention programme
by fertilising the experimental group learners’ minds by explaining to them what the PCTLA entails. As presented by the grade 8 mathematics CAPS
syllabus the following topics were covered during the intervention: Pythagoras’s theorem; perimeter, surface area and volume; numeric and geometric patterns; graphs; transformation geometry; ratio and rate and financial mathematics.
At the beginning of each lesson, the researcher would put an outline on the blackboard of what was going to be covered in the lesson. By seeing the outline on the board, learners can begin to think about the existing knowledge and concepts that are related to the day’s lesson. For this study, seeing an outline on the blackboard stimulated learners’ thinking about various topics and this helped to activate their prior knowledge about the topic of the day. The researcher created a classroom environment in which social interaction (see section 2.6.4) was highly valued. This is an environment in which learners believed that what was important was the effort they spent looking for solutions and that they would have learnt something even if they did not find the correct solution to the given problem.
Polya’s four-phase problem solving process (see section 2.10) was modified and used to solve the problems and tasks for this intervention. Figure 3.1 below shows the problem solving model that was used for this intervention.
Figure 3. 1 A cyclic and dynamic model for problem solving
Source: Wilson, Fernandez & Hadaway (1993)
Problem posing
Understanding the problem
Looking back Making a plan
Carrying out the plan
This model is dynamic and cyclic in nature and conveys that in some cases after learners understand a problem and devise a plan they may be unable to proceed. This may require the learner to make a new plan, to develop new understanding or to pose a new related problem (Wilson, Fernandez & Hadaway 1993). For this study, learners were also required to look back at the formulation of the given problem, to frequently revise the whole problem during the problem solving session and to be in a position to restart if required (Dendane 2009).
The researcher gave hints only during problem solving and there were no formal lessons illustrating methods to solve the problems. After grappling with unfamiliar problems, learners were required to place their solutions on the board and to fully explain their work to the class. Other learners in the class were encouraged to critique the solution and at the same time try to provide alternative solutions to the problem. As the class discussed the solution to the problem, the researcher guided the discussions as needed by asking questions to ensure that learners understood the solution before moving on to the next problem.
The researcher also employed teacher-led and learner-led groups in her teaching. This was done by forming three groups for example group A, B and C. On day 1, the researcher would tell learners in group B and C to work independently while working with group A, which would be the target group of the day. After the lesson, the researcher would assign follow-up work to group A. On day 2, the researcher would work with a new group, which would be B, while group A worked on teacher follow-up centre and the third group C would still work independently. On day 3, the researcher would work with group C and in that way a three-day rotating system was established.
During the intervention programme, the researcher implemented the strategies for developing mathematical problem solving skills as explained in section 2.12. She developed the experimental group learners’ thinking skills and useful habits of mind as set out by Cuoco et al (1996) by encouraging learners to always look out for patterns, perform thought experiments, explain
the solution methods to the teacher or their peers and develop conjectures. Learners were also required to visualise the problem, make mental pictures and think out aloud while solving problems.
The researcher structured learning situations (see section 2.12.2) to develop the experimental group’s mathematical problem solving skills by implementing the following:
When given a new problem, learners were encouraged to identify how it was similar or different from previous problems and how this could influence their approach to solving the problem.
Learners were required to use different problem solving strategies (see section 2.9) to solve given problems and were encouraged to compare the effectiveness of the different strategies. Discussions were held towards the end of lessons to compare all the different strategies and valid solutions generated by all learners.
The researcher encouraged learners to value the problem solving process by implementing the techniques stated in section 2.12.2.3.
Learners were encouraged to generate their own problems. The advantages of having learners pose their own problems were given in section 2.12.3. The action of having learners generate their own questions transforms their relationship with authority and tests (Holt 1968) and at the same time affords them the opportunity to develop mathematical problem solving skills.
Learners were given enough time to think before responding to questions. Providing learners with waiting time before answering questions helps to develop their mathematical problem solving skills since they have the opportunity to think deeply about the problem at hand.