PROBLEM-SOLVING HEURISTIC INSTRUCTIONAL METHOD

The teaching treatment was conducted from July to October 2014. The researcher negotiated with the school authorities of the two experimental schools to allocate one hour per week as it was impossible to conduct the study after school hours for operational reasons. In total 9 hours were used in teaching each of the Grade 6 classes in the two experimental schools.

60 3.5.1 Designing the problem-solving heuristic instructional method

The problem-solving heuristic instructional method is underpinned by two theories, namely the APOS theory and the modelling and modelling perspectives approach. Adopting them as guide enabled the researcher to develop a preliminary genetic decomposition, which is a specific mental construction learners may make as they develop a conception in algebra, and this was implemented flexibly. Problem-solving entailing MEAs informed the pedagogical approach used to teach Grade 6 learners algebra. The APOS theory was used as a framework to develop learners’ understanding of algebra. The researcher developed MEAs with which learners are familiar and commonly experience in their daily lives. Most importantly, the modelling- eliciting activities featured components of all six principles of the modelling and modelling activity. The modelling-eliciting activities create the necessary environment for learners to develop a more comprehensive understanding in algebra. When learners create meaning from their own symbolic representation it could be hypothesized that meaningful learning (Ausubel, 1962) is promoted, as opposed to imposing a system of symbols and notations on learners (Chamberlin & Coxbill, 2012). The activity designed had components that guided the development of learners’ conceptual understanding in algebra as it occurs through the mathematization of the modelling-eliciting activity.

3.5.2 Implementation of the problem-solving heuristic instruction approach 3.5.2.1 Who implemented the problem-solving heuristic instructional method? The researcher took up the role of researcher-educator and implemented the intervention himself. There was a myriad of reasons why the researcher decided to implement the problem-solving heuristic instruction approach by himself. Firstly, there was a challenge in training the mathematics educators in the experimental schools on how to implement the problem-solving heuristic instructional method because these educators had their own professional schedule and the researcher could not find an appropriate time to provide the training. Pursuing data collection in this direction could have delayed the data collection process. Secondly, the study had to ensure there was only limited variation in the implementation of the problem-solving heuristic instructional method across the two experimental schools, in order for the true impact of this approach to be measured. In experimental research, variables that are exposed to the experimental group and which are likely to change the dependent variable must

61 be similar (Gay, Mills & Airasian, 2011). Lastly, the researcher wanted to ensure that the problem-solving heuristic instructional method was implemented correctly as per its design. The study was wary of the effect that the researcher-educator role, played by the researcher, could have on the internal validity of the study. However, through the prolonged process of the researcher teaching the learners in the experimental group, they gradually became accustomed to the researcher as being one of their educators. The researcher was already an educator in the education department in the same district and understood all polices and rules of the KZN education department with regard to teaching and learning. Hence it was not a challenge for the researcher to accustom himself to the learners and their classroom situation.

3.5.2.2 What was the classroom setting in each experimental school?

Ninety-two learners in the two experimental group schools participated in the problem- solving heuristic instructional method. Experimental Group 1 was divided into 9 groups of five members each and one group of four members. Experimental Group 2 consisted of 8 groups of five learners each and 1 group of three learners. According to Wessels (2014, p. 4), “MEAs are solved in groups of three to five people”. The APOS theory has a social component that relies on cooperative learning, as the context of group-work is more likely to give rise to more explicit questions, doubts, and explanations by learners than what would typically transpire in individual contexts (Vidakovic, 1993). Arnon et al. (2013, p. 107) suggest that “The APOS theory functions under the premise that working in groups makes a difference in the affective domains of the individuals”.

3.5.3 Instruction and learning

The researcher used the MEAs as a medium of instruction to develop and explain concepts in algebra through the preliminary genetic decomposition as explained in section 3.5.3.1(see section 2.3.2.1). Learners responded positively to the problem- solving heuristic instructional method by developing and linking their understanding to the algebraic concepts being taught with the everyday situations they experience in their daily life, which translates to learners developing a deep conceptual understanding of the algebraic concepts being taught.

62 Learners were also given activities to do at home individually. The activities were based on selected algebra questions from Grade 6 learners’ mathematics work book. This was to enable them to reflect on the various stages of the preliminary genetic decomposition. In the following teaching session learners in each group were supposed to discuss, reach agreement and then present a common answer to the whole class.

3.5.3.1 Preliminary genetic decomposition

i. Action stage: The learners were guided to reflect on their understanding of the MEA to formulate a rule or relationship among the elements in the MEA in line with their understanding of the MEA. The learners were guided to test their rule by substituting input values to give output values in line with the objectives of the MEA.

ii. Process stage: As the learners repeated and reflected on these actions several times, they gradually began to interiorise the actions into a mental process, where they could conceptualise a rule as an expression that dynamically transforms an input value into an output value without performing any extensive calculation. Once this has been established, learners are guided to manipulate the rule in order to be able to substitute output values to predict input values in accordance with the objectives of the MEA. iii. Object stage: As the learners reflect on the operations applied in this process, they gradually encapsulate the process-conception into an object-conception where they conceptualize a rule or relationship as a static object that can itself be transformed. At this stage the learners are guided in thinking of parallel problems similar to the MEA, and modify and manipulate the original rule to explain the goals of that activity.

iv. Schema: The learners are given activities that will enable them to reflect and interconnect the various stages of actions, processes and object conceptions in a structured manner.

3.5.4 Evidence of performance after engaging in the problem-solving heuristic instructional method

The conceptual understanding learners gained through the implementation of the problem-solving heuristic instructional method translated to learners being able to tackle correctly algebra questions they have not encountered before.

63 3.6 MEASURING EFFECTS OF THE PROBLEM-SOLVING HEURISTIC