Act it out: in this strategy the learners act out the problem situation in order to find the solution Acting out the problem in reality extends their understanding of the situation at hand.

Example: the Principal and six teachers met on the first day of school. They shook hands with each other once. How many handshakes were there altogether?

The mentioned strategies and skills (teaching methods and strategies) are acquired over a period of time and with experience, therefore to teach the subject teachers need to discover appropriate teaching methods and strategies in their formative years to make the content understandable. 2.5 PROBLEM SOLVING AND METACOGNITIVE KNOWLEDGE

Problem solving is one of the most important reasons for studying mathematics as it is regarded as a dynamic thinking and imaginative invention (Luneta, 2013). Mathematics problem solving has moved away from the drill and practice method as a result the demand of the mathematics curriculum on learners metacognitive and cognitive ability has increased tenfold. Metacognition warrants special attention due to its role it plays in problem solving. Sharma (2016:1) stated that “problem solving in any setting is a complex cognitive activity” therefore learners need greater metacognitive knowledge to investigate complex problems and make the connections between mathematical ideas. Kribbs and Rogowsky (2016:65) stated that the combination of the following metacognitive skills (comprehension, mental representation,

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solution construction and solution execution) when used in a strategic manner allows the learners to engage in solving the most complex problems.

One of the earliest goals of problem solving was the development of metacognitive skills. Metacognition warrants special consideration as it involves thinking, choosing the appropriate strategy to solve the problem and evaluating the chosen strategy to see if the solution made sense. Metacognition is often studied and related to problem solving as it involves the ability to think, read and write. According to Gurat and Medula (2016) metacognition was developed between the 1970s through the 1990s and it is during these years that metacognition became a dominant tool involving the thinking process. Gurat and Medula (2016:6) studied the use of metacognitive strategy knowledge involving mathematical problem solving amongst pre-service teachers and they stated that “metacognition refers to one‟s knowledge concerning one‟s own cognitive processes…..or anything related to them”.

Studies have shown that there is a strong link between metacognition and problem solving. Many researchers have identified that metacognition is a key aspect in the problem solving process. According to Gurat and Medula (2016:4) there are three types of metacognition, namely, metacognitive skills, metacognitive experience and metacognitive knowledge. Posamentier and Jaye (2006) indicated that as learners develop their metacognitive skills become more successful in problem solving. Learners needed to ask themselves, “What technique did I use to solve a similar problem in the past?”; “How do I find the derivative?”; “Is there anything I don‟t understand?”; “Am I headed in the right direction?”; “Have I made any careless mistakes?” (Posamentier and Jaye, 2006:80). In metacognitive knowledge there are three broad types which are of importance, namely, strategy knowledge – this refers to the learner‟s knowledge of general strategies for learning, thinking and problem solving; task knowledge – this refers to understanding of cognitive tasks as well as the when and the why to use these strategies; person knowledge – this refers to familiarity about the person (self), cognitive issues and the motivation to perform. This is all indicated in Figure 7.

Metacognitive strategy knowledge involves issues of when and where to use cognitive and metacognitive strategies and “also involves the skills needed to solve a problem such as prediction/orientation, planning, monitoring and evaluation” (Gurat and Medula, 2016:2). The metacognitive strategy knowledge of Isagani is discussed in Gurat and Medula (2016:12- 15). In the Isagani process (Figure 7) the problem needs to be presented in written form and then orally. The problem is first read and reread to bring about understanding. When analysing the problem the learner visualizes the problem by creating drawings or detailing the needed details. Certain details not related to the problem are discarded. According to Alexander

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(2015) when drawings are used learners are less likely to be bogged down as the excess language is removed from the problem. The problem is broken down into pieces so that it can be examined individually and a relation is created to real life situations.

Alexander (2015:2) stated that the teachers must always try and relate the problem to reality as much as possible and to events that are current in the learner‟s life. In this way learners can make a connection to real life situations. Various illustrations are created using the details provided in the problem and the strategy to be used is chosen. The strategy to be used will depend if the problem is recognizable or was encountered previously. If the problem is recognizable then the learner uses the known formula or considers the several strategies and subsequently chooses the strategy that suits the problem. In the event of the problem not been familiar it is read repeatedly and the trial and error method. If this fails, the learner then requests assistance from others to verify their understanding against their own. When the problem is understood and the learner is positive of using the correct strategy, he uses the steps methodically to arrive at the solution. On arriving at the solution he reverts to the problem to determine if the answer is apparent and there is no need to go through the entire process again. If time permits he reflects on the problem by rereading the problem, analysing it thoroughly, devising a plan and carrying it out.

Analyse _{Devise a plan}

Carry out the plan Re-read (Rehearsal Read and understand the problem-ask question to self (elaboration/ mentoring Visualize – make illustrations, drawing table, etc (organisation) –relate to real life situations (critical thinking)

–breakdown the problem into pieces and look at the problem part by part (predictions/orientation/ organisation

–ask question to self (elaboration/mentoring) Familiar -recall formula/strategy used before (rehearsal) –ask question to self (elaboration/ mentoring Solve

Get the answer

Check (if it has still time) -evaluation (self regulation) -ask self (elaboration) Not familiar

trial and error (strategies in solving) –ask self (elaboration/ mentoring) Ask and compare

understanding (social)

Compare answer with others (social)

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Figure 7 Metacognitive Strategy Knowledge of Isagani (Gurat and Medula, 2016:13)

According to the Ministry of Education (2007:4-5) when the learners participate in problem solving, they will engage in an extensive range of cognitive experiences that will be of assistance to them and prepare them for the many problem solving situations they will encounter in their lives. They will be trained to discover and learn mathematical concepts with understanding and perform skills in context; reflect on the nature of inquiry in the mathematics world; develop strategies that can be useful to new circumstances; connect the mathematics they study at school with its relevance in their daily lives; make associations between the concepts in mathematics; represent mathematical ideas and replicate (model) situations using concrete materials, pictures, diagrams, graphs, tables, numbers and symbols; move from one representation to another and recognize the connections linking them to other representations; through collaboration communicate their explanations and take note of the explanations of their peers and persist in tackling fresh challenges (Ministry of Education, 2007). Taking the aforementioned into consideration one can deduce that the focus is on the learner. According to Debrenti (2013:88) it is important to make the learner part of the problem as it will “involve him in the solving procedure, offer him the possibility of self-expression or manifestation, help him to experience success and for him not to be afraid of failure, make him understand that mistake is allowed and lead him towards the pleasures of solving a problem”.

Kuzle (2013:258) sums it up nicely by stating that “mathematics teacher education programs should allow pre-service teachers with opportunities to learn about a variety of pedagogical and learning issues, and means for implementing problem solving within the lessons, as well as to also experience them with respect to (meta)cognitive and non-cognitive aspects of problem solving”.