Analysis of the lesson observations

The pre-service teachers were randomly placed in groups of six. As a group they had to prepare a lesson and present the lesson to their colleagues. These lessons were classified as teach back lessons or co-teaching. I also saw this as didactic teaching and learning. In the absence of microteaching in their training I used this opportunity to expose the pre-service teachers to the intricacies of teaching mathematics and to learn mathematics. During their co-teaching they were exposed to mathematical knowledge for teaching (MKT) and pedagogical content knowledge (PCK) used by their colleagues. This was to gauge their content knowledge and how they developed their teaching using visualization and problem solving strategies. This is supported by Boonen et al (2016) who argue that besides professional teacher training the pre- service teachers need to be exposed to pedagogy and methodology.

The pre-service teachers were randomly placed in groups. In their groups they had to choose a topic from the mathematics curriculum. Although all groups presented their lessons I chose the first group for the purpose of this study.

Mars

This group, Mars, presented a lesson on Data Handling. Two pre-service teachers working in tandem introduced the lessons to their colleagues. Using a Powtoon presentation combined with a video clip they presented a clip on transportation in South Africa. As the video clip was presented the pre-service teachers had to indicate the concepts alongside key aspects on the worksheet to give meaning. What was very interesting during this introductory phase was the manner in which they discussed the concepts. They pointed out to their colleagues that certain concepts repeated themselves and using Powtoon the pair reinforced the concepts in a fun filled manner (figure 25). The use of the video clip and Powtoon brought the concepts alive creating a better understanding.

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Figure 25 Pre-service teacher presentation of a teach back lesson

The next pair was responsible for the teaching phase. As the concepts such as tally, bar graph, mean mode and median were already discussed in the introductory phase, they presented a case study relating to travel in South Africa. The pre-service teachers were placed into groups of six. Using the learner centred approach they had to gather data from amongst themselves. According to experiential learning, learners need to experience learning as learning takes place through action (Beaudien and Quick, 1995) and I expected the same from the pre-service teachers. The pre-service teachers were required to organise the collected data in a table, then into a tally table and then a graph. Once this was completed questions were presented to them. The questions were both open ended and closed. They were quite ingenious in producing their answers or solutions to these questions. The use of problem solving strategies as using a diagram and working backwards was evident. Tables, sketches and diagrams were used to represent their understanding of the questions. According to Bal (2014:2) representation is a formation of a mathematical concept and it is a way to show an actual situation from a different view. This group had to choose a group leader to present their solutions. The group leader, by using a chart, presented the solutions in a very logical manner. The pair responsible for the teaching phase posed additional questions to the group leader on how certain solutions were arrived at. The pre-service teachers that were observing the lesson was asked to comment or request further explanations on how solutions were reached. The group leader, obtaining assistance from her group, used the white board to explain the manner on how they arrived at the solutions. I deduced from this presentation that the representations used to find the solution followed the structure of the question. The pre-service teachers translated the problem

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algebraically in a sequential order. Furthermore the idea of collaborative engagement during the discussion opened up avenues for innovative learning among the pre-service teachers. They were in a position to support each other learn mathematics.

The pre-service teachers displayed adequate MKT and PCK during their teach back lessons. This indicated that they had the confidence to teach mathematics.

Mala

The school where Mala was observed is situated in a suburb on the North Coast. It has large population of predominantly Indian learners from within the suburb and a small population of Black learners who travel from the nearby townships. This school receives a proportionate funding based on its roll but majority of the funds are received via school fees levied on the learners. It is a well-resourced school, has a team of well experienced teachers and often produces excellent results.

Mala was observed teaching a grade 4 class. The mathematics lesson began orally with a quick mix of bonds and tables. As prescribed in the CAPS document and the Jika Imfundo tracker „mental maths‟ must be done daily as it is forms an important pre-requisite in all mathematics lesson. This provides a vital foundation in mathematics as the learners are provided with opportunities to use their mental counting strategies to develop their proficiency with numbers. The lesson was based on the measurement of time. Mala introduced a problem related to the times of television programming schedules. She fostered the idea on learners that they must try and relate all problems to themselves and their environment. She discussed a problem solving strategy on the chalkboard as a means to guide her learners to solve the problem. Using Polya‟s steps of problem solving (discussed in the literature review) she initiated the discussion. She read the problem to her learners and asked them to respond to what the problem required. In this manner she provided a start to the problem. She used appropriate questions (Do you think you are on the correct path?) and prompts (try an alternate step) to initiate responses from her learners. She led the learners through the steps in a logical manner using simple rules working from the known to the unknown. The Structural Learning Theory (discussed in chapter 3) supports this method as learners need to be taught the simple steps and rules in order to proceed to more complex steps.

Mala mentioned a mathematical concept from within the problem and asked her learners to draw what they thought it meant or represented. In this manner she was determining if her learners displayed mathematical understanding (discussed in chapter 3) of the concepts. This is an important part of problem solving as learners construct their own meaning of the concept.

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The learners must be given opportunities in the mathematics classroom to represent their understanding of mathematical concepts in a variety of ways. Mala had an understanding of Polya‟s problem model and briefly mentioned to her learners what would have occurred in steps one and step two of the problem model. It must be noted that she never mentioned the model by name or diagrammatically showed the learners the steps involved. The learners responded positively to the teacher. They were eager to show off to the other learners in the classroom what they had drawn. She randomly chose volunteers and asked them to explain their diagrams to the class. The discussion although noisy was encouraging. The learners used their basic drawings to explain and justify their steps of the problem. When she was satisfied that her learners were coping with the basic parts of the problem she encouraged them to work towards a solution. At first the learners worked individually. On completing their work the Mala asked her learners to discuss their efforts with their peers. When learners assist each other, they will feel more successful, empowered, and confident about their learning. I found the discussion to determine if their friend‟s solution was any different to theirs was very constructive. During this activity Mala walked around the classroom and the learners openly engaged her to give her decision if their answer was the correct one. She tactfully cajoled them to find out for themselves. Once the entire class was done with this activity the she asked for volunteers to put up their solutions on the chalkboard. These solutions were discussed with the learners. They were given an opportunity to determine which of the solutions on the chalkboard were correct. Mala consolidated the learner‟s efforts by discussing the concepts within the problem and also showed them how each step was related to each other.

For the written activity of the lesson the learners were given a problem solving worksheet. They were instructed to use any method that they were comfortable with as long as they showed their working on the worksheet. Although encouraged to work individually learners were observed talking to their peers discussing certain problems on the worksheet. Instead of curtailing this, the pre-service teacher allowed this to continue. I thought that this was a useful change this as this is a constructive approach to collaborative learning. This collaboration allowed the learners to progress with the problems especially in the stages they felt they were facing a challenge.

On the completion of the written activity Mala asked for a volunteer to present the first solution to the class. Thereafter she randomly asked the learners for any alternative answers. Intermittently during the classroom discussion she engaged the learners by asking questions or making general statements, example, „Do you think your answer is right?‟ or „I think something is missing‟ and also putting an incorrect solution on the board. I found this to be an interesting ploy on her part as this created doubt and some of the learners were forced to go back and check

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their solutions. Some of the learners were confidently stating that their solutions were correct and the other learners were very convinced the second time around that their answers were correct.

Some of solutions were presented numerically, others schematically or diagrammatically and a few had a mixture of numerical and diagrammatic solutions. The solutions that were represented arithmetically showed that the learners merely used the numbers that were given in the problem to calculate the solution. They did not comprehend the association between the numbers in the problem and the concepts within the problem. The other arithmetical solutions were long and the learners themselves were confused while trying to articulate the answers to their peers. The learners who represented their solutions schematically, diagrammatically or involving the mixture of arithmetical and diagrams, showed that they understood problem. The representations showed their visualization skills as they illustrated their understanding of the concepts from the problem. What is important to note here is that not all learners constructed the same visual images due to their own prior knowledge and experience. I observed that these representations indicated the learners had heeded their teacher‟s advice to use their prior knowledge and experience when solving the problems. It was indicative that it was their mathematical understanding drawn from using their own previous knowledge now been presented as their ideas in the solutions. This is relevant to Kolb‟s Experiential Learning Theory and Kolb‟s Experiential Learning Cycle. It is expected that the learners utilise their previous experience, process the concepts and apply it to the problem. By using this learning cycle learners were able to apply their knowledge and understanding to significantly provide their solutions.

Mala possessed the relevant knowledge of content and teaching. Her readiness for the lesson was evident as she had come prepared with her charts as a teaching resource. The creative use of the charts enhanced the lesson as the majority of the learners were able to engage in the lesson. The use of concrete representation on the chart allowed the learners to make direct connections to concepts and relate to them. This was very evident when she asked her learners to explain where they had come across what was shown on the chart. A notable feature in this lesson was the confidence in which Mala used her pedagogical content knowledge, common content knowledge and specialized content knowledge to explain the mathematical content and explain the concepts to the learners. The pre-service teachers need to acquire the necessary types of knowledge (discussed in Chapter 3) before they enter the classrooms. According to Killen (2015:30) teachers need knowledge of their subject and must understand the concepts to engage the learners. In this way it will allow the pre-service teachers to feel secure about their knowledge, understanding and skills and their capability to assist learners learn (Killen,

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2015:33). They will be able to teach in a logical manner and make the subject understandable to the learners (Killen, 2015:30).

Tina

Tina‟s school is located on the periphery of a suburb, surrounded by a low cost development area and informal settlement. The school population is predominantly Black with majority of the learners from within the area and some travel from the nearby township. This is a quintile two school making it a no fees paying school. The school receives all funding from the Department of Basic Education and is well resourced and has the basic amenities. The school has an experienced mixed teaching staff.

I observed a mathematics lesson based on 2D shapes and their respective properties. Tina went straight into the lesson without delving into the learner‟s prior knowledge. According to Structural Learning Theory, a theory used within this study and discussed in Chapter 3, prior knowledge is essential. Determining the learner‟s prior knowledge lends itself to developing the learners existing knowledge and building understanding. According to Ikegulu (1996) making a well-structured knowledge association allows for easier retrieval of prior knowledge and the facilitation of new knowledge.

In the observed lesson Tina presented certain concepts on the chalkboard as the lesson progressed. She explained these concepts verbally. I saw this as a mere superficial explanation of the concepts. According to Killen (2015:50) teachers need to build their lessons around the primary mathematical concepts in the classroom. The learners were placed at a disadvantage during this lesson as some of the concepts mentioned were not within the grasp of the learners. The CAPS document sets out specific content and concepts that needs to be taught in each strand in each grade. This is to ensure conceptual progression through the grades in the various phases. Tina had chosen concepts that were not within this grade. If she had engaged with the CAPS document in her planning, where the order and progression of topics are carefully stated, she would have discovered this important aspect. Furthermore her lesson plan indicated scant content material. To be truly effective as a teacher and to ensure active learner engagement the lessons must be thoughtfully pre-planned and then presented. All the pre-service teachers are supposed to engage with their mentor teachers when planning their lessons. If Tina had done so then she would have prevented any short comings in her planning and presentation of the lesson.

Furthermore she missed an ideal opportunity to utilise concrete models or visual means to show the learners the connection between the mentioned concepts and their properties. By using the

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concrete models or allowing learners to visualize and represent their thoughts would have allowed the learners to perceive the relationship between the representations and the mathematical concepts. This would have developed their conceptual understanding and this knowledge would have boosted their mathematical thinking and understanding.

According to Kolb‟s Learning Theory it is important that that the learners have proficiency and understanding of the subject matter in order to progress further. Reference to abstract conceptualization is made in Kolb‟s Learning theory. This is the thinking phase and understanding phase. During this phase assimilation occurs. The learners make use of existing ideas to new understand new ideas. The learners learn to make the connections with what mathematical aspect is been in taught with what they know. This overlaps with the structural learning theory. According to the structural learning what is inside the learners head (prior knowledge) is important. This in turn is linked to the mathematical understanding as learners use this prior knowledge to make connections to understand what is been taught.

Very early in the observation I determined that Tina was using the teacher centred approach as she was taking centre stage in this entire learning process. Very little opportunities were been afforded to the learners to become active participants. Mathematics teachers need to realise that if learning is to take place then the lessons are not to be teacher oriented. They need to move away from being a dispenser of knowledge to becoming a facilitator of knowledge as mathematics is less about the teacher and more of what the learners are doing. When this type of methodology is applied in the classroom then the learners will become the producer or constructors of their own knowledge.

During the lesson Tina merely stated the properties of the 2D shapes without making any direct reference to the shapes in a visual form. The relevance of these properties was lost on the learners.

She should have directed the lesson enabling the learners to arrive at an understanding of what was been taught. According to the theory of Mathematical Understanding, learners cannot develop a level of understanding if they do not engage actively in the lesson. According to Killen (2015:66) learners should not be merely given information but the teacher should rather guide their learning to bring about a better understanding. In this lesson the use of visual or representation means would have satisfied the learning objectives of this lesson. According to the Curriculum and Policy Statement (2011) the use of representations in whatever form is an essential learning tool and it lends itself to the development of important mathematical skills.

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In this lesson the learning activity entailed the learners copying a summary from the chalkboard. The teacher summarised the properties and the learners were asked to copy the summary from the chalkboard. No mathematical skills were taught or reinforced during this lesson. This is a dangerous manner to teach a difficult subject like mathematics. Failure to teach this lesson effectively was her lack of knowledge of teaching methodology. Teaching and learning suffers in the classroom if there is a mismatch between the learners‟ learning styles, teacher‟s teaching styles and teaching methodologies. With a more skilful teaching approach and using a variety of ideas together with effective teaching resources she would have created a better mathematical understanding environment for her learners. In this regard I do not think that Tina was aware of the mathematical ability and proficiency of her learners.

A homework task was given to the learners and they were asked to commence with it once they had completed copying the summary from the chalkboard. This exercise was given from the workbooks supplied by the Department of Basic Education. This book had colourfully illustrated 2 D shapes. Partial drawings of the 2D shapes were provided and the learners were