VISUAL THINKING AND MATHEMATICAL LEARNING

The manner in which our learners utilise their thinking skills in the mathematics classroom can be viewed with trepidation. Sometimes, as mathematics teachers there is caution that is non- existent. The reliance on the teacher, textbooks and workbooks in the primary school has resulted in information overload leading the learners to resort to memorization of mathematical procedures rather than using their thinking abilities. This can be dangerous to a subject like mathematics as the learners will lack the basic thinking and learning skills to understand and apply the content imparted to them. Therefore we as teachers need to encourage our learners as well as expose them to utilise their thinking ability. To support their higher level thinking ability teachers can alternatively encourage their learners to start thinking visually as we all have the innate ability for visualization.

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Using visualization as a starting point, the learners will be in a better position to demonstrate their understanding to learn visually and also make inferences when confronted with mathematical information. This will indirectly contribute to their mathematical thinking.

Visual thinking is the ability to turn all types of information into different forms of visuals that aids the communication of the information. According to Martins (2014) thinking visually is powerful and highly efficient and does not require much effort. This allows the learners to develop their conceptual skill of insight using their visual insights to unlock knowledge (Martins, 2014). All learners have different ways to interpret and learn visually based on their intellectual experience, example, seeing a policeman. When the learners see a policeman different thoughts conjure up in their mind. Based on their experience or prior knowledge, some may associate him with trouble and others may associate him as someone who can be approached when one is in trouble.

The sight of things when the learners are exposed to different teaching mediums in the classroom example, a diagram will elicit a response in their minds. Focussing visually on the image (diagram) may result in them thinking and making their own associations. Inadvertently a mental outcome is produced in the learner‟s mind. I make reference to an example of an incident I observed in a teacher‟s classroom. A geography teacher wanting to teach his learners the cardinal points drew a diagram (Figure 13) to denote the four cardinal points. He asked his learners to name the four cardinal points referring to them as directions. One response from a learner was top, down, left and right. Obviously this was not the answer the teacher was expecting but it was what was perceived in the learner‟s mind.

Figure 13 Teacher representation of the cardinal points

According to Puphaiboon and Woodcock (2005) understanding a diagram (whether it is drawn on paper or manipulated in some other manner) is part of the thinking process. Through mathematical thinking visual images are drawn on paper or created using visual tools of concepts. Boaler (2016) suggested that the teachers ask the learners how they see mathematical ideas and illustrate what they perceive. It is in this ways that ideas are germinated in the brain. When the learners learn through these visual means mathematics changes for as they require a

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deeper and better understanding of the concepts in the problem. These conceptual images consolidate their understanding and capture the most important aspects of their mathematical thinking which leads to mathematical learning (Figure 14).

Mathematical thinking, according to Kilpatrick et al (2001), must be seen in the same vein as learners having proficiency in mathematics. Proficient learners are those who are skilful in the five strands, namely, conceptual understanding where the learners are able to comprehend mathematical concepts, understand the purpose and their relations; procedural fluency when the learners are able to carry out procedures accurately, efficiently and in appropriate context; strategic competence where the learners are able to formulate, represent and provide solutions to mathematical problems; adaptive reasoning when the learners are able to think logically and reflect, explain and justify their answers; productive disposition when the learners are able to see mathematics as sensible, useful and worthwhile in their lives coupled with one‟s efficacy (Kilpatrick et al, 2001:116).

According to Figure 14 mathematical thinking occurs during the processing of concepts. Learning takes place from linking procedures and concepts. This occurs in amalgamation with compression. Compression is a thinking procedure used to elucidate the development of concepts. This mental process can be considered to be the means of a disciplined problem solving procedure from which concepts are developed (Sangpom, Suthisung, Kongthip and Inprasitha, 2016:74). Aptly described visual thinking (mathematical thinking) leads to visual learning (mathematical learning). This can be translated to talking, reading and writing their thoughts. This allows the learners to organise and consolidate their mathematical thinking. They gain insight of the meaning of the concepts making a solution to the problem possible. This is also evident in the statement made by Yilmaz et al (2009) who asserted that through the process of visualization a mental transformation takes place that eventually leads to successful understanding of concepts.

Learning is not merely talking or absorbing content knowledge. It is a complex process in which conceptual knowledge is created, recreated and understood. The learning process depends on the learner‟s personal perception, previous knowledge and skills which they use to create new knowledge. As teachers we must be aware that cognitive structure of all learners and their individual learning styles they have will vary. We also need to recognize that learners due to their backgrounds will attribute dissimilar meanings to the imparted content knowledge and concepts. The learners who do not comprehend the vocabulary or words in the problem or have difficulties to understand the language of instruction can fall back on their visual ability to learn. Learners observe directly. The visual will trigger a response thus making an association

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possible. Also through observation they will learn from the visual cues of the other learners and teachers in the classroom. Therefore the pedagogical value of using illustrations and demonstrations in teaching and learning cannot be overemphasised.