LANGUAGE, COMMUNICATION AND VISUALIZATION

Unknown to many outside the education sphere, mathematics as a language is learnt by many from an early age albeit at different levels and varying experiences. Mathematics as a language used in many forms to communicate and it is important for the pre-service teachers to understand what the learners have learnt before they enter the schooling environment. To understand the mathematics language better during the communication process, visualization steps in as a provider of clarity of the spoken word. Thus a symbolic relationship is created between language, communication and visualization. The language that is used to communicate the visual has to be apt and within the level of the learners. Visualization in written or oral form and communication in visual form is essential for the learners to construct meaning. Communicating through visualization makes it a powerful learning strategy as it becomes easier for the teacher to teach challenging mathematical concepts. It becomes easier for the learners to understand the concepts as they will be able to organise their thinking by seeing the relationship between the concepts and the visual mean.

The pre-service teachers need to understand that success in mathematics is dependent on the quality of language communication. The Department of Basic Education (2018:83) acknowledges that “language plays an important and critical role in the teaching, learning and understanding of mathematics” and that the learners need assistance to build their “mathematical language so that it is easier for them to explain their mathematical thinking” (Department of Basic Education, 2018:47). Figure 16 indicates the ten principles of mathematics pedagogy, the role of the teacher and learner in classroom teaching (Anthony and Walshaw, 2009) with mathematical language and mathematical communication prioritised as necessary for effective learning.

These principles will inter discussed within the language, communication and visualization context. Lerman (2002:107) defined learning mathematics as “learning to speak mathematically” and if the learners are to make sense of mathematical ideas then they require to understand the mathematical language used in the classroom including terms and expressions (Anthony and Walshaw, 2009:153). The language of mathematics is often a barrier to

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understanding the concepts within problem solving. In the context of South Africa, which has nine official languages and other dialects, mathematics teachers are sometimes forced to code switch in order to clarify a concept. The Department of Basic Education (2018:7) refers to code switching as the “conscious switching from one language to another language during teaching and learning” and makes mention of “translanguaging” which refers to “a flexible use of language seen as an internal strategy by which speakers use all of their linguistic resources to communicate”. This can be problematic in that certain words in a certain language can have a different meaning compared to the words (concepts) used in mathematics. It is in this context that visualization that can be used to provide learners with a sound understanding of words used in the mathematical language. The difficulties of relating culturally learnt words to mathematical concepts can be alleviated by using visual representations to clarify concepts which will assist the learner‟s mathematical understanding. According to Presmeg (2006) when the medium of instruction is in a language that is not the home language of the learner then having visual elements as part of the lesson can assist in the comprehension of the material been taught. Whilst problem solving in the classroom is dependent on the teacher‟s mathematical knowledge to teach concepts, it is equally important that teacher‟s knowledge of visual representations is necessary to assist the learners understand the concepts in problem solving (Boonen et al, 2016).

Figure 16 Principles of effective pedagogy of mathematics (Anthony and Walshaw, 2009:148) One of the goals of mathematical problem solving is to develop oral and written communication. Lampert and Cobb (2003:237) described communication and language as “as a primary means by which mathematics is taught and learned”. Communication in the mathematics classroom provides a teacher with proof of what the learners know and also

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remediating their weaknesses (misconceptions). Hyde (2006:7) stated that for the learners to comprehend mathematical concepts they need to utilize language. It is therefore important that the correct mathematical language is used effectively in the classroom for communication so that the learners can understand the language of mathematics. Therefore the teachers and the learners need to use and understand language to enhance a variety of communication skills. Hyde (2006) stated that people deny the importance of language in mathematics as a means of communication. Moyer (2000) emphasises the strong link between the use of language and learning mathematics. The learners can negotiate the meaning of concepts, expressions and procedural knowledge with their peers and teachers as well as also add their new ideas (Anthony and Walshaw, 2009). Since language is versatile, it can be rearranged and combined limitlessly in order to communicate further (Jalongo, 2000:50). It also plays a vital role in mediating and negotiating learning therefore it is essential to engage the learners in various forms of oral and written communication as it will develop a permanent record of the development of their knowledge (Luneta, 2013).

According to the Department of Basic Education (2018:79) “the use of language should not interfere with the learner‟s ability to speak about what they are doing” and “the spoken language needs to be used in such a way that learners are able to express their thoughts as clearly as possible”. Through oral communication the learners develop a better understanding and it allows the teacher to gauge the learners understanding. Oral communication in the form of storytelling with images offers explanations to concepts. The learners listen to the story and absorb what is being said and understand the concepts that they were not familiar with previously. The use of visual representations of concepts in the story can be an effective method to greatly influence the learners understanding. It is important that the teachers utilise appropriate visuals to match the concepts used in the story because the learners put a label to the concept when displayed visually, example, a circle is associated with something round or a revolution. The teachers need to ensure that the visuals provided are as close to reality as these visuals by association are stored in either their short or long term memory. Later in their learning when confronted with a similar concept they will be able to recall and utilise it to share a variety of solutions (verbally and visually) and also re-evaluate their ideas (Anthony and Walsh, 2009:152).

Discussion through questioning between learners and teachers create a better understanding of concepts. By teachers asking questions based on the conceptual images (key words), the learners develop their own creative train of thought. When asked by the teacher how they arrived at the answer, the learners by making their connections with prior knowledge, are able

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to justify their thought process leading to a solution. The teachers can use this advantageously during discussion by allowing the learners to grow their ideas off each other. They, the teachers, can intervene intermittently by guiding the discussion by asking pertinent questions. This will create open dialogue in the classroom and allow them to learn at their own pace therefore the teachers need to allow the learners to about how they arrived at the answer. Through open discussion and constant dialogue within the classroom validates their conceptual understanding. These types of proactive lessons assist in shaping the learners thought process leading to successful knowledge construction as they use their ideas to build their thinking with the new ideas obtained from their peers. Discussion as a form of dialogue between the learners indicates to the teacher their understanding or misconceptions. In the event of misconceptions, discussion in the social construct of the classroom allows for its undoing. The learners have the power to support each other in their knowledge construction. By giving individual responses, a consensus can be reached on the understanding of mathematical concepts thus overcoming any misconceptions. The learners can also voice their problems that they are experiencing, allowing the teacher to address them instantaneously.

Lerman (2002) stated that it is not only oral communication that is important in mathematics but written mathematical communication is its equal. Luneta (2013) stated that written communication entails more reflection than oral explanation and should not be underestimated. Written communication stimulates a learner‟s imagination and creativity (Luneta, 2013) and should be encouraged in the classroom. Written communication, as an essential collaboration of both oral and written (text and visual) is critical as it lays the foundation for the development of mathematical skills. Written mathematical work allows the learners to be expressive. It is a concrete manner in articulating what they have learnt in a creative manner. The learners create meaning from the text and through visualization one can see their thoughts in writing. How and what they communicate will depend on their prior knowledge of the given information or their knowledge of the concepts. The written work provides an overview of the learners understanding as they write and represent key mathematical concepts. As teachers, we must acknowledge that no two learners will interpret concepts the same way due to their environmental experience and prior knowledge. Any work in any type of written form (text or visual representations) allows the teacher to evaluate the learners understanding and progress. According to a mathematics teacher (in Silver, 2017) “Writing in mathematics gives me a window into my student‟s thoughts that I don‟t normally get when they just compute problems. It shows me their roadblocks, and it gives me, as a teacher a road map”. As an educator I have observed that writing helps improve the learners understanding in problem solving. The written aspect aids their reasoning skills as they are able to see the greater picture in front of them.

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They are able to focus their attention on the various steps allowing them to examine and re- examine their solutions. On assessing the learner‟s tasks, the teachers read the learners written work. They are able to see how the learners justify how they arrived at their answer. The teacher is then in a position to provide constructive feedback or describe strategies for improvement (Anthony and Walshaw, 2009:154).

The mathematics classroom is moving away from being more text oriented. Modern mathematics teaching is no more reliant only on the textbook. The teachers are exposing their learners to a multitude of learning material and workbooks with illustrations in mathematics lessons. They are now using technology to support their teaching. Many learners in today‟s classrooms are having difficulty with language and the visual mediums assist to fill the vacuum. The teachers have recognised the importance of the visual component in mathematics and it can be regarded as a mathematical language on its own. Visualization as a language allows the teachers to utilise a vast range of models, visual mediums and representations to support their lessons in the mathematics classroom as these are valuable tools to provide concrete explanations when language fails. Visualization allows for visual representations to overcome the language barrier in the classrooms. These visual mediums as visualization tools, together with speaking, allow the learners to externalise their thinking. According to Novick (2004:307) diagrams are among the oldest preserved examples of written mathematics. It can be used constructively by the learners to communicate effectively when they cannot find words to communicate their thoughts. Hence using visual representations can be an effective way to support problem solving. This allows the learners to convert words from the problem into pictorial or schematic representations to communicate their thinking. Thus using the combination of oral and written communication together with visualization allows the learners to incorporate their mathematics literacy strategies by using the varied opportunities in the classroom to link their language and ideas. In this manner the teachers are able to see the learners understanding of the problem before they solve it.

The pre-service teachers need to realise that the mathematical language they use to communicate to their learners must be correct. The mathematical language they use will hinge on their mathematical knowledge (discussed in chapter 3). Mathematical knowledge is also indicated as one of the principles of effective pedagogy (Figure 17). It is therefore imperative that they have a good grasp of the mathematical concepts and pedagogical content knowledge to increase the learner‟s mathematics knowledge using sound communication and visualization skills.

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