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Mathematics Through Visual Nuance

Date Sumbitted: 6th June 2020

Elements of the mathematics syllabus are entrenched in visually nuanced requirements. As children grow and mature, they learn to use all the senses provided them in a multitude of combinations11, 14. Children's learning is also enhanced by the groups of people surrounding them and how these groups of people teach them through various experiences3, 12. Thus, the challenge is how to include all people with the senses they have in a mathematics syllabus which has the potential to be interpreted through teachers' personal experiences7, which often include interpreting the world around them through sight. These inherent styles of teaching may not be the best method for learning by students with visual impairments6. This means that teachers of all classrooms should create flexible teaching plans and learning experiences for those with varied levels of vision4. However, there are some aspects of the current state-based mathematics syllabus2 which should be scrutinised before implementing relevant learning experiences in the classroom. Primarily, equal access to mathematical learning is an imperative for all students2, 6, 15, 16, 20. Secondly, the ability to estimate should not be relegated to the visual senses only. Thirdly, the ability to write, draw, or graph should not be relegated to the standardised realm of reams of paper marked with pencil. Thus, students with visual impairments should necessitate teachers' accommodation in their practices.

Equal access to the best education possible is the pinnacle under which all curriculum and pedagogy should reside4. However, there are potential hindrances which, left unchecked, are loopholes for those with visual impairments to be left behind8. Moreso, many contemporary methods of teaching mathematics may provide sub-par mathematical education to all bar those with a disposition towards rote learning15. Textbooks and worksheets provide prescriptive lessons with little chance for students to question or think outside the metaphoric box20. Moreover, this linear form of working provides few opportunities for students to collectively experience mathematics18 as students are required by design to complete their own worksheet in front of them. One small step in levelling the field for this kind of learning is the acquisition of electronic, large print and Braille textbooks10. These textbooks can be useful as part of the lesson if built upon in other aspects of classroom life7 such as practical work20 and groupwork15. However, these experiences can have limitations. Most stage outcomes surrounding the outcome 8NA2 have sub-outcomes rooted in recognising patterns. Learning experiences surrounding these outcomes and content points should rely on more than written, drawn, or coloured patterns for students to achieve their outcomes. Visually impaired students may find similar success while recognising patterns in textured materials, in various sized goods, in musical intonation, or in spoken or written language. One example of the latter for those with severe visual impairment is the pattern of written Braille; letters 'a' through 'j' are almost the same as letters 'k' through 't', with the latter set adding one extra dot in the 3-dot position9. Another sighted learning experience easily remedied is that of Position, such as in Early Stage 12. For all students to find inclusion in experiences in which they, "describe the position of an object in relation to themselves using everyday language,"2, p.73, these experiences should meld with visually impaired students' mental map of the classroom17. This may look like using the classroom as it is as the reference point and make a noise with the item as it is placed. This way, all students are able to use phrases like between, next to, to my left, or inside2, 15. Students should work together in these kind of situations, thus all students can have equal access to mathematical learning experiences. This provides all students with opportunities to develop skills surrounding problem solving, empathy, and social health, all through the Zone of Proximal Development18. However, there are two aspects of the NSW Mathematics Syllabus2 which require greater preparation to mitigate potential vulnerabilities for visually impaired students: the skills surrounding estimation, and the skills developed from graphing.

The skills surrounding estimation are littered throughout the NSW Mathematics Syllabus2. Estimations of length, distance, area, volume, mass, and temperature are often linked to visual tasks and learning experiences20. While there are government-funded programmes providing accessible geometric equipment10, this equipment provides an exacting solution rather than a way for visually-impaired students to estimate. This means that both teachers' preparations and their fundamental pedagogies surrounding how estimation should work in the classroom need to change to meld to the needs of every student in their classroom7. The initial step should be to redirect the learning experiences' foci from guessing then finding a solution, to problem solving towards a solution15. The ability to estimate should hinge on students' previous experiences6. For example, a student with a severe vision impairment who may have never sighted a thermometer may be able to estimate the temperature in the playground based upon a previous experience of a similar day where they were told the temperature. Similarly, both a visually-impaired and non-visually-impaired student may be more familiar with the distance between two points if they walked between them. This kind of estimation is useful in the ramifications of graphing, mapping, and using the Cartesian Plane, which will be discussed later. Lengths and areas which are traceable with students' hands may be estimated by comparing them to body parts such as arms, or to previously measured objects which may also include the teacher's yardstick. Students may be able to compare volume and mass by holding the item for estimation in one hand while they hold a labelled weight in their other hand. However, many syllabus outcomes between 9MG and 12MG for all stage groups use the words, "measures, records, compares, and estimates,"2, p.23, so mathematical learning experiences must not cease at estimation but continue to find an exact solution20 so students of all visual capabilities can have more experiences to draw upon for future estimations. Furthermore, there are usually multiple ways for students to estimate certain items before a specific measurement is bestowed upon them. One student may estimate how much water is in a glass by picking up the glass to feel how heavy it is. Another student may feel inside the cup to investigate how far the level of the water is. Another student may even drink the water in order to see how many mouthfuls of water there was. Although the results from comparing these estimations can be seen as beneficial, there may be more useful ways for continuing towards the syllabus requirements of measuring and recording. By students finding different but useful information, comparing their found and estimated data, they are able to work towards accurately measuring. Once estimation of a particular item's properties has concluded, a collaborative dialogue to find the best way to exactly measure the requisite property can be undertaken. By providing the methods and the solutions to the posed questions in an inclusive environment1, students of all visual abilities are able to demonstrate their prowess for estimation in their findings and conclusions. These findings and conclusions should inform and hone students' future estimations.

However, reaching the developments which some curricular outcomes imply appear to require use of a pen and ruler, which some visually-impaired students may seem incapable of. Much of the content surrounding 2D-Space and 3D-Space appears to reference the use of paper and pictures in aiding in development towards the curricular standards. For example, the Stage 1 course content titled Three-Dimensional Space 1 asks students to recognise that, "objects look different from different vantage points,"2, p.121 to, "identify cones, cubes, cylinders and prisms when drawn in different orientations,"2, p.121 and to, "recognise familiar three-dimensional objects from pictures and photographs,"2, p.121. To read these course outcomes literally would infer that all students would use their eyes to complete the teacher-developed experiences at hand. Similarly, the Syllabus also informs teachers that maps and the Cartesian plane should have their place in the classroom. Memory maps are of utmost importance to vision-impaired students as they learn and continue to navigate the classroom17 and in fact their whole world. However, to ask these students to commit to memory an entirely new map for one or two lessons' work is more than impractical, it should not be expected. As previously mentioned, the content of learning experiences should not revolve around the curriculum; rather the curriculum should revolve the needs of the students7, 16. The answers to the questions, "Who are the students? ... [and] How do they learn?"12, p.xvi should be the first informants of learning experiences5. Students with severe visual impairments may never need to write or draw in this modern and technological age. The widespread use of screen readers19, and similarly accessible things such as global navigation satellite systems, tactile paving, and the development and training of echolocation techniques can benefit students with severe visual impairments more than recognising shapes on a page. Students should have lessons applied to their world around them3, 13, 15. This means that curricular outcomes should be interpreted from the viewpoint of what students can do and will do in the greater world4. Akin to this, students with minimal visual impairments may be able to participate in activities on paper with various visual aids like large texts worksheets, magnifying glasses, and various digital enhancers attached to a computer10. However, these changes may not be considered sufficient for fulfilling all curricular criteria for mathematics. Students are also asked to, "record halves of objects using drawings,"2, p.55, to, "record volume and capacity comparisons informally by drawing,"2, p.111, and to, "use drawings to record findings from a pan balance,"2, p.113, amongst other similar outcomes. It may seem difficult to amalgamate seemingly opposite sides of a potentially visual learning experience. However, as previously discussed, students working together can be seen as the solution to the visual quandary15; 18. By students sharing their own methodologies and experiences, each student will be able to view 2D and 3D objects in their own way. Where a visually-able student may see a shape on a page, a visually impaired student may be able to use a magnifying glass to trace the shape, where a blind student may know how to get an electronic device to describe the shape on the page in various ways. Student collaboration and corroboration should be a driving force behind learning18. Furthermore, by using familiar experiences and settingswith14, students may be able to build upon already existing knowledges and feel comfortable doing so15. Examples of these may be using an area of the playground with distinctive features as pointers on a map, providing students with opportunities to understand maps, the Cartesian plane, and even 2D and 3D space on a small scale, thus providing the scaffolding for future experiences in the world around them.

In conclusion, students should be able to engage with learning experiences because these experiences draw ona dn extend upon knowledge already present in their lives, no matter their level of sight. By providing relatable and interesting content, teachers may be able to sculpt the future development of students' mathematical understanding for years after they have left the classroom. It is not by rote which many students learn; it is by engaging with real world mathematics in real world scenarios, where students can see how the mathematics works and why the mathematics is useful and beneficial. This leads towards students working together as standard. A large portion of mathematics is problem solving, and students are able to come to more ingenious solutions when they are provided the time and the space to work together. Students with all levels of vision can benefit from working together and understanding the application of mathematical thinking on their's and others' ways of life. This attitude means that mathematics no longer has to be entrenched in visual nuances, but can be wrestled with and relished.

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