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Mathematics Practically

Date submitted: 23rd October 2020

Keep Calm & Count On.

Maths is one of the most important areas because it can help us understand aspects of every other part of life.

Whether we're reading a recipe, working out how long it will take us to get somewhere, or budgeting our money so we can get the next holiday, mathematics is integral to how we interact with the world around us.

But if we don't have all the knowledge we'd like, we go and find out. If we don't know how to find out, we go and find someone else who might know how to find out. If we do not know how to find someone - or they give unhelpful information - our understandings are hindered. This flows onto the rest of our lives!

So too with students with visual impairments; there can be a few ways in which these students receive sub-par information about their surroundings. This needs rectifying!

Many researchers over recent years have identified ways in which students with a visual impairment may not be receiving all the information their sighted compatriots are:
Firstly, researchers from across the US and Canada (Rosenblum, et al., 2020) found that many students with a visual impairment struggle with graphs, charts, and maps because they are pictures which potentially require further description than just looking at them.
Secondly, Emerson and Anderson (2018), researchers from Western Michigan University, further claimed that some graphs and charts cannot be verbally decoded.
Thirdly, seeing researchers come out with these claims may make teachers feel nervous, or even feel the same way described just above, as Pritchard and Lamb (2012) claim in their research paper.
But lastly, Sandra Lewis and her colleagues (2014) identified three key area which can inform teachers and families alike on how to approach maths from a helpful point of view for all students, not just those with a visual impairment:

Independent LivingThis may look as simple as being able to fill the dishwasher effectively, or as complex as being able to use a computer for work, recreation, and organisation of life. By knowing which aspects of maths are applicable to one's life helps people live an independent life.
Social InteractionBy knowing what the right questions to ask and how to ask them, all students should learn how to get information as well as inform others - both of which are key skills in mathematics: to know the answer, and to know why the answer is what it is.
Self DeterminationMore than just self awareness, self-determination is fulfilment in life regardless of ability. This happens no matter peoples abilities, so it is imperative we as teachers and families provide as many quality engagements and experiences as possible for all students.


Bed Time Maths: Useful as brain break tasks; has to be read out to students by sighted person. Website not easily accessible by screen readers (not labelled well, and have to scroll through the same headings each page!). An example of how not to programme a website for an individual's use.

Blindfold Games: Do need a username and password for most of this developer's games; useful for all students, not just students with a visual impairment; great for teaching functionality of tablets.

Desmos Calculator: Screen-reader accessible scientific calculator. No clutter on page (yay!). Remembers previous calculations which are easily accessible too.

Kate's Maths Lessons: Good for students with low-vision; abominable for those using a screen reader. Coloured, clearly labelled pictures, big text. However, the pictures are not audio described well for those relying on audio description, with descriptions such as, "area of a rectangle image", for a picture with graph lines and a blue rectangle covering two lots of five rows of squares.

Math Share: Useful internet programme for figuring out algebra and similar equations set by teacher. Has typing and voice interfaces, and some accessibility by Microsoft Narrator.

Wolfram Alpha: Fabulous graphing and Cartesian plane website for those with screen magnifiers and the like to provide typing input without having to rely on pen and paper. For certain equations, does give screen-reader friendly results - for example, click here for the results of a circle.

Lewis, S., Savaiano, M.E., Blankenship, K., & Greeley-Bennett, K., (2014). Three areas of the expanded core curriculum for students with visual impairment: research priorities for independent living skills, self-determination, and social interaction skills. International Review of Research in Developmental Disabilities, 46, 207-252.

The Expanded Core Curriculum (Lewis, Savaiano, Blankenship, & Greeley-Bennett, 2014) is an integral part of including all students from all backgrounds. This especially includes students with all types of visual impairment since they may require specific instructional methods which vary from teachers' other instructions.

The notion of the expanded Core Curriculum [ECC] includes nine addition key areas for teachers to conjugate upon for their lessons. The first is inclusion of assistive technologies for anyone who would benefit from their implementation on any level. The second is career education, which directs students to career opportunities which they may not be aware. Compensatory access for students' ease of access is important; however, specific instruction may be required for understanding pictures, graphs, and maps. Furthermore, these forms of access should point towards students living as independently as possible. This can include specific instruction about orientation and mobility around their everyday environs. This includes their recreation and leisure time to maintain a healthy and balanced lifestyle. All these areas should culminate in three final areas: social interaction, sensory efficiency, and self-determination.

These three areas are the end goal for how the individual interacts with the world around them, thus their sense of belonging and understanding of self. According to Lewis, et al. (2014), students with a visual impairment complete tasks around the house less efficiently and less independently than students with normative vision, which Lewis and her colleagues imply points towards a need for additional initial instruction for students with a visual impairment for their future autonomous life. Furthermore, social interaction skills may be challenging with varied levels of engagement or responsiveness for students with a visual impairment. This means opportunities for groupwork within the classroom and in future life may require additional attention on the part of the teacher and of others for people with a visual impairment to fully participate. Lastly, self-determination as the collection of all aforementioned disciplines should focus on more than self-awareness. Self-determination is fulfilled in life regardless of ability, so students of all visual suasions should have access to the instructions and actions which best suit them.

Pritchard, C.K. & Lamb, J.H., (2012). Teaching geometry to visually impaired students. The Mathematics Teacher, 106(1), 22-27.

The apprehension from teaching any student with any impairment may cause nervousness, doubts, apathy, or any ill-guided emotion in teachers' minds. Teachers may not understand the value of specific, descriptive instruction to students with a visual impairment, especially with regards to geometry in 2D and 3D spaces. These potential misunderstandings may be heightened when worksheets or textbooks are used, even when digital and other supposedly accessible media.

In instances of working in groups or with worksheets, students with visual impairments should be given opportunity to take notes in their preferred way. These methods may include a electronic device with a refreshable Braille display, an electronic device with software allowing for large text or reading a screen, or a scratchpad. Furthermore, mathematical symbols should be specifically taught to every student in their own context. For students with a visual impairment, use of these symbols may appear useless for conveying a desired meaning. For example, the Braille symbol for the word 'angle' is '|O' [literally translated as '|O'], but there are many symbols without Braille equivalent. Other methods for conveying meaning may include writing the words for the symbol instead, such as for the word 'angle', writing 'angle' instead of '|O'. The end goal for all learning experiences should be students understanding the content, not confused by it.

Furthermore, physical representations for students may be beneficial than hearing descriptions to use imagination. Examples in a learning experience on 2D or 3D shapes may be to have objects which represent the shapes and solids being looked at, such as holding a 10s or 100s MAB block for observing squares and cubes in a real space. Similarly, an example in a learning experience on mapping and graphing may include a rubber band board to make and observe lines and graphs made by the teacher, others, or themselves.

In conclusion, context for future use is necessary. As Pritchard and Lamb (2012, p.27) say, Problems can be represented 'visually' in many ways; they can be spoken, read, drawn, and felt."

Rosenblum, L.P., Cheng, L., Zebehazy, K., Wall Emerson, R., & Beal, C.R., (2020). Teachers' descriptions of mathematics graphics for students with visual impairments: a preliminary investigation. Journal of Visual Impairment & Blindness, 114(3), 231-236.

"Many teachers of students with visual impairments also report that their students are often not able to use mathematics graphics independently" (Rosemblum, et al., 2020, p.231). Graphs, charts, and maps can be challenging for students with visual impairments to visualise without seeing directly. Moreover, these students also frequently report their falling behind in classwork or grades when compared with their sighted compatriots.

Description of these graphs, charts, and maps can be useful for students with a visual impairment to understand their content. However, the journal article by Rosemblum, et al. (2020) found there was great variation in the descriptions used by teachers to benefit these students. While one interpretation may view varied descriptions for varied people, a better analysis may be teachers' misunderstanding how to best interpret information to make it accessible for students with a visual impairment. The study also measured the number of questions used by teachers, also with great variation. The general consensus of this finding was that the more questions a teacher asked, the better the results of the student or students in question were.

However, the study's main findings pointed towards the need for teacher training to better understand how to follow existing guidelines in order to make graphs, charts, and maps accessible and understandable by students with a visual impairment. Furthermore, teachers should also learn how to ask questions which encourage thinking beyond the factual answer, for students to understand reasoning behind the information in graphs, charts, and maps. Roseblum, et al. concluded that understanding the underlying reasons behind a data set could aid in students' understanding of the data. Moreover, the active engagement and critical thinking which stems from these kind of questions can be more pivotal for lifelong mathematical ability than mere comprehension.

Wall Emerson, R. & Anderson, D.L., (2018). Using description to convey mathematics content in visual images to students who are visually impaired. Journey of Visual Impairment & Blindness, 112(2), 157-168.

Mathematics in certain contexts is heavily laden with graphs, maps, and other pictures. Wall Emerson and Anderson (2018) suggest this reliance on visual media makes access to mathematics inaccessible to students with a visual impairment. Further use of these types of media in the classroom should be monitored and carefully used to facilitate the participation of all students to the best of their ability.

In their findings, Wall Emerson and Anderson (2018) found that for most images of graphs and maps, the description provided alongside the image - either by the teacher or by the accessible book - was just usable by the student with a visual impairment. However they also found for some pictures no amount of description was sufficient for covering the content conveyed in the graph or map. Moreover, over any period of time, students with a visual impairment may be unaware of the breadth of content they miss through unintentional ignorance. Thus, the implication from this finding is that content taught by the classroom teacher should have the capability to be described and viewed by all students in the classroom, including those with a visual impairment.

Furthermore, Wall Emerson and Anderson (2018, p.157) say, "Materials should be provided in several formats simultaneously so students can approach material in the mode that fits their needs in a variety of contexts." This means content surrounding graphs and maps should not just originate from a paper or digitised graph or map; these maps and graphs can originate from real-world experiences which are familiar to the students. Also, Braille, Large Print, and Digitised versions of graphs and maps often contain detailed errors which can inhibit students' participation. So all students should have access to the same graph or map in the way they view and understand best.

In conclusion, teachers should plan to provide all students with multiple ways of viewing the same source of graphs and maps, with a focus on the students' initiative and growth.

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