Case Study – Mathematics, Statistics and Operational Research and Visual Difficulties
Case Study 1 – Source: Bowers, D. Maths Support for Students with Visual Impairment"
This article discusses the experience of teaching mathematics to a student with visual impairment. The contribution of education technology to support the student is highlighted.
Providing support for students in mathematics concerns not only those who find the subject daunting but also those who have special needs of other types. This article describes the experiences of A, a school-leaver with fairly severe visual impairment, who embarked on a mathematics A level course at Suffolk College.
The First Term
A arrived with a range of GCSEs, including mathematics, and the desire to take A level maths to support an A level in computing, his main subject. His visual impairment was such that he could read printed text only with the help of a powerful magnifying glass, which only focussed on a few words at a time and was of little help in viewing larger diagrams such as graphs or geometric constructions. He had a small telescope with which he endeavoured to see the whiteboard, although this was still difficult even when care was taken by the lecturer to make the writing particularly large and clear. He could not see well enough to write or draw confidently himself with pen and paper. He could, however, read fairly comfortably from a computer screen (being backlit), and had well developed keyboard skills for touch-typing text.
The first few weeks of class were a challenge, not only for A but also for the lecturers, who had no recent experience of students with this kind of special need. It soon became apparent how much mathematics is a visual process. Simple algebra operations, such as the two-step transposition of y = ax + b into x = (y-b)/a – rely as much on visual identification of the terms and pattern recognition as on a breakdown of the composite functions and their inverses. Try explaining to students with no prior knowledge how to go about solving equations such as 4(2 + 3x) – 3(1 – 2x) = 95 without waving your hands about, pointing at the negative signs, underlining like terms in different colours, etc! all of this would have been a distant blur to A.
We also quickly noticed just how much the exposition of mathematical techniques relies on the use of verbs such as "look at", "look for", "can you see", "identify", and so on. How frustrating this must be for a student who is partially sighted.
A brought a small cassette recorder into classes to tape the lessons, which he would transcribe on his computer at home. Who has ever listened to an audiotape of their own maths teaching? It is a salutary experience! Rarely is a sentence fully formed and there is continual use of words such as, "this", "that", "here", "down there", etc. as reference is made to written statements or diagrams. And do we realise just how imprecise we often are in the description of what we are doing? Ask yourself how you would read the expressions; 2(x + 1) or 1/(x – 3). Of course, when pointed out in isolation like this, it is easy to spot the potential ambiguities but in the maths classroom they crop up at every turn.
Equally worrying, we felt, was our own inconsistency in reading mathematical expressions. We tend to use alternatives which we consider equivalent (such as ‘two into x plus 1’, ‘two times brackets x plus one’, and ‘two….(pause)….x-plus-one.) and rely on the students looking at what we have written to make sense of our utterances. Too often (I claim) what we say when standing at the whiteboard serves more the purpose of proving to our class that we are still awake than of providing a sufficient verbal statement of what we are doing. The analogy of football commentary on television compared with that on radio seems appropriate. The recognition that a student such as A was in the class and would be relying on an audiotape of the lesson to understand the subject matter heightened our awareness of this matter. But is it not a worthwhile aim for all of our teaching to endeavour to keep ambiguity and inconsistency to a minimum? If we do not do so, are we not inadvertently compounding the problems students have in understanding what is going on? Imagine A is sitting in on every class!
The Second Term
In the meantime, A’s special needs had been diagnosed more clearly by the college’s student services. He now had his own laptop computer, which he brought into classes. His keyboard skills were such that he was able to make a reasonable attempt at taking notes directly – provided of course that what we as teachers said was clear enough for him to do so. The computer had a piece of specialist software which allowed sections of the screen to be magnified. He could now read what he was writing fairly comfortably (the backlit computer screen was essential for this), and refer back to it immediately. A could now contribute more to class discussion, for example by recalling key formulae or steps involved in a calculation.
An extra hour of one-to-one tuition in mathematics was provided for A from special needs funding. This was generally given by the same maths lecturer immediately after the main lesson of the week. It allowed the lecturer to ‘proof-read’ on the computer what A had noted in class, to ensure that he had an accurate set of notes and to clarify any details of methods. The lecturer could also load into A’s machine the word processor file of any handouts issued in class and at the same time take away on floppy disk a copy of the work A had done at home to check through. An obvious requirement was to ensure that the software was common – in our case Microsoft Word running under Windows.
It was during these computer based sessions that both A and the lecturer learned to exploit a range of features of the word processor, such as short-cut character allocations, for example, ALT+a for alpha and CTRL+SHIFT+= for superscript. Judicious use of copy and paste allowed A to concentrate on the overall structure of mathematical methods rather than spending time retyping the routine explanations. We also discovered that it is possible to copy and paste simple arithmetical expressions into the Windows calculator, press the "=" button to get the answer and then copy and past this back into the document. (Not a lot of people know that!) In short, the use of modern computer technology (A’s first love) provided considerable motivation to grapple with the sometimes complicated mathematics.
The Third Term
As the algebra in the course became more demanding and the concepts of calculus were developed, we introduced A to the computer algebra software Derive. This excellent program can "do" algebra. For example, symbolically expand and factorise polynomials, solve equations, differentiate and integrate analytically. It also has a very versatile built-in graph plotter. An invaluable feature is how it displays algebra in ‘pretty print’, with fraction lines, superscripts, brackets and the like in their proper positions. A no longer had an excuse for keying in 1/x + 3 when he meant 1/(x + 3).
Using Derive, it is possible to annotate your work and save it to disk. A soon learned to produce solutions to standard A level style problems as a Derive file. Admittedly, he used Derive to perform the differentiation and solve the equations but we do not consider that that is "cheating". The technology is there to be used by those who need it despite what the exam boards say! A student can demonstrate understanding through clear explanation and comments throughout and it is also possible for the lecturer when viewing the Derive file on screen to verify that each step is properly structured and the correct syntax used.
The Second Year
Support by the technology, A continued to progress with algebra, calculus an related areas. He also used his knowledge of spreadsheets to look at sequences and series, tables of function values, and graphs. At the same time, there were some topics which he still had difficulty coping with, such as geometry and trigonometry, which rely on visualisation and diagrammatic constructions. (There were simply not enough hours in the day to start playing with geometry packages such as Cabri or Sketchpad.) Also, we made the decision not to embark upon any mechanics or statistics work. Instead, A spent time putting together a portfolio of work on the topics of pure maths he had covered, ending up with an impressive folder of documents generated using Word, Excel, and Derive. His pride in this achievement was considerable. In June, he sat the AS exam in pure maths. Sadly, though not unexpectedly, the examining board did not make many concessions to A’s visual impairment. An exam paper was sent with slightly larger type than usual, which A still had difficulty reading. He was permitted to produce his answers on a word processor under strict supervision but not to use any of the other software tools he had mastered. The board would only allow an extra 25% of time, which was inadequate for A’s special needs. He failed to achieve a pass grade.
Technology is often the key to providing appropriate maths support for students with special needs. Visual impairment is one scenario when meaningful communications can take place via keyboard and screen. Such students can also ‘do’ mathematics, although the process might rely on the mastery of specialist software rather than on the mastery of pen-on-paper. The experiences and lessons learned when working with students such as A can make us aware of issues in understanding and delivery which should allow us to enhance our delivery to all students.
A has been accepted onto a Higher National Diploma course in Software Engineering, where his experience of doing mathematics in a computer-based environment has already given him a head start over most of his peers.
Case Study 2 – Beevers, C. (2002). The Experiences of a University Teacher with a Visual Impairment.
I was diagnosed to have Retinitis Pigmentosa (RP) in 1964, as I came towards the end of my first year as an undergraduate in the Mathematics Department at Manchester University. RP is the second largest cause of blindness in the developed world and it affects some 1 in 2000 of the population worldwide. RP can take a number of forms with a range of severity usually dependent on age. Most of those affected are spotted in their teens or twenties, with the prognosis that although sight loss may be gradual it will be inevitable. The deterioration rate depends on a variety of factors. RP is a disease of the retina in which either the light sensitive rod or cone cells, or both, gradually die. In the former case, rod cell death leaves the individual seeing as if
When diagnosed I was advised by the ophthalmologist to leave University and do something more suitable for employment. This is advice I did not take, thought doubtless generations of students at Heriot-Watt University in Edinburgh do wish I had taken it! However, other people were more helpful and the personal example of Professor Bickley, a blind professor at Imperial College in London, was inspiring. He found time to see me following a request from my Manchester tutor, and allowed me to feel that my goals were achievable. Role models like him can play a more important part than might be thought.
I had always wanted to teach, so when I graduated with a first class honours degree in Mathematics in 1966 and was given the chance to stay on, I took the opportunity with alacrity. I gained a PhD in Applied Mathematics in 1969 and joined the Mathematics Department at Heriot-Watt. I have taught there since 1969, gaining promotions to Senior Lecturer and the Professor. I enjoy my job and have taught thousands of engineering and science undergraduates, as well as honours mathematicians, through many years.
Over the years the Employment Service has been helpful with access to work initiatives, although it still needs persistence and assertiveness to make the Service work best for the individual. A particularly useful period in the middle 1980s saw an outstanding Employment Service worker facilitating a group of visually impaired professionals meeting in Edinburgh four times a year to discuss the solutions we had each found in our workplaces. We invited manufacturers of equipment to attend and saw the latest inventions before they hit the marketplace. This was a time when aids like closed-circuit television, scanners and talking computers were in their infancy. Currently, the range of equipment I use includes a small tape recorder for note taking, a flat bed scanner, and a screen reader – though for many years the closed circuit television was a real bonus. With the CCTV, written material placed under the camera can be viewed on a screen. Magnification is increased very simply at the turn of a knob, and in my case the reversal of the image was invaluable in that black print on white paper becomes white words on a black screen. I used the CCTV from the early 1980s until the middle 1990s, when it became necessary to start to use a screen reader for the computer. JAWS can cope with word processing, spreadsheets, email and well-designed websites. The scanner too has been most useful as it can take printed pages and convert them into readable Word files. Mathematical display is a problem and neither JAWS nor the scanned image can cope with the layout of fractions, powers and more complex equations, though there may be some progress on this front using MathML tagging in the future.
Good as the electronic aids have been, nothing can compensate for the role humans and animals play in the life of someone with severe sight loss. A succession of human readers have played a vital part in helping me cope with the job by reading research papers, mathematical texts and examination scripts. Some of this was achieved by support from the Employment Service, though often my wife and daughters have had to go the extra mile by reading something late at night or early in the morning in case the contents were needed that day. I have also been blessed with two excellent guide dogs: Kimmy from 1988 served me for eight years and Tenko is now in his sixth year of work.
Issues and Reflections
Loss of vision is not easy to accept but there are silver linings. It was my disability that first led me into the area of computer aided assessment. I could see that the computer in the early 1980s could contribute to the learning process for undergraduates but, in addition, its introduction meant that I could engage more meaningfully in student tutorials, since I could read their wok on screen in a way I could not read their handwriting. Later, when I was finding it difficult to read at all, I asked the undergraduates to talk through what they had written, and this ensured a much more interactive and empathetic style of teaching. For lecturing, I have used prepared overhead projection sheets and I memorise the content of my lectures, perhaps easier in Mathematics than in other subjects. My colleagues have been very supportive over the years and I like to think I have helped them along too.
Various collaborative efforts have resulted in CALM, CUE and SCHOLAR, which provide computerised tutorials and online learning and assessment resources. I am now co-director of the recently formed Scottish Centre for Research into On-Line Learning and Assessment (SCROLLA).
Case Study 3
Source http://www.rnib.org.uk/xpedio/groups/public/documents/publicwebsite/public_studentstoryjames.hcsp (information accessed and extracted September 2008)
Summary: James, a university student and developer of a website for others affected by sight loss, talks to us about his experiences.
Choosing a course
I had decided that I wanted to take a course in Mathematics at university due to my strong interest in the subject throughout my teenage years. As I progressed through school I found even more aspects of the subject fascinating, this is when I knew that maths was the subject for me. I chose to accept the offer made by the University of Bath after visiting the department. I was made to feel most welcome and was impressed by the whole day, but by far, the most impressive thing was the fact that they had arranged for me to meet someone from the learning support department to discuss support that could be provided and to give me advice on applying for the disabled students’ allowance.
When I visited another university on a departmental open day and had asked about whether I would have easy access to lectures notes and other materials, the representative told me that some lecturers made their notes and available and others did not and that it would be for me to persuade them to provide notes for me if I needed them!
My lecturers are most understanding of my visual impairment. The Director of Studies for Mathematics kindly emailed all my lecturers to let them know I was in their lectures. Since then I have felt extremely comfortable conversing directly with the lecturers face to face or via email to convey any difficulties I may have or to make requests. All of the lecturers have made adjustments where I have requested them, and several have gone out of their way to enlarge problem sheets and handouts for me.
I would say that my course has not presented any difficulties that weren’t already thought about by the university and dealt with before they occurred. The main problem would have probably been making notes in lectures, but the Department of learning support organised typed notes for those modules where notes in an accessible form were not already available. They also produce reformatted exam papers in large print with greater spacing between lines; this can be very problematic, in particular with producing maths formulae in large print. I also know that if I were to come across any other problems, all I have to do is mention it and the staff will do their best to help me overcome it.
Looking to the future
I always find it difficult to answer questions on my future ambitions because I always seem to change my response every time I am asked it! I certainly plan to finish my degree course (which is four years), and then maybe go into finance in the city (good transport links). Alternatively, I have always had a keen interest in computers and I can see myself working in IT industry somehow.
Wish list of improvements
I think I have to say they helped as best as they could, although some teachers at school were not understanding of my needs. At the school where I took my A levels, there were no problems to speak of and everyone, staff and students alike, were very considerate. So far, all the staff I’ve met at my university have been very helpful and most importantly, I have felt them to be approachable. As yet, I fail to find any serious weak points.
James’ tips for other students
Apply for the disabled students’ allowance earlier rather than later (it took 3 months for me) and be in contact with your university’s learning support department throughout the process. Also do not hesitate to speak up about any problems you can foresee or you have when you get there, after all, if you don’t let anyone know, then they can’t help.