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 Alan (not his real name), a school-leaver with fairly severe visual impairment, who embarked on a mathematics A level course at Suffolk College.

The First Term

Alan arrived with a range of GCSE's, 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 white board, 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 touchtyping text.
    The first few weeks of class were a challenge, not only for Alan 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 Alan.
    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.
    Alan 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 blackboard 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 Alan 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 worth while 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 Alan is sitting in on every class!

The Second Term

In the meantime, Alan'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 or 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. Alan 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 Alan 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 Alan 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 Alan'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 Alan had done at home to check through. An obvious requirement was to ensure that the software was common - in our case Microsoft Word v. 6 under Windows 3.1.
     It was during these computer based sessions that both Alan and the lecturer learned to exploit a range of features of the wordprocessor, such as short-cut character allocations, for example, ALT+a for alpha and CTRL+SHIFT+= for superscript. Judicious use of copy and paste allowed Alan to concentrate on the overall structure of mathematical methods rather than spending time retyping the routine explanations. We also discovered that is is possible to copy and paste simple arithmetical expressions into the Windows calculator, press the "=" button to get the answer and then copy and paste this back into the document. (Not a lot of people know that!) In short, the use of modern computer technology (Alan'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 Alan to the computer algebra software Derive. This excellent program (discussed in Issue 3 of this newsletter) 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. Alan 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. Alan 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

Supported by the technology, Alan continued to progress with algebra, calculus and 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, Alan 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 Alan's visual impairment. An exam paper was sent with slightly larger type than usual, which Alan 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 Alan's special needs. He failed to achieve a pass grade.

Conclusions

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 Alan can make us aware of issues in understanding and delivery which should allow us to enhance our delivery to all students.

Footnote

Alan 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.

© David Bowers 1997