Volume I Number 4, November 1994

A Graphical Calculus Course for Blind Students

Albert A. Blank
College of Staten Island, CUNY
Karen Luxton Gourgey
Baruch College, CUNY
Michael E. Kress
College of Staten Island, CUNY

For blind students seeking education and a career in science, engineering and mathematics, the calculus has presented a formidable barrier. This is not simply due to the intrinsic difficulty of the subject, which is obstacle enough for most students. The additional hurdle for blind students is the substantial graphical component of the typical calculus course. There are two major aspects of that graphical component: primarily, there is the representation of geometrical objects, especially the graphs of functions; then, there is the presentation of mathematical formulas as a graphic display.

Under a grant from the National Science Foundation, the Computer Science Department (CSD) of the College of Staten Island and the Computer Center for the Visually Impaired (CCVI) of Baruch College are developing text materials and providing an environment to offer blind and visually impaired students technologically assisted access to the graphical content of the calculus. The goal of the project is to equal or exceed the quality of courses for students with unimpaired vision. We are installing facilities for reading mathematical text and graphics directly without the help of sighted readers. It is only quite recently that any such technology has become practical and affordable for institutions. We can expect that it will become so for individuals before long.

We wish to bring such a course into being for the students who need it right now. For that reason, we are trying to base our system, insofar as possible, on "off-the-shelf" technology and courseware, rather than try to invent much of it.

For basic course content, we are using the successful self-paced mastery course in calculus developed at Carnegie-Mellon University for students of science, engineering and mathematics, developed by Albert Blank assisted by Raymond E. Artz. (1) This is being supplemented by essential prerequisite materials that visually impaired students may lack. The original text was oblivious to the special needs of these students and many small adaptations will be made. For example, the original text stated in its second paragraph, "The axes divide the coordinate plane into four quadrants which are traditionally labeled with Roman numerals as shown in Figure 1-1." No further clarification was given.

The use of a course designed for self-pacing is basic to the program. Our students will work with new multisensory media that present the course materials and will augment their skills with new language and new techniques of expression for presenting their own work. They will generally need extra time, effort and training. Furthermore, thorough mastery is vital because the calculus is fundamental to most of science and engineering. Moreover, mastery is important to give the students confidence that they can become proficient in areas that were hitherto largely inaccessible to them.

Many of the problems of presentation to students with visual impairments can now be addressed with existing technology in a multimedia multisensory environment:

-- audio-tactile tablets can be prepared and programmed beforehand to present graphics.

-- scanners with optical character recognition (OCR) software can be used to read conventional printed text into ASCII files. Braille printers with appropriate translation software can render those files in Braille.

-- for those who do not read Braille or even those who do, screen reading systems provide access to ASCII encoded text files.

-- enlarged display screens are available for those with lesser degrees of impairment.

-- Hypermedia techniques can be used to provide easy access at will to information in the courseware.

1. Presentation of graphics.

We are using the touch-sensitive NOMAD audiotactile tablet(2) to present graphic images. Consider, as an example, the problem cited above as Figure 1.1 in the calculus textbook. The graphical version of this purely verbal and visual statement is embossed on a soft plastic sheet 16.5 inches wide by 11.75 inches high. (3) A tactile grid, each cell measuring 1 inch by 1 inch, is displayed in relief on this sheet. The more important details of the figure are presented on the graphic as heavier and in higher relief than the less important ones in order to give tactile expression to their varying importance. At the bottom margin of the graphic, at the base of each vertical grid line, there is a round button. When any of these buttons is pressed, NOMAD voices the x-coordinate of that line. Similarly, along the left margin, there is a row of buttons that voice the y-coordinate of the horizontal gridlines attached to them. Heavier lines or double lines mark the x-axis attached to the button, "y=0," and the y-axis, attached to, "x=0." Pressure at the intersection of the axes will voice, "origin". Pressure at any other point of the x-axis will voice "x-axis," etc. Along the upper right margin of the graphic there is a row of diamond-shaped buttons. A press of one of these will voice an associated keyword from the text. (3)

Any feature of the graphic can be programmed to voice three levels of information in succession. This is especially useful when applied to the rectangle on the lower right. At that location, the first level of information gives the figure number and name. The next level lists features of the graphic that can then be located through NOMAD's search capabilities. The third level can give any information about the graphic or the lesson that the instructor desires. NOMAD lacks high level editing capabilities but it is easy, though sometimes tedious, to program.

The grid spacing and button placement of the figure shown here will be maintained for all the graphs the student will use in the course. The keywords will differ depending on the lesson. The origin and axes may lie anywhere or even be located outside the picture frame. The coordinates of successive grid points on the axes can differ by some other constant than one. At the same time, the constant structure of the graphics will offer a consistent, familiar environment in which the student can operate securely.

The hope for the future is that the process of making a graphic will be automated so that a blind person can operate interactively at a work station to create and analyze such a graphic without requiring the assistance of a sighted person.

2. Presentation of formulas.

Formulas introduce special problems that technology has not resolved in simple ways:

a. Optical character recognition. OCR programs offer great promise. However, they are not yet completely reliable readers even of straightforward literary text. Moreover, technical print containing formulas is still far beyond their capabilities.

b. Braille. Braille systems for rendering formulas exist and others are in development. The Nemeth code was developed specifically for the purpose of rendering mathematical expressions in Braille. It uses standard six dot Braille and, by virtue of adroitly constructed combinations of Braille characters, is able to represent very complex expressions. The Nemeth system needs to use compound characters to represent many of the symbols that are single characters on a keyboard. A more significant difficulty is that the code lacks the graphical elements of complex mathematical expressions that enable the learner to develop an intuitive grasp of the material.

Computer Braille is a six dot system which represents letters, numerals and punctuation (including parentheses, brackets and braces). It is most useful for communicating between ASCII based and Braille based devices without the need for a great deal of translation. Computer Braille can be used for mathematical formulas but its use doesn't make it easier or faster to understand them.

A hybrid system is being developed by Prof. John A. Gardner at Oregon State University under an NSF grant. This is the DotsPlus system which combines eight dot Braille with tactile display of some of the graphical elements in technical formulas. With eight dots per Braille cell, a true one-to-one match could be made between Braille cells and ASCII's eight bit bytes. For blind programmers, this would be even more useful than computer Braille. We plan to test the DotsPlus system in the course of our program. DotsPlus, in common with other Braille systems, requires preliminary training of our volunteers and cannot be deployed immediately. Anything but the limited use of Grade 1 Braille will have to be postponed.

c. Voice presentation. Until we are able to install more advanced methods of presenting technical text, we are simply using the time-honored method of preparing audio cassettes made by a trained reader.

The application of voice synthesis to read ASCII encoded text files appears to be the most promising method in sight. In our multimedia laboratory, we shall soon install AsTeR, T.V. Raman's program described earlier in this issue of _ITD_. AsTeR has great parsing and expressive capabilities. It can auditorily render the structure and content of a mathematical formula in ways analogous to a graphical display. AsTeR has excellent hypertext facilities that permit sophisticated random search for information. These capabilities exceed those of a trained mathematical reader, as demonstrated by cassette tapes prepared by Recording for the Blind, Princeton, NJ. AsTeR reads technical ASCII files written with the LaTeX macro package for the mathematical typesetting language, TeX. The combination of LaTeX and AsTeR has a special advantage: it is possible to use a command set that expresses the semantic content of a symbol as well as its typographical form. For example, the symbol (a,b) for an ordered pair of entities is used in mathematics in many different contexts and interpretations. If it were used to represent the coordinates of a point in a plane, say, it would be possible to use a special LaTeX macro that would cause the symbol to be printed as usual but cause AsTeR to speak, "point a click b".

As yet, AsTeR requires substantial hardware and software resources that would usually be available only in an institutional setting. Until AsTeR is installed, mathematical formulas can be written out in English for synthesis by screen reading programs. For example, the formula


could be voiced as "fraction a plus b over c plus d", where the extra spacing is to be read as brief pauses.

3. Presentation of student work.

The LaTeX macro package is a comprehensive word processing program for literary text enhanced by special facilities for processing technical formulas. It is already the most common form for the computer processing of mathematical text and extensions of LaTeX are being developed for other sciences. LaTeX offers a special benefit to our students, who generally have keyboarding skills: they can present their work in LaTeX. A LaTeX source file uses only keyboard symbols. Since LaTeX expresses the semantic content of a formula through simple macros that can be interpreted either by print graphics, Braille, or voice synthesis, it can be used as a common basis for all computer assisted presentation of technical text. The effort to learn the few LaTeX commands appropriate to a particular course of study is about the same that any student would devote to earning the symbolism of the subject. It would be unnecessary for a student to learn many, if any, of LaTeX's visual typesetting commands.

Work executed in LaTeX could be printed in typeset form by the student for submission to the instructor or the LaTeX source file could be viewed in that form on the instructor's screen. With appropriate software and a Braille printer, the student's work could be saved in hard copy for future use. Wherever AsTeR is installed, the student could review his work in audible form with the assistance of AsTeR's search facilities. Without AsTeR, we would expect that audible review could be done by standard screen reading systems "trained" to render the LaTeX commands. For example, "$ \sin (x) $," would be voiced as, "sine of x."


It is our hope that others will act along the lines explored by us. The technology for education and training for careers in science, engineering and mathematics of people with visual disabilities is already here. As an added bonus, much of that same technology can do double duty and serve people with certain kinds of learning disabilities. We need not and should not wait for the technology to reach a higher state of perfection as, surely, it will. We can upgrade component-by-component as the technology improves. For now, we can enjoy the marvelous advances that permit us to do what would have been impossible a scant two years ago when our group first contemplated instituting such a program.


We are grateful to Julio C. Perez for reviewing the section on Braille and contributing his knowledge. This work was supported in part by the National Science Foundation, Experimental Projects in Human Resources and Education, Grant No. HRD 9450166, "Multisensory Calculus for Teaching Students with Visual Impairments," 1994.


1. Supported in part by a grant from the Carnegie Foundation.

2. Available from The American Printing House for the Blind (APH), Louisville, Kentucky.

3. On request, print copies of this graphic on letter-sized paper (reduced about 50%) are available. For our visually impaired readers, we can provide copies on swell paper. We regret that we can distribute swell paper copies only to those readers who are visually impaired because of the associated costs. Please write Prof. Blank at the above address for a copy, one per reader.

Blank, A. A., Gourgey, K. L., & Kress, M. E. (1994). A graphical calculus course for blind students. Information Technology and Disabilities E-Journal, 1(4).