Close up of a boys hands working with pegs on an arithmetic board or ciphering slate, circa the 1940s.
Close up of a student’s hands working with pegs on an arithmetic board or ciphering slate, circa the 1940s.

Math for students who are blind

Mathematics has always been a core subject in the Perkins curriculum. The 1836 Annual Report described the students’ coursework in the early years of the school: “Aside from learning to read, write, spell and ’cipher’, they studied English, grammar, geography, arithmetic, natural philosophy, astronomy, natural history, composition, algebra, geometry and French.”

Mathematics, like other subject areas, required the development of tactile teaching tools. Without such tactile aids, people who were blind used “mental arithmetic” or calculated with the aid of everyday objects, e.g., dried beans, buttons, etc. Mental arithmetic techniques are quite sophisticated and effective, and many people who are blind still rely upon them daily. However, for more complex mathematical problems, tactile devices take the place of pencil and paper and speed up calculations greatly.

Mathematical tools

Nicholas Saunderson (1682-1739), an English mathematician who became blind in his later years, invented the first arithmetic board. In the following decades, arithmetic slates and counting grids appeared in many variations. The slates often had movable counters with raised numbers, symbols, or braille markings on the ends. Other approaches used angled counters that fit in correspondingly shaped cells or counters that were turned slightly to indicate different values.

In later decades, braille slates and braille writers were popular and useful tools for teaching math. In the 1940s, Benjamin Smith, Perkins’ principal and later director, did a great deal of “original work on teaching arithmetic on the braille writer.” These innovations affected the design of the Perkins Brailler, which was introduced in 1951. Dr. Edward Waterhouse, a Perkins director who had earlier worked on the development of the brailler, wrote that some of its “most popular features were introduced for the convenience of mathematics pupils.”

Nemeth code

In 1956, the Braille Authority of North America adopted Nemeth Code for Mathematics and Science Notation. Developed by Abraham Nemeth, the code allowed for linear representation of mathematics problems. Nemeth Code uses the alphabet of Standard English Braille, but nearly every other cell is assigned a different meaning. A student learns to interpret certain cell configurations as punctuation marks when reading literary braille and as numbers or mathematical symbols when reading Nemeth Code.


The abacus was always a promising tool for tactile math, but it was difficult to use because the slightest movement dislodged the beads and ruined the calculation. Terrence V. Cranmer, a leader in the field of blindness who was blind himself, introduced a modification that solved this problem. The Cranmer abacus has a felt backing that stabilizes the beads, permitting the device to be moved or stored in mid-calculation. This innovation has made the abacus a popular and useful tactile mathematics device since the 1970s.

Perkins uses a variety of technologies in the mathematics classroom. The abacus and the Perkins Braillers are still reliable aids, while geometry requires tactile two- and three-dimensional representations. Geometric-two-dimensional charts are easily depicted tactilely, but three-dimensional problems must be represented with three-dimensional objects, such as spheres, strings attached to various points in a box, etc.


Computers were introduced into Perkins classrooms in 1968, and math teachers today employ modified computer applications, tactile graphics, talking calculators capable of graphing calculus problems, tactile measuring tools, and talking money identifiers. The future will undoubtedly bring further innovations, and Perkins will incorporate appropriate new technology to give its students the most challenging and supportive classroom possible.

Suggested citation for scholars

McGinnity, B.L., Seymour-Ford, J. and Andries, K.J. (2004) Math. Perkins History Museum, Perkins School for the Blind, Watertown, MA.

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