Project Math Access

Teaching Mathematical Concepts

Basic Number Facts and Operations

Collaborative and Inclusive Strategies

The Personal Perspective of Abraham Nemeth

(taken from: NEW BEACON, Vol. XVIII, No. 210, June 15, 1934, pp. 146-148)

Editors’ Note: We have decided to include this article because of its historical value to demonstrate that for decades, persons who are blind and their teachers have striven to develop better methods and strategies for the study of mathematics.

There are two essentials for the blind student of mathematics. The first is a comprehensive system of notation, capable of expressing all mathematical relationships neatly and concisely, for until a system is devised the student is obliged to improvise his own method, and such improvisation is often clumsy and apt to prove incapable of expressing all the niceties required of it. The second is apparatus, primarily to take the place of the pencil and paper which enables the seeing student of such a subject as geometry to draw the picture of the problem that he seeks to solve, and so to have something concrete before him.

Although it is true that the higher mathematician more and more dispenses with the concrete as he comes to move in those realms that are not far removed from philosophy, there is a long and arduous journey to be traveled before such heights are reached, and on that journey the blind student needs apparatus as truly as the seeing. If the blind lover of mathematics persists, it is possible that in time he may be more at home in these higher reaches of mathematics than his seeing rivals, and may dispense even more readily with external aids (“Geometry is the proper science for the blind because no help is needed to carry it to perfection,” said an eighteenth century blind mathematician), but such heights are attainable only to a chosen few.

There have been great blind mathematicians in the past, long before any recognised system of notation for the blind student existed, notably of course Nicholas Saunderson, Lucasian Professor of Mathematics at Cambridge in the early eighteenth century. He worked out his arithmetical, algebraical, and geometrical problems on a square board, itself divided into smaller squares, with a hole in each square in which Saunderson placed pegs. Less outstanding than Saunderson, but worthy of mention, was Herr Weissenberg (born 1756), who was fortunate in having a very enterprising tutor, who taught him algebra, trigonometry, and geometry, and who modified Saunderson’s board for his young pupil, introducing certain improvements. Later on, the board was still further modified by Valentine Hauy, of the Institution des Jeunes Aveugles.

And lest we should be tempted to think of skill in mathematics as a purely masculine achievement, we read of Mademoiselle de Salignac (born 1741), whose conversation with Diderot is thus described by Diderot himself: “I said one day to her: ‘Mademoiselle, figure to yourself a cube,’ ‘I see it,’ she said. ‘Imagine a point in the centre of the cube.’ ‘It is done.’ ‘From this point draw lines directly to the angles: you will then have divided the cube—’ ‘Into six equal pyramids,’ she answered, ‘having every one the same faces: the base of the cube and the half its height.”’

In face of this erudition, it is comforting to read later that Mademoiselle de Salignac was not lacking in more feminine graces, for it is stated that she made “garters, bracelets, and collars for the neck, with very small glass beads sewed upon them in alphabetical patterns.”

One of the outstanding figures in the history of the early education of the blind in this country was the Rev. William Taylor, first Superintendent, nearly a hundred years ago, of the Wilberforce Memorial School, York, and later one of the founders of the College and the Blind Sons of Gentlemen at Worcester. He is remembered in schools for the blind today as the inventor of the Taylor Arithmetic frame, with its star-shaped eight-angled holes, and metal type. For many years the Taylor frame was the only piece of apparatus used for the teaching of mathematics, but because, in spite of its undoubted ingenuity, it is rather a cumbrous appliance, comparing very unfavourably with the pencil and paper of the seeing mathematician, it was only rarely that the blind boy or girl progressed further than a working knowledge of elementary arithmetic. Even today, most blind people of average education will admit that when they leave school days behind them they also discard the Taylor frame, though it is hoped that the recently devised cover for the frame, which enables the board to be carried about without disarranging the type, may make it of more practical service.

As we have said above, various systems of notation have been devised from time to time by the blind student of mathematics, and by the teachers in various schools, but for many years there was no uniformity in this respect, so that the sign which for one student stood for plus might conceivably stand for minus somewhere else. One Braille notation was devised by the eminent Cambridge mathematician, Henry Martyn Taylor, who was overtaken by blindness in 1894, when engaged in the preparation of an edition of Euclid for the Cambridge University Press. By means of his ingenious and well thought out Braille notation he was enabled to transcribe many advanced scientific and mathematical works, and in 1917, with the assistance of Mr. Emblen, a blind member of the staff of the National Institute for the Blind, he perfected it. It was recognised as so comprehensive that it was soon adopted as the standard mathematical and chemical notation, and is universally used by English-speaking people.

In 1914, Mr. G. B. Brown, himself a mathematician, was appointed Principle of Worcester College for the Blind, and his enthusiasm infected his pupils, so that under his direction they improved the school apparatus, and among other things devised a graph board, enabling them to do algebraical and trigonometrical graphs.

A few years later, when Mr. Emblen was engaged in coaching Miss Sadie lsaacs (a brilliant blind girl who in 1924 took her London degree with honours, and was awarded a scholarship as the student who gained first place in the University), he was brought forcibly up against the lack of apparatus for the blind student of mathematics. The toothed wheel pencil and compasses, enabling pupils to make their own geometrical figures, had been invented many years before by Mr. Guy Campbell, but otherwise there was little available. As a result, Mr. Emblen invented a mathematical demonstration board, which is now very generally used for the study of geometry and the plotting of graphs. It consists of a baize-covered board, marked in half-inch and centimetre squares, and into it pins are inserted. Geometrical figures, such as the triangle or parallelogram, are made by slipping rubber bands over these pins, while circles are made by means of flexible steel bands, slotted at one end to allow the insertion of the other; quite elaborate figures, such as the nine-point circle, can be rapidly and easily made by means of this board.

About 1918, Mr. Taylor introduced algebra type for use with the Taylor Arithmetic frame (the invention of his namesake many years before), and together he and Mr. Emblen compiled a pamphlet “How to write Arithmetic and Algebra by means of the Joint Type Method.” This is a companion volume to Mr. Emblen’s “Guide to the writing of Arithmetic and Algebra, with Mathematical and Chemical Formulae,” a study of which will enable the Braillist, whether he is a mathematician or not, to transcribe into Braille any scientific or mathematical book. The fact that books as widely varying as Godfrey and Bell’s “Winchester Arithmetic,” Godfrey and Siddons’ “Elementary Algebra,” Darwin’s “Tides and Other Phenomena of the Solar System,” Marr’s “Introduction to Geology,” Ashford’s “Electricity and Magnetism,” Fletcher’s “Elements of Plane Trigonometry,” Smith’s “Conic Sections,” Eggar’s “Mechanics,” Jeans’ “Universe Around Us” (all illustrated with diagrams, where these are to be found in the ink-print versions) have been published in Braille, is an indication that the system of mathematical and chemical notation devised by Mr. Taylor and Mr. Emblen is able to meet the very heavy strain laid upon it, and can justly claim to be comprehensive.

More recently, Mr. Emblen has been responsible for a List of Tables of Weights and Measures, with the metrical equivalent in every case given on the same line of Braille, and for “A Text Book of Mathematical Tables,” including 4 figure logarithms, trigonometrical ratios, and various formulae.

An International Committee was appointed at the Vienna Conference in 1929 whose aim it is to secure uniformity of mathematical and scientific notation. The English representative on this Committee is Colonel Stafford, who, while he is keenly alive to English interests, is even more keenly alive to the importance of securing a measure of uniformity, if it can be done for the mutual good of all. The task of the Committee is not an easy one, as no country every lightly discards the system it has adopted as its own; but just as uniformity in Braille music notation has broken down frontiers, and brought the music of many nations within the reach of all those who have adopted the code, so it is hoped that a similar measure of uniformity may be achieved in the realm of mathematics and science.