Project Math Access

Teaching Mathematical Concepts

Basic Number Facts and Operations

Collaborative and Inclusive Strategies

The Personal Perspective of Abraham Nemeth

With training and use of appropriate equipment, blind students can produce print math expressions which can be easily interpreted by their sighted teachers who do not read braille. Without the recommended training and equipment, a student who uses braille must submit written math assignments to a sighted individual who can read Nemeth Code; the sighted person then must produce a print version of the assignment by interlining the braille with handwritten math symbols which can be read by the math teacher. Elimination of the interim step in which a sighted individual interlines braille results in a more efficient, independent method by which the student is able to communicate with mathematics teachers. The strategy described in the following paragraphs requires the use of a Braille Lite, a braille input note taker equipped with synthetic speech and a refreshable braille display, and an inkprint printer. For more detailed information regarding the Braille Lite, the reader is referred to the following website for the Braille Lite user’s manual: http://www.freedomscientific.com/fs_support/doc_notetakers.asp.

To use the Braille Lite to produce math symbols in print, create a new file, and turn off the braille translator by moving to the Status Menu. Select "t" and, at the prompt, braille the letter, n. Exit the Status Menu with the command, e-chord. Input information using the computer braille code. That is, use Nemeth Code numbers (in the lower part of the braille cell with no numeric indicator) and computer code punctuation. Indicate uppercase with u-chord preceding any uppercase symbol rather than the capital sign (dot 6). The reader is referred to Code for computer braille notation (1987) and Computer braille code made easy (1998) for additional information regarding the symbols which comprise the computer braille code.

Braille symbols to use in Braille Lite math files are listed in Table 1. These characters will be accurate representations of the print symbols when printed on an inkprint printer. For other mathematical symbols, use the ASCII code number for the corresponding symbol; a list of the common symbols can be found in Table 2. To enter the ASCII code, press the "ALT" command (dots 3-5-chord), then the ASCII number that represents a particular character, followed with e-chord. For example, follow these steps to print “ten degrees” with the numeral ten and the print symbol for degrees (10°):

- Create a new braille file without Grade 2 translation as described above;
- Type the number,10, in Nemeth Code, without the numeric indicator;
- Press dots 3-5-chord; the Braille Lite will say, "ALT";
- Type the number 248 in Nemeth Code, without the numeric indicator;
- Type e-chord; the Braille Lite will say, "small circle."

Using this method, the print version of some expressions will be somewhat different than the standard print version, but will be quite easy to interpret. For example, to print the math symbols representing the algebraic expression, x squared plus two x minus one equals zero, follow these steps.

- Turn off the braille translator as described above.
- Braille the letter, x, without a letter sign.
- Braille the superscript, 2, using the following modifications.
- Because it is not possible to produce a number which actually appears in the superscripted position in print using this method, we have chosen to use the caret, or up arrow, to indicate that an expression is at the superscript level. This symbol is brailled by pressing the u-chord followed by dots 4-5.
- Follow that symbol with the superscripted expression. In this example, it is the numeral, 2.
- To indicate that the superscript portion of the expression has ended and there is a return to the baseline, braille the dot 5, the Nemeth Code baseline indicator. In the computer code, the dot 5 represents the quotation mark.
- In print, the expression will appear as follows: x^2”. The sighted math teacher should be informed that this expression means x squared.

- Braille the remainder of the algebraic expression: the plus sign (dots 3-4-6); computer code numeral 2 (dots 2-3); the letter, x; the minus sign (dots 3-6); the computer code numeral 1 (dot 2); the equals sign (spacebar, dots 1-2-3-4-5-6, spacebar); and the computer code numeral 0 (dots 3-5-6).
- When this expression is printed, it appears as follows: x^2”+2x-1 = 0.

There is no ASCII character for "not equal." To produce this symbol in print, braille the equals sign and then x-chord, h (the backspace character for printers). Follow this with a slash (dots 3-4). The printer will print the equals sign, and then back up one space so the next character, the slash, will be printed on top of the equals sign.

To produce the more complex mathematical expressions found in calculus, we recommend the methods used to input math expressions in computer math programs such as Mathematica (http://www.wolfram.com/) which are, by necessity, limited to symbols on the baseline. Symbols such as the integral, which require more than one linespace in print, are represented by alternative symbols in order to maintain a one-line, horizontal format which can be inserted into a document using symbols on the standard computer keyboard. These expressions are easily read by sighted math teachers who are familiar with these programs. Following are examples of an integral, a summation, and a limit, written in a comprehensible fashion.

The integral from zero to one of v of x with respect to x equals f of one minus f of zero is written as the following using a Braille Lite: Int[0,1]v(x) dx = f(1)-f(0). The left bracket is produced using the u-chord followed by dots 2-4-6; the right bracket is produced by brailling u-chord followed by dots 1-2-4-5-6.

An expression indicating summation, the sum from zero to six of 3 to the power i is written as follows using the Braille Lite: Sum]I= 0, 6]^i. The example of a limit is written in words as follows: the derivative of f of x equals the limit as h goes to zero of f of x plus h minus f of x all over h. Using the Braille Lite, this expression is written in the following manner: F’(x) = lim[h → 0 ](f(x+h)-f(x))/h. The arrow is produced by brailling two sets of dots 3-6 followed by the greater than sign, dots 3-4-5.

The authors believe that this strategy enables blind students to achieve higher levels of independence. They do not have to depend on sighted individuals to produce print mathematics. In addition, they can review their work tactually on the braille display and easily make changes in their work by using regular Braille Lite editing commands. Finally, they can send the same file to a braille embosser to produce a paper braille copy of their work.

Symbol Name | Symbol | Dot Configuration | Braille Symbol |
---|---|---|---|

ampersand | & | 1 2 3 4 6 | |

apostrophe | ' | 3 | |

backslash | \ | u-chord, 1 2 5 6 | u+spacebar |

opening brace | { | 2 4 6 | |

closing brace | } | 1 2 4 5 6 | |

opening bracket | [ | u-chord, 2 4 6 | u+spacebar |

closing bracket | ] | u-chord, 1 2 4 5 6 | u+spacebar |

opening parenthesis | ( | 1 2 3 5 6 | |

closing parenthesis | ) | 2 3 4 5 6 | |

colon | : | 1 5 6 | |

comma | , | 6 | |

decimal point | . | 4 6 | |

divide (forward slash) | / | 3 4 | |

dollar sign | $ | 1 2 4 6 | |

equals | = | 1 2 3 4 5 6 | |

exclamation point | ! | 2 3 4 6 | |

exponent (caret) | ^ | u-chord, 4 5 | u+spacebar |

fraction line | / | 3 4 | |

greater than | > | 3 4 5 | |

less than | < | 1 2 6 | |

minus | - | 3 6 | |

multiply (star or asterisk) | * | 1 6 | |

number sign | # | 3 4 5 6 | |

percent | % | 1 4 6 | |

period | . | 4 6 | |

plus | + | 3 4 6 | |

question mark | ? | 1 4 5 6 | |

quote | " | 5 | |

root sign (tilde) | √ | 4 5 | |

semicolon | ; | 5 6 | |

underline | _ | 4 5 6 | |

vertical line | | | 1 2 5 6 |

Symbol Name | Symbol | ASCII Value |
---|---|---|

vertical line | │ | 124 |

cent mark | ¢ | 155 |

pound sign | £ | 156 |

alpha | α | 224 |

beta | β | 225 |

gamma | γ | 226 |

pi | π | 227 |

uppercase sigma | Σ | 228 |

sigma | σ | 229 |

theta | θ | 233 |

omega | ω | 234 |

delta | δ | 235 |

infinity | ∞ | 236 |

epsilon | ε | 238 |

intersection | ∩ | 239 |

equivalent | ≡ | 240 |

plus or minus | ± | 241 |

greater than or equal | ≥ | 242 |

less than or equal | ≤ | 243 |

top half of integral | ⌠ | 244 |

bottom half of integral | ⌡ | 245 |

division symbol | ÷ | 246 |

approximately equal | ≈ | 247 |

degrees | ° | 248 |

radical (square root) | √ | 251 |

to the nth power | ^{n} |
252 |

squared | ² | 253 |

Braille Authority of North America (2000). Code for computer braille notation. Louisville, KY: American Printing House for the Blind.

Dixon, J. & Gray, C. (1998). Computer braille code made easy. Boston, MA: National Braille Press.