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 Note or MPower; the reader is referred to the website of the manufacturer of the Braille Note and MPower, HumanWare Group, Inc. for more information about the devices (www.humanware.com). Strategies for using a Braille Lite to print math were described in a previous article (Kapperman & Sticken, 2003).

To use the Braille Note or MPower to produce math symbols in print, create a new file. Turn on computer code mode by pressing o-chord followed by the letter, g. Press c for computer code. Next, turn on eight-dot mode: while in the file, press o-chord to move to the options menu, press k, then, repeatedly press the spacebar to move to “use six or eight-dot computer braille?” To choose eight-dot braille, press dots 1-2-5. Press e-chord to return to the file. Eight-dot mode is necessary to enable the tactile display on the Braille Note to show dot seven under any symbol which was preceded by u-chord.

When eight-dot mode is operative, the enter key is dot eight. Therefore, the spacebar must be pressed simultaneously with the enter key in order to move to a new line. Likewise, the backspace key becomes dot seven, so the spacebar must be pressed simultaneously with the backspace key in order to move backwards through a line while in eight-dot mode.

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), or press the backspace key (dot 7) and the symbol simultaneously. 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. The reader is referred to Table 1 for a list of the math symbols, including computer code punctuation marks, which the Braille Note is capable of producing.

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 x2 + 2x – 1 = 0, follow these steps:

- 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.

The degree symbol is another example of a symbol which appears above the baseline in print. As an illustration, we have chosen the expression, 10° (ten degrees), which can be printed using the following modifications:

- Braille the numeral, 10 (dot 2, dots 3-5-6).
- Braille the upward-pointing caret to indicate that the expression is slightly above the line of print in the superscript position (u-chord, dots 4-5; or dots 4-5-7).
- Braille the letter, o.
- If the degree symbol is followed by another mathematical symbol, braille the dot 5, the Nemeth Code baseline indicator. If no mathematical symbol follows immediately, the dot 5 is not necessary.
- The expression will be shown as 10^o in print.

To modify expressions which include subscripted symbols, we recommend following the expression on the baseline with the letter, v, followed by the subscripted expression. The letter, v, has been chosen because it can be viewed as a downward pointing symbol, easily interpreted by a teacher reading print, rather than the potentially confusing use of the Nemeth Code subscript indicator, dots 5-6, which is the semicolon in computer code. When the subscripted expression is ended, insert the dot 5 to denote the end of the subscripted expression, even if it is not required according to the rules for Nemeth Code. The expression na+nb+nc = nd (read as “n sub a plus n sub b plus n sub c equals n sub d) would appear in print as nva”+nvb+nvc = nvd.

The representation of arrows presents another challenge for a blind math student; we recommend the following modifications. The shaft of any arrow is represented by two sets of dots 3-6 (which will be printed as two hyphens) rather than the Nemeth Code arrow shaft, dots 2-5 (which will appear as the numeral 3), because they will actually look like the print arrow shaft, easily interpreted as an arrow shaft by a sighted teacher. For the right-pointing arrow head (→), use the computer code “greater than” sign (dots 3-4-5). For the left-pointing arrow head (←), use the computer code “less than” sign (dots 1-2-6).

Following this convention, to represent a left pointing arrow:

- Braille the “less than” sign (dots 1-2-6).
- Braille two sets of dots 3-6. In print, this will appear as <--.

To represent a double-headed arrow (↔):

- Braille the “less than” sign (dots 1-2-6) for the left-pointing arrow head.
- Braille two sets of dots 3-6 for the arrow shaft.
- Braille the “greater than” sign (dots 3-4-5) for the right-pointing arrow head. In print, this will appear as <-->.

The representation of arrows above the line of print which are located over the major expression begins with the upward pointing caret as described above in the discussion of superscripts. Arrows which are to be shown on the line of print do not have any type of position indicator preceding them. Following is an illustration of how we would recommend representing a line segment A B with the right pointing arrow over the expression, AB.

- Braille an uppercase letter, A.
- Braille the upward-pointing caret (u-chord, dots 4-5).
- Braille two sets of dots 3-6 (arrow shaft) followed by the “greater than” sign (dots 3-4-5) (right-pointing arrow head).
- Braille an uppercase letter, B. In print, this expression will appear as A^-->B.

Following are examples of an integral, a summation, and a limit, written in a comprehensible fashion.

Not all math symbols are available using the computer braille code. To produce such symbols, and to print the more complex mathematical expressions found in calculus, we recommend the methods used to input math expressions in computer math programs such as Mathematica (Wolfram, 1999) 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.

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 Note: 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 Note: Sum[i= 0, 6]3^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 Note, 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 use of this strategy will enable blind students to achieve higher levels of independence. They can communicate easily in the language of mathematics with sighted individuals who cannot read braille, review their work tactually on the braille display, and edit their work using regular Braille Note 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 |
---|---|---|---|

accent (spoken as “single quote”) | ` | 4 | |

ampersand | & | 1 2 3 4 6 | |

apostrophe | ' | 3 | |

asterisk | * | 1 6 | |

at sign | @ | 4 7 | |

backslash | \ | u-chord, 1 2 5 6 or 1 2 5 6 7 | u+spacebar or |

brace (opening) | { | 2 4 6 | |

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

bracket (opening) | [ | u-chord, 2 4 6 or 2 4 6 7 | u+spacebar or |

bracket (closing) | ] | u-chord, 1 2 4 5 6 or 1 2 4 5 6 7 | u+spacebar or |

caret (exponent) | ^ | u-chord, 4 5 or 4 5 7 | u+spacebar or |

colon | : | 1 5 6 | |

comma | , | 6 | |

decimal point (period) | . | 4 6 | |

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

dollar sign | $ | 1 2 4 6 | |

double quotation | “ | 5 | |

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

exclamation point | ! | 2 3 4 6 | |

fraction line (division slash) | / | 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 | |

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

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

percent | % | 1 4 6 | |

period | . | 4 6 | |

plus | + | 3 4 6 | |

question mark | ? | 1 4 5 6 | |

semicolon | ; | 5 6 | |

tilde (root sign) | √ | 4 5 | |

underscore | _ | 4 5 6 or 4 5 6 7 | or |

vertical bar | | | 1 2 5 6 |

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.

Kapperman, G. & Sticken, J. (2003). Printing math symbols using the Braille Lite. Journal of Visual Impairment and Blindness, (97), 710-713.

Wolfram, S. (1999). The Mathematica Book. (4th ed.). Boston, Ma: Cambridge University Press.