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Investigating Braille Investigating BRaille A MATHEM ATTICAL Phase 1: Generating Patterns and Presenting Results. (a) ... Pupils can record hand-drafted patterns on squared paper - maybe with pro-forma (3 x 2) rectangles to represent RESOURCE FOR the 6-dot cell. In which case, the relevant pattern can be filled in as appropriate. Figure 3 illustrates some material produced "MAIN STREAM" by a partially sighted pupil. TEACHERS The Braille Cell by John Siret Here is a selection of ideas which have been generated over the past few years by pupils at Dorton House School, together 1 4 with some background information which should enable pupils and teachers to develop their own ideas in relation to the Braille Code Investigation. This can be used at Key Stages 3 and 4 - and beyond! Introducing The Investigation (a) Open-ended approach (appropriate for high ability pupils): The Perkins Brailler is set out as in Figure 1 and it produces a Fig. 2 6-dot Braille cell as shown in Figure 2. Investigate. The Layout of the Perkins Brailler (b) ... Patterns can be generated on a computer. Figure 4 illustrates the work of a boy who was looking at a possible 5-dot cell configuration. (c) . At some stage, pupils will need to tabulate their 000 ~i) O~ results: Input Keys Linee-feed Back-space Space Bar I-- --aI! m Fig. 1 _.. _a _ (b) Or, more suitable for pupils of average ability: i . ........ See Figures 1 and 2. Try to find out how many different patterns can be made up using the 6 Braille dots. First, find out how many different patterns can be made using 1 dot. Then find out how many patterns can be made using 2 dots. Carry on to find the number of patterns that can be made using 3, 4, 5 and 6 dots. Now add up all the patterns to find the total. And possibly: Investigate other possible systems which use a dzfferent number of dots. Fig. 3 20 Mathematics in School, November 1997 1 2 3 4 5 6 7 8 9 10 "- . o oo o, o 11 12 13 14 15 16 17 18 19 20 *OL~ 0i~ 0 ~ i 01 r 0 0 $ $ $ 21 22 23 24 25 26 27 28 29 30 31 32 **. *I Fig. 4 6-Dot Braille Cell (ii) ... explain what they do and why. No. of Dots Used No. of Patterns Made (iii) . predict results and carry out checks. (For example, by relating their results to Pascal's triangle, pupils can predict O 1 1 6 results for bigger cell sizes and carry out appropriate testing). 2 15 (iv) ... develop their understanding of combinations. 3 20 (v) ... generalize, along the lines: 4 15 5 6 Cell Size No. of Patterns 6 1 20 TOTAL 64 0 1 21 2 22 3 23 (Whatever cell-size is being studied, a table like this is worth 4 24 arriving at). (d) . More capable students will naturally wish to use spreadsheets to present their material: n 2n (e) ... Some pupils prefer to generate results numerically: E.G. Ways of choosing 3 from 6: Phase 2: The emphasis here is on encouraging pupils to Phase 1 activities provide pupils with plenty of opportunities explore new lines of enquiry as independently as possible. to: Discovering the existing 6-dot Braille code 123 124 125 126 234 23 5 236 By this time, students will already know that there are 64 possible patterns. Information Sheet 1 shows how 26 of them 145134 135146 11364 5 62 52 6 are allocated to letters of the alphabet to make up the (simple) 156 125 code known as Grade 1 Braille. A more complicated system, (Grade 2 Braille), employs a system of 'contractions' - which helps to reduce the volume of material substantially. Full details of the Braille Code are given in the RNIB's Braille Primer, but to give readers some idea of what they entail, a small selection have been included 34 4 5 in Information Sheet 2. Of course, there are many other aspects of the code - such as 'Wordsigns', 'Punctuation' and 'Braille Abbreviations' - together with a complicated set of rules which govern the way everything is used! Fig. 6 On top of all that, there is a much more complex system which makes up the mathematical Braille code. A few 'tasters' (i) ... set work out logically. For example, an interesting are included in Information Sheet 3. Of course, pupils do not sub-task is to list the combinations (see (e) above) in such a really have to know any of this information to be able to way that there are no repeated patterns and no omissions. develop a coherent code of their own .... Pascal's Triangle 1 1 0 dot Braille System 1 dot Braille System 1 1 2 2 dot Braille System 1 2 1 4 3 dot Braille System 1 3 3 1 8 4 dot Braille System 1 4 6 4 1 16 5 dot Braille System 1 5 10 10 5 1 32 6 dot Braille System 1 6 15 20 15 6 1 64 7 dot Braille System 1 7 21 35 35 21 7 1 128 8 dot Braille System 1 8 28 56 70 56 28 8 1 256 9 dot Braille System 1 9 36 84 126 126 84 36 9 1 512 1 10 45 120 210 252 210 120 45 10 1 1024 10 dot Braille System ,,Ya Iaa Fig. 5 Mathematics in School, November 1997 21 Making Up Your Own Code Pupils may wish to stick with the 6-dot cell and try to 'improve' it in some way by developing an alternative code of their own. Any serious attempt to do this would need to retain a sensible balance between increased complexity of the code (more difficult to learn) on the one hand, and any space-saving properties that might accrue. Pupils can do this 1 4 by writing out coded messages using their own codes - and comparing them with the standard 6-dot version. There is plenty of scope for pupils to 'invent' codes based on different cell-sizes. For example, 5-dot Braille (see Figure 7) has enough single-cell patterns to sustain an alphabet, and there are sufficient left over to develop multi-cell contractions. (However, representing mathematical Braille would be very difficult - especially fractions, which require 'upper' and 'lower' cell positions for numerators and denominators). 7-dot Braille (see Figure 8) would provide many more patterns and reduce the need for contractions. Fig. 9 O ( Symmetry Braille users often mistake 'e' for 'i', or 'm' for 'sh', or 'with' therefor 'of'. are ....symmetry there are properties many such in the possibilities. relevant patterns In each which case, make life confusing. Figure 10 shows some of these, (pro- O~ O duced by a year 10 girl). II I 1 I I 1 t , I / IIll IIII~T I~ i 1 !!i11 1-I 1 IIIII Ir Fig. 7 3 7 Fig. 10 Conclusions The Braille code investigation works extremely well as a GCSE coursework assignment. AT1 Level Descriptors are adequately covered up to Level 8 and there are ample oppor- tunities to show 'excellent performance' beyond Level 8 ... (for example, by producing a critically thoughtful analysis of Fig. 8 the current scheme - leading to suggested improvements that Cell Configurations could be made). I- Figures 2, 7, 8 and 9 show two actual and two hypothetical cell-configurations. Different cell configurations can be ex- References plored without necessarily worrying about any practical or (i) Braille Primer, RNIB, London (ii) Braille Authority of the United Kingdom, Braille Computer Notation, tactile considerations - (i.e. by concentrating on the purely RNIB, Peterborough mathematical properties of the code). Spatial considerations Footnote can be particularly interesting: Is there a 'best' configuration for Louis Braille 'invented' the 6-dot code in 1825 at the tender age of 16. storing information on a standard sized sheet of Braille paper? Today, his system is recognised all over the world and each year, hundreds of books, magazines and newspapers are produced in this medium. There is 8-Dot Computer Braille also a growing variety of computer-based Braille technology which provides Visually Impaired people with even greater access to information. See Figure 9. Louis Braille died in 1852 at the early age of 43 - but he left a priceless Details of the 8-dot computer Braille are available from legacy to blind people. His name will be remembered well into the next the RNIB. In the 8-dot code, the subset in which dots 7 and millennium. 8 are absent agrees with the standard 6-dot code. Hence the numerical layout with the 6-dot scheme at the top, and with Author dots 7 and 8 shown on the bottom line. John Siret, 27 Birchwood Ave, Sidcup, Kent DA14 4JY. 22 Mathematics in School, November 1997 INFORMATION SHEET 1 THE ALPHABET IN BRAILLE CODE (The dots used for each symbol are shown in brackets). = a (1) " =b (1,2) "" =c (1,4) =d (1,4, 5) *=e (1,5) "'=f (1,2,4) ;- g (1,2,4,5) ; - h (1,2,5) a *= i (2,4) * =-j (2,4,5) " =-k (1,3) S = -I (1,2,3) : = m (1,3,4) :==n (1,3,4,5) . =o (1,3,5) i" p (1,2,3,4) =:= q (1,2,3,4,5) a .=r (1,2,3,5) : =s (2,3,4) t (2, 3,4, 5) :. =u(1, 3,6) :: = x (1,3,4,6) a .=v (1,2, 3,6) a" = w (2,4,5,6) : y (1, 3,4, 5,6) : z (1, 3,5,6) Mathematics in School, November 1997 23 INFORMATION SHEET 2 A SELECTION OF BRAILLE CONTRACTIONS (The dots used for each symbol are shown in brackets) Single cell: ." -'ar' (3,4,5) ".-='gh' (1,2,6) .:.= 'and' (1,2,3,4,6) a = ('er' (1,2,4,5,6) ..='in' (3,5) a.
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