THE DESIGN AND DEVELOPMENT OF AN ELECTRONIC ABACUS FOR THE VISUALLY IMPAIRED
A.R.Barker and M.A.Hersh
Department of Electronics & Electrical Engineering,
University of Glasgow, Glasgow G12 8LT, Scotland.
Tel: +141 330 6137/4906. Email: abarker@elec.gla.ac.uk,
m.hersh@elec.gla.ac.uk
Abstract | Introduction | Existing Work | The
Modern Abacus | Initial Specification |
Input device – Design Features | Testing and
Results | Conclusions | References
Abstract:
This paper outlines the arguments for and against using the abacus as a
mathematical tool, and its relevance in a world dependent on electronic
calculators. It also describes the design of an accessible electronic abacus
interface, and its uses as a stand-alone calculating device or as part of a
larger system. Finally it discusses the development and programming of the
input stage.
Keywords: blind, visually
impaired, representation of mathematics, abacus.
1. Introduction
Mathematics forms the basis of
all engineering and science. The development of mathematical notation has been
a crucial part of the evolution of the subject, but mathematical notation is
very visual, making it largely inaccessible to blind people. Research into the
range of technology and software allowing a visually impaired person to perform
complex mathematical tasks leads to question whether the more basic aspects of
a solid mathematical education and understanding have been properly considered.
A visually impaired student has
no equivalent to the pen and paper method used by the sighted. Using a Braillewriter
has been claimed to take up to two extra hours for students to complete
homework (Osterhaus, 2004). A visually impaired person cannot have a
simultaneous perception of a mathematical expression, chart or diagram as a
sighted person would, and will need to read over the information a number of
times before it will be committed to memory. It is easy to become lost or
confused in the middle of an equation. Also, many people who become visually
impaired over the age of twenty find it extremely difficult to learn Braille,
particularly the specialised codes such as those for maths and music.
A popular solution is to use a
‘talking’ electronic calculator for problems too difficult to be solved by
mental arithmetic. These are widely available in a range of complexities and
prices from those with a simple tactile numeric keypad to software graphics
calculators, as well as those with enlarged print and displays for users with
residual vision. However if sighted students are discouraged from forming too heavy
a reliance on calculators should this not hold true for their visually impaired
peers? With a calculator no record is kept of any working, and a teacher cannot
identify if a fundamental lack of understanding has caused a mistake, or simply
that the student has pressed the wrong button.
It is an important consideration
that accessibility design should be open to all visually impaired people. The
design should cater for a wide range of visual acuities, from blindness to
blurred vision, as well as considering impairments such as colour blindness and
dyslexia. We should remember that many visually impaired people also have
another disability, most commonly a hearing impairment (CSSR Statistics, 2003).
It is important to provide visually impaired students with a sense of
independence, while making the content of their work understandable by teachers
who, especially if they are themselves sighted, may be finding it difficult to
adapt to new representations and technology. Finally it is important for the
proposed system to integrate as fully as possible with existing technology and
software, as the design is redundant if nothing will work alongside it.
2. Existing Work
Most existing tactile
representations are Braille-based, however the number of single cell dot
combinations is limited. As a result most mathematical symbols are represented
with multiple cells, which can be difficult to remember. Recent developments
have produced more computer-friendly eight-dot codes. The increased number of
dots allows for a one-to-one representation (Schweikhardt, 2000). Conversion
becomes a direct translation from one font to another, rather than deciphering
a code. Recent research has combined Braille symbols for letters and numbers
with raised print forms of mathematical symbols (Gardner, 1993). Tactile
representations allow the user to understand and reproduce the mathematical
language. They do not consider how a visually impaired student comes to reach
an understanding of the concepts behind the notation, which are taught to
sighted students using mostly visual examples (especially at a younger age).
Research into mathematical
learning and development has shown high dependence on the use of small uniform
objects (for example unit cubes and cuisenaire rods) to practise and learn the
fundamentals of arithmetic (Edwards, 1999). Such tactile methods of notation
have existed for over 5000 years.
3. The Modern Abacus
The modern abacus takes three forms,
Chinese (suan-pan), Japanese (soroban) and Russian (schoty).
Figure 1: Bead layouts of the 3 modern abacus forms, from Abacus: The Art Of Calculating With Beads
The Chinese arrangement is
preferred for this design, as it is often useful to represent an intermediate
total on a single rod that exceeds nine. Also the number representation is closer
to how we would represent numbers on a page, from left (most significant digit)
to right (least significant digit), than the Russian design.
Figure 2: Detailed layout of a Chinese (suan-pan) abacus, from Abacus: The Art of Calculating With Beads
3.1 Using The Abacus
The most popular method of
counting is derived from the Chisombop hand-counting method. Each lower deck
bead represents a unit and each upper deck bead represents five units. Each rod
represents a single digit, so when representing integers the rightmost rod
counts units, the next rod to the left counts tens, and so on in increasing
powers of ten. One increases the total displayed on the abacus by moving beads
of the desired value towards the beam (the bar separating the upper and lower
decks). Lower value beads are exchanged in numbers for a bead of higher value
when necessary. The beads furthest from the beam on each deck in the Chinese arrangement
are not used in the counting process, but to keep track in substitutions. A
standard abacus with 10 rods, for example, can represent positive integers up
to (1010 – 1). If the 2 rightmost rods are
dedicated to numbers after a decimal point it is possible to represent positive
integers up to (108 – 1).
Addition and subtraction are simply extended forms of counting. Multiplication
and division consist of repeated additions and subtractions respectively.
3.2 Further Developments
Cranmer recognised the
possibility of the abacus as a mathematical device for the blind in 1956 “if
the beads could be made a bit more stable” (Gissoni). The soroban-style Cranmer
abacus has been widely accepted as a quiet and efficient way for a blind person
to perform calculations. The black plastic frame with contrasting white beads,
and tactile markings make it easy to navigate. The frame is fitted with a layer
of foam rubber covered with felt under the beads to prevent them from slipping.
3.3 Comparison With Calculators
The introduction of the talking
calculator has fuelled the opinion that the abacus is “archaic, obsolete and a
waste of time” (Osterhaus, 2004). Arguments from visually impaired students and
their teachers do not reflect this.
“For the child with a visual
impairment the abacus is comparable to the sighted child’s pencil and paper.”
“A very tangible way to keep track of the various steps in more complicated
maths problems.”
“A person using an abacus properly is doing more thinking than those only
using a calculator.”
Sewell (2004) outlines the three
main disadvantages to over-use of a calculator:
1. A calculator doesn’t allow a student to learn problem-solving
skills.
2. There is no backup if the battery goes dead.
3. A deafblind student cannot hear the replies of a talking calculator.
And another blind abacus user
states:
“With a calculator it is just
typing buttons. They (students) don’t have to do anything – they don’t even
know the steps.”
Dependence on a calculator does
not impede understanding of the concepts of number and place value, but does
not give any incentive to understand. Students often obtain an answer out by a
factor of 10, 100, etc. due to a misplaced decimal point. The ability to
approximate an answer and identify such discrepancies is not developed.
The speed and the accuracy of the
abacus has been tested against that of the computer on many occasions, the most
famous being a contest in Tokyo, November 1945, between a Japanese soroban
champion and an American expert calculator operator (Kojima, 1954). Time taken
to complete the problems was taken into account as well as the number of
problems calculated successfully. Over five rounds of increasingly complicated
calculations the abacus lost only the third round (multiplication). Calculators
have become faster and almost unbeatable, but speed is not the only important
consideration. An experienced abacus user can still perform calculations
quickly and efficiently. There are other issues; the range of operations
performable on the abacus is small compared to that of the standard scientific
calculator. A sufficiently developed and capable abacus could even assist in
the understanding of symbolic manipulations as well as numerical calculations.
3.4 Evaluation
The abacus is essentially a
tactile representation of numbers to the visually impaired. Schweikhardt
identified a number of criteria to evaluate existing representations and
improve his own (Schweikhardt, 2000).
A. The notation must be readable by finger.
B. The number of characters in an expression should be at a
minimum.
C. Tactile characters should be understandable intuitively.
D. The notation should be usable in a computer-based learning and
working environment.
E. The joint education of blind and the sighted should be
supported.
Further criteria were identified
from previous work
F. The representation should be systematic and easy to understand.
G. The representation should be accessible to visually impaired
people of all ages, abilities and backgrounds.
The abacus fulfilled all criteria
with one exception, that it was not compatible with a computer-based
environment.
4. Initial Specification
The aim was to design a
stand-alone calculating tool with optional interface to a personal computer. The
system can be considered as an alternative to numeric keypad input and an
electronic calculator, and has the added bonus that it will function with no
power. This in turn opens the door to the representation of abacus numbers
alongside letters and symbols in maths packages. With such modifications the
electronic abacus will fulfil all the requirements previously set out for a
tactile representation. A further aim was to eliminate the disadvantages listed
earlier by reworking and improving the general abacus design. It is hoped that
in testing the electronic abacus will prove to be a more enjoyable way of
learning mathematics and will encourage visually impaired students to become
more enthusiastic about the subject. This paper concentrates on the design of
the input abacus interface and its initial connection to a serial port of a
personal computer.
Figure 3: Basic block diagram of proposed system layout.
5. Input device – Design Features
Essentially a stand-alone
calculating device with an expansive number range. An internal microprocessor
allows the device to work independently without connection to the personal
computer. Decimal, binary and hexadecimal bases will be accompanied by more
irregular modes such as time, money and imperial measurements. Later work will
investigate the representation of complex numbers, coordinates and vectors. Two
rods are reserved for powers. A toggle switch will select either straight
powers or powers of 10.
Figure 4: Power representation, the above arrangement could be read as 532 or 53 x102.
A tactile decimal point slides
along the abacus beam. This will not be sensed and is for reference only.
Finally a single switch denotes whether the number displayed is positive or
negative. A variety of solutions were discussed for this feature including modified
bead layouts. However it was decided these deviated too far from the original
abacus design.
5.1 Bead Sensing
Bead positions are read using
micro switches set into notches in the base of the abacus (to prevent
slipping).
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Figure 5: Open switch. |
Figure 6: Closed switch. |
5.2 Switch Reading
The switch matrix has 8 row
inputs (direct) and 10 driven column outputs (via 4 to 16 line decoder) to and
from the microcontroller. Diodes connected in series with each switch ensure
its output does not affect that of any others. All rows are pulled low in turn
and the output lines are read. A closed switch will give logic zero output on
the corresponding column.
Figure 7: Generic switch matrix. Diagram adapted from Schematic for MIDI Keyboard Controller Design for PC Soundcard (Petkov).
5.3 Programming
Using the MPLAB programming
environment the switch scanning process was developed from a simple program to
output a count to the decoder of zero to (n – 1), where n is the number of
columns. Each output pin of the decoder is pulled low in turn.
The user is given a choice
between 2 scanning modes. Discrete scanning steps through each column once and
then enters SLEEP mode. The processor ‘falls asleep’ until a reset button is
pressed and the scanning begins again. Continuous scanning steps through the
columns and repeats until another command is issued. The processor will
immediately recognise any error made by the user, for example an unidentified
bead arrangement, give a signal and stop scanning until the error can be
corrected and the processor reset. The processor will also recognise when a
number of beads need to be exchanged for those of a higher value. The number
displayed is still valid and this can happen often in the intermediate stages
of a calculation, so this is not an error. The processor will alert the user
but then returns a value to storage and continues scanning.
5.4 Scanning Program
The scanning program reads the
switch states by sending a value COUNT to a 4 to 16-line decoder, with outputs
connected directly to the rows of the matrix. The main program increments
COUNT, reads and stores switch states and then loops back until all matrix rows
have been pulled down. The switch configuration is converted to an actual
value. The output of each set of switches represents the position of the 8
beads on a rod. For example in the decimal base, the switch positions are shown
in figures 8 and 9:
The innermost unit bit is tested
first (bit 4), then the other unit bits are then tested in turn by rotating
left into the same position. A value of 1 is added to the total for each bit
set. The process is repeated for the innermost five bit (bit 5), rotating the
other bits right into position in turn. Once all bits have been tested the
total is added to the program counter which steps through another lookup table
to determine the returned value.
Figure 8: Switch positions and corresponding values for decimal base Figure 9: Layout of switch outputs in temporary storage.
6. Testing and Results
To date two phases of testing
have been completed. The program was tested in the MPLAB debugger environment
by directly inputting switch values into the program. All possible correct
switch combinations were simulated, with examples of every type of error. Tests
have been successful in decimal, binary and hexadecimal modes.
Testing of the real circuit was
carried out using a connection to serial port COM1 of a personal computer. The
Windows application Hyper Terminal was used to view the output of the circuit
in discrete scanning mode. Further modifications were made to the program in
order to align number outputs and cut off leading zeros.
7. Conclusions
Future work on the input device
will concentrate on adding more bases to the abacus and investigation into
using existing speech engines, screen magnification software and Braille
embossing technology to obtain an output accessible to the visually impaired.
The next step is to investigate the integration of the abacus with existing
maths software packages, and developing CAL software to instruct in the use of
the abacus and introduce mathematical concepts using the abacus. Further
developments will also consider an output to the user in the form of an abacus
with motorised beads.
References
Department of Health, Personal
Social Services, CSSR Statistics. (2003). Registered Blind and Partially
Sighted People Year Ending 31 March 2003, www.doh.gov.uk/public/blindandpartiallysighted03.htm
Edwards, S. (1999). 100 Maths
Lessons. Scholastic Ltd.
Fernandes, L. (2003). A Brief
History of the Abacus, Abacus: The Art of Calculating With Beads, www.ee.ryerson.ca:8080/~elf/abacus/history.html
Fernandes, L. (2003). A Brief
Introduction to the Abacus, Abacus: The Art of Calculating With Beads, www.ee.ryerson.ca:8080/~elf/abacus/intro.html
Gardner, J. A. (1993). DotsPlus –
Better Than Braille? Proceedings of 1993 Conference on Technology and Persons
with Disabilities, Los Angeles.
Gissoni, F. History of the
Cranmer Abacus for the Blind, home.europa.com/%7Epaulg/abacus.history.html
Ifrah, G. (2000). A Universal
History of Numbers, Wiley Press.
Kojima, T. (1954, reprint 1987).
The Japanese Abacus, Its Use and Theory, Charles E. Tuttle Company Inc.
Osterhaus, S. and Sewell, D. (2004).
Abacus Versus Talking Calculator, Texas School for the Blind and Visually
Impaired, www.tsbvi.edu/math/abacus.htm.
Petkov, J. D. MIDI Keyboard
Controller Design for PC Soundcard, www.geocities.com/JDPetkov/Hardware/midikeyb/midikeyb.htm.
Schweikhardt, W. (2000).
Requirements on a Mathematical Notation for the Blind,
www.vis.uni-stuttgart.de/ger/research/pub/pub2000/icchp00-schweikhardt.pdf