New approaches in access to mathematics

1
New approaches in access to mathematics

• Alistair D N EdwardsHeather McCartney
• University of York, UK
• http//www-users.cs.york.ac.uk/alistair/
• http//www.lambdaproject.org/

2
The Problem

• Mathematics relies on visual notations
• Blind people do not have access to those
  notations
• Blind people cannot do mathematics
• ?

3
Addressing the problem

• The Lambda Project
• Linear Access to Mathematics for Braille Device
  and Audio synthesis
• Successor to the Maths Project
• and the lessons learned

4
Written notations

• For communication with other mathematicians
• As a form of external memory
• For symbolic manipulation

5
A replacement for pencil paper

• but it turns out not to be that easy

6
The problem

• Mathematical notations
• unambiguous
• non-redundant
• Must show the brackets
• 3x 4 24
• 3(x 4) 24

7
The Lambda Project

• EU Information Society Technologies
• Partners in 8 European countries
• Including
• Arca Progretti, ITL
• University of York, GBR
• RNIB, GBR

8
LAMBDA

• Linear Access to Mathematics for Braille Device
  and Audio Synthesis
• Linear
• Braille
• Audio

9
Lambda architecture
10
Braille and speech
11
Editor

• Full demonstrations available
• Copies of the software also available
• In return for evaluation

12
Example

• 3(x4) 24

13
(No Transcript)
14
Features of the editor

• Minimize interaction
• automatically close the current group
• insert separator
• menus shortcut keys
• Structural simplification

15
Complex example

• One plus compound root of fraction, numerator x
  to the power of two minus y to the power of two,
  denominator x plus y, end fraction, the quantity
  x y end quantity, close compound root equals
  zero

16
Complex example
17
Evaluation

• A series of Lambda Days
• People use the software
• give feedback
• take it away
• give more feedback
• Current results are encouraging

18
Lambdas objectives

• To give access to the same notations as sighted
  mathematicians
• A real need
• But does it teach mathematics?

19
Problems

• Cognitive effort of using the software distracts
  from the mathematics
• Symbolic manipulation is not supported
• The external representation is part of the
  cognitive representation?

20
Simple example

• 2x y 4 y

21
The need?

• A form of direct engagement with the mathematics
• Without a symbolic intermediary

22
A soundgraph
23
Soundgraphs

• A long history
• Mansur, Blattner Joy (1985)
• Edwards Stevens (1993)
• Brown Brewster (2003)

24
Basic set-up joystick control
25
Input

• Specify graph as an equation
• Add, subtract, multiply divide graphs
• Differentiate, integrate
• Zoom in and out
• Press fire for current x, y

26
Two graphs
27
Combining graphs
28
How much maths could we do?

• Simultaneous equations
• Roots of equations
• linear
• polynomial
• Trigonometry
• Would need to teach maths differently

29
Summary

• Blind students need access to mathematics
• Lambda provides access to most symbolic notations
• but teaching maths to children needs a new
  approach
• possibly based on soundgraphs