I am blind [stitched photographs]
by
Andy Warhol, 197686










Adapted
educational aids are
a necessary
component of any
mathematics class.
They are especially
needed to supplement
textbooks that have
omitted tactile
graphics or contain
poor quality ones.
However, they are
also needed to help
in interpreting
mathematical
concepts  just as
their sighted peers
benefit from various
manipulatives. It is
very beneficial to
the entire class
when the Braille
student's aid is a
fun and useful tool
for the sighted
students and teacher
as well.

Math teachers
need to verbalize
everything they
write on an overhead
or blackboard and be
precise with their
language. If the
Braille learner
still has difficulty
keeping up, the math
teacher should be
encouraged to give
the student/vi
teacher a copy of
their overhead
transparencies prior
to class if
preprepared or
immediately after.
Another alternative
might be for a
classmate to make a
copy of their notes
to share.

Math teachers
need to give
worksheets, tests,
etc. to vi teachers
to transcribe into
Nemeth far enough in
advance, so that the
Braille student can
participate with
their fellow
students in class 
not later alone.

Relate various
mathematical
applications to
student activities
enjoyed by blind
students as well as
the sighted students


Put various
mathematical
concepts to song
or at least
teach similar to
an athletic
cheer.

The FOIL
method for
multiplying
binomials F
 O  I  L:
First,
Outside,
Inside,
Last!!!!

Quadratic
formula sung
to the tune
of Pop Goes
the Weasel

Be sure to
include athletic
experiences that
a blind student
can relate to;
include the
parabolic curve
of a diver, as
well as the
football
quarterback's
pass.

Math teachers
need to realize that
it is their job to
teach the
mathematical
concepts to their
students. This is
not the job of the
VI teacher. The vi
teacher can be very
helpful by insuring
that all materials
are in proper Nemeth
code and all
graphics are of good
quality if the math
teacher is able to
supply these in
print in a timely
manner. However, any
math teacher will
tell you that there
is always that
teachable moment
that you cannot
anticipate. This is
when it is
imperative that the
math teacher has
some tools at
his/her disposal. It
is the
responsibility of
the VI teacher to
expose the math
teacher to the
various tools and
aids available to
him/her. Math
teachers can be
quite creative, as
many VI teachers
have discovered.

Blind students
should not be
excused from
learning a math
concept because they
are blind: "Blind
students can't
graph." "Blind
students can't do
geometric
constructions." Not
only can they graph
and draw geometric
constructions, with
the right tools,
they can often do so
better than their
sighted peers.
Consideration should
be taken into
account however with
regard to number of
problems assigned.
It is permissible to
shorten the
assignment, as long
as the student can
demonstrate
competence in the
content area.

It is very
important for all
students (and
especially for the
VI student) to use
as many senses as
possible when
learning a new math
concept. They need
to read a new math
problem, write it,
listen to it,
tactually explore it
through
manipulatives, and
when possible move
their body and/or
manipulative through
space. If it's a
fractional problem
involving food for
example, they can
even taste and eat
the problem.

There is an
ongoing need for
fourway
communication among
the math teacher,
the VI teacher, the
family, and the
student. Braille
textbooks, materials,
and aids need to be
ordered early. The
source of a problem needs to be
discerned as quickly
as possible  is it
the math concept,
the Braille, or the
quality of the
tactile graphic?
Vocabulary in itself
can be a problem.
Fractions have
numerators and
denominators in
print and Braille;
however, they have
"tops" and "bottoms"
in print and "lefts"
and "rights" in
Braille.

For classroom
test taking, the
student should be
given the test in
Braille (with an
option for partial
oral administration;
for example, in the
case of students
with learning
disabilities who
need word problems
read) and supplied
with appropriate
tactile graphics,
aids, abacus, and/or
talking calculator.
Blind students
should be given at
least twice the time
to complete tests.
At times, it may be
desirable for the
blind student to
take the test
separate from the
group due to the
needed extra time,
use of aids (especially
those involving
speech), and/or
partial oral
administration.
II.
Challenges in
Teaching Mathematics to
the Visually Impaired
A college student
working on her
bachelor's degree in
mathematics education
asks questions about
teaching a visually
impaired student.
(1) What are some of the
challenges that you are
faced with when teaching
the vi mathematical
concepts?
Susan
replies: One of
the most difficult
challenges has been
teaching concepts
involving
threedimensional
objects. 3D problems
are found in all levels
of mathematics. They are
often difficult for
students with vision to
understand, especially
when trying to create
3D objects in a
twodimensional drawing.
Such a drawing, even
when tactually raised,
makes little sense
without sighted
"perspective." Yet, the
textbooks continue to
draw these 3D raised
line drawings that seem
to contradict what the
math teacher has just
taught the student. For
example, a teacher may
have just explained to a
student that a cylinder
has two bases which
consist of two congruent
circles and their
interiors and let them
examine several real
cylinders. Then, when
the homework is assigned
or the test is
administered, they are
given a twodimensional
drawing that would seem
to indicate that a
cylinder only has one
base and it consists of
an ellipse and its
interior. Sometimes my
students would be better
off without the "picture."
Whereas, it may help the
sighted student, it
often causes confusion
for the blind student.
In addition, the blind
student has to learn
what the 3D object
really feels like, and
then what it "feels"
like as a sighted person
would see it. Talk about
extra work! In addition
to solid geometry,
algebra can also cause
similar problems. For
example, when solving
linear systems with
three variables, many
sighted students have
difficulty visualizing a
threedimensional graph.
Most mathematicians
would agree that it is
impractical to use a
twodimensional graphing
display to solve a
system of three
equations in three
variables, and this is
for people with vision!
The study of vector
calculus and the
calculus of space create
an even greater
challenge; however, I
leave this to others.
The next most
immediate challenge is
keeping up with the
advancement in math
technology tools for the
sighted. The scientific
graphing calculator is
becoming a required tool
in more and more math
and science classrooms.
Once not allowed, they
are now becoming a
requirement for
coursework and even
standardized tests.
There is no such
equivalent to the TI8?
series for the blind.
The GRAPHIT software
program from Freedom
Scientific does graph
certain functions, but
again, it is limited,
and it is not a
standalone calculator.
It requires a PC (or
notetaker) and an
embosser. ViewPlus
Technologies has created
the Accessible Graphing
Calculator program which
is intended to have
capabilities comparable
to a fullfeatured
handheld scientific and
statistical graphing
calculator, but as yet,
it cannot graph multiple
functions at the same
time nor work with
matrices. The blind
student can work the
majority of these
problems without a
scientific graphing
calculator, but the
point is that they are
at a disadvantage if
they must do everything
"manually."
The Nemeth Code
allows the blind student
to braille all the
necessary mathematical
symbols for the highest
level of mathematics,
but often the Nemeth
Code is not taught to
the blind student as
they progress through
their lower level math
classes. (Although I
feel Nemeth Code is
relatively easy to learn
for students, most
sighted vi teachers seem
to have a great fear of
it, possibly due to lack
of proper instruction in
their college training
program and roadblocks
for selfteaching.) This
creates great
difficulties as they
progress into Algebra
and most students MUST
use the Nemeth Code (or
some other tactual code)
to be successful in
higher mathematics.
Often, remediation must
take place while trying
to learn new concepts.
For many years,
translation software has
been available to
convert literary print
to literary braille, but
converting print math to
Nemeth Code proved much
more difficult. It is
just in the last few
years that three Nemeth
translation software
products have come on
the market, as well as a
computerized Nemeth
tutorial to assist
teachers in producing
Nemeth materials.
(2) Much of the
language of mathematics
relies heavily on visual
reference hence, how
does this challenge the
vi student?
Susan
replies: I have
already touched on some
of this in answering
your first question.
However, I have some
specific pet peeves I
can address here. Over
the years, many new
symbols have been
created to supposedly
make it easier for a
sighted student to learn
mathematics or to save
print space. One of
these is the raised
negative sign which
usually appears in
elementary school and
disappears at the start
of Algebra I. This
symbol creates confusion
and takes up
considerable space in
Nemeth Code. For example,
to write (3, 4) (with
the negative signs
raised) one must use 12
cells, whereas using the
regular minus sign uses
8 cells. In Geometry, we
have the print symbols
for line, ray, and line
segment which consist of
a picture of a line, a
ray, or a line segment
drawn above two points,
such as line AB. These
pictorial abbreviations
help a sighted student
remember the definition
of a line, ray, and line
segment and save space.
They merely cause
confusion for a blind
student, make him/her
learn the picture
symbols which only help
a sighted student, and
take up considerable
more space than merely
writing out the word.
For example writing "line
AB" in braille would
take up 8 cells, and
writing the pictorial
symbol takes up 12 cells.
In addition, the symbols
representing the picture
of the line follows the
AB, so the student has
to read all of the cells
before they can figure
out whether AB is a line,
a ray, or a line segment.
Nevertheless, advanced
high school and college
mathematics contains
even more "pictorial"
symbols, which the vi
student needs to
assimilate, right along
with their sighted peers
if they are to succeed.
Yes, the language of
mathematics does rely
heavily on visual
reference, and the
teacher of the visually
impaired is challenged
to be quite creative at
times. Creative teachers
can help their vi
students learn to be
creative as well.
Braille students usually
need to learn the print
way and the braille way;
the print way to
communicate with their
sighted peers and
teachers and the braille
way for their own
understanding. Although
this is often double the
work, sometimes it can
be double the
understanding and double
the creativity.
Our new algebra book
this year really
stressed the visual
concept of "shadow" to
lead into the section on
solving systems of
inequalities. Rather
than skip over such a
seemingly difficult
concept to teach a blind
student, we jumped in
with both hands (literally)
making birds and animals
and trying to explain
how our hands could
block the path of light
to a surface, and define
a region of darkness.
Everyone could remember
when we went on the last
field trip in the hot
Texas sun, and someone
said "Let's get out of
the hot sun and into the
cool shade." The
building had created a
nice shaded region by
blocking the heat of the
sun in that area. Later
on as we were graphing
our inequalities on our
graph boards, one
student really liked and
understood why that side
of the boundary line
should be shaded, but he
was having difficulty
with the boundary line
being dashed or not
included in the solution.
In his mind, he couldn't
see how we could exclude
the boundary line (or
wall casting the shadow).
I said "The wall was
just painted and it's
still wet, so you can
get as close as you want,
but just don't touch it."
He really liked that
answer, and I don't
think he'll ever forget
the concept.
In Geometry when
teaching the concept of
symmetry, textbooks and
teachers often use
examples in nature (including
the human body) and
twodimensional pictures.
These are all good
examples to use. Paper
folding can be a lot of
fun and makes a lasting
impression as well.
However, one needs to
very careful with using
the alphabet, which most
textbooks do use. If you
use raised line drawings
of print letters, these
may just "look" like
pictures to the braille
students (which is fine)
but one needs to
designate them as such.
If you simply state "Which
letters have a vertical
axis of symmetry?" you
will have different
answers from your
braille students because
the braille letters have
different lines of
symmetry from the print
letters. One year on our
staterequired test for
graduation, they asked
how far a certain letter
of the alphabet had been
rotated. The braillist
wisely drew a raised
print letter on its side.
The problem was that the
blind student didn't
know what the print
letter looked like
before rotation!
A vision
assistant asks: How important is it
for our elementary kids
to do a subtraction
problem in the brailler
(example: 3 digit
subtraction with
cancellation signs) if
they are using the
abacus? When you do a
subtraction problem, on
the brailler, do you
have them solve the
problem from right to
left?
Susan replies:
I'm going to answer
your question from a
secondary math teacher's
viewpoint. I believe
that elementary students
need to be exposed to
working addition,
subtraction,
multiplication, and
division problems on the
braillewriter in a
spatial arrangement 
not necessarily using
cancellation signs 
until they understand
the concept. They should
solve the problem very
similar to the way a
print student would  in
the case of subtraction
from right to left.
Although the textbook or
worksheet may give
examples using correct
Nemeth Code including
cancellation signs, you
want the students'
calculation procedures
to be easy and quick.
Your students can always
use the abacus to check
their work on the
braillewriter. All the
while, they should be
learning mental math
techniques as well. Once
the concept is learned,
speed, accuracy and
flexibility are more
important, and we should
see the student quickly
progressing to the
abacus and mental math,
basic calculators, and
eventually scientific
calculators as they
begin higher mathematics.
If a blind student has
never had to work a math
problem in a spatial
arrangement (on a
braillewriter, with
TACKTILES, or other
manipulative) and has
only used an abacus,
they will most likely
have difficulty in
algebra with the concept
of adding, subtracting,
multiplying, and
dividing polynomials
when presented in a
spatial arrangement.
This is especially true
with division. You just
can't manipulate
variables on an abacus.
IV. Solving
Quadratic Equations
Graphically, by
Factoring, and by Using
the Quadratic Formula
A vision
teacher asks: I have a braille
using student in 11th
grade math. He and his
class are going to be
solving quadratic
equations with graphing
calculators next week.
He has Graphit on a BNS.
My question is: is there
a way either using
Graphit or the
scientific calculator on
the BNS to reveal the
roots of an equation. If
not, is there something
you would recommend,
preferably so he can do
the work independently?
Your help would
be much appreciated.
Susan
replies:
The ability
to "see" the connection
between a graph and its
equation can be helpful
to both visual and
tactual learners. I
still do this the old
fashion way with my low
vision and braille
students; they manually
graph selected quadratic
functions on large print
graph paper or graph
boards. The xintercepts
are revealed to be the
roots of the related
quadratic equation. Then
we move on to using the
Accessible Graphing
Calculator (AGC) from
ViewPlus Software.
Graphing calculators
simply allow students
many more opportunities
to make that connection
in a brief period of
time.
To solve a particular
quadratic equation in
standard form (reveal
its roots), your student
should be able to
instruct GraphIt (or
the AGC) to graph the
related quadratic
function. Then, the
zeros will appear as the
xintercepts. In other
words, the real roots of
the quadratic equation
will be the values of x
where the function
crosses the xaxis.
For example: Graph
y=x22x3 (y=x^22x3)
to find the roots of
0=x22x3 (0=x^22x3).
The graph crosses the
xaxis at x=1 and x=3.
Therefore the roots of
0=x22x3 (0=x^22x3)
are 1 and 3.
If the roots are not
integers, you will
probably not be able to
determine the exact
value of the roots in
this manner, but solving
quadratic equations
graphically is still a
quick way to determine
the NUMBER of real roots,
and this is extremely
valuable information. I
might add that when my
braille students
manually graph a
quadratic function with
integral zeros, they get
exact answers. When a
low vision student uses
his TI82 scientific
graphing calculator and
the trace feature, he
gets decimal
approximations of the
correct zeros! For
example, if x=1, the
graphing calculator
might say x=1.0021053.
We often get similar
approximations on the
AGC.
Since we can only
find approximate
solutions to quadratic
functions by using the
graphing method, the
math teacher will next
teach your student how
to solve SOME quadratic
equations by factoring.
Finally, the teacher
will introduce your
student to the quadratic
formula which will allow
him to solve ANY
quadratic equation.
With the right tools
and your guidance, your
student should be able
to complete all of the
above work independently.
V. Solving Systems
of Equations in
Three
Variables
A private
tutor for a state
rehabilitation
department asks:
I tutor a
visually impaired
individual in college
who has just
successfully completed
elementary and beginning
algebra. He is currently
taking intermediate
algebra. What would be
the best approach in
solving systems of
equations in three
variables for a visually
impaired student? I
would greatly appreciate
some suggestions on how
I should go about
teaching such problem
solving.
Susan replies:
Even most sighted
students will have
difficulty trying to
visualize a
threedimensional graph.
So, these suggestions
will work for these
students as well. I
mention this because
this method of
instruction allows a
better integration of
the blind student into
the regular math
classroom. It is more of
a kinesthetic approach,
and many sighted
individuals prefer this
learning style.
Use a corner of the
classroom as that part
of space where the x, y,
and z axes are all
positive. This simulates
the first octant (When
graphing in space, space
is separated into eight
regions, called octants.)
Then place three braille
rulers to represent the
x, y, and z axes. Ask
your student to locate
(1,0,0), (0,2,0), and
(0,0,3) [using units of
1 inch or 1 cm]. Then
ask him to plot (1,2,3).
If he has been using a
graphic aid for
mathematics (rubber
graph board) or other
coordinate plane to plot
2dimensional
coordinates, it may take
him some time to get
adjusted to the fact
that he needs to think
of moving to the front
or back along the xaxis.
He moves right or left
along the yaxis, and
now he will move up and
down along the zaxis.
Next, place a box in the
corner and ask your
student to find the
coordinates of each of
its vertices. Then
rotate the box 45
degrees or place the box
on its side. Did the
coordinates of the
vertices change?
At this point you
could move to a
twodimensional graph
board or raised line
graph paper divided into
4 quadrants and placed
on a table. Then graph
the first two
coordinates on the graph
board and have your
student raise his finger
up to illustrate going
up the zaxis into space
or down (beneath the
table) to illustrate
going down the zaxis.
At this point, he is
really having to do a
lot of visualization,
but hopefully he is
starting to locate the 8
octants in his mind's
eye.
Remind your student
that just as a system of
two linear equations in
two variables doesn't
always have a unique
solution of an ordered
pair, neither does a
system of three linear
equations in three
variables always have a
unique solution that is
an ordered triple. Just
as the graph of ax+by=c
on a coordinate plane is
a line, the graph of
ax+by+cz=d is a plane in
coordinate space. These
three planes can appear
in various
configurations similar
to the way two lines in
a coordinate plane could
intersect in one point,
in infinitely many
points (actually the
same line), or in no
points (parallel lines).
This is the time to
pull out three planes
(actually several sets
of three sturdy sheets
of paper  braille paper
perhaps or cardboard).
First show your student
an example of the three
planes intersecting at
one point, so that the
system has a unique
ordered triple solution.
(You may be able to find
a nice cardboard box
that contained a set of
8 glasses nicely
separated (by the
perfect manipulative) to
nestle in the 8 octants. If so, this really helps
the student retain the
"picture" in his mind.)
Next, have the three
planes intersecting in a
line, and therefore,
there are infinitely
many solutions to this
system. (This is
reminiscent of a paddle
wheel.) You could then
show him various ways
that three planes would
have no points in
common, and these
systems would have no
solutions. (Form a
triangle with the three
planes. Find a cardboard
box arrangement for six
glasses. In the
classroom, use the
floor, the tabletop, and
the ceiling.) If all
three planes coincide,
there are again
infinitely many
solutions. If two of the
planes coincide and the
third plane intersects
them in a line, there
are infinitely many
solutions.
At this point, some
teachers will simply
state that it is
impractical to use
graphing to solve a
system of three
equations in three
variables, and have
their students use
linear combination or
substitution to solve
the system, after first
reducing the system to
two equations with two
variables. Then the
student can use the
familiar techniques for
2x2 systems. Usually
textbooks provide
systems that can be
solved relatively easily
by linear combination
and substitution, but
even they can often be
quite timeconsuming.
One has to be very
careful to avoid
computation errors,
since one mistake early
on may not be detected
until the final check of
your answer, and many
pages of work may have
already been recorded.
However, if the student
has suitable technology,
he can use matrices to
solve a 3x3 system
rather easily.
Unfortunately, a
graphing calculator with
this type of
sophistication (which is
userfriendly) does not
exist for the blind, and
finding the inverse of a
3x3 matrix by hand
involves a great deal of
computation. It is only
an attractive solution,
if calculators can carry
the burden. (My students
and I have developed a
tedious technique using
Scientific Notebook and
JAWS.) None of this will
still mean anything to
the student unless you
can relate it to
realworld problems. Be
sure to include such
problems that perhaps
involve banking and
consumer awareness. (For
example: If a business
sells three kinds of
snacks by the pound, how
many pounds of each
makes up the magic
combination? How much
should a parent invest
in three different
investment tools paying
different yields to
accumulate a college
fund for their infant?
If a factory has three
levels of pay (based on
productivity), how many
hours at each pay scale
are required to complete
a particular order?)
Other teachers may
feel that it is
important to include
even more manipulative
activities because they
offer students an
excellent opportunity to
bridge the gap from the
concrete to the abstract.
Depending on your own
philosophy, the
curriculum requirements,
your student's learning
style, visual memory (if
any, and time
constraints, you may or
may not wish to try the
following activities.
Take a piece of print
isometric dot paper and
make a "raised dot"
version [For example,
xerox it onto a piece of
capsule paper and run it
through one of the
tactile imagining
machines. (See Math Graphs Made by
Others for Students)]
or use a geoboard. Next
you or the student can
create a
threedimensional axis
system using raised
lines or rubber bands. (If
using the paper, be sure
that the student can
still tactually discern
the dots from the axis
lines.)
Then have your
student graph an ordered
triple such as (2,5,1).
Locate 2 on the positive
xaxis. Then move 5
units along in the
positive direction,
parallel to the yaxis.
From that point, move 1
unit along in the
negative direction,
parallel to the zaxis.
You have arrived.
To graph a linear
equation in three
variables, let's graph
3x+2y3z = 6. First find
and graph the x, y,
and zintercepts. To find the xintercept,
let y = 0, and z = 0,
and solve for x, and
continue in a similar
manner for the other
intercepts. Connect the
intercepts on each axis
and a portion of a plane
is formed that lies in a
single octant. [Solution:
The three intercepts
are: (2,0,0), (0,3,0),
and (0,0,2).]
VI.
Linear
Measure, Perimeter, and
Area
A college
student working on her
bachelor's degree in
mathematics education
asks: In
teaching the topic of
Measurement to a blind
student, I have a
concern: How should I
approach teaching him
Perimeter and Area?
Susan replies:
I would
teach linear measurement
very similarly to the
way one would teach a
sighted student. In the
United States we have
two systems of units
that we use to measure
length.
I would allow my
students to measure
several real world items
using both customary and
metric braille rulers,
emphasizing the concept
of precision. We would
also work on several
problems requiring
estimation and use of
the most "sensible" unit
of measure within each
system. In addition, we
would convert from one
customary unit of length
to another, and from one
metric unit of length to
another. The student
should also be exposed
to raised line drawings
and be required to
measure these as well.
From here we could
move on to the concept
of perimeter. For a
beginning student we
could define perimeter
to be the distance
around a shape (later, a
polygon). We might have
the student walk around
the outside of the
school building, the "perimeter"
fence of the campus, or
around the track and
count the number of
paces. A student on the
track team would soon
learn how many times
around the "perimeter"
of the track resulted in
a kilometer, a mile, 100
yards, etc. Then I would
present the student with
a raised line drawing 
perhaps of a square.
Using string, we could
trace the perimeter of
the square and snip it
to be exactly the same
distance. Then the
length of the string
would equal the
perimeter of the square.
We could then examine
and determine the
perimeters of raised
line drawings of a
rectangle, triangle,
trapezoid, pentagon,
etc. with each side
appropriately marked in
braille with customary
and/or metric units.
Having calculated the
perimeter of many
different figures, the
student can eventually
discover the formula for
the perimeter (or
circumference) of a
circle.
When learning about
area, we can say that
just as we can measure
distance around shapes,
we can also measure how
much surface (area) is
enclosed by the sides of
a shape (or polygon).
Luckily, my classroom's
floor is composed of
square foot tiles, and
we go about determining
how many such square
tiles are required to
cover the surface area
of this floor. Everyone
is delighted when we
find a much easier way
to determine this by
multiplying the length
and width of the room.
Then one can progress to
various manipulatives.
Paper shapes made out of
raised line graph paper
can be cut into pieces
and reassembled to form
new shapes with the same
area. Rubber graph
boards can be
partitioned with rubber
bands to form shapes,
and grid squares can be
counted to determine
area. Wooden tiles can
be assembled to form
various shapes and
determine area as well.
This knowledge can then
be transferred to raised
line drawings
illustrating area. The
student should advance
through finding the area
of a square, rectangle,
parallelogram, triangle,
and complex shapes.
Eventually, the student
can investigate and use
the formula for the area
of a circle.
VII.
Transformations,
Line Symmetry, and
Tessellations
A VI teacher
writes: I
have a seventh grade
braille student who will
soon be studying a math
chapter in a regular
classroom. Among the
topics are the following:

Translations
(slides)

Reflections

Line
Symmetry

Tessellations
I have some ideas
for the teacher. However,
being blind myself, I
know these concepts can
be very difficult to
grasp. I would
appreciate any ideas
which I might share with
the classroom teacher.
Susan
replies: I
usually introduce
translations,
reflections, and
rotations (sometimes
called transformations)
together. As a firm
believer in the use of
manipulatives (for the
sighted as well as the
blind), I pull out my
box of assorted
triangles and
quadrilaterals. I select
two congruent
nonregular polygons and
place one on top of the
other; two scalene
triangles are my
favorite. I then proceed
to slide, flip, or
rotate the top
manipulative to
demonstrate a
translation, reflection,
or rotation. The bottom
manipulative remains in
place as the original
figure. This correlates
well with most print
textbooks which may show
the original figure in
red and the transformed
figure in black. If you
wish the student to
translate a figure to a
given point, rotate it
to a new position, and
reflect it over a given
line, you could use four
congruent figures. I
would probably want to
use magnetic
manipulatives or ones
with velcro in a
confined space, to keep
things in place. Be sure
to show the student the
textbook tactile
graphics illustrating
the same transformations,
so they will become
familiar with what the "average"
textbook furnishes them.
If these graphics are
not of high quality,
make your own using some
type of Stereocopier and
capsule/swell paper.
Furthermore, I show my
students examples of
test questions on
transformations from one
of the many TAAS
mathematics release
tests in braille 
produced by Region IV,
Houston, Texas. Region
IV has superb tactile
foil graphics.
When we reach the topic
of line symmetry, I
remind my students of
when they were younger
and made valentine
hearts by cutting a
folded piece of paper.
Believe it or not, my
high school students
have fun folding a piece
of braille paper and
cutting out hearts or
some other symmetrical
design. I tell them the
folded edge is a line of
symmetry. Then, I get
out my manipulative box
again, selecting two
congruent right
triangles. After placing
one on top of the other,
I flip (reflect) the one
on top over the line
segment formed by one of
the legs to create a
larger isosceles
triangle with a line of
symmetry (altitude) down
the middle. You can also
have your student use
paper folding to
determine symmetry lines
for figures studied so
far (rectangles,
hexagons, etc.). Again,
be sure to show the
student the textbook
tactile illustrations of
symmetry and/or make
your own graphics as
outlined above.
Tessellations or tiling
patterns is an
arrangement of figures
that fill a plane but do
not overlap or leave
gaps. In a pure
tessellation, the same
figure is used
throughout. I usually
begin with having my
students check out my
classroom floor, which
is composed of square
tiles. I also have a set
of tables in the shape
of isosceles trapezoids,
which create a
tessellation. Then I
move to textbook or
homemade tactile
graphics of
tessellations using
rectangles, equilateral
triangles,
parallelograms, right
triangles, regular
hexagons, etc. Let the
students explore to find
that any triangle or
quadrilateral can be
used to tessellate a
plane, but that only
certain polygons with
more than four sides
tessellate a plane.
Tessellations that use
more than one type of
polygon are called
semipure tessellations.
At this point, I get out
my wooden Discovery
Blocks from ETA (various
and duplicate sizes of
triangles, squares,
rectangles, and
parallelograms) and let
them design their own
tessellation. One young
man designed an
incredibly beautiful
tessellation and placed
the blocks inside a
frame. It was quite a
magnificent piece of
parquetry.
VIII.
Geometric
Constructions
A teacher
writes: The
student I work with is a
ninth grade braille
reader who is in
advanced classes. Since
she does not like to use
foil or the Sewell
raised line drawing
technique, I was hoping
you might have
information on how my
student can learn to
bisect angles tactually.
Susan Replies:
For
constructions, my
students don't use foil
or the "usual" Sewell
raised line drawing
technique either. We use
some type of rubber on a
flat surface  whatever
you have available. Some
of my students and I
happen to like an old
Sewell raised line
drawing board which has
rubber attached to a
clip board so that I can
clip my braille paper to
this to keep it from
sliding. But, others use
a rubber pad on top of a
regular wooden drawing
board or table. Still
others might like a
similar rubber on wood
board from Howe Press
because it too has a way
of clipping the paper
down.
Next, you will need a
braille compass from
Howe Press. The compass
has a regular pointed
end, but the other end
has a small tracing
wheel attached. I have
not been able to find
these compasses anywhere
else. Should you find
another source, please
let me know. Next you
will need a straightedge
 any "print" ruler will
do if you don't have a
plain straightedge,
since the student is a
braille reader. Finally,
you will need a tracing
wheel. Use one from the
homemaking department,
or Howe Press, or the
APH tactile drawing kit,
or the local hardware/hobby
shop.
For your student to
bisect an angle you
would first take a piece
of braille paper (not
the flimsy Sewell
plastic) and place it on
your rubberized surface
(board). Draw the angle
you wish the student to
bisect using a
straightedge and tracing
wheel. Remove it from
the board. Label the
angle with an "A" at the
vertex using slate and
stylus or your
braillewriter. Return
the braille paper to the
board. Ask the student
to bisect angle A. The
student should first
reverse the paper. Place
the compass point on A
and draw an arc,
locating two points B
and C on the respective
rays of the angle.
Reverse the paper. Place
the compass point on B
and draw an arc in the
interior of the angle.
With the same compass
setting, place the
compass point on C and
draw an arc, locating
point D  the
intersection of the two
arcs. Reverse the paper.
Draw a ray, AD, which is
the angle bisector of
angle A. Voila!!
Using a similar
technique with only a
compass and straightedge,
a blind student (or
anyone else) can also
copy a line segment,
bisect a segment, copy a
triangle, copy an angle,
construct the
perpendicular bisector
of a segment, etc. These
are the same basic
techniques that the math
teacher would use except
that the braille student
would usually prefer
reversing the paper so
as to take the most
advantage of the raised
drawing on the reverse
side. The end product is
easily graded by the
math teacher  allowing
the student to stay in
the regular classroom
setting throughout the
construction.
See the Resources Pages if
you need to order any of
the items mentioned
above.
IX. Teaching a Blind
Student
How to Graph on a
Coordinate Plane:
No Tech, Low Tech, and High
Tech Tools
Susan Osterhaus,
Secondary Mathematics
Teacher, TSBVI
Editor's Note:
In the
author's words, "Although
graphing calculators are
mainstays of most
secondary math
classrooms, it is
important for all
students to understand
the concept of graphing
on a coordinate plane
before they move to the
graphing calculator."
This is especially
important for visually
impaired students, and
Susan Osterhaus, math
teacher at TSBVI,
ensures that her
students learn to do so
manually _ they must
physically plot points,
graph lines, and find
slope. Below are her
answers to questions
about how to teach this
skill, and her
suggestions for students,
teachers, and parents.
1. How
can blind students graph
linear equations,
inequalities, and
systems of inequalities
independently and
efficiently? Or is this
the time when the VI
student doesn't
participate because of
the visual nature of the
task?
Most academic blind
students, even those
with spatial orientation
problems, are quite
capable of graphing, and
as one of my students
exclaimed, "Not only can
we do it, it's fun!"
There are several tools
they can use to do so:
The Graphic
Aid for Mathematics,
from the American
Printing House for the
Blind (APH), is
excellent for graphing
algebraic equations. It
can also be used in
geometry, trigonometry,
etc. It consists of a
cork composition board
mounted with a rubber
mat, which has been
embossed with a grid of
1/2inch squares. Two
perpendicular rubber
bands, held down by
thumbtacks, can create
the x and yaxes.
Points are plotted with
pushpins at the
appropriate coordinates.
Points are connected
with rubber bands (for
lines), flat spring
wires (for conic
sections), or string
(for polynomial
functions). I like for
my students to graph
extensively, and they
can do so incredibly
fast on the APH Graphic
Aid. In fact, many print
students also like using
it because it is fast,
fun, and allows them to
learn graphing skills in
another modality. You
can make your own graph
board by affixing a
piece of raised line
graph paper (also
available from APH) to a
cork board
and proceeding
as described for the
Graphic Aid.
If a student needs to
turn in copies of graphs
for homework, he can use
Wikki Stix
and
High Dots on
APH graph paper.
This method can be quite
expensive, however, and
is very time consuming.
It also tends to be more
of a test of artistic
ability than a
demonstration of
understanding of
graphing concepts.
The ORION
TI34 talking
scientific calculator (from
Orbit Research) and the
Accessible
Graphing Calculator
(from ViewPlus
Technologies) are
examples of more high
tech solutions for
graphing activities. I
described them in a
previous See/Hear
article (Winter, 2002),
but strongly recommend
that students be able to
graph manually as well.
It is important for
visually impaired
students to be able to
use a variety of tools,
and know when to use
each of them. For
example, a former
student decided to graph
a quadratic function
manually because it was
"too easy to bother with
the computer." Yet, he
will use the AGC to
graph an exponential
function.
2. How do
students represent
inequalities that
require a solid line or
a dotted line on the
graph?
The APH Graphic Aid
described above works
well. Plot the points
with pushpins and
connect them with a
rubber band when the
boundary line is to be
included in the solution
(a solid line in print).
Leave off the rubber
band when the boundary
line is not included in
the solution (dotted or
dashed line in print).
3. How
can VI students show
shaded parts on a graph?
When graphing one
inequality in two
variables, I simply have
my students place their
hand on the shaded side.
I check each graph as my
students complete them.
When graphing a system
of two inequalities, the
student places one hand
on the shaded side of
the first inequality.
Then they place the
other hand on the shaded
side of the second
inequality. Where the
two hands overlap (including
the boundary lines where
applicable) is the
solution. Pretty soon
most of my students are
able to handle three or
more inequalities
without multiple
overlapping of hands. We
even progress to linear
programming problems
involving four or more
inequalities. In these
problems, a bounded area
with vertices is often
found, and it is pretty
obvious where the shaded
portion (the solution)
is located.
4. Is
there a way for them to
do multiple problems on
a piece of paper? What
if they need to be
turned in to another
teacher, or you can't
check each graph as it's
completed?
If a studentmade,
manually produced paper
copy is required, the
student could use APH
graph paper attached to
a corkboard. She could
plot her points using
stickon high dots, puff
paint, etc. and could
form solid lines using
Wikki Stix. She could
actually use a colored
pen, pencil, or crayon
to color the shaded area
of the solution. This
would take much longer,
however, and would be
very labor intensive. It
will be important to
know the purpose of the
assignment and the
concept(s) being taught.
A paper copy of a single
function can be created
on the AGC (it can't
graph multiple functions
on the same graph).
Often, it is possible
for a sighted person (teacher,
peer, parent, teaching
assistant) to make a
print copy of the
student's graph _ the
visually impaired
student graphs on the
Graphic Aid and someone
copies it exactly onto a
piece of paper to turn
in. You can also divide
the Graphic Aid into 4
to 6 small, separate
coordinate planes for
multiple problems. If
you have a digital
camera, you could even
email or print a
picture of the student's
graphs. Better yet, have
the student take her own
photos!
Please be sure that
visually impaired
students are allowed to
participate in all kinds
of graphing activities
and that they are
supplied with the proper
tools. I would rather
see them become
proficient using a
rubber graph board
because they will learn
so much more with this
method, and they can do
so independently.
Creative exploration
should begin in the
early grades and allowed
to blossom. Remember,
the beauty of a tactile
graphic is found in the
fingertips of the
beholder. And there can
be no more beautiful and
meaningful graphic than
one created by those
very same fingertips.
ϟ
Susan
Osterhaus has been
teaching secondary
mathematics for 29 years
at the Texas School for
the Blind and Visually
Impaired in Austin,
Texas. She has a
bachelor's degree in
Mathematics, a master’s
degree in Mathematics
Education, and
certifications in
Secondary Math, English,
and Teaching the
Visually Impaired from
the University of Texas
at Austin. Susan has
taught or consulted with
VI students from grades
6 through college, at
achievement levels
ranging from 3rd grade
to talented and gifted.
Δ
9.Abr.2012
publicado
por
MJA
