Teaching Math to Students Who are Blind or Visually Impaired
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Teaching Math to Students Who are Blind or Visually Impaired


NARRATOR: The name “Perkins”
carved in stone. Below a gothic tower,
a boy navigates with a cane. A title… I actually started
in the fall of 1978. I did have a background
in teaching mathematics. I have a bachelor’s degree
in math, a master’s degree
in mathematics education, and I have a Texas certification
to teach secondary mathematics. But I had no knowledge at all of teaching blind and visually
impaired students, and I just kind of happened
across… There was an opening, and I
decided to just check it out and basically I went in for,
I thought, my first interview at the Texas School for the
Blind and Visually Impaired in Austin, Texas. Actually it was TSB
in those days, Texas School for the Blind. And they kidnapped me. (laughing):
No, they wouldn’t let me go. You know, I walked in and the
principal who interviewed me said, “We need you desperately. Your credentials are fantastic.” And I was there… you know, normally when you’re
in interview you’re supposed to be telling
them the best things about you, and I’m going, like,
“But I’ve never taught the blind and visually
impaired.” “No problem,” you know. “You will have to go back to
school, get your certification, but you can do it.” And so I said yes, and here it
is 36 years later, so I must have liked it. I really thoroughly still enjoy,
you know, what I did. But again, I knew, you know,
virtually nothing about teaching the blind
and visually impaired. And, in fact, in those days,
unbeknownst to me, a lot of people really didn’t
feel that a blind person could go on into higher
mathematics… Let me put it this way,
the average person. We have our geniuses, you know, that just happen to be blind
and so forth. But the average student who was
blind was thought to, you know, not really have any hope of going into higher mathematics
and so forth, that it was such
a difficult subject. Well, anyway, I didn’t know that
and so I just jumped in. And bottom line, when I started
in 1978, the highest level of mathematics taught
at our school, at least, was a kind of a two-year,
I’d say equivalent to pre-algebra nowadays. And now we have students taking
calculus, scoring fives– five, that’s the highest you can
score on an AP calculus exam. So we proved them wrong,
those other people. NARRATOR:
Fade to black. What I had to do was, first,
I had to learn braille. I didn’t know literary braille,
much less the Nemeth code. The Nemeth code is the braille
for learning mathematics and science notation. And so I had to start there. I had to learn Nemeth code, and I’ll tell you a little story
about that. I came to the school for
the blind, and again, I thought I was told, you know, get all
kinds of help any assistance I needed. We had a lot of teachers
who themselves were blind. So I’m asking
for the teachers’ help. I go to the braille teacher,
she is the teacher of braille, and I ask for help and she goes,
“I don’t know Nemeth.” And I went, “Okay.” And then I went to the social
studies teacher, who had… He was a social studies teacher but he had to have taken math in
college, so I said, “Can you teach me
Nemeth?” and he laughed. And I went, I said,
“What’s going on here? I’m missing something.” And what he told me was that
they were too old. I was young for them. NARRATOR: We see a page
with column headings, “Symbol” and “Nemeth.” In the “Symbol” column
on the left are common math symbols such as
plus, minus, multiplication and division signs, greater
than, less than and equal signs. In the column to the right are
the symbols as they would be displayed in Nemeth code
using six-point braille cells. OSTERHAUS: And so a man named
Dr. Abraham Nemeth decided to create
this special code, and he was a professor
of mathematics himself and he wanted to be able
to read and write, you know, in all these symbols
in a code. So he invented the Nemeth code,
and I ended up teaching myself, my students and the rest
of the staff the Nemeth code, and as I was teaching myself
and learning, I saw how beautifully it was
done, how logical it was. NARRATOR: We see a photograph
of Dr. Nemeth on the occasion of his induction
into the Hall of Fame for Leaders and Legends
of the Blindness Field. Dr. Nemeth is holding a bronze
plaque of his likeness. OSTERHAUS: And Dr. Nemeth
has passed away now, but I want to say,
“Thank you, Dr. Nemeth,” because I just don’t know how I really could have done
what I did without the Nemeth code. I’m not saying that I feel
like I’m a good teacher, but having that Nemeth code, that ability to give
these students the higher mathematics using
these higher-level math symbols was just a real necessity. And so I think that’s
the main thing that has really expanded
this world of mathematics to, I’m going to just say,
the average student. I’m not saying I didn’t have
some very brilliant students, but the average student can now
take mathematics and enjoy it and, I hope, have as much fun
with it as I have over the years. NARRATOR:
Fade to black. OSTERHAUS: So I’m going
to actually start with the low-vision students. Remember, this was 1978, okay? I had to enlarge something
by hand-enlarging it. So I would just get out
my large print paper or, you know, lined paper,
whatever, and make everything large
and then copy that, because we didn’t have copying
machines that would enlarge. So we had to do all that
by hand, and then all the tactile
graphics, again, had to be done by hand. So there I was, with my… I had a Sewell raised line
drawing board. It’s just a clipboard that’s got
a little rubber padding on it. And I would put braille paper
on top of that. I would use a Howe Press
compass, by the way, which is from Perkins. And I would go ahead and draw
my tactile graphics using that and just a ruler
to guide me. NARRATOR: In a photograph,
we see a person using a Howe Press compass to trace and produce
raised-line drawings on a sheet of braille paper. The braille paper is
on a raised-line drawing board. OSTERHAUS:
So very, very basic. Now, I’m not going to say that
some of those tactile graphics are still very good. I don’t want to throw out
all the old stuff. But basically everything was
done by hand. All the brailling was done,
again, on a Per… I feel like I’m advertising
because… for the Perkins Braille Writer. We would put everything… And so it was one copy,
and then we had a machine called a thermoform machine
that would make our copies, the kind of plastic paper. That’s how it started, okay? That was the original method. And then over the… And if you’re thinking, well,
why wasn’t I using high tech? It’s not… believe me, if there
was any higher technology, I would have been using it, but at that time,
that’s all we had. And then later on, you know, there were many, many
improvements. NARRATOR: We see displayed
on a black background several thermoform pages depicting a variety
of geometric shapes, such as triangles and circles,
as well as a page with plot points
of intersecting lines. In addition, there are two
green plastic protractors with raised lines
and markings. OSTERHAUS:
But there’s so many… there’s just so many more tools that you can create tactile
graphics with these days. When you create
a tactile graphic, let’s say with Microsoft Word,
you get a nice print copy… And by the way, again,
I’m always thinking in terms of low-vision students
and the braille student. So I make one graphic, let’s
say, on the Microsoft Word, and you can do it any way
you want. Some people are more artistic
than I am, so they’ll use Corel or some
other type of drawing program. But you create a black line
master, and then you can go in and where
you would normally put print, you can still put print– you can put large print font
for your low-vision students, but then you can change the font
to a braille font, and then what you have to do
with the copy that is in braille,
it’s not really raised… It’s just, you know,
you’re printing it out and there are braille dots. So what you have to do is copy
that onto this special paper called swell-touch paper
and it does what it says, swell. So everywhere there’s black,
including the braille, you put it through
this special machine and there are like three
manufacturers of these machines and… three vendors,
and you put it through and it comes out the other end and all of the black lines
are raised and the braille is raised. So that’s probably… I would
say that is about the fastest method of getting a very good
quick graphic. NARRATOR: A photograph shows
a sheet of thermal paper coming out of the machine
that heats the paper, causing any black line or image
reproduced on that paper to swell and become
a raised line, dot or shape. In this case, a line arcs upward
on raised graph paper. OSTERHAUS: Now when I’m
preparing my math materials, I actually use… I’m not saying everybody has to
use, but this is my method, I use a product
called Scientific Notebook. And it’s a software that’s kind
of like Microsoft Word. When you look at it you think you’re maybe
in something like that except it has a special little
extra icon that you switch back and forth
between text and the actual math. So, anyway, I just get on my
computer and keyboard and I type in all of my math, and the description or the text
portion is one thing and then the mathematics
is in… actually even in a different
color when I’m looking at it. And I can change that font size
to any size font, any type of font, so for my
low-vision students, they get a perfect copy
in the font… If they want Comic Sans 24,
they get Comic Sans 24. And then I take that particular
document after I’ve created it and import it into something
called Duxbury, and that’s DBT Win. I think we’re up to 11.2 now. And it translates it into
very good Nemeth code and then I do need to do
a little formatting. So that’s how I create, you
know, all of my math work now. So we’ve come a long way
in 36 years. NARRATOR:
Fade to black. OSTERHAUS: I was teaching
very much like I teach today even when I taught sighted
students. In fact, I’ll go back
to my student teaching days. They used to have a nickname
for me. I hope it was… (laughing) They called me
the Tinker Toy Lady, because I was teaching geometry
and I would come in and I would make all these 3D
models and come in… for the sighted students. So truthfully, I want to say,
when people ask me how do I think mathematics
should be taught, I want to say, I think all
students should be taught like I teach blind students. And I’ve learned this
over the years. When I was, you know,
growing up, I was taught totally visually, I think,
mathematics. And I thought I was
a visual learner myself. Now that I’ve been teaching
for 36 years, I think I’m more of a
multisensory learner myself and I’ve learned so much while, you know, teaching
these students. NARRATOR: In a video clip,
we see a boy who is blind in a math class at Perkins. Today, he and his teacher
are working on fractions using segments of wood
in various sizes that are labeled both in marker
and with braille tape. BOY: Four-twelfths
equals one-third. TEACHER:
You got it. Nice job. OSTERHAUS: The way I approach
everything, I found out that it’s called
the multisensory approach. When I started out, I just did
my own thing, but then people later on
told me, “Wow, you use the
multisensory approach.” I said, “Oh, I do? Glad to hear that.” “And you believe
in universal design.” So let me explain a little bit
about that. As far as multisensory approach, when students are
in my classroom I really try to get them, at
least, if they have some vision to look at it. Basically, we use
as many senses as possible. If they can see a little bit,
some of them, even if they’re a braille
student, can have a little vision and we want them to use as much
of that as they can, even if it might just be color. And then we want them to,
you know, of course, if they’re a braille student,
to feel it. But even if they’re
a low-vision student, I still have them in there
doing a lot of tactile work. With the… I’ll even
have them eating math. You know, if you make a pizza and they have to cut it
into pieces, into fractions, all kinds of things. I can still remember doing,
you know… I don’t think they do them
anymore, they do too much work
for us now, you used to be able
to break the crackers into four pieces, and so we would do the fractions
that way and then I would say, “Okay, eat one-fourth of your
big cracker” and so forth. And it’s amazing. They learn a lot better when
they get to eat their math. NARRATOR:
Fade to black. So I was desperately constantly
telling people, “Find me a better calculator.” And again, over the years they’ve had several of the basic
calculators. You can buy them anywhere
these days. But finally, I was really
in need of a… at least a talking
scientific calculator, and I started doing a lot
of research to find the perfect one. Went all over the place, had the students evaluate each
one of these. And finally we found, just as I was about to make
a bad decision, a new calculator came out,
and it was from Orbit Research and it was called the ORION TI-34 talking
scientific calculator. It was, like, a third of the
price of all the other ones and had more functions, and again, it was based
on Texas Instruments’ product. So I have… I am from Texas,
so I have confidence in Texas Instruments. And that ended up being
our calculator of choice. NARRATOR: We see displayed
a photograph of the TI-34, a Texas Instruments
talking calculator. OSTERHAUS: However, at a certain
point, TI decided to stop making the TI-34, so then we went to… and actually Orbit Research
asked me, by that time I had gotten
involved and had helped them actually
with the TI-34, so they asked me for my input. And I am actually the one who
put the stamp of approval on them doing the TI-36X. So currently we have
the ORION TI-36X, which has, as I said,
many more functions. I think it has 122 functions. And then ViewPlus came out with
the audio graphing calculator, which was a software product
that was on… basically used on a PC. And I, you know, learned
how to use that and again asked them to, you
know, continually update that. And we used that for many,
many years. Bottom line, I was involved, and we got this fantastic
collaboration between Texas Instruments,
Orbit Research and the American Printing House
for the Blind, and we now have the ORION TI-84 Plus talking
graphing calculator. APH came out with something
called Math Flash, which is a cute little program
of teaching… helping… Well, it’s not really so much
teaching, it’s giving sample problems,
but it’s in such a cute way. If a student gets the problem
correct, you know, it gives them all this great
talkative feedback and praises them and so forth. And if they get it wrong, it does things like
flush the toilet. The people that I worked
with were Touch Graphics, who do the Talking Tactile
Tablet, and since I field-tested that,
you know, that’s the one that we got
to keep. The IVEO is the competition. It’s ViewPlus. I just want to mention them,
though. You know, but it’s the fact that
Touch Graphics got to us first and we field-tested that. And the Talking Tactile Tablet, I thought what
we were going to do is that it was going to mainly
be for my blind students. NARRATOR: A raised-line graphic
of a right triangle sits within the frame
of a Talking Tactile Tablet. The tablet is connected
to an open laptop computer that displays an X-Y axis on a
background field of graph paper. OSTERHAUS:
But that particular year that we were first
field-testing it, I happened to have a student
who had achromatopsia, which is basically
real color blindness, no… just seeing basically
in black and white and grays. And he also had dyslexia. And it turned out that this was the most fantastic
thing for him. As it turned out,
the contrast was best black on canary yellow– not
that he could see canary yellow, but the contrast was the best. So I did his graphics that way
with the black line masters but on the canary yellow paper. I did the whole
Talking Tactile Tablet with him. I was so pleased. He was absolutely ecstatic. The iPad, yes. What had happened there… I’d been working with
the University of Arizona, and they are taking
AnimalWatch Vi Suite, and they are basically opening
it up to the Vi population. And this is, I think, one
of the first apps for the iPad that is truly, you know,
accessible for our students. It has the… again, it’s on the
iPad, so you can listen to it and so forth, but in addition
to all of that, we have a braille script
that goes with it– a hard-copy braille script– hard-copy tactile graphics… We even have
three-dimensional objects. NARRATOR: A fourth-grade boy
who is blind is using the AnimalWatch app
on an iPad. This particular math problem
involves determining the amount of weight that
a cheetah gains per month over its first year of life. The boy can hear the problem
read to him using the voiceover feature
on his iPad. He also has a refreshable
braille display that allows him to read
the problem. On the screen of the iPad,
we see the text of the problem and a picture of a cheetah. On the desk to the right
of the iPad is a small three-dimensional
plastic figure of a cheetah. COMPUTER VOICE: …to find the
average weight gained per month. (boy laughing) BOY: 60 divided by… COMPUTER: Divide 60 by 12 to find the average weight
gained per month. OSTERHAUS: And the little
three-dimensional objects are the actual animals
themselves. We came up with, I think,
good tactile graphics, but there’s still nothing better than they really need
to kind of feel… even though it’s, of course,
a much smaller version of what it’s going to be. NARRATOR:
Fade to black. OSTERHAUS:
My first… and this was old. This is the oldest tool I… Well, maybe not. But if not the oldest,
one of the oldest tools. In my closet when I got there there was something called–
it’s got a long name– Graphic Aid for Mathematics. We just call it
the rubber graph board. It looks like a coordinate
plane, and you put the X and Y axis on
with rubber bands and you use push pins. And I’m going to tell you
that we… It’s changed over the years. They’ve made adjustments
and so forth. But it is still
the greatest thing. So I don’t believe in throwing
out the old with the new. We keep the old that is good
and add the new is what we do. So we still have that and I
still just absolutely love that particular tool. NARRATOR: We see a photograph of
a Graphic Aid for Mathematics. In the lower left corner, three
push pins with rubber bands stretched between them form
the X and Y axes of a graph. Two other plot points are
noted by push pins, and a thin piece of flexible
black plastic describes a line that passes from the 0,0 point
on the graph through the other plot points. To the upper right on the board, rubber bands stretched
between push pins create a triangle shape. The thin, flexible black plastic forms a circle
around this triangle, intersecting at the points
of the three angles. OSTERHAUS: Some people complain
that we’re, like, “Well, Susan, you know, you can
only do one graph at a time” or, you know, “and then if the
student has 12 graphs to do, what do they do?” So I was, like, thinking
very hard, “How am I going to do this?” And the light bulb finally went
off because I have a motto that anything I can do
my students can do better. So I had been taking
digital pictures. I would do something and I would
take a digital picture for a presentation,
for a PowerPoint. And I thought, “If I can take
a picture, they can take a picture.” So we teach blind students, our totally blind students
and the low-vision students, they can take a digital picture
of their graph and put that in their math
teacher’s shared folder or however they want it
and hand their homework in. So I have brought an old, old
tool into the modern world. There’s something
called Geometro that has not been around
as long. It is a Canadian…
a Canadian vendor created these. They’re… if you can imagine,
polygons with a Velcro edge and you can make something
we call nets and then you take the nets
and you fold them up to this three-dimensional model. They are the most fun thing ever
for geometry. There’s something called
a braille print protractor that… actually I had something
similar to it in my classroom but we didn’t have a vendor
for it anymore and I asked APH to kind of
reinvent it. And this braille print
protractor has braille and print on it, so again,
it’s universal design. And it’s got this little wand, and it’s really what people
out there, a geologist, would call a goniometer. They would use it for measuring
the angles on crystals because it has this wand, and you use it in a very
different way. It’s kind of like you turn it
almost upside-down and you actually, the wand actually forms
the angle that you want and then its supplement. So it’s another teaching tool. You get such a good tool for
teaching supplementary angles. NARRATOR: We see an example
of the braille print protractor with the APH logo as described. The wand portion, which pivots
from the center of the compass’s
horizontal base, comes to a point on the end
that sweeps over the compass arc just below raised markings
that denote a distance of five degrees on the arc. This allows the student
to measure an angle. OSTERHAUS: And then if you think
about, as I mentioned, like just a ruler. So we have at least… we have an English measurement
flexible ruler that’s both braille and print. We have a metric one. We have a braille print
yardstick. So all of those kind of tools. And I know I’ve… Oh, there is another one that’s
another one of my favorites. It’s called Omnifix Cubes. This is not available
at the blind store. This is just something
you can purchase at Didax, which is one
of the math education type online stores that you can
buy from. And they’re just these cute
little… Actually they come as a net. You fold them up into a cube. But the cubes fit together
and they’re not unifix cubes. They fit on all sides. NARRATOR: In a photograph,
we see many Omni cubes stacked in various
configurations. In the center of the picture we see one of the cubes
unfolded, its six sides flat on the table. OSTERHAUS: So when you’re trying
to create this three-dimensional drawing
of squares or cubes… which they love to put
on standardized assessments. This is the real big thing. And we used to try to do this
with regular cubes, and the kids…
if you can see it and maybe keep the cubes
together, you’re okay. But to explore them tactilely,
your cubes would all fall apart and so forth, so this was just
a wonderful thing, again, to have, and so I use
that with students. NARRATOR:
Fade to black. OSTERHAUS: There are obviously
problems with online testing, but that’s where they
would like to go. And I think that they’ve been
more successful in, let’s say,
with English language arts. Now, so what are the problems
with math? Okay. Well, I’ve already addressed
a little bit about that talking about the iPad and needing that extra hard copy
additions. So as far as what we can do and whether we think
it’s good or bad, first of all, you can listen
to math, but listening to math… what I
try to tell sighted people is, “You try to do it.” Because a lot of these testing
organizations say, “Well, they can just listen. “They can just listen
to the math. “I mean, they can listen to…
they can listen “to an English passage, a
nonfiction or fiction passage. Why can’t they just listen
to math?” Well, what I like to say
then, “Okay, if you think listening
is the way to go, “then everybody takes
their online test in math “by listening. “There will be no print. “There will be nothing visual. You go ahead and take a math
test just by listening.” And then they kind of go,
“Oh, I get it.” The other thing is with… let
me go to low-vision students. The way that these online folks
are doing it, they are incorporating
a zoom feature so people can enlarge things. They are coming up with
calculators that, again, you can zoom them and they are
actually on the test itself. They have contrast. You can choose. Do you want black
on canary yellow or black on white
or white on black? You know, you can do
that type of thing. So, they are coming up
with a lot of features, that type of features. And even math tools that these
low-vision students are going to be able to use
and manipulate. Again, when you get
to the braille reader, some of them have said to me, “Well, what about
refreshable braille”? And that’s… and basically
a lot of students are using
refreshable braille now. But I’ve heard… not everyone. But, you know, just about all
of the students that I know, they have some type
of a refreshable braille. However, at the present time, they get one line
of refreshable braille. Well, when I do math, yes,
I may do one line at a time, but I look back at the line
before. I want kind of this bigger
picture. And there are some things that
you can create in math that require your looking at
more than one line at a time. For instance, a number line
graph can be created, and in fact this is
the standard way. Now it is considered
the standard way in the United States and Canada
that we make number line graphs. And they require three lines. You can’t do that on
a one-line refreshable. But again, I’m going
to tell you right now, teachers are still saying, “We still need the hard-copy
braille, the hard-copy tactile graphics
for now.” They don’t feel that the
technology is there yet. NARRATOR:
Fade to black. You know, when you’re studying
orientation and mobility, they do put us
under the blindfold and we kind of simulate it and I was amazed how all my
other senses started kicking in, things that I had never bothered
to notice, never heard before, never felt before,
never, you know… the sun coming in the…
just all of these things that I had never heard
or felt, et cetera, before until we blocked out
the vision momentarily to where I actually had to use
my other senses. So again, I’m not going
to say… there are certain aspects
that are more difficult that are just easier
to grasp if you can see it. But I still think that
there is no ceiling. I mean, and now with all
the new technology, it’s just becoming
much more accessible. Everything is still…
like I said, we’re behind. You know, the technology
continues to be behind, but from 36 years ago, boy, have we come a long way,
baby, as they say. So I really encourage
many students… You know, not everyone is
going to be a mathematician, but certainly at least explore
mathematics. I’ve had a lot of students
who were fantastic in math and, unfortunately, they didn’t
go on to become mathematicians, but they certainly went on
to do other, you know, fantastic things
and used their math. NARRATOR:
Fade to black.

About James Carlton

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5 thoughts on “Teaching Math to Students Who are Blind or Visually Impaired

  1. nice video. I want to ask a question, when a VI person search for solution for any mathematical equation, in that condition teacher who is explaning the question, he sad that we put the X here. he is explaning for a sighted student with the help of hand waving or text. how can a blind student gain where he is pointing x.please suggest me any link, YouTube channel and app which is accessible for Blind users. for maths solution. thanks sir

  2. I’m visually impaired I Went to public school all through high school I HATED math with the ever living breath of me. Couldn’t see what teachers were doing up there and as for nemeth I thought it was just evil because like she said the Braille teacher could not teach me so I was just clueless. I graduated in 2016 with barely a C in math

  3. My cousin is NLP ( completely blind) and excelled in math. She's gone on to a career as a chartered accountant. I want to both say that technology has helped the blind and visually impaired immensely but…braille is still important. My cousin learned Braille and Nemeth Code and teaches it to her kids.

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