In this webcast, Susan Osterhaus from the Texas School for the Blind and Visually Impaired shares her insights related to instructional strategies and resources for teaching math to students who are blind or visually impaired. She talks about the use of technology and the challenges of standardized testing for this population.
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Presented by Susan Osterhaus
Length of time to complete: approximately 30 minutes
CHAPTER 1: Introduction
OSTERHAUS: 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.
CHAPTER 2: The Importance of the Nemeth Code
OSTERHAUS: 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.
CHAPTER 3: Producing Accessible Math Materials
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.
CHAPTER 4: Applying a Multi-Sensory Approach and Universal Design to Math Instruction
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.
CHAPTER 5: Technology Tools for Students Who Are Blind and Visually Impaired
OSTERHAUS: 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.
CHAPTER 6: Helpful Teaching Aids for Students and Teachers
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.
CHAPTER 7: Issues and Challenges with Standardized Testing
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 cantt 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.
CHAPTER 8: Final Thoughts
OSTERHAUS: 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.