A parent writes: I have been working with my daughter on math, and I know math reasonably, but it is visual in nature and a challenge to know the best way to present it. My daughter is not exactly "resisting" Nemeth, but rather until last year, she was able to pretty much do everything in a print medium, but lost more of her vision making that impossible. She went to a residential school for the blind where she learned Braille reasonably efficiently, and she knows Nemeth to "read" it, but writing it is often slow and she makes occasional mistakes- which, of course, makes it difficult.
The school she is in now is a "regular" school that has no experience in dealing with blind students. They have provided the math text (as well as her other textbooks) in braille.
The problem comes in attending classes, where blackboard work to the class is effectively useless, and taking tests, etc where translating back and forth between braille and print to have effective communication between her and the teacher is proving very difficult. She has traditionally done everything in her head in math (she can do amazingly complex calculations in her head) but obviously, at some point that is an unworkable strategy.
She likes math, she is very good at it, and would like to continue in it. My goal, I suppose, is to try to find the best way to go about this...should we concentrate on Nemeth alone? or is there other technologies that might make this easier? I, of course, don't have a clue, and rather than "reinventing the wheel" here, I am hoping to research to find the best way for her to achieve the best she can.
Susan replies: I teach secondary mathematics at the Texas School for the Blind and Visually Impaired in Austin. In my opinion, learning to read and write Nemeth Code is absolutely essential for your daughter to be able to continue in higher mathematics. I am surprised that she is better able to read than write. My adventitiously blind students are usually faster at writing than reading. Of course, they do all of their homework for me in Nemeth, so I guess they get LOTS of practice! They use Perkins braille writers and can therefore easily read their own work - especially with all those steps in Algebra. They use either an abacus or a talking calculator to perform long computations. Using the braille writer for computations is too time-consuming. Previously, our standardized tests did not allow any students to use calculators. Now, the TAAS (our state test required for a high school diploma), SAT, and ACT are allowing braille students (and sometimes all students) to use calculators. I still value the use of the abacus as a braille student's equivalent to paper and pencil for a sighted student.
Here in Texas, a blind student in elementary or secondary school should be able to obtain instruction in Nemeth Code. After high school graduation, they are on their own, and I get frequent calls from college students and their professors on how they can learn Nemeth Code. There are few opportunities for blind college students to learn Nemeth code. So, try this as an incentive for your daughter to learn it now while she still can - assuming of course that she would like to go to college.
I am a user of technology for preparing materials for my students and for correspondence, but the field is way behind for blind individuals, especially in the areas of math, science, and engineering. Although it is easy to translate print into Grade II literary braille, research is still continuing on perfecting how to get from mathematical print equations to Nemeth Code and vice versa. (Go to Current Research) Three print to Nemeth translation software packages are currently available: MegaMath, DBT (Duxbury Systems, Inc.) and Scientific Notebook/Nemeth Filter (MacKichan Software, Inc./MAVIS at New Mexico State University). I beta-tested all three products. The Scientific Notebook/Nemeth Filter (SN/NF) is very user friendly for secondary and higher mathematics, especially for producing Geometry materials, and suits my needs best. I can obtain a regular print, large print, and braille copy from one document. MegaMath does not provide a useable print copy and is less user friendly; however, with practice, one can become quite proficient at producing all levels of Nemeth materials. MegaMath might be preferred for producing elementary level mathematics materials, as it allows for the spatial arrangement of addition, subtraction, multiplication, and division problems, whereas SN/NF does not. DBT is the least user friendly at the present time, but their new beta version has a LaTeX importer, which imports Scientific Notebook files for translation to Nemeth. This importer was developed at, and is copyrighted by, New Mexico State University.
I do allow one type of technology, if the braillewriter is not acceptable in the mainstream classroom. Several of my students have used a Braille-Lite, which has one row of refreshable braille. The student doesn't use the translation mode and simply brailles in Nemeth Code and outputs in Nemeth. However, they can always go back a line and re-read their last step as they are progressing through an algebra equation or a trig identity, for example. The key features here are that it is a braille device and it has a row of refreshable braille. Other manufacturers have similar notetaking devices. A regular computer with a refreshable braille display is also acceptable. I do not advocate the Braille 'N Speak (made by the same company) as the student only receives voice-output as they make entries into the equipment.
There are many tools, aids, and supplies for teaching math to blind students, and I hope your daughter has had (and will continue to have) the opportunity to use them. Does she know how to graph on a number line? Does she know how to graph on a rubber graph board (Graphic Aid for Mathematics by APH) or raised line graph paper on a cork board independently? Does she know how to measure an angle using a braille protractor (Braille/Print Protractor from APH)? Can she (or will she) learn how to do constructions in Geometry using a braille compass (from Howe Press) and straightedge? Is she provided manipulatives, especially in Geometry?
An opposing view:Hi. I have been totally blind from birth. I remember math being one of the most difficult subjects because of its visual nature. There are a couple suggestions I would have to help deal with this problem. First, it is my opinion that Nemeth code is an absolute nightmare. It looks like jumbled up nonsense under the fingertips. I took a course just so I could learn to read my math books, and it was still ridiculously difficult. I realize this is going to stir up some controversy, but I feel that private tutoring in math is the best way to approach this, and it gives your daughter the best chance for really understanding the concepts. I recommend the use or what is known as a raised line drawing kit to help your daughter attempt to visualize how math problems are arranged. This is particularly important when dealing with fractions. You can obtain the raised line drawing kits from suppliers of blindness-related equipment. I learned the shape of the numbers so that sighted folks could demonstrate concepts for me with the raised line drawing kit. There is also something called a cube slate which also can be helpful. I don't know if the cube slates are sold anymore, but they have cubes with all the braille number combinations and a rubber board so that the cubes can be arranged to help keep track of what one is doing. Maybe a combination of these tools would be the best bet.
Susan replies: I'm sorry to hear that you had such a negative reaction to Nemeth Code. I do not find it to be a "jumbled up nonsense under the fingertips"; on the other hand, I think for the most part that it is very logical, systematic, and an absolute miracle for braille readers wishing to continue in higher mathematics. I am not a tactual reader though. As a math teacher with visually impaired students, I taught myself to read Nemeth Code visually (and braille it) out of necessity to be able to teach my students. There were no courses at the university in Nemeth above the basic numbers and operations, and I needed to be able to teach Pre-Algebra, Algebra I, Informal Geometry, Geometry, Algebra II, Math of Money, Trigonometry, etc. As I would introduce each new print mathematical symbol, the students and I would learn the corresponding Nemeth symbol; as I said earlier, I really learned to appreciate the logic of why Dr. Nemeth did what he did. Perhaps the key here is that students learn Nemeth Code most easily if they learn each new symbol as they progress through the mathematics. Learning Nemeth as a separate course from mathematics is as logical as a sighted person learning all the print mathematical symbols in a separate course. However, sometimes lack of time necessitates the Nemeth Code class.
I do agree that tactile graphics made using the tactile graphics kit by APH can be extremely useful - especially when created by certain people more artistic than I am, such as the Region IV Service Center in Houston, Texas. I was on a panel of experts called in to help facilitate the improvement of such graphics for our TAAS (state test required to graduate from high school) and for our math textbooks.
I do not like the graphics produced from the Sewell raised line drawing kits, except for emergency situations. They are too flimsy when using the plastic wrap type film that comes with the kit. However, when a piece of braille paper is placed on the drawing board and a tracing wheel and/or writing implement (regular pen or pencil) and braille compass are used along with a straightedge, even I (no artist) can make an excellent quick-fix graphic that any Math teacher (non-VI certified) can use to communicate with a blind student.
If you have access to a tactile imaging machine ("toaster"), you can transform black-lined print graphics into raised line drawings within a matter of seconds. (Go to Math Graphs Made by Others for Students)
I have also had great success using sturdy manipulatives to introduce many math concepts.
A successful blind Nemeth Code user replies: Actually, I had no trouble with the Nemeth code at all. I was first introduced to it in second or third grade. (When do we start doing math these days?) Anyway, my itinerant teacher did not know Nemeth at all, so it was up to me to learn it. And learn it I did, as I went along. I had very little difficulty with it, and math in general was no trouble (until I reached trig in 12th grade). Algebra was only minimally annoying with the graphed equations, but trig has lots and lots of them, and I'm sorry to say, that is the first math class I did not get at least a B in. Oh well. That's ok though, because if I need something like that done now, I just use my computer. *grin* Well, guess that's it. Nemeth isn't all that bad, it just takes some time. It's actually not all that different from regular braille (whatever that is) and I found it very easy to learn.
A Network Specialist in a data communications group replies: I read your messages to the list with much interest. I fully agree with your statements about the Nemeth Code and wonder what sort of educational hick up occurred which broke the learning process for the person who did not do well with it.
I find it alarming and totally unnecessary that so much of the blindness community seems to think that science and math are to be avoided at all possible cost. There certainly are problems in communicating mathematical ideas using tactile methods, but it is sure not impossible by any means. I know that there are blind engineers and people should think of at least one blind mathematician every time they use natural logarithms. There is just no excuse for a blind kid graduating from high school without even having had Algebra.
Yet another supportive user replies: I really enjoyed your messages! Would you consider giving a summer crash course in Nemeth and Math. I've done the Hadley course, read the BANA computer code, but really have little confidence in my math skills such as Algebra, and the stats I took in Grad school. You ought to consider a math camp for adult blind--I know you'd get a result. I'd come!
A returning student replies: I lost my sight 7 years ago as a result of diabetic retinopathy. In January I will be returning to school at the University to pursue simultaneous bachelor's and master's degrees in computer science (I already have about 3/4 of my EE degree, but haven't been to school in over 15 years), and for the first couple of semesters I will be concentrating mostly on my math courses. After talking with many people about this, I have decided to approach this by using nemeth braille -- I have talked to a few who have managed to "pass" their math requirements without braille, but most of them admit that it was a struggle, and once the course was completed, they quickly forgot about it. I want more than that; I want mastery, and I'm convinced that braille is the only way to go to get to this level. In case you're wondering, yes, I do read grade II braille, and do have enough sensation in my fingers to do the job -- not very fast, but that will come with more practice.
Once again, I want to say thanks for your positive approach to math and technology for blind students -- the more things like this that I read, the more convinced I am that I am making the right choice.
A college instructor writes:Hi all. I know of a blind college student who would prefer to do his algebra in braille. If he uses the computer to do any calculations, can they be printed out in braille as well as in print? Does the braille version come out in Nemeth? What advice would you give this student in terms of successful strategies for completing beginning and advanced Algebra? I will be pleased to pass on any tips or answers.
Susan replies: I am glad that he wants to do his algebra in braille. I am definitely an advocate for using the Perkins Braillewriter so that all the steps can be shown in Nemeth Code (both for the teacher who reads Nemeth and the student). Listening to steps in algebra doesn't work for 99% of the population (including me). However, I finally decided that technology had advanced sufficiently to satisfy me when they came out with refreshable braille displays. My students use Blazie's Braille Lite with one row of refreshable braille, but any computer or notetaker with refreshable braille is acceptable. They input all their algebra steps in Nemeth Code and can easily go back and check any previous step in braille as they continue toward their solution. My students input in Nemeth, don't translate, and output in Nemeth. It works for me; it works for them. The problem comes in trying to produce something in print for the teacher who doesn't read Nemeth. Many students try to input in something they invent (half-way between Nemeth and print) which both they and the print reader can decipher. It makes me cringe. Unfortunately (at the present time), Nemeth Code does not translate into print with the touch of a button - as with literary braille, but I'm helping with the alpha testing. *CAUTION* If you input in Nemeth Code and run it through the translator, you will output garbage! (Go to Current Research)
Stand alone talking scientific calculators or a good calculator software package can be used for computations. I don't know that most people would need to print out the answers in braille or print, if the plan is to plug these computational answers into an ongoing equation, working toward a final solution. However, there is now a braille scientific calculator, if one is willing to pay the price for the refreshable braille display. Did you catch my previous recommendations on calculators? (Go to Calculators)
He'll also need some other aids, tools, and supplies (especially for tactile graphics), but I'll send you that information directly. (Go to Susan's Math Packet)
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. We are just refreshing some basics of algebra, when I came across a problem. I don't know the Nemeth symbols for putting together roster notations, set builder notations, solution sets, empty sets and subsets, my question to you is what are the codes? or if you may know what sources I can look into in order to find them it would be very helpful of you to let me know.
Susan replies: First, I'll give you some sources that will help you with this and more things to come.
On my math homepage, go down to the very last item on the contents page and click on: Download Computerized Nemeth Code Tutor. I worked with RDI on this for three years, and you can download it for FREE!! Once you have the tutorial on your computer, go to Lesson 10, Section 6:Braces. This section covers your questions on: roster notations, set builder notations, solution sets, and empty sets. It also has information on intersections and unions of sets. Lesson 11, Section 2 covers your question on subsets. We also incorporated a lot of math hints.
Ruth H. Craig, Learning the Nemeth Braille Code, A Manual for Teachers, Brigham Young University, 1979, 1987 (available in braille and print from the American Printing House). This is the most user-friendly resource book, but I'll warn you that it doesn't have the highest level math. It does have a small section on set notation on pages 65-66.
The Nemeth Braille Code for Mathematics and Science Notation 1972 Revision, American Printing House for the Blind, Louisville, Kentucky, 1979 (available in braille and print). This is THE code book. It has everything of course, but it's a bit difficult to navigate.
Helen Roberts, et al., An Introduction to Braille Mathematics, Based on The Nemeth Braille Code for Mathematics and Science Notation, 1972, Library of Congress, Washington (available in braille and print from the American Printing House). This is very similar to the code book and is similarly difficult to navigate, but it gives more examples.
I have all of the above references, and use them all.
Now, if you don't already have it, I highly recommend that you purchase Scientific Notebook (SN). You can prepare all of your student's materials on this in print and run it through the Nemeth filter. It's ready to go as is, but I like to bring the document up in either Duxbury or MegaDots (not for translating because it's already translated) to check the dots and change the formatting to the way I want it. Although I can, and always used to, use six-key entry on the computer to prepare my Nemeth materials because I'm fast, I'm even faster at typing into SN. Also, this way I have a beautiful print and braille copy. In addition, the back-translator is in alpha testing at the moment. Here are your contacts for these products:
MacKichan Software, Inc. 600 Ericksen, Suite 300 Bainbridge Island, WA 98110 Phone: 1-877-SCI-NOTE Fax: 1-206-780-2857 E-mail: info@mackichan.com Website: http://www.mackichan.com/ With Scientific Notebook, create attractive documents with text, mathematics, and graphics, have it compute the solutions, import data from your graphing calculator, connect to the Internet and download documents, then translate to Nemeth Code and/or convert to large print. (See MAVIS below.)
Mathematics Accessible to Visually Impaired Students (MAVIS) Chris Weaver, MAVIS Program Coordinator New Mexico State University Math Department MSC 3MB P.O. Box 30001 Las Cruces, NM 88003 Phone:(505) 646-2664 Fax: (505) 646-1064 Email: chrweave@nmsu.edu Website: http://people.rit.edu/easi/easisem/mavis.html Nemeth Code filter that translates Scientific Notebook documents containing mathematics to Nemeth Code. "The fast production of Scientific Notebook files on a Braille embosser means that visually impaired students can obtain class handouts, syllabi, exams, and other course materials in the sciences in real time. And it means that institutions can comply more easily with federal disability regulations." (see MacKichan Software, Inc. above) (Chris also has two large print fonts if you are interested.)
A computer access specialist writes: I read an article a while back that talked about the difficulty of doing advanced mathematics in Braille. According to this article, Braille doesn't include all of the symbols that you might want to use, such as the sigma or integral, making more advanced algebra difficult to use.
I am interested in finding ways to present advanced mathematics to blind students.
Susan replies: Nemeth Code does include symbols for advanced mathematics, including the integral, sigma, etc. The problem is that advanced mathematics texts, articles, etc. are written in print. Most of this information can be tapped into electronically these days, but the blind reader cannot access it. As John Gardner states: "For many decades, people in organizations providing services for the blind have been transforming this kind of information into some form that blind people could use. Such transcribing is an art, and an enormous amount of loving human labor is required for every equation, every table, chart, graph, picture, etc. The need far outstrips the resources of the organizations that do this work." The state of Texas is required to braille all state adopted textbooks for students in K-12 grades. There are no such requirements at the college level.
Be sure to check out John Gardner's articles and work in progress:
The "Bumpy Gazette", Volume 2: Issue 1, June, 1996, is also devoted to math and science. It contains articles by: John Gardner, David Schleppenbach, and Albert Blank. Contact Repro-Tronics for your copy. They are able to provide computer disk and braille copies of the gazette upon request.
A professional writes: How does one prepare a recently blinded (but fairly competent braille reader) to do the math sections of the GED? Are there any commercially available Nemeth materials (the equivalent of "Read Again" for math) for adventitiously blinded teens?
Susan replies: The CAI Nemeth Code project (I was on the Research and Development Institute, Inc., Sycamore, IL, panel of experts.) is completed, and The Computerized Nemeth Code Tutor is now available to the public. I wish it were available for blind clients as wellas the sighted teachers it was designed for; however, they are currently working on a new tutorial for Braille users to access with a refreshable Braille display. (Go to Current Research)
I have had several students in recent years who are adventitiously blind due to trauma (mostly gunshot victims). First, we try to give them at least a quick introduction to literary braille and an abacus class (perhaps during summer school), so they know some literary and the basic Nemeth Code symbols (learned while reading and answering the abacus problems). Many students find the Nemeth easier than the literary braille. However, you say that this student is a fairly competent braille reader. Excellent!
If the student is at the pre-Algebra (possibly paced Algebra level - 2 years to take Algebra I) level or below, I can place him/her into that class, give them a braille book, and teach the math and the Nemeth Code as we go along. However, if they are at a more advanced level (Geometry, Algebra II, etc.), I can do one of two things:
Place them in a lower level math class as a review and teach them the Nemeth Code as we go along.
Teach a separate Nemeth Code class similar to the Computerized Nemeth Code Tutor.
I have always felt that Nemeth Code was best taught as the students were learning the math and the new symbols; however, one year I happened to have five students who were past Algebra II, had completed their math requirements, were planning to go to college, and were between mediums. They could all still use print by spending hours huddled under a CCTV and/or using other forms of magnification, but they realized this was not going to be efficient for them in college, and their vision continued to deteriorate. I warned them that once they entered college, there would be no lessons in Nemeth Code. I get calls from colleges around the state. No one is teaching Nemeth - including the Texas Department of Assistive and Rehabilitative Services (formerly known as Texas Commision for the Blind) - for the student over 22. So, I had all of these students in a class where they were learning Nemeth following a plan similar to the Computerized Nemeth Code Tutor, were learning how to read raised line or tactile graphics, and were reviewing for the TAAS, ACT, or SAT. The TAAS, ACT, and SAT people provide practice materials in braille (also on cassette with tactile graphics).
If the GED people have practice materials in braille (or cassette with tactile graphics), I would order these for the student in question. (We also have a braille copy of Contemporary's GED transcribed by APH in our LRC.) Then based on the level of mathematics contained therein and the mathematics level of the student, I would choose one of the methods outlined above. Don't expect this to be an overnight accomplishment. However, if the student is highly motivated and intellectually capable of learning the material, amazing things can happen.
Now, I suppose you're going to tell me that this teacher is "just" a regular math teacher and knows no Nemeth Code. If so, give him/her a copy of the Computerized Nemeth Code Tutor and wish her/him good luck!
An expert on braille literacy writes: A gentleman called me who is 52, adventitiously blind, and has been a braille user "off and on" for about 10 years, although he says he is quite slow. He has diabetes and prefers jumbo braille, but can use regular braille slowly. He has never learned Nemeth code. He is now in college and taking an algebra class. He is having a great deal of trouble with this class. He uses the computer a lot with speech (he doesn't have a refreshable braille display) but his book is not available on e-text.
He really likes using the cube slate because he can move the pieces around easily. He finds this much easier than having it on paper. He wanted to know if someone has invented something similar for Nemeth that he could learn to use. I think I had read somewhere that Tack-Tiles is developing a set for Nemeth? Anyone have any information about that or any other movable system?
Susan replies: I teach secondary mathematics at the Texas School for the Blind and Visually Impaired in Austin, and I have taught at the college level. I have consulted with blind math students and their instructors at both levels.
In my opinion, learning to read and write Nemeth Code is essential for being able to continue in higher mathematics for 99% of the blind population. Although there are a few people who can do higher mathematics mentally, most of us need to "see" the process, whether it be in print or braille. Since your gentleman is having a great deal of trouble and is using speech on a computer, I do not believe he falls within the 1% category. I am also disappointed that he doesn't have a refreshable braille display because this is the most appropriate use of technology to do algebra in my opinion. But with his tactual impairment, Nemeth on paper or on a refreshable braille display may not be appropriate.
TACK-TILES Braille Systems has developed two sets of Nemeth TACK-TILES(R). One is a more general set and the other is configured to teach calculus and chemistry. I did participate as a consultant in their development. Perhaps the Nemeth Tack-Tiles can meet both his concerns and mine. Contact Dr. Kevin Murphy for more specifics:
P.S. Gentle readers: I think these Nemeth TACK-TILES(R) would be very appropriate for elementary students and/or students with learning disabilities, as well as those who suffer vision loss in adulthood and/or may also lack fine tactual discrimination. Your thoughts and eventual experiences would be greatly appreciated.