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Nemeth Code: Frequently Asked Questions (1 of 4)

 

 

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The catalog of products for the visually impaired and hard of hearing.

LS&S specializes in products for the blind, visually impaired, deaf, and hard of hearing. Here you will find a great collection of low vision aids, hearing helpers, daily living aids, and information designed to help you or a loved one regain independence. Adjusting to life-altering changes can be difficult, but in the case of vision or hearing loss LS&S can help you find useful products that will make a difference in your life.

 

Orion TI-36X, Talking Scientific Calculator

Item #: 241002

Price: $249.95

This modified Texas Instruments calculator is based on the TI-36. It has all the mathematical features of the TI-36, but it has also been enhanced for greater usability. Mathematical functions include 127 scientific functions including one and two variable statistics and trigonometry. Usability features include special ergonomic design, high quality natural speech, speech playback when each key is pressed, repeat playback, rechargeable battery with 6 hours of operation, and a learning mode function to help locate and confirm keys. Includes AC adapter and charger, earphone, and instructions in large print, cassette, and disc for screen readers. One year warranty.

 

8 Digit Talking Calculator

Item #: 6638

Price: $13.95

Low vision calculator with tilt display features a high contrast, multi-colored keyboard and 5/8" LCD display digits. Calculator has standard mathematical calculations and is solar powered, with battery back up. Calculator measures 5" high and 6-7/8" wide.

 

Large Magnetic Numbers

Item #: 471005

Price: $12.95

These bright colored, extra large (2.5") letter and number magnets are a great way to introduce children to the alphabet and numbers. With their large size, children learning basic skills can have tactile feedback and can trace the shape of the letters with their hands. 42 pieces, either set. Math Symbols

 

Number Braille Blocks

Item #: 451015

Price: $18.95

These wooden blocks are a great way to introduce a child to Braille. They're also colorful and fun for the sighted child. Child can trace the print letter tactually or feel it in Braille. Alphabet has 27 blocks and includes numbers. Math set has 16 blocks. (Math Blocks)

 

Time Timer Audible 6 inch timer

Item #: 471014

Price: $34.95

Great audible and visible 8" timer is a wonderful timing and teaching aid. Helps teach the concept of time as well as time management. Red indicates the remaining time. Alarm sounds when time is up.

 

Wikki Sticks

Item #: 803

Price: $4.95

Bright, flexible, bendable pieces that adhere to almost any surface. Shape and re-shape them to demonstrate shapes of continents, letters, animals, or even basic circle and squares. Basic Package: 48 8" pieces in primary colors and black and white.


Collaborative/Inclusive Strategies

  1. 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.
  2. 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 pre-prepared or immediately after. Another alternative might be for a classmate to make a copy of their notes to share.
  3. 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.
  4. Relate various mathematical applications to student activities enjoyed by blind students as well as the sighted students -
    1. Put various mathematical concepts to song or at least teach similar to an athletic cheer.
      1. The FOIL method for multiplying binomials F - O - I - L: First, Outside, Inside, Last!!!!
      2. Quadratic formula sung to the tune of Pop Goes the Weasel
    2. 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.
  5. 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.
  6. 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.
  7. 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.
  8. There is an ongoing need for four-way 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.
  9. 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.

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 three-dimensional objects. 3-D problems are found in all levels of mathematics. They are often difficult for students with vision to understand, especially when trying to create 3-D objects in a two-dimensional drawing. Such a drawing, even when tactually raised, makes little sense without sighted "perspective." Yet, the textbooks continue to draw these 3-D 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 two-dimensional 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 3-D 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 three-dimensional graph. Most mathematicians would agree that it is impractical to use a two-dimensional 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 TI-8? series for the blind. The GRAPH-IT software program from Freedom Scientific does graph certain functions, but again, it is limited, and it is not a stand-alone 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 full-featured hand-held 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 self-teaching.) 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 two-dimensional 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 state-required 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!

 


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 x-intercepts 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 Graph-It (or the AGC) to graph the related quadratic function. Then, the zeros will appear as the x-intercepts. In other words, the real roots of the quadratic equation will be the values of x where the function crosses the x-axis.

For example: Graph y=x2-2x-3 (y=x^2-2x-3) to find the roots of 0=x2-2x-3 (0=x^2-2x-3). The graph crosses the x-axis at x=-1 and x=3. Therefore the roots of 0=x2-2x-3 (0=x^2-2x-3) 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 TI-82 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.

 


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 three-dimensional 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 2-dimensional 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 x-axis. He moves right or left along the y-axis, and now he will move up and down along the z-axis. 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 two-dimensional 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 z-axis into space or down (beneath the table) to illustrate going down the z-axis. 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 time-consuming. 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 user-friendly) 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 real-world 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 three-dimensional 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 x-axis. Then move 5 units along in the positive direction, parallel to the y-axis. From that point, move 1 unit along in the negative direction, parallel to the z-axis. You have arrived.

To graph a linear equation in three variables, let's graph 3x+2y-3z = 6. First find and graph the x-, y-, and z-intercepts. To find the x-intercept, 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).]


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.


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.

 


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 non-regular 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 home-made 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 semi-pure 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.

Please visit the Resources Pages for more information on any of the resources I have mentioned above.

Currently, research is in the works to make mathematics more accessible to the blind, and the following projects are just completed, undergoing beta testing, or still in the planning/prototype stage. For more information on their work in progress contact:

Design Science, Inc.
140 Pine Avenue, 4th Floor
Long Beach, CA 90802
Phone: 800-827-0685 (no technical support calls at this number)
Main: 562-432-2920
Fax: 562-432-2857
Contact: Neil Soiffer
E-mail:
Website: http://www.dessci.com

MathPlayer's Math-to-Speech Technology. MathPlayer 2.0 makes math in web pages accessible to visually impaired readers.

Dr. Neil Soiffer, Senior Scientist at Design Science, will be presenting a session at CSUN 2005 entitled, "Advances In Accessible Web-Based Mathematics." He'll discuss how the recent reauthorization of the IDEA (Individuals with Disabilities Education Act) and ongoing standards and technology development will make mathematical expressions on web pages and other media accessible to everyone. He will also be demonstrating how math in web pages can be made accessible today. The talk takes place noon Wednesday, March 16th, in Marriott's Boston Room. A session abstract is available from CSUN:

For more on math accessibility(http://www.dessci.com/accessibility/)


Dotless Braille
Contact: Susan Jolly
E-mail:
http://www.dotlessbraille.org

Nemeth Back Translator

They are almost finished developing open source software, BackNem, to back-translate all linear (but not yet spatially-arranged) Nemeth braille expressions to MathML. BackNem is written in Java so it can be installed on any computer. The generated MathML can then be displayed as print math by using any number of freely available browsers. Examples include Microsoft Internet Explorer in conjunction with the MathPlayer (see Design Science above) plug-in as well as Netscape.

They plan to make a beta test version of BackNem available starting in September 2004. Please watch Dotless Braille for details. This work is supported by the National Science Foundation under Award No. IIS-0312487.


gh, LLC
1305 Cumberland Avenue
West Lafayette, IN 47906
Phone: (765) 775-3776
Toll Free: (866) MY-3-DOTS [693-3687 ]
FAX: (765) 775-2501
E-mail: gh  
Website: http://www.ghbraille.com 

MathSpeak" Initiative
http://www.gh-mathspeak.com/index.php

A grant from the Indiana 21st Century Research and Technology Fund was awarded to gh, LLC in the Summer of 2004 to develop MathSpeak". gh was chosen by Indiana's Department of Commerce to research and develop a standard for the production of Digital Talking Book versions of Math and Science Books (using MathSpeak"), and a software player (the gh PLAYER") module that can properly render the math and science both aurally and visually. Specifically, several mathematics textbooks, one science textbook, and a standardized test involving math and science are being prepared in this format by gh and tested among print disabled students in a variety of settings.


Henter Math, LLC
Phone: 888-533-MATH (6284) or 727-347-1313
FAX: 727-302-9422
E-mail:
Website: http://www.hent ermath.com

Virtual Pencil

VPAlgebra is now available! Henter Math is pleased to announce the release of Virtual Pencil Algebra, computer software for interactive access to algebra for students who are blind or visually impaired. This standard Windows application presents the equations visually for the sighted teachers, and audibly for the blind students.



MathMonkeys, LLC
26 Church Street, Harvard Square
Cambridge, MA 02138
Phone: 617.497.2096
FAX: 617.497.2116
ht tp://www.live math.com
http://www.math speak.org

Math Speak. In MathEQ Expression Editor highlight an expression and MathSpeak This to hear your computer read the expression as a human would read it aloud.


National Institute of Standards and Technology ( NIST)
NIST, 100 Bureau Drive, Stop 3460,
Gaithersburg, MD 20899-3460
(301) 975-NIST (6478)
TTY (301) 975-8295
E-mail:
Website: http://www.nist.gov

NIST Prototype Tactile Visual Display

Computer scientists and engineers at NIST have created a tactile graphic display that brings electronic images to the blind and visually impaired in the same way that Braille makes words readable.


Tactile Dynamics, Inc.
110 Commerce Drive, Suite 210
Fayetteville, GA 30214
Tel: 770-716-9222
Fax: 770-716-9599
Website: http://www.wyfiwyg.com/ no longer active

TDI Full-Page Braille Display

Their patent was awarded in November 2004.  They have worked out most of the manufacturing issues and should have demo units available mid spring 2005.

Many innovations evolved as the development process progressed.  Originally, they anticipated a bdm (braille display module) 30-dots by 30-dots as a basic component out of which larger displays could be assembled.  A lot of time and little funds encouraged them to simplify the assembly steps further.  A standard industrial technology, which they discovered recently, permits the manufacture of displays as 1 unit as big as 24-inches by 24-inches.  They can consequently offer 2 initial models, text-only with 6-dot and 8-dot characters, 20 cells by 6 lines and 40 cells by 25 lines at retail prices $2600 and $5200 approximately.

The current design does permit braille text as well as tactile graphics.  Dot elements are spaced uniformly on the grid so this is possible.  They want braille product developers  to create the necessary driver software before we enter the tactile graphics arena.

They will be presenting at CSUN in March 2005 and anticipate a prototype will be available for show and tell.


Touch Graphics
330 West 38 Street Suite 1204
New York, NY 10018 USA
Phone: 212-375-6341
FAX: 646-452-4211
Contact: Steven Landau
E-mail:
http://www.touc hgraphics.com

TTT: Talking Tactile Tablet

TouchGraphics Company has begun work on a sophisticated Authoring Tool that will allow teachers of blind and visually impaired students to create their own talking tactile pictures for the TTT, a new computer peripheral device. This site will offer news on the product as it is developed, and will provide special information to members of the Teachers' Design Collaborative, a group of experts in the field who are participating in the Research & Design process. Check in regularly to learn of our progress, and to find out about an exciting opportunity to receive free materials in exchange for your participation in the development process.


ViewPlus Technologies, Inc.
1853 SW Airport Avenue
Corvallis, Oregon 97333
Phone: 541.754.4002
Fax: 541.738.6505
e-mail:
Website: http://www.viewplus.com/

The Accessible Graphing Calculator (AGC) is new and improved.

AGC is a scientific calculator that provides voiced feedback, and can even make those graphs accessible by audio. Extremely versatile it can import data from Excel ® and a host of other applications. Sighted users can use the access the AGC through a slick visual user interface. And, of course, it can print perfect tactile copies of graphs in seconds by printing directly to any Tiger Embosser. The AGC is accessible to anyone who can use a computer, regardless of ability, allowing the user to concentrate on math, not on learning the tools to access it.

Ink & braille - plus ink & tactile graphics. The Tiger TM Pro embosser now has the option to print ink text and emboss braille concurrently, and even overprint.

IVEO Software with Touch Pad - allows you to touch, hear, and see electronic documents simultaneously.

These reference sheets have been created to assist you in familiarizing yourself with the proper Nemeth code for the common symbols found in Algebra I, Algebra II, Geometry, and Set Notation. 

The following versions are designed for sighted users in Adobe Acrobat format. You must have the latest version of the Free Adobe Acrobat Reader. Once you have opened the file, you can print directly. The page is formatted in four columns consisting of the print math symbol, the braille dots, a description, and the ASCII equivalent.

The following versions are designed for braille users in Braille Formatted File (BRF) format. The pages are formatted with the proper Nemeth code symbol and a description. You can send the file directly to your embosser or open the files in your braille translation program and emboss from there. 

The following versions are designed for braille users in MegaDots format. You will need to have a MegaDots program on your computer for it to work properly. The links below will open a braille file that you can send to your embosser. The page is formatted with the proper Nemeth code symbol and a description.

The Set Notation reference sheet is a web page.

The following is a general math Nemeth reference sheet.

American Foundation for the Blind (AFB)
11 Penn Plaza, Suite 300
New York, NY 10001
Phone: 1-800-AFB-LINE (232-5463) or (212) 502-7600
E-mail:
The American Foundation for the Blind (AFB) is a national nonprofit that expands possibilities for people with vision loss. AFB's priorities include broadening access to technology; elevating the quality of information and tools for the professionals who serve people with vision loss; and promoting independent and healthy living for people with vision loss by providing them and their families with relevant and timely resources. AFB's work in these areas is supported by the strong presence the organization maintains in Washington, DC, ensuring the rights and interests of people with vision loss are represented in our nation's public policies.

In addition to its New York City headquarters and Public Policy Center in Washington, DC, AFB maintains offices in Atlanta, Dallas, Huntington, WV, and San Francisco. AFB is also proud to house the Helen Keller Archives and honor the over forty years that Helen Keller worked tirelessly with AFB to expand possibilities for people with vision loss.

AFB National Literacy Center (NLC)
American Foundation for the Blind
100 Peachtree Street, Suite 620
Atlanta, GA 30303
Phone: 404-525-4845, FAX: 404-659-6957
E-mail:  
brl-help listserv, DOTS (Development of Teacher Support) for Braille Literacy , and Braille Literacy Mentors in Training: The Next Generation workshops: way of sharing ideas among practitioners, consumers, parents, and others interested in braille literacy including Nemeth Code and tactile graphics.

AssistiveTech.Net
A diverse resource for assistive technology (AT) and disability-related information. Their searchable database helps you target solutions, determine costs and find vendors of AT products for:

  • People with disabilities
  • Family members
  • Service providers
  • Educators
  • Employers

Association for Education and Rehabilitation of the Blind and Visually Impaired (AER)
1703 North Beauregard Street
Suite 440
Alexandria, VA 22311
Phone: 703-671-4500
FAX: 703-671-6391
E-mail:
International membership organization dedicated to rendering all possible support and assistance to the professionals who work in all phases of education and rehabilitation of blind and visually impaired children and adults. 
AERBVI Publications
Computerized Nemeth Code Tutor
Strategies for Developing Mathematics Skills in Students Who Use Braille

BRAILLE AUTHORITY OF NORTH AMERICA (BANA)
Mathematics Technical Committee
Betsy McBride, Chair
Mary Archer, Board Liaison

Committee Members:

  • Anthony (Tony) Evancic
  • Susan Osterhaus
  • Helen McMillan
  • Jean Simpson
  • Allison O'Day, Consultant

Tactile Graphics Technical Committee
Lucia Hasty, Chair
Carol Morrison, Board Liaison

Committee Members:

  • Constance Craig
  • John McConnell
  • Allison O'Day
  • Diane Spence
  • Susan Osterhaus, Consultant

Please address all questions to the Chairperson of BANA, who will then see that they are forwarded to the proper technical committee.
BANA has established "technical" committees composed of transcribers, braille readers, and educators whose purpose is to update the braille codes, formats, or techniques. These changes, which may be suggested by readers, transcribers, and/or producers are carefully studied. Proposed new codes or revisions are submitted to other technical committees to avoid conflict with existing braille codes. The BANA Board issues final approval for adoption and dissemination.

BRL: Braille Through Remote Learning Homepage
Ron Broadnax., Computational Science Educator
The Shodor Education Foundation, Inc.
923 Broad Street Suite 100
Durham, NC 27705
(919) 286-1911 (Voice/TDD) (919) 286-7876 (fax)
E-mail: 

CALIFORNIA TRANSCRIBERS & EDUCATORS OF THE VISUALLY HANDICAPPED (CTEVH)
Braille Transcribers and Proofers:
Computer-Generated Tactile Specialist - Raphael Camarena,
Mathematics Specialist - Mary Denault
Hand-Drawn Tactile Specialist - Art Benitez

CPB/WGBH National Center for Accessible Media
WGBH Educational Foundation
125 Western Avenue
Boston, MA 02134
Phone: (617) 300-3400
TTY: (617) 300-2489
FAX: (617) 300-1035
Email:
Making Educational Software and Web Sites Accessible Design Guidelines Including Math and Science Solutions by Geoff Freed, Madeleine Rothberg and Tom Wlodkowski
The Access to PIVoT Project, January 2003

EASI - Equal Access to Software and Information for Persons with Disabilities
latest research in the field (especially pertaining to math, science, and engineering), excellent publications can be downloaded from here, also has a listserv.

Eisenhower National Clearinghouse for Mathematics and Science Education (ENC)
ENC Online is a K-12 math and science teacher center.   goENC.com will combine the best of ENC Online with useful new tools to save you time.

The Hadley School for the Blind
700 Elm Street
Winnetka, IL 60093-2554
Phone: 800-323-4238, FAX 708-446-8153
Email:
Offers a tuition-free home study course entitled Essentials of Nemeth which offers exercises in both reading and writing Nemeth Code and covers the basic arithmetic operations, fractions, some algebra, and a little bit of geometry.
Offers several math courses including: Mathematics Diagnostic Test, Essentials of Mathematics I, Essentials of Mathematics II,  Mathematics I: General, Mathematics II: Pre-Algebra, Applied Mathematics, Algebra, Geometry, Doing It the Metric Way, Abacus I, Abacus II.

Mathematics Internet Library
Free Internet Mathematics Books from Quick Notes, K-8 Sites, Specific Subject Sites, Advanced Mathematics Sites, Advance Placement Test Help, Advanced Mathematics and Tests from SOS Mathematics, free Computer Software, Special Purpose Sites, Improving Math Grades and Test Scores, and much more. 

The Math Forum at Drexel University
Website: http://www.mathforum.org
one of the most useful online resources for K-12+ math teachers. Forum features include: Ask Dr. Math, discussion groups, Internet newsletter, Teacher2Teacher, search engine, and more.

Dr. Peter B.L. Meijer
Building WAY 41
High Tech Campus 5
5656 AE Eindhoven
THE NETHERLANDS
E-mail:
The vOICe Math Functions
Accessible Graphing Calculator for the Blind

National Braille Association, Inc.
3 Townline Circle
Rochester, NY 14623-2513
Phone: 585-427-8260, FAX 585-427-0263
E-mail:
Mission: To provide continuing education to those who prepare braille, and to provide braille materials to persons who are visually impaired. (For example: Nemeth Code Reference Sheet, publications and workshops on Nemeth Code, tactile Graphics, and Nemeth translation software).

National Council of Teachers of Mathematics (NCTM)
1906 Association Drive, Reston, VA 20191-1502
TEL: (703) 620-9840 | FAX: (703) 476-2970
articles, curriculum, classroom and professional development resources, and a search engine. NCTM has also developed a set of principles and standards important for teaching and learning mathematics.

National Library Service for the Blind and Physically Handicapped
Library of Congress,
Washington, DC 20542
Phone: 202-707-9275 or 202-707-5100, or 800-424-8567
FAX: 202-707-0712
E-mail:
Administers a free library program of braille and recorded materials circulated to eligible borrowers through a network of cooperating libraries. Also provides the publication: Sources of Custom-Produced Books: Braille, AudioRecordings, Large Print (2001).

The NASA JSC Learning Technologies Project Information Access Lab
NASA Learning Technologies
Johnson Space Center Education
Where Innovation Meets the Classroom
The Math Description Engine (MDE)
The Math Description Engine Software Development Kit is freely available to and can be used by software developers to make computer-rendered graphs more accessible to blind and visually-impaired users. It has a simple API that lets you easily add alternative text and sound descriptions to your graphs. Version 1 of the MDE SDK generates text descriptions for 2D graphs commonly seen in math and science curriculum (and practice). The mathematically rich text descriptions can also serve as a virtual math and science assistant for blind and sighted users, making graphs more accessible for everyone. Use the Math Description Engine to make graphing calculators, data analysis programs, computer simulations, or webite displays more accessible.

MathTrax
MathTrax is a graphing software that is accessible to blind and vision-impaired students. It supports critical math, science and technology learning objectives, and makes teaching and learning about science and math more accessible, fun, and easy for everyone! It describes graphs three ways - with "smart" text descriptions, nonverbal sound descriptions, and custom graphics descriptions - to suite different learning styles. Teachers and students can use the Equation panel to evaluate equations, the Data panel to import and plot tabular data and the Simulation panel to model roller coaster physics and rocket launches. MathTrax is fully accessible to screenreader users. These users can conveniently and immediately access MathTrax's alternative text and sound graph descriptions.

Research and Development Institute, Inc.
Gaylen Kapperman, Director
P. O. Box 351
Sycamore, IL 60178
Phone: 815-895-3078, FAX 815-895-2448
E-mail:
Computerized tutorial for Nemeth Code and instructional strategies manual on teaching math to visually impaired students. RDI has produced an accessible Nemeth Code tutorial which operates on the Braille Lite. The tutorial can be downloaded at no cost from http://www.freedomscientific.com.

Science Access Project Homepage
John Gardner, Director
Department of Physics
Oregon State University
Corvallis, OR 97331-6507
Phone: 541-737-3278, FAX: 541-737-1683
E-mail:
The purpose of this group is to develop methods for making science, math, and engineering information accessible to people with print disabilities. These include: Tactile Display Methodologies, Audio/Visual Display Methodologies, Access to Object-Oriented Graphics, Research and Development of Major Software Utilizing Display-Independent Information and Multimodal Display, and Overview of Research and Development of New Hardware for Non-Visual Information Display

T. V. Raman
IBM Research
Almaden Research Center
San Jose, CA
Phone: (408) 927-2608 (Work), (408) 268-1660 (Home)
Email:
Inventor of AsTeR - Audio System For Technical Readings - a computing system that produces audio documents from electronic documents. It is especially adept at handling technical documents with complex content - math.

TIRESIAS
Information resource for people working in the field of visual disabilities
Contact: Dr. John Gill
Scientific Research Unit
Royal National Institute for the Blind
224 Great Portland Street
London W1N 6AA
United Kingdom
Telephone +44 20 7391 2371
Fax +44 20 7388 7747
Email
This website contains technical and price information on devices that are currently commercially available, and provides contact details of the manufacturers of the devices. The manufacturers or their agents have provided the data, and the authors have not verified the accuracy of their claims. Prices are quoted exclusive of sales tax, value added tax, shipping costs or import duties unless otherwise stated.
ITC Devices
Prices for products in the above-listed chapters are quoted in the currency supplied by the manufacturer. Use the universal currency converter to calculate the price in your preferred currency unit.

Download Duxbury version (2K)
Download BRF version (1k)

Roster method of representing a set:

A = {-1, 0, 1, 2}

,a .k .(-1, 0, 1, 2.)

Set builder notation:

B = {x | x is an integer and x > 1}

,b .k .(x | x is an 9teg] & x .1 1.)

The empty set (two versions):

 { }

.( .)

 Ø

_0

Subset:

Ais a subset ofB (subset) or

,a _

Ais a proper subset ofB (proper subset)

,a _

Union:

A union B

,a.+,b

Intersection:

A intersection B

,a.%,b

Other Math Pages

 

Nemeth & Adventitiously Blind High School Student

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.

Nemeth and College Algebra

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)

 

SET NOTATION

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.

  1. 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.
  2. 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.
  3. 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.
  4. 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:
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.)

(The Nemeth Code filter from MAVIS is now incorporated into Duxbury's DBTWIN as their LaTeX Importer)
Mathematics Accessible to Visually Impaired Students (MAVIS)
Chris Weaver, MAVIS Program Coordinator
New Mexico State University
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)

See the cheat sheet on set notation.

Nemeth and Advanced Math

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:

Science Access Project Homepage
John Gardner, Director
Department of Physics
Oregon State University
Corvallis, OR 97331-6507
Phone: 541-737-3278, FAX: 541-737-1683
E-mail:
Website: http://dots.physics.orst.edu/

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.

Repro-Tronics Inc.
75 Carver Ave.
Westwood, NJ 07675
Phone: 800-948-8453, FAX: 201-722-1881
E-mail:
Website: HTTP://WWW.REPRO-TRONICS.COM/

 

Nemeth and GED Students

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:

  1. Place them in a lower level math class as a review and teach them the Nemeth Code as we go along.
  2. 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!

"Moveable" Nemeth

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:

TACK-TILES Braille Systems.
P.O. Box 475
Plaistow, NH 03865-0475
Phone: 800-822-5845, FAX: 603-382-1748
E-mail:
Website: http://www.tack-tiles.com/

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.