Teaching Students with Sensory Impairments Teaching Activities in Science

(Dion, M., Hoffman, K., & Matter, A. at Videncenter for Synshandicap, Hellerup, Denmark; This handbook provided by

the Elmwood Resource Centre, Christchurch, New Zealand)

Teaching Mathematics

Elmwood Visual Resource Centre, Christchurch, NZ, and

Susan Osterhaus, Texas School for the Blind and Visually Impaired, Austin, Texas

A. Strategies (Wendy Montgomery, M.A.T., Trinity University)

A-1. Use multi-sensory techniques when introducing new math concepts

(Osterhaus, 2002b; ADE, 1996). Students with VI “need to read a math problem,

write it, listen to it, tactually explore through manipulatives, and when possible,

move their body and/or the manipulative through space” (Osterhaus, 2002b,

Collaborative/Inclusive Strategies #7).

A-2. Relate math to enjoyable activities and teach math concepts in songs

or chants (Osterhaus, 2002b). For examples, see Susan Osterhaus’ website at

http://www.tsbvi.edu/math/teaching.htm.

A-3. “Raised pictures, diagrams, and concrete objects are necessary to

develop concepts” (ADE, 1996, p. 15).

A-4. When teaching number sense, keep the concrete objects small so

that they can all fit within the student’s hand(s) or arms. Otherwise, the student

will have difficulty developing the concept of “three-ness”, for example (Hatlan,

2005).

A-5. There is in the USA a separate Braille for math and science notation

called Nemeth code (ADE, 1996). This code is especially important for higher

levels of maths, but should be taught in lower levels as well, in order to facilitate

smooth transitions into higher math. Teachers of the VI will be responsible for

teaching Nemeth code, but the math teacher should be aware of it and provide

all assignments, handouts and notes in a timely fashion to the VI teacher so that

they can be transcribed into Nemeth (Osterhaus, 2002b).

See information below for specific techniques to teach Nemeth code.

A-7. Help students connect math to daily life by starting a daily math

journal, in which they create and solve daily math problems (Krebs, 1995).

A-8. Students with VI can have difficulty determining which strategies to

use and developing a plan to solve the problem. Help students improve the

efficiency of their problem-solving strategies by discussing the steps they use to

solve different problems, why they choose a particular method, and what works

and does not work (Krebs, 1995).

B. Technology (Wendy Montgomery, M.A.T., Trinity University)

B-1. DOS and Windows provide a free math software program called

Triangles, to be used with students who are blind or vision impaired. For more

information and how to download, go to http://dots.physics.orst.edu/triangle.html

(Science Access Project, 2005) or Win Triangle at

http://tap.oregonstate.edu/WinTriangle/WinTriangle.htm (Stewart, 2004).

B-2. Some assistive technologies and materials for math include a

pegboard for tactile graphing, talking scientific and graphic calculators. For more

information refer to http://www.tsbvi.edu/math/graphing.htm (Osterhaus, 2002a).

Other alternatives include geometric shape kits (preferably 3-D and sturdy),

clocks with raised or Brailled numerals, Wikki Stix (kit), and Braille and large print

measuring devices (ADE, 1996).

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 -

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

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

ii. Quadratic formula sung to the tune of Pop Goes the Weasel b. 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! go to top

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. go to top

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).] go to top

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. go to top

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. Go to top of page

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.

Teaching Nemeth Code A new teacher of the visually impaired writes: How do you teach Nemeth? Susan replies: Here is an outline of how I believe in Teaching Nemeth Code

A. Long Term 1. Ideal learning environment - The braille reader learns each new

Nemeth symbol as the print math symbol is introduced in each sequential math course from elementary to middle to high school to adulthood from their math/vi teacher.

2. Next to ideal and more realistic - The braille reader learns each new Nemeth symbol prior to being taught the print math symbol in each sequential math course from their VI teacher. As the student matures, and if a braille textbook is provided, the student can anticipate the new symbol on their own by reading ahead. The student should also be encouraged to be their own self-advocate.

3. Newly blinded young adult or Adult who is new to Nemeth - a. Have the student begin to learn or review literary braille if

necessary. Then slowly introduce Nemeth as they take an abacus class. Following this, have them go into slightly more difficult math class(es) and continue progressing to the level desired.

b. Have the student fall back one semester or a year from their present math level. Let them review (for one semester or a year) the math concepts they know in print, while the VI teacher teaches them how to replace the print with Nemeth.

B. Short Term 1. Nemeth Code Class a. Teacher follows Craig or RDI tutorial sequence b. Supplement with excellent tactile graphics c. Use assessment release or practice tests in braille d. Personalize for their needs 2. On Their Own a. Resource books from APH b. Craig book c. Nemeth Code Reference Sheet in braille d. Hadley School for the Blind Course e. Local college course f. Online course g. Local tutors who are blind and know Nemeth

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. <sigh> 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. Go to top of page

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)

Checklist To Determine If a Graphic Should Be Brailled

• If the actual object is unavailable, consider a tactile graphic. • If the object is too small to examine by touch and recognize details, consider a

tactile graphic. • If the object is too large to examine, consider a tactile graphic. • If the object is dangerous to touch, consider a tactile graphic. • When it is necessary to show size relationship between objects, consider tactile

graphics. • When the student needs information from a map/figure/graph to participate in

classroom discussions, answer questions, etc., a tactile graphic should be done.

Checklist For Making Decisions About A Tactile Graphic

• Why is this picture/map/figure important? • What are the most important elements to communicate? • Who will use this material? o age group o mental and/or physical condition o students ability or experience with reading graphics • How will this figure be used? o with or without help from a sighted teacher o with other children who are sighted or blind o with actual concrete objects • Where will the material be used? o in a classroom setting o at home for leisure reading or games o as part of a test instrument o as an orientation map • How will the map be produced? o to be used for one copy, one time o to be used as a master from which many copies can made

Basic Principles For Preparing Tactile Graphics

• Make the tactile graphic as clear as possible. Always keep in mind the point of view of the braille reader. It is up to the producer to present the information in a clear, concise manner for the student.

• Know the important facts to be kept in mind when creating the graphic. • Determine if the original shapes and textures are necessary to convey the concept,

or can simple geometric shapes or braille signs be used to illustrate the concept. • Omit unnecessary parts of the diagram (i.e. unreferenced or irrelevant sections of

a map) so that the original shapes and textures can be presented on a larger and clearer scale.

• Keep in mind the knowledge level, skill base, and age level of the reader. Use age appropriate language.

• Determine if the text requires measurements to be made or an operation to be performed, or if the original shapes, textures and total form are necessary to convey the concept. If so, the lines and angles are reproduced retaining a proper scale.

• Remember to keep it simple; unnecessary information, clutter, may prohibit the student from gaining relevant information therefore making the graphic useless.

• Edit/proofread the graphic with your fingers, not your eyes, before showing it to a student. Beware, if someone says your graphic is "pretty" or "beautiful", take a second look, your student may not be able to understand it at all.

Report on Braille Adaptations of the Texas Assessment of Academic Skills BACKGROUND INFORMATION The Division of Student Assessment at the Texas Education Agency (TEA) facilitated a meeting on October 14, 1993 to review the current procedures for brailling and producing large-print versions of the Texas Assessment of Academic Skills (TAAS) tests. In attendance were staff representing the following agencies or groups:

• Division of Student Assessment, TEA • Texas School for the Blind and Visually Impaired • Services for Visually Handicapped Students, TEA • National Computer Systems Psychological Corporation • Teachers of Visually Impaired Students (School districts throughout Texas) • Braille Readers • Braille Specialists

MAJOR CONCERNS IDENTIFIED Major concerns were identified at the initial meeting, solutions were recommended and a plan of action was outlined. The three main areas of concern were in test administration, braille test item adaptations, and the acquisition/reporting of data. Test Administration: Meeting attendees provided the following information:

• Teachers want explicit guidelines as to how much or little assistance they can provide a student during the test.

• Teachers want a list of necessary materials prior to the day of the test. Teachers want practice materials available for sighted students to also be available in braille.

• Teachers want clarification on how much time a student can spend on a test. • Teachers want a toll-free help line for questions or problems on test day i.e., a

defective booklet, no list of materials received, etc. • Teachers want specific instruction as to what is allowable in the transcription of

the writing response from braille to print.

Braille Test Item Adaptations: The following issues relating to test item production were discussed;

• The types of visuals/graphics represented in braille.

• Test items require multiple scanning tasks, (i.e. large tables, long reading passages, etc.)

• Testing instructions/directions to the student. • Spatial concepts such as 3-D concepts and textures that are not appropriate to

braille. • Participation of VI teachers during the item development process.

Test Reporting Meeting attendees offered the following concerns:

• Availability of raw data for braille test results. • Access/availability of test results. • Affect of omitted items on test results

RECOMMENDED SOLUTIONS Test administration:

• Revise the general and special instructions to clearly define the parameters for teacher intervention during braille test administration.

• Provide a toll-free number in the instructions. • Provide braille measurement specifications.

Braille test item adaptations:

• Establish an interim standards committee for the convention of graphics and test item adaptations in braille for the TAAS and end-of-course tests.

• Involve specialists, in the field of vision, on item advisory review committees. • Involve TEA Special Education /VI staff in dissemination of information related

to the interim standards. • Establish an external quality review committee.

Test reporting:

• Outline access issues in the Coordinator Instructions manual. • Provide a method of coding braille and large print test documents.

PLAN OF ACTION As a result of the October 14, 1993 meeting a committee was established to review measurement specifications sample TAAS and end-of-course test items for accuracy, clarity, braille appropriateness, graphic representations, format, style and content. The first meeting of the Interim Standards Committee for the Convention of Graphics and Test Item Adaptations met on November 16 and 17, 1993Each sample test item was brailled in advance and provided to the committee members for reference during the item-by-item analysis.

The Committee met again in December 1993 with a final meeting in February 1994The goal of the committee was to complete the sample test item analysis and reach agreement as a group, on decisions for braille test adaptations for the Spring 1994 braille TAAS and end-of-course tests. Following is a report from the Committee outlining their recommendations.

Interim Standards Committee Recommendations

Assessment Specialists:

• Laura Ayala, Division of Student Assessment, Texas Education Agency • Nan Bulla, Diagnostician Texas School for the Blind and Visually Impaired,

Austin, Texas • Jenny Kile Russell, Division of Student Assessment, Texas Education Agency • Phyllis Stolp, Division of Student Assessment, Texas Education Agency

Braille Reading Professionals:

• Olivia Schonberger: Consultant for Student with Visual Impairments, Region XIX Education Service Center, E1 Paso, Texas

• Marilyn Williams: Certified Braille Proofreader, Region IV Education Service Center, Houston, Texas

Teachers with multiple years experience in administering braille tests:

• Lorinda Heslip, Teacher of Students with Visual Impairments, Klein Independent School District

• Pat Knox, Teacher of Students with Visual Impairments, Garland Independent School District

• Rita Livingston, Principal, Texas School for the Blind and Visually Impaired • Susan Osterhaus, Teacher of Students with Visual Impairments, Texas School for

the Blind and Visually Impaired • Christy Shephard, Teacher of Students with Visual Impairments, Cypress

Fairbanks Independent School District • Renee Shepler, Teacher of Students with Visual Impairments, Austin Independent

School District

Braille Test Specialists

• Priscilla Harris, Certified Braille Transcriber, New York Department of Education, National Braille Association

• Diane Spence, Braille Production Specialist, Region IV Education Service Center

The Interim Standards Committee for the Convention of Graphics and Test item Adaptations met to make recommendations on the appropriate adaptations for brailling test items from the Texas Assessment of Academic Skills (TAAS) and

end-of-course tests. All grade levels and subject area items were carefully reviewed with recommendations made by the committee based upon teacher input, student braille-reading needs, test protocol, and national standards for brailling test material. National standard formatting guidelines were followed for the majority of the print test transcription. The committee made specific decisions on individual test items which are outlined in the following sections of this report. There were, however, basic concepts discussed and recommendations made as follows: Omitted Items If a print item is not adaptable in braille, it should be omitted on the braille test and noted in the special instructions to the test administrator as well as referenced on the braille test. Order of Presentation The decision was made to maintain the print order of presentation as much as possible (question?-graphic? answer choices) in an effort to provide consistency for the student throughout the test. Previous versions of the test had been brailled showing the questions, followed by the answer choices, ending with the graphic. There was a concern that the answer choices are not always the last item presented on all questions and that this might cause some confusion to the student. Picture Descriptions The committee members felt very strongly that if the test included pictures, either as a focusing technique or to set the stage for the upcoming passage or question, that these pictures should be described. There was discussion about the additional reading that would be required of the braille student as compared to the sighted student because of the added picture descriptions. The recommendation was made to include a copy of all picture descriptions in the special administration instructions and instruct the student to either read the picture descriptions themselves or have the test administrator read them. Boxed Material The committee recommended that boxed print material also be boxed in braille even if the question and answer choices pertaining to the boxed material had to be moved to a new page. Special Instructions to the Student The committee recommended that when a table or graphic appears on a different page than the question, that the question be modified to indicate its location. For example: "...Use the graph below to answer question 13 on the next page." Graphic Preferences The committee had specific opinions about the types of graphics that were prepared for their review. A clear distinction was made as to the types of graphics that could be done using the computer graphics method and the ones that should be hand drawn. Committee members recommended that the embossed computer graphics method be used for simple shapes such as, squares, triangles, bar graphs, but not for more complex graphics requiring

multiple textures, circles, angles, X-Y coordinate planes, maps, shaded groupings, etc. Scanning Tasks Simplified There were several test items which involved scanning tasks. In print reading passages, words were underlined and the students asked, for example, "the underlined word in the passage means..." In braille, the paragraphs containing underlined words were numbered. The test item was modified to tell the student: "In paragraph (3), print page a24, the word means.." General Notes:

• If sample questions are unnumbered in print, the braille sample test item should begin in cell one with no number.

• When a picture is used as a focusing technique and is randomly placed on the page, the picture description should be placed after the centered heading of the story.

• If picture descriptions are included in a test, the following note should be placed on the transcriber's note page: "When a test item contains a picture, the picture has been described in braille. If you wish to have the description read aloud, ask your teacher."

• All words that are double capped or bold faced to show emphasis, will be italicized in braille.

• Charts and tables should be kept on one page, even if this means leaving large empty spaces on the previous page.

• Short passages and questions should be kept with answer choices on the same braille page, even if this means leaving large empty spaces on the previous page.

• Periods should be placed after all question numbers and answer choice letters even if there are no periods in print.

• If the print test instructs the student to "Mark your answer ." The braille version will say "Write your answer."

SUBJECT AREA INFORMATION: READING

• When a reading passage contains underlined words, the word should be italicized in braille. A paragraph number should be inserted, and enclosed in parenthesis at the margin before the paragraph containing the underlined word. A line should be skipped before the paragraph number in all instances except when a cell 5 heading precedes the numbered paragraph. In this case the cell 5 heading, will be followed by the number enclosed in parenthesis followed by the cell three paragraph entry.

• The underlined (italicized) word should fall on the same braille page as the number of its paragraph. If a reading question refers to an underlined word in the passage, the question should be modified to reflect the paragraph number and the print page containing the word (i.e. In paragraph (3), print page a24, the word ...)

• If the reading selection contains a table, chart or graph, the entire table, chart or graph must be on one braille page and the question referencing the table, chart or graph should be modified to say, On print page a53 the table...

Subject Area Information: MATH

• If answer choices are to be displayed in 4?corner style on an page, they should be in the following order: AB. CD.

• If the 4-corner style of placing answer choices is used, consider drawing lines separating the four areas on the braille page.

• All right angles on graphics need the right angle notation. • In brailling tables where the full braille cell "=" is used to represent an amount for

counting, place one braille space in between each full braille cell. • If the print table contains a key: e.g. "Each ? represents 10 puppies." The braille

equivalent "=" the braille equivalent = should be substituted for the print symbol and the key should be moved under the title of the table/graph and placed in cell 5If the print does not include this statement, enclose the statement in \\tn symbols and place in cell 7 with runovers in cell 5.

• When tally marks are used for counting, use underscore marks in braille should be used "_" with no space in between them unless they are shown in groups of five in print. If that is the case, a space should be placed in between each set of 5 tally marks.

• Number lines in tests for third through fifth grade will be hand drawn. • Number lines in tests for grades 6 through exit level will be done on computer

using the mathematical number line designations. If only one or two number lines appear in a test, insert the number line transcriber's note explaining all the symbols used, just before the question where the number lines are presented. If several number lines are found throughout the test, the number line transcriber's note and symbol descriptions should be placed on a special symbols page.

• For test items that instruct students to count the number of blocks shown (one hundred, strips of ten, and units of one), a transcribers note will be included stating the h stands for hundred, t for tens, and o for ones.

• Test items showing base ten blocks with shaded squares will be hand drawn with raised dots showing the shaded areas.

Subject Area Information: WRITING .

• A note on the transcribers note page will be included to, Numbers, without number signs, appear in the right margin on the line in which numbered items begin.

• When a writing passage contains underlined words, the words will be italicized in braille and the question number will be placed on the line where the underline (italics) begins.

Guidelines for Braille Test Adaptations Texas Assessment of Academic Skills developed by Diane Spence, Director, Braille Services Region IV Education Service Center Houston, Texas The Interim Standards Committee for the Convention of Graphics and Test Item Adaptations met from November 1993 through February 1994 to make decisions on the appropriate adaptations for brailling test items from the Texas Assessment of Academic Skills (TAAS) tests. All grade levels and subject areas were carefully reviewed and decisions made by the committee were based upon teacher input, student braille-reading needs, test protocol, and national standards for brailling test material. Standard national formatting guidelines were followed for the majority of the print test transcription. The committee made specific decisions on individual test items. These decisions and other basic concepts are outlined as follows:

Order of Presentation The decision was made to maintain the print order of presentation as much as possible, question-graphic-answer choices. This was done in an effort to provide consistency for the student throughout the test. Previous versions of the test had been brailled showing the questions, followed by the answer choices, ending with the graphic. There was a concern that the answer choices were not always the last item presented on all questions and that this might cause some confusion to the student.

Picture Descriptions The committee members felt very strongly that if the test included pictures, these pictures should be described for the student. If pictures were being used as a focusing technique or to set the stage for the upcoming passage or question, the braille-reading student should have the same information. There was a great deal of discussion about the additional reading that would be required of the braille student as compared to the sighted student because of the added picture descriptions. The decision was made to include a copy of all picture descriptions in the test administrators specific braille instructions and instruct the student that they could either read the picture descriptions themselves or the teacher would read it for them.

Boxed Material The committee decided that if material was boxed in print, it should be boxed in braille. They felt the setting of boundaries around material was very important, even if it meant the question and answer choices pertaining to the boxed material had to be moved to a new page.

Special Instructions to the Student Some test items referred to tables of information or graphics that were placed on one page and the question(s) on another page. In this situation, either the directions or the question was modified to tell the student where the graphic was located. For example:"...Use the graph below to answer question 13 on the next page."

Graphic Preferences The committee had specific opinions about the types of graphics prepared on the braille test. A clear distinction was made as to the types of graphics that could be done using the computer graphics method and the ones they wanted drawn by hand. The embossed computer graphics method was appropriate for simple shapes such as, squares, triangles, bar graphs, etc. This method was not appropriate for more complex graphics requiring multiple textures, circles, angles, X-Y coordinate planes, maps, shaded groupings, etc.

Scanning Tasks Simplified There were several test items that involved scanning tasks. In reading passages, words were underlined and the students asked, for example, "...the underlined word XXX in the passage means..." In braille, the paragraphs containing underlined words were numbered. The test item was modified to tell the student: "In paragraph (3), page a24, the word XXX means ..."

General Test Item Adaptation Recommendations: • If sample questions are unnumbered in print, the braille sample test item will

begin in cell one with no number. • The running head for all tests will include the abbreviated acronym for the test in

all caps followed by the initial cap grade level and the grade number. In all literary tests, the grade number will be in literary braille. If the math test is bound with a literary test, the running head grade number will be in literary braille. If the math test is transcribed in a volume by itself, or bound with other material which is transcribed using the Nemeth braille code, the grade number will be in Nemeth braille. (i.e.TAAS Grade 3)

• Tables and charts that are boxed in print will be boxed in braille, even if this causes the question and answers to fall on another page.

• STOP will appear in all caps at the margin when shown in print. There will be a skip line before the word stop and the next section of the test will start on a new page.

• Whenever possible put questions and answers on the same braille page. If, and only if, a graphic is so large that it takes up one entire braille page, then and only then should the question and answer be moved before the graphic. Whenever possible follow the print order of presentation: Question, graphic-table-chart, answer choices. If the question has two parts and only the first part of the question and the graphic, table-chart, will fit on one page, move the second part of the question with the answer choices to a new page.

• Modify the directions and/or the questions to tell the student where to look for the graphic or question. Add a transcriber's note if necessary. (i.e. Use the graph on the next page to answer question 5.; From the table on the previous page, which of the following best describes..., Transcriber's Note: Use the graph below to answer the question on the next page.)

• When a picture is used as a focusing technique and is randomly placed on the page, place the picture description after the centered heading of the story.

• All picture descriptions should be enclosed in "TN" note symbols and placed in cell 7 with runovers in cell 5.

• If picture descriptions are included in a test, the following note should be placed on the transcriber's note page: "When a test item contains a picture, the picture has been described in braille. If you wish to have the description read aloud, ask your teacher."

• All words that are double capped or bold faced to show emphasis will be italicized in braille.

• Keep charts and tables on one page, even if this means leaving large empty spaces on the previous page.

• Keep short passages and questions with answer choices on the same braille page. Never separate the passage from the question and answer choice, even if this means leaving large empty spaces on the previous page.

• Place periods after all question numbers and answer choice letters even if there are no periods in print.

• The word "DIRECTIONS" will be capped as in print and centered in braille. • The words "SAMPLE A, B, C" etc. will be capped as in print and centered in

braille.

Subject Area Information: READING • When a reading passage contains underlined words, the word should be italicized

in braille. Insert a paragraph number enclosed in parenthesis at the margin before the paragraph containing the underlined word. Skip a line before the paragraph number in all instances except when a cell 5 heading precedes the numbered paragraph. In this case present the cell 5 heading, followed on the next braille line by the number enclosed in parenthesis, followed on the next line by the cell three paragraph entry.

• An underlined (italicized) word should fall on the same braille page as the number of its paragraph.

• If a reading question refers to an underlined word in the passage, modify the question to reflect the paragraph number and the page containing the word. (i.e. In paragraph (3), page a24, the word ...)

• If the reading selection contains a table, chart, or graph, be sure that the entire table, chart or graph fits on one braille page. Modify the question referencing the table, chart, or graph to say: On page a53 the table ....

• Directions to the student will be blocked in cell 5. • Questions begin in cell 1 with runovers in cell 5.Answer choices begin in cell 3

with runovers in cell 7.

Subject Area Information: MATH • If the math test is placed in a separate volume from the other tests, use a Nemeth

number in the running head for the grade level. If the math is bound with literary parts such as reading, writing, etc., maintain the literary number throughout the test.

• Directions to the student will begin in cell 5 with runovers in cell 3. • Questions begin in cell 1 with runovers in cell 5.Answer choices begin in cell 3

with runovers in cell 5.If the question contains two parts, the second part of the question will begin in cell 7 with runovers in cell 5.

• Displayed expressions within questions will begin in cell 7 with runovers in cell 9 with no line skipped before or after the expressions.

• All answer choices need letter signs and punctuation indicators (i.e. ;,A_4). • If answer choices are to be displayed in 4-corner style on an page, they should be

in the following order: A. B. C. D.

• If the 4-corner style of placing answer choices is used, consider drawing lines separating the four areas on the braille page.

• All right angles on graphics need the right angle notation. • In brailling tables where the full braille cell "=" is used to represent an amount for

counting, place one braille space in between each full braille cell. • If the print table contains a key: "Each ? represents 10 puppies.", substitute the

braille equivalent "=" for the print symbol and move the key under the title of the table/graph and place in cell 5.If the print does not include this statement, enclose the statement in \\tn symbols and place in cell 7 with runovers in cell 5.

• When tally marks are used for counting, use underscore marks in braille "_" with no space in between them unless they are shown in groups of five in print. If that is the case, place a space in between each set of 5 tally marks.

• Number lines in tests for grades 3-5 will be done on foil. • Number lines in tests for grades 6 through exit level will be done on computer

using the mathematical number line designations. If only one or two number lines appear in a test, insert the number line transcriber's note explaining all the symbols used, just before the question where the number lines are presented. If several number lines are found throughout the test, place the number line transcriber's note and symbol descriptions on a special symbols page.

• Test items showing base ten blocks with shaded squares will be hand drawn with raised dots showing the shaded areas.

• All number lines are to be considered as spatial and will have a blank line before and after them.

Subject Area Information: WRITING • When a writing passage contains underlined words with question numbers, the

words will be italicized in braille and the question number will be placed in the right margin on the line where the underline (italics) begins.

• A note on the transcriber's note page will be included to say: "Numbers without number signs appear in the right margin on the line in which numbered items begin."

• All underlined words are transcribed in grade two braille unless the item is specifically asking about spelling and then the underlined words are presented in uncontracted, grade one braille.

Omitted Items If a print item is not adaptable in braille, it will be omitted on the braille test. A list of omitted test items is included in the specific braille instructions to the test administrator and referenced on the braille test.

Susan Osterhaus' Math Packet • APH flyers of products relevant to math instruction • American Thermoform Corporation Product Catalog featuring the Swell-Form

Graphics Machine II, Swell-Touch paper, EZ-Form and Maxi-Form Thermoform Machines, and Brailon.

• Association for Education and Rehabilitation of the Blind and Visually Impaired (AER) Publications and Resources Flyer

• Betacom Corporation flyer on the VisAble Large Display Scientific Calculator • Community Advocates, Inc. flyer on their standard and metric click rules • Computers to Help People, Inc. Accessible Mathematics and Science Books flyer • Duxbury Systems flyers including information on DBT WIN and MegaDots'

MegaMath Mathematics Translators, and using Scientific Notebook to produce advanced Nemeth code and braille graphics

• EASI (Equal Access to Software and Information) flyer • Exceptional Teaching Aids, Inc. flyer • Freedom Scientific Blind/Low Vision Group Product Catalog of products

relevant to math instruction • ghIIc flyer describing their math braille translation services, tactile graphics, and

other products and services • Goldstein Educational Technologies TI-83 Trainer flyer • The Hadley School for the Blind flyer describing their tuition-free math courses

including Nemeth and abacus instruction • Howe Press Catalog of Products including drawing and measuring devices for

geometry • HumanWare flyers on PIAF (pictures in a flash) tactile image maker and the

BrailleNote and BrailleNote QT with refreshable braille • Independent Living Aids, Inc. flyer on their educational resources for teaching

mathematics • Iowa State University Media Resources Center order form for videotapes on

Preparing Tactile Adaptations for Math and Science and Using Adaptations for Math and Science

• MacKichan Software, Inc. CD Booklet which includes a trial version of Scientific Notebook

• "Mathematics Their Way" by Mary Baratta-Lorton, Addison Wesley, 1995, outline of contents

• MAVIS (Mathematics Accessible to Visually Impaired Students) technical description of projects (including Nemeth Code filter for Scientific Notebook)

• Metroplex Voice Computing, Inc. flyers on their voiced mathematics products and voiced math demos on CD

• National Braille Association List of Publications, Order Form, and Membership Application

• Nemeth Code Reference Sheet by Gloria Buntrock • Orbit Research flyer on the ORION TI-34 talking scientific calculator

• Repro-Tronics Tactile Image Enhancement Products Catalog • Robotron Group flyers on the Leo Braille Display Scientific Calculator • TAAS and Algebra I End-of-Course Braille Release Tests Order Form • TACK-TILES(R) flyer • Tactile Vision Inc. flyer describing their services and product list of raised

graphics • "Teaching Math to Visually Impaired Students" a website by Susan A. Osterhaus,

outline of contents • Texas Instruments flyer on the TI ViewScreen Graphing Calculator Packages and

the Logan Electric flyer on the True-View Light Box • ViewPlus Technologies flyer on the TIGER Advantage tactile graphics and

braille embosser for network & personal use and ViewPlus Software flyer on the AGC (Accessible Graphing Calculator)

• Wikki Stix flyer from Omnicor, Inc.

This packet is available from: Susan A. Osterhaus 1100 W. 45th St. Austin, TX 78756 512-206-9305 email: susanosterhaus@tsbvi.edu