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.
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.
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.
Relate various mathematical applications to student activities enjoyed by blind students as well as the sighted students -
Put various mathematical concepts to song or at least teach similar to an athletic cheer.
The FOIL method for multiplying binomials F - O - I - L: First, Outside, Inside, Last!!!!
Quadratic formula sung to the tune of Pop Goes the Weasel
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.
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.
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.
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.
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.
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.
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).]
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.
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.
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.