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II. 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.
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.
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!
A vision assistant asks: How important is it for our elementary kids to do a subtraction problem in the brailler (example: 3 digit subtraction with cancellation signs) if they are using the abacus? When you do a subtraction problem, on the brailler, do you have them solve the problem from right to left?
I'm going to answer your question from a secondary math teacher's viewpoint. I believe that elementary students need to be exposed to working addition, subtraction, multiplication, and division problems on the braillewriter in a spatial arrangement - not necessarily using cancellation signs - until they understand the concept. They should solve the problem very similar to the way a print student would - in the case of subtraction from right to left. Although the textbook or worksheet may give examples using correct Nemeth Code including cancellation signs, you want the students' calculation procedures to be easy and quick. Your students can always use the abacus to check their work on the braillewriter. All the while, they should be learning mental math techniques as well. Once the concept is learned, speed, accuracy and flexibility are more important, and we should see the student quickly progressing to the abacus and mental math, basic calculators, and eventually scientific calculators as they begin higher mathematics. If a blind student has never had to work a math problem in a spatial arrangement (on a braillewriter, with TACK-TILES, or other manipulative) and has only used an abacus, they will most likely have difficulty in algebra with the concept of adding, subtracting, multiplying, and dividing polynomials when presented in a spatial arrangement. This is especially true with division. You just can't manipulate variables on an abacus.
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.
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?
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 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:
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.
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
quadrilaterals. I select
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
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
manipulatives or ones
with velcro in a
confined space, to keep
things in place. Be sure
to show the student the
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
Furthermore, I show my
students examples of
test questions on
transformations from one
of the many TAAS
tests in braille -
produced by Region IV,
Houston, Texas. Region
IV has superb tactile
VIII. 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.
In the author's words, "Although graphing calculators are mainstays of most secondary math classrooms, it is important for all students to understand the concept of graphing on a coordinate plane before they move to the graphing calculator." This is especially important for visually impaired students, and Susan Osterhaus, math teacher at TSBVI, ensures that her students learn to do so manually _ they must physically plot points, graph lines, and find slope. Below are her answers to questions about how to teach this skill, and her suggestions for students, teachers, and parents.
Most academic blind students, even those with spatial orientation problems, are quite capable of graphing, and as one of my students exclaimed, "Not only can we do it, it's fun!" There are several tools they can use to do so:
The Graphic Aid for Mathematics, from the American Printing House for the Blind (APH), is excellent for graphing algebraic equations. It can also be used in geometry, trigonometry, etc. It consists of a cork composition board mounted with a rubber mat, which has been embossed with a grid of 1/2-inch squares. Two perpendicular rubber bands, held down by thumbtacks, can create the x- and y-axes. Points are plotted with pushpins at the appropriate coordinates. Points are connected with rubber bands (for lines), flat spring wires (for conic sections), or string (for polynomial functions). I like for my students to graph extensively, and they can do so incredibly fast on the APH Graphic Aid. In fact, many print students also like using it because it is fast, fun, and allows them to learn graphing skills in another modality. You can make your own graph board by affixing a piece of raised line graph paper (also available from APH) to a cork board and proceeding as described for the Graphic Aid.
If a student needs to turn in copies of graphs for homework, he can use Wikki Stix and High Dots on APH graph paper. This method can be quite expensive, however, and is very time consuming. It also tends to be more of a test of artistic ability than a demonstration of understanding of graphing concepts.
The ORION TI-34 talking scientific calculator (from Orbit Research) and the Accessible Graphing Calculator (from ViewPlus Technologies) are examples of more high tech solutions for graphing activities. I described them in a previous See/Hear article (Winter, 2002), but strongly recommend that students be able to graph manually as well. It is important for visually impaired students to be able to use a variety of tools, and know when to use each of them. For example, a former student decided to graph a quadratic function manually because it was "too easy to bother with the computer." Yet, he will use the AGC to graph an exponential function.
The APH Graphic Aid described above works well. Plot the points with pushpins and connect them with a rubber band when the boundary line is to be included in the solution (a solid line in print). Leave off the rubber band when the boundary line is not included in the solution (dotted or dashed line in print).
When graphing one inequality in two variables, I simply have my students place their hand on the shaded side. I check each graph as my students complete them. When graphing a system of two inequalities, the student places one hand on the shaded side of the first inequality. Then they place the other hand on the shaded side of the second inequality. Where the two hands overlap (including the boundary lines where applicable) is the solution. Pretty soon most of my students are able to handle three or more inequalities without multiple overlapping of hands. We even progress to linear programming problems involving four or more inequalities. In these problems, a bounded area with vertices is often found, and it is pretty obvious where the shaded portion (the solution) is located.
If a student-made, manually produced paper copy is required, the student could use APH graph paper attached to a corkboard. She could plot her points using stick-on high dots, puff paint, etc. and could form solid lines using Wikki Stix. She could actually use a colored pen, pencil, or crayon to color the shaded area of the solution. This would take much longer, however, and would be very labor intensive. It will be important to know the purpose of the assignment and the concept(s) being taught. A paper copy of a single function can be created on the AGC (it can't graph multiple functions on the same graph). Often, it is possible for a sighted person (teacher, peer, parent, teaching assistant) to make a print copy of the student's graph _ the visually impaired student graphs on the Graphic Aid and someone copies it exactly onto a piece of paper to turn in. You can also divide the Graphic Aid into 4 to 6 small, separate coordinate planes for multiple problems. If you have a digital camera, you could even e-mail or print a picture of the student's graphs. Better yet, have the student take her own photos!
Please be sure that visually impaired students are allowed to participate in all kinds of graphing activities and that they are supplied with the proper tools. I would rather see them become proficient using a rubber graph board because they will learn so much more with this method, and they can do so independently. Creative exploration should begin in the early grades and allowed to blossom. Remember, the beauty of a tactile graphic is found in the fingertips of the beholder. And there can be no more beautiful and meaningful graphic than one created by those very same fingertips.
Susan Osterhaus has been teaching secondary mathematics for 29 years at the Texas School for the Blind and Visually Impaired in Austin, Texas. She has a bachelor's degree in Mathematics, a master’s degree in Mathematics Education, and certifications in Secondary Math, English, and Teaching the Visually Impaired from the University of Texas at Austin. Susan has taught or consulted with VI students from grades 6 through college, at achievement levels ranging from 3rd grade to talented and gifted.
publicado por MJA