Introduction:

General Guidelines for Teaching Primary Math to Young Braille Users

Generally, teaching math to visually impaired children with no other handicapping conditions will be similar to teaching math to their sighted peers. Visually impaired students will be introduced to the same concepts in the same order and at the same time as other students. Like these children, visually impaired students will require many hands-on experiences with manipulatives to really understand math. For children, both sighted and visually impaired, it will be important for them to learn to do the following in math:

communicate math ideas both orally and in written format
relate math ideas to the real world
develop reasoning skills
solve real problems

Purpose of the Program

The purpose of this program is to supplement the math program you are using with your mainstream students and/or to provide a core program on which you can build with your visually impaired student. Teaching strategies are suggested which will work with visually impaired children, but which can also be used with the whole class. Likewise, many of the manipulatives provided in the regular primary curriculum can be used with little or no adaptation for visually impaired children.

Scope of the Program

This program has been designed for use in the kindergarten and primary grades. Because schools may have children in ungraded programs for these grades and they use math programs from a variety of sources, it is possible that you will want to teach something to the visually impaired child that is not indicated for the grade on which he or she has been working. You may feel that your student has missed some concepts or that he or she is functioning above or below grade level in some areas. Feel free to check the skills lists for other grades for the topic on which you are working to find a similar lesson which you can use or adapt. To provide added flexibility, this program is not limited to a single grade.

Contents of the Program

The program contains the General Guidelines for Teaching Primary Math to Young Braille Users and eight Primary Math Units. Each of these parts is bound individually in booklet form. The Primary Math Units are as follows:

Matching, Sorting, and Patterning
Number Concepts
Place Value
Number Operations
Measurement
Geometry
Fractions, Mixed Numbers, and Decimals
Data Collection, Graphing, and Probability/Statistics

Although this program stresses the use of manipulatives, only a limited number of manipulatives are included with the units. Many of the lessons draw from manipulatives used with children in the regular classroom and give a number of easily obtained manipulatives that are readily available around the classroom or home.

Likewise, adapted aids, both from the American Printing House for the Blind and other sources are mentioned in this program but are not included with it. These can be purchased independently of the program. They include such things as clock face models, number lines, and abacuses.

This program tries to encourage the use of manipulatives as a basis for understanding the concept being taught and to provide additional activities and worksheets for reinforcement and practice. Information has been included for adapting regular print worksheets for use with visually impaired students and special worksheets will be made available and sold separately for the topics in this program.

Program Development

The concepts incorporated in this program are based on a compilation of approximately ten state curriculum guides, comparison with the Texas School for the Blind and Visually Impaired Academic Mathematics, a review of five commercial primary textbook programs including the National Council of Teachers of Mathematics (NCTM) Addendum Series, NCTM Principals and Standards for School Mathematics, and information gathered from related NCTM Convention Sessions. The use of these sources helped to determine which concepts are being taught nationwide, which concepts might be different for blind and visually impaired children, and which are being recommended by teachers of mathematics to give students a firm foundation on which to build.

Research findings related to the best methods for teaching mathematics to visually impaired children were used in the selection of methods and materials for this program. A complete annotated bibliography is included at the end of this General Introduction. It may be noted that many of the articles cited were written in the 1960's and 1970's when more funding was available for research. Since that time, the educational setting has changed from residential schools and resource rooms to full inclusion in the mainstream classroom and technology has blossomed and taken over. Nevertheless, many of these findings are still relevant for blind and visually impaired children building a foundation for mathematics today and deserved a second look.

People who served on the advisory committee for this project included both sighted and blind, experienced and educators new to the field from a variety of programs throughout the country involved in education of the visually impaired and/ or teaching mathematics. The following served as advisory committee members for this project:

Mr. Anthony Evancic, Math Teacher (Retired), Western Pennsylvania School for Blind Children, Pittsburgh, Pennsylvania
Dr. Philip H. Hatlen, Superintendent, Texas School for the Blind and Visually Impaired, Austin, Texas
Dr. William E. Leibfritz, Associate Professor, Central Michigan University, Mt. Pleasant, Michigan
Mr. Ted Lennox, Teacher of the Visually Impaired, Carr School, Lincoln Park, Michigan
Mrs. Mary Mitchell, Itinerant Teacher of the Visually Impaired, Louisville, Kentucky
Mr. Andrew S. Papineau, State Consultant and Member of the Research and Development Committee of the American Printing House for the Blind, Madison, Wisconsin
Dr. LaRhea D. Sanford, Supervisor of Low Incidence Programs, Nashville, Tennessee
Ms. Merry Vahala, 1st Grade Teacher of a Mainstreamed Blind Child, Brook Elementary, Byesville, Ohio

In addition to the guidance provided by the advisory committee, this program was also evaluated by teachers of the visually impaired and mainstream teachers who are currently working with visually impaired students in kindergarten through the third grade. Their evaluations and comments served as the basis for revisions prior to production of this program.

Mathematical Framework for the Program

This program is designed to help teachers make curriculum decisions and give them some effective ways to help their students attain the mathematical power needed both for every day use and for careers in the 21st century. The foundation of the mathematics needed for personal and professional use must be well laid in the early school years. It is the hope that this program will do just that.

For over a decade, we have seen sweeping new directions in mathematics education. The impetus for this change came from the poor performance of our children especially when compared with results from other industrial countries and from the demands of our expanding technological society. No longer are basic computational skills sufficient for either personal or professional needs. Business and industry are having difficulty filling positions with persons possessing adequate thinking, problem solving, and quantitative reasoning skills, as well as the ability to communicate these abilities. The changes needed in mathematics education must begin with our youngest students. The mathematics curriculum for these young children must involve much more than just counting. Our young students must be helped to understand mathematical concepts and relationships and develop mathematical reasoning and problem solving skills.

Too often, mathematics was viewed solely as being about getting right answers. Too rarely was it seen as an area for clear, creative thinking. Often, the rules, algorithms, and procedures of mathematics were learned without any real sense of why they exist. And too often, mathematics was taught via teacher explaining and student merely listening and reproducing. These prevalent concentrations on recorded computational topics, resulted in a curriculum very narrow in scope, and hence, narrow in usefulness. Emphasis on rote computational activities also led children to believe that the mathematics they were learning wasn't usually supposed to make sense. Students became merely passive receivers of rules and procedures rather than active participants in constructing their own knowledge.

This program will help students develop meaningful procedures and algorithms from sound concepts. In doing so, this program will help teachers stimulate their students to learn. They will be able to help their students construct their own mathematical knowledge and understanding.

Problem solving will undergird this program. Personal experience is paramount in developing good problem solving abilities and must be incorporated in this effort. One of the major reasons we teach students mathematics is to develop the ability to solve problems. Mathematics curriculums at all levels must have a problem solving emphasis. This is especially true for our youngest students. A challenge of this program is to make problem solving a central focus of all children's mathematical learning so that they can develop the abilities and attitudes necessary to become confident and successful problem solvers.

Mathematics is the key to opportunity. Our young children deserve our very best efforts to ensure them this opportunity!

Accommodations for the Visually Impaired Math Students

A few simple accommodations will be required for visually impaired students who may have gaps in their concept development, may do some things in a different way such as using braille for reading and writing, and may even excel at some things such as doing mental mathematics. The few simple accommodations required for the visually impaired child may actually benefit others in the class as well. For example, the visually impaired child may not be able to see the illustrations in a math book or on a worksheet so manipulatives may be used longer--the better to cover that all important and often shortchanged concrete first step also benefiting some of the struggling learners. Also, the abacus, an excellent computational tool for visually impaired children, can be used with the whole class since mathematicians are advocating that students learn a variety of ways to compute. Such accommodations will help the blind child, but they will also add interest and enrich the learning of the whole class.

Concept Development

Since so much of what is learned is normally learned through the sense of sight, students with limited or no vision often have gaps in concept development. For children with normal vision, much incidental learning is taken for granted. Common objects given to children to count may be unfamiliar to blind children. It cannot be assumed that blind children will know what a particular object is or how it is used. For this reason, it is always a good idea to discuss the objects used in the lessons with the children to clarify this information for the blind child.

Another example of a gap in the concept development of blind children can be seen when numbers are introduced. Most children start their formal education already able to recognize numbers. After all, they have seen numbers all around them--on calendars, rulers, cell phones, clocks, the house, the car's license plate, money, pages in a book, etc. Visually impaired children, on the other hand, may see numbers in braille for the first time at school.

To help overcome this problem, parents and teachers need to make opportunities to introduce, identify, and teach blind children Braille numbers. If children are to be able to find braille numbers and bring them to school or tell about them as their sighted peers do, they will have to have things in their environment with braille numbers on them just as normally sighted children do. Parents will need to provide braille and/or tactile markings on things at home. The teacher of the visually impaired will be able to help parents with braille markings and sources for relevant items are included in appropriate places within the program.

Children with severely limited vision may use vision for most tasks, but be so limited in what they see that they may be unaware of things that others see easily. Examples might be the circular window above the door in the church or the triangular shape of the roof on the house across the street. Blind children may not even know there is a window above the door or a roof on the house. Special efforts must be made to give visually impaired children opportunities to have as many firsthand experiences as possible.

When objects are not readily available for visually impaired children to touch, it will be necessary to improvise to teach the same concept. For example, if you are giving examples of shapes students can find in their environment, the visually impaired child will be more apt to find the circle around the doorknob and the rectangle around the light switch than the circular window and the triangular roof mentioned above. On the other hand the child may be able to find shapes in a window in the classroom or at home and examine the roof on a dollhouse or on one built with toy blocks to get the concepts of a window and a roof.

It is tempting as visually impaired children get older and have a good command of language to describe things to them or to check on their concept development by having them describe things for you. This, however, can be very misleading. Just because an individual can describe something does not mean that he or she understands it or has developed a real concept of it. Blind children can be very good at using what they have heard to make it sound like they know about something when they really do not. This phenomenon is often referred to as verbalism. To avoid verbalism, do not take any concept for granted, give blind children as many firsthand experiences as possible, and only use description as a last resort, not as the primary method of teaching children about the world around them.

Altering Lessons for Visually Impaired Students

As a teacher, your first inclination when the class is doing a task normally requiring vision, such as writing numbers, marking an answer, or drawing a figure, may be to have the blind child do it orally or skip it. Although this may be the fastest and easiest option, it may not be the best option for the visually impaired student.

Instead, consider what you want students to get out of doing the task. If an answer is the only thing they are getting, then use the easy option. However, most of the time, a careful analysis will show that students are getting much more than just an answer. They are improving their writing skills, learning to work independently, or really having to think about the figure they are constructing. If this is the case, it will be necessary to find a suitable alternative method to use for the visually impaired child to do the task.

For some tasks there are obvious alternatives. For example, the visually impaired student who cannot write numbers with print will write them in braille with a braillewriter or a slate and stylus. For marking answers, blind children can find the braille answer or raised line drawing which answers the question, put their nondominant index fingers horizontally under the answer, and using the finger as a guide, mark under the answer with a crayon, pencil or graphic art tape.

With less clear-cut tasks such as drawing (constructing) a figure, there are many more alternatives from which to choose. Some examples of materials which can be used are waxy string which sticks to paper, graphic art tape, yarn which sticks to a flannel board, rubber bands on a geoboard, and crayon on a paper covering a screen board. Several products such as the Sewell Raised Line Drawing Kit, the Tactile Marking Mat, Draftsman Tactile Drawing Board, and the Swail Dot Inverter are commercially available for just this purpose. Whatever method is chosen, it is important that the visually impaired child be able to use the materials with relative ease and produce an acceptable and tactually discriminable figure (circle, square, or other simple shape) as required. The final figure need not be perfect, but the visually impaired child should have thought through all of its important features just as the rest of his or her classmates did.

Using Mental Math

Although many people do mental math, most have probably never been taught, but just figured it out on their own. Doing mental math is something most people learn to do to some extent because it is easier than getting a piece of paper and a pencil and writing down a problem to work. Think how much more valuable this skill is to a visually impaired individual who may require an abacus or a slate and stylus and braille paper rather than an easily obtainable pencil and scrap of paper to work an everyday problem.

There are strategies which you can teach students to help them do mental math and, interestingly enough, mental math is another area of math which mathematicians are recommending for children who are not visually impaired as well.

First, after students have had ample experiences with concrete objects, illustrations, and tactile graphics, be sure that the students memorize the addition, subtraction, multiplication, and division facts. This will make doing both written and mental math easier and is a good mental exercise.
Next, emphasize the associative, commutative, and inverse properties of math and point out how students can use these properties to make shorter problems with easier to compute combinations.
Stress that there is more than one way to arrive at a correct answer and some ways are faster and easier than others. For example, to arrive at the total number of spots on a group of dominoes, one can count, add, multiply, or group them and use a combination of these methods to arrive at the same answer. Students need to experiment to find the most efficient method for themselves.
Teach students to round numbers and estimate correct answers and to see if their answers make sense. This is especially important for them to do when they are working with decimals and fractions.
When a person is using mental math to compute an addition problem or a subtraction problem involving more than one column of numbers, it is easier to go from left to right as is done on the abacus rather than from right to left as has traditionally been done on paper. To help students do this for example, when a problem involves regrouping from the ones column to the tens column, have the students learn to concentrate first on the tens, only looking at the ones to see if the total will be greater than or less than nine. Then, when the total for the tens column has been determined, think about the number of ones there are in the problem.

Although visually impaired children often have well developed memories and may excel at mental math, this should never be considered as a substitute for using braille or learning to do computation other ways. Blind people have many other things that they must remember so it is not fair to expect them to do long and difficult math problems without giving them the appropriate tools with which to work.

Using Braille for Math

Braille will be used instead of print for some visually impaired students. If this is the case with your visually impaired student, the teacher of the visually impaired will probably have the primary responsibility for teaching the visually impaired student the various braille configurations needed for math. Nevertheless, the mainstream teacher who is working with the student on math needs to know that all print math symbols can be made in braille and have some idea of what is involved in using braille for math so that she or he will better be able to work with the child.

Although the braille used for math is written by the student the same way as it is for reading and other subjects the child will be taking in the primary grades, the braille used for math is different. The braille used for reading and other subjects is called the Literary Braille Code; this code deals with written words and does not contain math symbols such as the plus and the equal sign. The braille used for math is known as the Nemeth Braille Code; this code contains braille equivalents for all the symbols found in print math and science.

Because the two codes are each composed of the same dot configurations, the braille user will have to learn different meanings for the same symbols and slightly different ways to write numbers in each code. For example, the dot configuration for "ing" in the Literary Braille Code becomes the "plus sign" in Nemeth Braille Code, and "1" in Literary Braille looks like this , but in the Nemeth Braille Code, "1" may look like this or sometimes just a single dot like this . Even terms for the same symbol can vary from code to code. For example, the symbol used to signify a number, may be referred to as a number sign in Literary Braille and as a number indicator in Nemeth Code.

If the student has some usable vision, he or she may opt to use it for math even though he or she uses braille for most things. Since interpreting illustrations by touch is like putting together a puzzle in your mind a piece at a time, the braille reader will find this to be difficult. Reading print for long periods of time may be slow and difficult for the visually impaired student who finds it easier to read braille. On the other hand, this same student may find that he or she can interpret math illustrations more effectively by sight rather than by touch. This will, of course, depend on the amount of vision the student possesses and the learning style of the student. If the student opts to use vision in such a situation, a print book may need to be made available. It is entirely possible that this low vision student will wish to have both a braille book and a print book to make optimum use of both media.

If you have the primary responsibility for teaching math to a visually impaired student who is using braille, or if you would just like to learn more about the braille math code, several references are available. (See resource list at end of this guide.)

Computation and Adaptive Devices

Unfortunately, computation tends to be difficult in braille. Imagine trying to do computation on a typewriter, and not being able to cross out or squeeze in numbers for regrouping. Primary students benefit from learning to manipulate numbers in braille for computation. Solving problems on the braille writer gives students practice in using math facts. It also enables both the teacher and the student to see where in the process an error was made. But because of the difficulty some students have in using braille, other tools and techniques are often used for mathematics with the visually impaired.

Cranmer Abacus

The most important tool for computation is probably the Cranmer Abacus. This is an adaptation of the Japanese abacus called the soroban. The adaptation has a felt backing which holds the beads in place while the visually impaired user examines and manipulates them. There are 13 vertical rows of beads with 5 beads in each row--one bead above the separation bar and 4 beads below. The bead above the bar has a value of five. The four beads below have a value of one each.

This device is small, requires no batteries, and enables the user to manipulate numbers within columns, to learn about place value, and to perform all the basic computations at a faster rate than that of a normally sighted person using a paper and pencil. For students with more limited finger dexterity, a Large Abacus can be provided. Use of the abacus builds better understanding of numbers and aids in the development of mental math.

Beginner's Abacus

The Beginner's Abacus is a simple counting frame with two vertical rows of nine beads each made to resemble the Cranmer Abacus and the Large Abacus with a black frame, large white beads, and a red background. The Beginner's Abacus can be used for counting, learning abacus terms such as "set" and "clear," and building a concept of place value with tens and ones.

Brannan Cubarithm Slate and Cubes

Because it is difficult to write and work problems in braille, visually impaired students may use the Brannan Cubarithm Slate and Cubes instead of a braillewriter or a slate and stylus. The Brannan Cubarithm Slate is a rubber grid into which small plastic cubes with braille on them can be fit. Each braille cube has one of five dot configurations on each of five of its sides. (The sixth side is blank.) By turning a cube in different directions, the numbers 0-9 can be formed in braille on the grid. Number and operation signs are omitted when working on this grid.

Finger Math/Chisanbop

Finger math or Chisanbop is more than just counting on your fingers up to ten. Rather, it is a system whereby the left hand represents the tens column and the right hand represents the ones. In finger math, the fingers represent single units and the thumbs, units of five. To indicate a particular number, the fingers on each hand, which correspond to the digits in the desired number, are lightly pressed on any handy flat surface. In this way, any one- or two-digit number from 0 through 99 can be shown.

Finger math is very closely related to use of the Cranmer Abacus with the thumbs having a value of five and the four fingers each having a value of one. Finger math can be used to do the basic operations of addition and subtraction when all the numbers involved are smaller than 100. Although finger math is very handy, it is limited to showing only two-digit numbers with tens on the left and ones on the right. Therefore, it is important to help students relate finger math to the abacus, which can be used for much larger numbers.

Talking Calculator/Computer

Talking calculators and portable note taking devices provide a quick and easy way for the visually impaired individual to do computation. Although this would seem to be the obvious adaptive device for the visually impaired person to use for computation, it is rather limited in what it is actually teaching the visually impaired student. Once the student has learned to associate certain keys with specific numbers and functions and can use the keyboard on the calculator or computer to enter desired numbers and functions at will, very little additional learning really takes place. The calculator does the work and indicates the correct answer orally. This is fine for adult computer users, both sighted and blind, who already understand the mathematical processes involved in getting the answers, but for young children who need to learn these processes and not just produce an answer, use of the talking calculator or computer should be postponed until children have had numerous experiences with manipulatives of all kinds and have learned to calculate in more meaningful ways.

Tactile Illustrations

Early primary books and workbooks are not always available in braille because they are too pictorial. These print books rely heavily on pictures and illustrations. Although some pictures and illustrations can be made with raised lines and embossed symbols (tactile graphics), making them is not enough. The blind child has to be able to interpret them, and that is no easy task.

Interpreting Tactile Illustrations

It is often difficult to determine what a tactile graphic represents since tactile illustrations cannot be examined as a whole but only as one small part that can fit under the finger at a time. Then the student must remember that part while moving on to the next small part until some image of the illustration takes shape in his or her mind. This is a little like seeing one piece of a puzzle at a time and trying to discover what the picture is.

As the blind child traces around the illustration with his or her fingers, he or she comes to intersecting lines that form different shapes, depending on which direction he or she tracks at the intersection. Is the shape a triangle or a square? Is this the whole boy or just his head? Or, is this just the sleeve of his shirt?

Interpretation of tactile illustrations is also difficult for a blind child since he or she may not be familiar with the object being shown. For example, other children have probably observed someone using a hammer and so will probably recognize it in a picture. The blind child may only know that a hammer is something that makes a noise but may never have had the opportunity to examine one. This could also be referred to as a gap in the child's concept development.

Even if the blind child has had experience with the object being shown, three-dimensional objects are not the same as their two-dimensional representations. The familiar wagon, which everyone knows has four wheels, a place to sit, and a handle for pulling, does not look anything like the two circles, rectangle, and line shown in the raised-line illustration. After all, perspective is a visual phenomenon.

Then there is the problem of making an item with which you know the blind child is familiar into a tactile graphic and finding that the child cannot identify it. Perhaps you fasten one sock to the page, two pretzels to another page, and three Q-tips to a third page to make a counting book. Glue used to fasten the sock to the page makes it feel stiff and hard and immovable and very unlike the sock that the child squashes together to put on his or her foot. Likewise, pretzels and Q-tips feel quite different when they are on a page and are traced with a finger rather than held in the hand.

Unless interpretation of the illustrations is the primary purpose of activity, it is a good idea to identify what the raised-line drawing is supposed to represent. Then the child can look for (or you can point out) features that will help the child identify that particular illustration. Without this kind of help with the interpretation, the child may very well conclude that an elephant has six legs if someone does not point out the trunk and the tail and how they might differ from the four legs in the illustration.

Even though it can be difficult, it is important for visually impaired students to learn how to interpret tactile graphics. In math, there are times when only an illustration will do. Since raised-line drawings will undoubtedly be used for geometry problems and other subjects later on, simple illustrations now should help to prepare visually impaired students for more complicated drawings later on.

Substitutions for Pictures

Because blind children often do not have access to primary books and tactile illustrations, the teacher will have to provide suitable substitutes. Such substitutes for math illustrations take many forms.

This may mean that the blind child will use manipulatives instead of the pictures for some things. Since mathematicians often feel that children do not have enough experiences with concrete objects before moving on to the semiconcrete, this could actually have a positive effect. It has been suggested that blind children need to work on concepts with a variety of manipulatives to help them generalize the concepts in much the same way that pictures do for the sighted child.

The abacus can provide another good substitute for the semiconcrete or picture stage of math development. Using the beads to show the numbers in a problem is certainly more abstract than using objects and less abstract than writing numerals. Thus, the abacus also offers another way to generalize the concept being learned and helps with this difficult to adapt stage of development.

One other often overlooked way of making simple illustrations for math problems is to use the braillewriter. The visually impaired student can make a tally mark (dots 4, 5, and 6) on the braillewriter for each item to be counted or added in a problem much as the other students do with a pencil. A full cell (dots 1, 2, 3, 4, 5, and 6) could also be used if it is necessary to show two different items. If it is necessary to indicate different colors or other characteristics, the initial letters of the colors could be used to represent these objects. For example, "r" could be used for red and "b" for blue. If more than one color begins with the same beginning letter, such as blue, brown, and black, another significant letter from each could be chosen, "b" for blue, "n" for brown, and "k" for black for example. Such illustrations are quick and easy to make on the braillewriter, are something visually impaired students can do by themselves, and actually serve in the development of algebraic thinking.

Making Tactile Illustrations

Teachers may even find it desirable to produce some tactile graphics themselves. Here are a few simple suggestions for teachers to keep in mind when they are making tactile illustrations for visually impaired students:

Finished tactile graphics do not have to look exactly like their visual counterparts; it is more important that they be understandable tactually.
Keep graphics as simple as possible. Use simple geometric shapes rather than more complicated ones for math problems. For example, have the child count outline circles rather than raised-line drawings of bunnies. If necessary, minimize difficulty of counting shapes by presenting them in an orderly fashion (e.g., rows and columns) versus a scattered arrangement.
Always strive to keep the number and types of lines, point symbols, and textures at a minimum within a tactile graphic to avoid unnecessary clutter. Simplify the graphic without compromising the intended purpose of the task.
If tactile elements are too close together, they may not be perceived as two different symbols. For this reason, separate symbols at least 1/8-inch apart, increasing to 1/4-inch depending on the tactile method and elements used. Shapes with sides less than 1/2-inch long may be difficult to recognize.
Choose tactile elements that are easily distinguished from each other by touch when placed in close proximity to one another. For example, maximize tactile contrast by intersecting a solid line with a dotted line, or using a densely dotted texture next to a striped area.
Dotted lines and rough textures are typically used to convey the most important features of a graphic.
Embossed lines are easier to follow than incised lines. However, incised (i.e., indented rather than raised) tactile lines are sometimes used for background grid lines so they don't compete with the embossed data lines plotted on a graph.
Sometimes it assists a tactile reader to place raised dots at the corners of a pentagon, hexagon, etc., when asked to count the sides of a raised shape, or simply identify it. Example:
Rarely are tactile pictures recognized or understood by a tactile reader without verbal or written support. Remember to add needed labels within the graphic itself that help to identify and orient the student to what is presented on the tactile page.
If possible, present tactile graphics in combination with 3-dimensional models, using both side-by-side, so that the student can better understand how a 3-dimensional geometric shape, such a cube, is shown in a 2-dimensional manner.
After preparing a graphic, close your eyes and use your fingers to see if you can tell what the illustration is supposed to show. If possible, test it out with another tactile reader before presenting it to the student.
For additional information on the design of tactile graphics, consult the "Tactile Graphics Guidelines" posted on APH's web site at http://www.aph.org/ edresearch/guides.htm.

With these things in mind, there are a number of materials that can be used to make tactile graphics. Some of the perennial favorites are gluing actual items such as pieces of felt or sandpaper or string on a page, using a tracing wheel to make a perforated line or a shape, graphic art tape, tactile image enhancer, or simply using glue or puff paint to outline shapes or make lines which are allowed to dry before they are given to the visually impaired student to examine. (Appendix G lists a number of materials that can be used to make tactile graphics.)

Using the Program

The General Guidelines are meant to familiarize the reader with ways visually impaired children, especially those with very little useable vision who will probably be using braille, can be taught math in the regular classroom. This booklet would be good for the classroom teacher and the teacher of the visually impaired to go over together as soon as the visually impaired student is assigned to the class. Together, they can discuss materials that the student might need and order accordingly. The classroom teacher can read the guidelines and talk to the teacher of the visually impaired about any questions she or he might have. This will help the classroom teacher be better prepared to begin math instruction with the visually impaired student.

The Primary Math Units address concepts normally covered in kindergarten and the primary grades. The Primary Math Units are:

Matching, Sorting, and Patterning
Number Concepts
Place Value
Number Operations
Measurement
Geometry
Fractions, Mixed Numbers, and Decimals
Data Collection, Graphing, and Probability/Statistics

By binding the units separately, the classroom teacher and the teacher of the visually impaired are encouraged to work together. Rather than including the primary math units in one overwhelming book that the busy teacher might lay aside and forget, eight less formidable units are provided. The teachers of the student who is visually impaired can get together periodically to discuss the primary math unit the class will be working on in the near future and make plans to provide any special materials or instruction which might be needed.

The units vary in emphasis and length depending on the grade level and the skills needed to be taught at that level. For example, the Number Concepts Unit is very large at the beginning levels and the Number Operations Unit is rather small; at later levels, this is reversed.

Some units are closely related to other units. The Number Concepts and Place Value Units are good examples of this on the beginning levels, and the Place Value and Number Operations Units are examples on later levels, particularly when regrouping is involved. When two units are closely related, it may be most appropriate to look at both units together.

Each unit contains the following:

an introduction to the unit
an overview of concepts taught in the unit
a list of materials used in the lessons
manipulatives to teach concepts
lessons/teaching strategies for each concept to be taught for each grade level
worksheets to reinforce concepts

Supplementing the Mainstream Curriculum

The sequence for teaching the units can be determined by the curriculum being taught in the classroom of which the visually impaired student is a part. Units were selected for inclusion in this program based on what is being taught in popular mainstream programs.

Although some concepts must precede others, many are interchangeable and can be taught in several different orders. For example, some programs begin with Matching, Sorting, and Patterning while others start with Geometry or Number Concepts.

Because order is important for the introduction of many of the concepts within units, concepts have been numbered to indicate this order. Each unit is identified by a letter representing the content area, followed by the grade level, followed by the objective number. For example: G K-1 represents Geometry, Kindergarten Level, and objective 1.

Teaching the Visually Impaired Child in a Self-Contained Setting

If you are working with a visually impaired child and you have no other math curriculum, this program could be used as a core, bare-bones curriculum. You may go through the program in the order the units are listed under the section titled Using the Program or you may use any other order of your choosing. Do not feel that you must complete one unit before moving on to the next; you can always return to a unit after working for a while on another one.

When this is the child's only math program, it will be necessary to provide manipulatives, additional worksheets and activities, and other teaching tools for the student to get the most out of it. A number of materials have been listed in the appendixes and in the teaching strategies to provide ideas to help you. If suggested materials are not readily available, feel free to make appropriate substitutions and additions.

The Importance of Being Flexible

The key to using this program is flexibility. This should not be considered a rigid program when it comes to sequencing either the units or the concepts developed within them. Do not feel that you must do everything in the order in which it is shown. Rather, consider these as guidelines upon which to build. Read them through and match them up with the program you are using. Use them to give you additional ideas for presenting similar concepts to your class, which includes a visually impaired student. Find concepts which may not have been included in your curriculum, but which might be beneficial to your students, particularly the one who is visually impaired. Try to work these additional concepts into your math curriculum.

Communicating with Family

Communicating has always been important in education. Now some states, districts, and/or schools request that parents spend a specified amount of time working with their children on homework each day. Whether or not that is the case at your work site, do encourage family members to work with the visually impaired child at home. Some math concepts are very easy to incorporate into daily activities such as eating, setting the table, or other everyday activities with the family. This is invaluable in emphasizing the idea that math is part of the real world and not just something done in school.

To encourage communication a sample letter giving a general overview of this math program is included following this section. Because you know the individual children and their families, this letter is best used as a sample for you to use to compose your own letter which better fits each child's individual family situation, and the way you are using the math program at school. As you introduce and use each unit in this program, share with the family activities that are taking place in the classroom and suggestions for reinforcing the concepts in the home.

Sample Letter to the Family about the Program

Dear Family,

This year in your child's math program, we will be learning about concepts in the following areas:

Matching, Sorting, and Patterning
Number Concepts
Place Value
Number Operations
Measurement
Geometry
Fractions, Mixed Numbers, and Decimals
Data Collection, Graphing, and Probability/Statistics

Notice that your blind or visually impaired child will be taught the same concepts as the other children are taught, but providing extra hands-on experiences will be especially important. These can be as simple as involving your child in cutting a sandwich in halves or fourths, measuring ingredients in a recipe, or shopping with a set amount of money. By sharing these experiences with your child, you will be modeling the importance and usefulness of math in daily life.

Throughout the year, you will be receiving letters describing what your child is going to learn in the upcoming math unit. There will be suggested activities listed that you can do at home to reinforce and support the math concepts your child is exploring. Many of these activities can be done easily with everyday items found around the house. Please take time to read these letters and spend time working with your child on these concepts.

In addition to completing the various activities listed in the letters, talk to your child about what he or she is doing in math class. Review with your child the math work that he or she brings home. Make sure your child completes any math homework or special assignments.

By continuing your child's math education outside of school, you are not only helping to support your child's growth in math, but also showing your child that math is all around us in our daily lives and that it is important to you. With this program, learning math should be a positive and rewarding experience for you and your child.

If at any time you have questions, please do not hesitate to contact me. Working as a team, we can build your child's understanding of math concepts. I look forward to working with you and your child this year.

Sincerely,

Teacher

Assessment

Assessment is a very important and necessary factor in determining whether or not a student has achieved the goal he or she is working toward. Some of the types of informal math assessment that can be done just as effectively with a visually impaired child as with sighted children are anecdotal notes, checklists, and portfolios.

Observations

Anecdotal notes or teacher-recorded observations are an informal way of tracking students' performances. These notes can be recorded on sticky notes, on index cards, on paper, etc. and can be referred to later as progress reports are being filled out. Notes such as these are a great way to really watch students in their natural state during their thinking processes and problem solving. Consider only observing 1 or 2 students during a lesson. You might also want to choose just one objective or concept to focus on per day or work period for the selected student.

Checklists

Checklists are a quick and easy way to assess students on how they are achieving the objectives of the lesson. This is a formal one-on-one way to get a closer look at what each child has learned and what he or she needs to work on in the future. The student's performance can be recorded at various times of the year and compared to see how the student is performing overall throughout the school year. A sample checklist that matches the master objective list for Kindergarten, Unit 1 is included following this Assessment section.

Portfolios

Portfolios are a collection of student work over time. They are meaningful for students because students can have some choice as to what goes in their portfolio and they can see their own progress as their mathematical knowledge has increased. This allows for self-assessment. Examples of what can go in a math portfolio are projects, worksheets, writings, recordings, drawings, game scoring sheets, etc.

In addition to portfolios allowing for self-assessment for the student, they also give parents and teachers insight into the child's success. For this reason it will be important to have all the material done in braille by the visually impaired child ink-printed so that print readers can access it, too. Then parents are able to get involved in knowing what their child is learning and how their child is progressing. Classroom teachers are able to gain knowledge of how the student works to solve problems (i. e. what strategies he or she uses). Also, teachers can look to see if the student demonstrates an understanding of certain concepts, uses correct mathematical terminology and symbols, connects math to other subject areas or daily life, and if the student has made progress. This hard copy evidence, such as what is kept in the portfolio, is a great reference tool to use at parent/teacher conferences and for writing progress reports. The child's teacher of the visually impaired should be able to ink-print the child's papers for the portfolio or arrange for someone else to do it.

Analyses

Without analyzing the assessment results, a student might be left behind as the scope and sequence of the math curriculum progresses. Assessment allows us to know if the student has achieved the learning objective or not. After it is determined whether or not the student really understood the concept, then we know whether to go on to the next concept with the student or to go back and continue on the concept he or she struggled with before moving on further. Closely assessing students is a crucial ongoing procedure that can give teachers the best understanding of what students are achieving.

If the blind child is not performing on the same level as his or her sighted peers, work with the child's teacher of the visually impaired to find a solution. The teacher of the visually impaired may know of a special technique or an adapted material that would help. Many of these materials are available for use by the visually impaired child through textbook libraries or from the American Printing House for the Blind at no cost to the local school district. These techniques and materials may be useful with other members of your class, as well as for both remediation and for enrichment.

Sample Checklist for Kindergarten

Kindergarten	Fall	Winter	Spring	Notes
MA K-1 Matches pairs of identical common objects
MA K-2 Match by size
MA K-3 Match by shape
MA K-4 Matches by color or texture
MA K-5 Identify the one object in a set of objects that is very different
MA K-6 Identify the one object in a set of objects that is different in only one way
MA K-7 Recognizes patterns
MA K-8 Describes patterns
MA K-9 Identifies the object or shape that does not belong in a simple pattern of two or three objects or shapes
MA K-10 Extends a simple pattern of two or three objects or shapes by repeating the pattern at least two more times
MA K-11 Creates simple patterns of two or three objects or shapes repeated at least two times

Appendices

Appendix A:	Commonly Available Manipulatives
Appendix B:	Foods to Use for Math
Appendix C:	Commonly Available Containers
Appendix D:	Sources for Braille Numbers in the Environment
Appendix E:	Math Products from the American Printing House for the Blind
Appendix F:	Other Adapted Math Materials and Their Sources
Appendix G:	Materials for Making Tactile Adaptations
Appendix H:	Materials for Thermoforming (Early Math Tactile Graphics)

Appendix A
Commonly Available Manipulatives

Balloons
Balls
Barrettes
Beads
Beans, various types and sizes
Bingo markers
Books
Bottle caps
Boxes (jewelry, paper clip, etc.)
Bracelets/Necklaces
Bubble-blowing wands
Building blocks
Buttons
Cake candles
Chalk
Checkers
Clothespins
Coins--Pennies, Nickels, Dimes, Quarters
Combs
Connecting cubes/links
Corks
Cotton balls
Cotton swabs/Q-tips
Crayons
Dice
Dollar bills
Domino tiles
Erasers
Flowers/petals (real or silk)
Forks
Gloves/mittens
Golf tees
Hats
Ice cubes
Jacks
Jingle bells
Keys
Knives, plastic
Lids, jar
Leaves
Lego's
Macaroni in a variety of shapes (pasta)
Marbles
Measuring cups and spoons
Nails
Napkin rings
Nuts (with or without bolts)
Nuts (with shells)
Ornaments
Packaging (peanuts)
Paint brushes
Paper clips
Paper rolls, empty (paper towels, wrapping paper, etc.)
Pencils/pens/markers
Pieces of paper
Pipe cleaners
Pizza circles
Playdough (worms, pancakes, balls, etc.)
Playing cards
Popsicle sticks
Puzzle pieces
Ribbons/yarn strips
Rings, jewelry
Rings, rubber canning
Rocks
Rubber bands
Scissors
Screws
Seashells
Shapes
Shoe laces
Shoes
Snap blocks
Soap bars/beads
Socks
Sponges
Spoons
Squeeze bottles
Straws
String/cord
Thimbles
Tickets
Toothbrushes
Toothpicks
Toy cars
Washers
Yarn

Appendix B
Foods to Use for Math

Candy canes
Cereal (Cheerios, Life, etc.)
Chewing gum
Cookies
Crackers (gold fish, saltine, etc.)
Fruits/fruit slices
Gingerbread men
Licorice sticks
Lollipops (suckers)
M&Ms
Marshmallows
Nuts (without shells)
Popcorn (popped)
Pretzels
Raisins
Seeds (pumpkin, watermelon, etc.)
Sugar cubes
Vegetables (peas, carrots, celery sticks, etc.)

Appendix C
Commonly Available Containers

Bowls
Cake pan
Cans and small containers
Cookie sheets--with sides for magnets
Cupcake holders
Cups
Egg carton
Gift boxes
Ice cube tray
Jars
Jewelry boxes
Lunch bags
Meat carton
Muffin tin
Paper cups
Pie pan
Plates Saucers

Appendix D
Sources for Braille Numbers
in the Environment

Automated teller machines (ATM's)
Bus schedules for local routes
Covers on fast-food, carry-out drinks
Elevator buttons
Labels on talking book cassettes
Menus
Page numbers in braille books and magazines
Room numbers, especially in government buildings and hotels

Appendix E
Math Products from the
American Printing House for the Blind

Matching, Sorting, Patterning	Catalog Number
APH Insight Calendar	5-18971-06
Chang Tactual Diagram Kit	1-03130-00
Classroom Calendar Kit	1-18970-00
Draftsman Tactile Drawing Board	1-08857-00
Focus in Mathematics	1-08280-00
Game of Squares	1-08430-00
Giant Textured Beads	1-03780-00
Graphic Art Tape	1-08878-00
Hundreds Boards	1-03105-00
Individual Calendar Kit	1-18971-00
Large Textured Block	1-03820-00
Peg Kit	1-03420-00
Puzzle Form Board Kit	1-03721-00
Seated Parquetry Set	1-03650-00
Sensory Cylinder Kit	1-03670-00
Shape Board	1-03710-00
Small, Medium, and Large Circles, Set II	1-08600-00
Sound Matching Board I	1-08710-00
Sound Matching Board II	1-08720-00
Work-Play Trays
Small Work-Play Tray	1-03660-00
Dividers for Small Work-Play Tray	1-03770-00
Large Work-Play Tray	1-03740-00

Number Concepts	Catalog Number
APH Insight Calendar	5-18971-06
APH Number Line Device	1-03480-00
Baseball Game	1-08100-00
Beginners Abacus	1-03180-00
Card Chart	1-03550-00
Number & Math Symbol Cards	1-03551-00
Classroom Calendar Kit	1-18970-00
Cranmer Abacus	1-03150-00
Digital Clock Model	1-03126-00
Focus in Mathematics	1-08280-00
Game Kit	1-08440-00
Giant Textured Beads	1-03780-00
Hundreds Boards	1-03105-00
Individual Calendar Kit	1-18971-00
Large Abacus	1-03170-00
Nemeth Code Reference Sheet	7-87500-00
Peg-A-Circle Set	1-08550-00
Peg Kit	1-03420-00
Sound Matching Board I	1-08710-00
Sound Matching Board II	1-08720-00
Textured Pegs	1-08930-00

Place Value	Catalog Number
APH Number Line Device	1-03480-00
Beginner's Abacus	1-03180-00
Cranmer Abacus	1-03150-00
Large Abacus	1-03170-00

Number Operations	Catalog Number
Analog Clock Model	1-03125-00
APH Number Line Device	1-03480-00
Baseball Game	1-08100-00
Beginners Abacus	1-03180-00
Braille 'n Speak	1-07320-01
Brannan Cubarithm Slate and Cubes	1-00320-00
Card Chart	1-03550-00
Addition (Math Drill Cards)	1-03552-00
Subtraction (Math Drill Cards)	1-03753-00
Multiplication (Math Drill Cards)	1-03574-00
Number and Math Signs (Drill Cards)	1-03551-00
Cranmer Abacus	1-03150-00
Digital Clock Model	1-03126-00
Focus in Mathematics	1-08280-00
Giant Textured Beads	1-03780-00
Hundreds Boards and Manipulatives	1-03105-00
Large Abacus	1-03170-00
Math Flash	D-19910-00
Multiplication and Division Table	5-82700-00
Nemeth Code Reference Sheet	7-87500-00
Peg-A-Circle Set	1-08550-00
Quick Pick Math: Addition	1-03570-00
Quick Pick Math: Subtraction	1-03571-00
Quick Pick Math: Multiplication	1-03572-00
Quick Pick Math: Division	1-03573-00
Quick Pick Counting	1-03574-00
Sound Matching Board I	1-08710-00
Sound Matching Board II	1-08720-00
Take Away Game	1-08900-00
Ten Spot Game	1-09820-00
Textured Pegs	1-08930-00

Measurement	Catalog Number
Analog Clock Model	1-03125-00
Brannan Cubarithm Slate and Cubes	1-00320-00
Classroom Calendar Kit	1-18970-00
Clock Face Sheets	1-03111-00
Digital Clock Model	1-03150-00
18-Inch Flexible Ruler	1-03050-00
English Measurement Ruler	1-03070-00
Focus in Mathematics	1-08280-00
Individual Calendar Kit	1-18971-00
Meterstick	1-03000-00
Metric English Measurement Ruler	1-03100-00
Nemeth Code Reference Sheet	7-87500-00
Sensory Cylinder Set	1-03670-00
Stick-on Tactile Ruler	1-03081-01
30-Centimeter Flexible Ruler	1-03670-00
Tactile Demonstration Thermometer	1-03020-00

Geometry	Catalog Number
Chang Tactual Diagram Kit	1-03130-00
Game of Squares	1-08430-00
Geometric Forms	1-03410-00
Geometry Tactile Graphics Kit	1-08841-00
Graphic Aid for Mathematics	1-00460-00
Nemeth Code Reference Sheet	7-87500-00
Peg-A-Circle Set	1-08550-00
Puzzle Form Board Kit	1-03721-00
Shape Board	1-03710-00
Small, Medium, and Large Circles, Set I	1-08590-00
Small, Medium, and Large Circles, Set II	1-08600-00
Textured Matching Blocks	1-08950-00

Fractions and Mixed Numbers	Catalog Number
APH Number Line Device	1-03480-00
Draftsman Tactile Drawing Board	1-08857-00
English Measurement Ruler	1-03070-00
Focus in Mathematics	1-08280-00
Fractional Parts, of Whole Set Game Kit	1-03290-00
Hundreds Board and Manipulatives	1-03105-00
18-Inch Flexible Ruler	1-03050-00
Meterstick	1-03000-00
Metric English Measurement Ruler	1-03100-00
Stick-on Tactile Ruler	1-03081-01
30-Centimeter Flexible Ruler	1-03031-00

Graphing Data Collection	Catalog Number
Bold Line Graph Sheets	1-04061-00
Brannan Cubarithm Slate and Cubes	1-00320-00
Focus in Mathematics	1-08280-00
Game Kit	1-08440-00
Graphic Aid for Mathematics	1-00460-00
Hundreds Board and Manipulatives	1-03105-00
Tangible Graphs	1-08860-00

Reference and Related Books from APH

Abacus Basic Competency: A Counting Method by Susan Millaway (7-00219-00)

FOCUS in Mathematics (7-42700-00)

Hands-On Experience with the Cranmer Abacus (Video) (1-30004-00)

Learning the Nemeth Braille Code, A Manual for Teachers by Ruth Craig (7-68653-00)

The Abacus Made Easy by Mae Davidow (4-00100-00)

The Nemeth Braille Code for Mathematics and Science Notation, 1972 Revision (7-87430-00)

Using the Cranmer Abacus for the Blind by Fred Gissoni (4-27110-00)

Appendix F
Other Adapted Math Materials and Sources

100 Board and Pegs (ILA, RNIB)
Attribute Block Activity Set (MA)
Attrinks--Shapes and Sizes (ET)
Beads and Laces (MA)
Checkers (MA, AME)
Clocks (MA, NFB, ILA)
Connect Four (LH, ILA, MA)
Discovery Trays (ET)
Dymo Tape Labels (LS&S)
Feel the Beat (MA)
Fill it Shapes (ET)
Games--Bingo, Connect Four, Monopoly, etc. (ILA, LH, LS&S, MA, NFB)
Jumbo Plastic Lacing Beads (ET)
Jumbo Tactile Mat Pattern Blocks (ET)
Magnetti Spaghetti (ET)
Measuring Cups and Spoons (MA, NFB)
Multilink 100 Pegboard (MA)
Multilink Math (MA)
Othello (LH)
Out-Of-Sight Poker Chips (LH)
Pedometer (AME)
Playing Cards (MA, AME, NFB, LH, ILA)
Protractor (MA)
Scale (AME)
Shape and Abacus Set (MA)
Sladecolour Indicating Buttons (ILA, RV)
Sock Clips (ILA, RNIB)
Sorting Shells (ET)
String-A-Bead Jumbo Foam Beads (MA)
Tape Measure (MA, LH)
Thermometers (MA, AME)
Tic Tac Toe (MA)
Token/Counters (ET)
Touch Game (ET)
Unique Shaped Poker Chips (LH, AME)
Watches (MA, NFB, ILA)
Yardstick (MA)

Sources & Abbreviations

Ann Morris Enterprises, Inc. (AME)
P.O. Box 9022
Hicksville, NY 11802
800-737-2118
http://www.annmorris.com

Exceptional Teaching Incorporated (ET)
5673 N. Las Positas Blvd
Suite 207
Pleasanton, CA 94577
800-549-6999
http://www.exceptionalteaching.com

Independent Living Aids, Inc. (ILA)
(ILA and Can-Do Products)
P.O. Box 9022
Hicksville, NY 11802
800-537-2118
http://www.independentliving.com

The Lighthouse Inc. (LH)
111 East 59th Street
New York City, NY 10022
800-829-0500
http://www.lighthouse.org

LS&S Group Inc. (LS&S)
P.O. Box 673
Northbrook, IL 60062
800-468-4789
http://www.lssproducts.com

Maxi-Aids (MA)
42 Executive Blvd.
Farmingdale, NY 11735
800-522-6294
www.maxiaids.com

National Federation of the Blind (NFB)
1800 Johnson Street
Baltimore, MD 21230
410-659-9314
www.nfb.org

Royal National Institute for the Blind (RNIB)
105 Judd Street
London
WC1H 9NE
020-7388-1266
www.rnib.org

Appendix G
Materials for Making Tactile Adaptations

From General Sources

Beads
Clay
Flexi-paper
Glue
Graphic art tape
Hot glue gun
Magnetic tape
Magnets
Paint, artist (in tubes with different tips to make different thickness)
Pipe cleaners
Playdough
Puff Paint
Sandpaper of various textures
Self-laminating sheets (You can braille on them and still read the print.)
Sewell Raised Line Drawing Kit
Styrofoam shapes
Textured materials with as much variety as possible
Thermo pen
Tracing wheel (sewing)
Velcro
Wikki-Stix
Yarn

From the American Printing House for the Blind	Catalog Number
Aluminum Diagramming Foil	1-04090-00
Braille Transcriber's Kit: Math	1-04100-00
Chang Tactual Diagram Kit	1-03130-00
Crafty Graphics	1-08844-00
DRAFTSMAN Tactile Drawing Board	1-08857-00
Embossed and Bold-Line Graph Paper (see catalog for various sizes)
Feel 'n Peel Stickers	1-08843-00
Geometry Tactile Graphics Kit	1-08841-00
Picture Maker	1-08838-00
Quick Draw Paper	1-04960-00
Swail Dot Inverter	1-03610-00
Tactile Marking Mat	1-03331-00
Tangible Graphs	1-08860-00
Teaching Touch	1-08861-00
Textured Paper Collection	1-03275-00

Appendix H
Materials for Thermoforming

Barrettes
Basic Shapes: circle, square, triangle, rectangle, diamond
Beads/string
Bottle caps
Buttons
Candy canes
Cheerios/cereal
Coins
Combs
Cookies
Crayon
Cups
Dog biscuits (bone-shaped)
Drinking straws
Gingerbread men
Heart outline
Keys
Lollipops
M&Ms
Nuts
Paper clips
Pencils
Popsicle sticks
Pretzels
Rings (for fingers)
Scissors
Screws
String
Toothbrushes
Zipper

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Hussey, S. R. (1981). Mathematical notation: The Halifax Math Code. Nova Scotia: Sir Frederick Fraser School. [Halifax Math Code]

Hussey, S. R., & Legge, L. (1956). The "Halifax Method" of arithmetical calculations. International Journal for the Education of the Blind, 6(2), 36-40. [Halifax Code] [number operations/computation]

Isley, A., & Anthong, E. (1981). Visually impaired learners: A handbook. Raleigh, NC: North Carolina State Department of Public Instruction. [curriculum]

Jackson, G., & et al. (1970). Developing mathematical concepts in visually handicapped pupils in secondary school. Cincinnati, OH: Cincinnati Public Schools. [number concepts]

Jolly, W. M. (1975). How can a blind person do math? Guide Dog Magazine, 10(3). [general]

Kang, Y. W., & Masoodi, B. A. (1978). Abacus instruction for moderately retarded blind children. Education of the Visually Handicapped, 10(3), 79-84. [adaptive devices--abacus]

Maddux, C. D., Cates, D., & Sowell, V. (1984). Fingermath for the visually impaired: An intrasubject design. Journal of Visual Impairment and Blindness, 78(1), 7-10. [adaptive devices--fingermath]

Mao, L. (1976). Efficacy and generalizability of mental abacus techniques in preparation of teachers of the visually handicapped. Dissertation Abstracts International, 37(4), 2113-2114A. [adaptive devices--abacus]

McCrimmon, S. (1974). Programmed instruction as a means of teaching blind children addition and subtraction on the abacus. Education of the Visually Handicapped, 6(3), 72-79. [adaptive devices--abacus]

Melrose, S., & Goodrich, G. L. (1984). Evaluation of voice-output calculators for visually handicapped users. Journal of Visual Impairment and Blindness, 78, 17-19. [adaptive devices--calculator]

Moore, M. (1973). Development of number concept in blind children. Education of the Visually Handicapped, 5(3), 65-71. [number concepts]

Morgali, R. R., & Lamon, W. E. (1976). Using the Papy-Lamon Minicomputer to teach basic addition facts and related concepts to visually handicapped children: A pilot study. Education of the Visually Handicapped, 8(2), 33-43. [adaptive devices--computer]

Morgan, J. M. (1975, April). Computer-assisted instruction for the blind and deaf. Cincinnati, OH: Cincinnati Public Schools. [adaptive devices--computer]

Murr, M. J. (1971). Frequency of geometric illustrations in mathematics, Louisville, KY: American Printing House for the Blind. [geometry]

National Council of Teachers of Mathematics. (2000). Principals and Standards for School Mathematics. Reston, VA. [general]

National Instructional Materials Information System. (1977). Learning disabled. Number concepts, primary grades. Columbus, OH: Ohio State University. [number concepts]

Nazarova, T. P. (1972). Some characteristics of the mental activity of partially-sighted school children. Defectologia, 2, 8-16. [mental math]

Nemeth, A. (1966). The Nemeth Code of braille mathematics and scientific notation 1965. Louisville, KY: American Printing House for the Blind. [Nemeth Code]

Nemeth, A. (1981). The Nemeth Code for mathematics and science notation. Louisville, KY: American Printing House for the Blind. [Nemeth Code]

Robicheaux, R. T. (1993). Mathematical connections: Making it happen in your classroom. Arithmetic Teacher, 40(8), 479-481. [measurement--money]

Rossi, P. (1986). Mathematics. In G. T. Scholl (Ed.), Foundations of education for blind and visually handicapped children and youth: Theory and practice. New York: American Foundation for the Blind. [number concepts]

Royal National Institute for the Blind. (1970). The teaching of science and mathematics to the blind. Report to the Viscount Nuffield Auxiliary Fund. London: Author. [number theory]

Rozanova, T. (1985). Conference on "Experience with Mathematics Instruction in the Lower Grades of Special Schools." Defektologiya, 2, 93-94. [number concepts]

Sharpton, R. E. (1977). An experimental study to measure the effects of the English Language Grammar method of teaching mathematics performance of the visually impaired. Dissertation Abstracts International, 38(3), 1206 (Order No. 77-78, 545). [curriculum]

Smith, B. F. (1967, August). Teaching of numbers: The modern approach to numbers for blind. Presented at ICEBY, Perkins School for the Blind. [number concepts]

Smyrnov, V. N. (1985). Microcalculator with magnifying glass for visually impaired school children. Defektologiya, 5, 73. [adaptive devices--calculator]

Staroscik, K. (1972, July). Modern teaching of mathematics in the school for blind children. Paper presented at the Fifth Quinquennial Conference of the ICEBY, Madrid. [number theory]

Steinbrenner, A., & Becker, C. (1982). Current status of abacus training in teacher education institutions. Journal of Visual Impairment and Blindness, 76(3),107-108. [adaptive devices-- abacus]

Steinbrenner, A. H., Becker, C., & Kalina, K. (1980). A survey on the use of the abacus in residential schools. Journal of Visual Impairment and Blindness, 74(5), 186-188. [adaptive devices--abacus]

Struve, N. L., & Cheney, K. M., & Rudd, C. (1979). Chisanbop for blind math students. Education of the Visually Handicapped, 1(4),108-112. [adaptive devices--fingermath]

Tapp, K. L., & et al. (1991). A guide to curriculum planning for visually impaired students. Madison, WI: Wisconsin State Department of Public Instruction. [curriculum]

General Guidelines for Teaching Math