Future Reflections Special Issue: A Celebration of Braille

by Carol Castellano and Dawn Kosman

**Editor’s Note: **The following article is a partial reprint of the math chapter from the book, *The Bridge to Braille: Reading and School Success for the Young Blind Child *by Carol Castellano and Dawn Kosman. It has been one of the hottest selling books at the NFB Independence Market since it was originally published by the National Organization of Parents of Blind Children (NOPBC) in 1997. At $12 plus shipping and handling, it is also a bargain. The book may be ordered online at <http://www.nfb.org/nfb/Independence_Market.asp> or by calling the Independence Market at (410) 659-9314, extension 2216. (When you call, be sure to also ask about Castellano’s most recent book, *Making It Work: Educating the Blind/Visually Impaired Student in the Regular School*.) We have indicated in the text below where we have omitted graphics, illustrations, and the examples of simulated Braille and print math problems. We also omitted the Nemeth Code cheat sheet which is in the book. Here’s how Carol and Dawn explain the fundamentals of doing math in Braille:

**Nemeth Code**

Nemeth Code is the system of writing math in Braille. It was developed by a blind professor of mathematics, Dr. Abraham Nemeth, to make it possible to write any kind of mathematical notation, even the most complex, in Braille. Nemeth Code contains Braille symbols for every possible kind of mathematical and scientific notation. All math in Braille math books is written in Nemeth Code.

Nemeth Code numbers are shaped just like their literary counterparts, which blind children generally learn first, so the Nemeth numbers are easy to recognize and read. The Nemeth numbers, however, are formed using the lower part of the Braille cell, dots 2, 3, 5, and 6.

The Nemeth Code Cheat Sheet at the end of this chapter explains how to write the math signs your child is likely to encounter in elementary school. [Note: This sheet has been omitted from this reprint. It is, of course, available in the original book.]

[Illustration omitted.]

**Getting Started with Numbers**

Children will be introduced to literary numbers first, but they need to be familiar with reading and writing both literary and Nemeth Code numbers by the end of Kindergarten, so that when they begin addition and subtraction (which will be written in Nemeth Code in their math books and worksheets), they will be ready.

Children can begin learning to recognize numbers as preschoolers. If you feel your child is ready to learn numbers, you can make flashcards for him/her. Here’s how: Mark the top of each card with a line of Braille *g*’s so the child will know which way to hold the card. Then Braille both the literary and Nemeth number on the card. Use the number sign (dots 3, 4, 5, 6) before the numbers; there is no space between the number sign and the number. Write the number in print above the Braille.

[Illustration omitted.]

A number line taped to the child’s desk in school can also be useful. A number line for literary numbers can be made with a Braille labeler (Braille labelers can only form literary numbers, not Nemeth numbers). To write Nemeth numbers, you can use the labeling tape attachment for the Braillewriter or a slate with slots for labeling tape. You can also Braille the numbers onto regular Braille paper or a self-stick plastic sheet, such as a laminating sheet, rolled into the Braillewriter. Cut off the strip of numbers and attach it to the child’s desk.

**Math Readiness**

Just as there are many skills that lead up to reading, there are also readiness skills that lead up to a child’s being able to add, subtract, and perform more complex math operations. Learning to recognize the numbers and to count are important readiness skills, but in addition, your child will need lots of experience with other concepts:

- “One of,” “two of,” “a few,” “several,” “many,” etc.
- How things compare with each other--more vs. less, a lot vs. a few, longer vs. shorter, thinner vs. wider, heavier vs. lighter, etc.
- Understanding that the number one refers to one thing, two refers to two things; three refers to three, etc.

Give your child lots of practice. For example, make up number games. Count out the forks for dinner together--”One for Mommy, one for Daddy, one for you, one for Brother;” or when you’re cooking together, say, “Give me one potato and two carrots;” or as you are playing, say “I’ll put two blocks in the container; you put three in.”

Count many things around the house--how many barrettes are in the container, how many cups are on the table, how many toy cars are in the basket, etc. You can also work with Unifix Cubes, a wonderful math teaching aid available at educational toy stores that can be used to introduce or practice many math concepts. These interlocking cubes will stay put when the child stacks them. They are color coded and can easily be tactually coded, too, with small pieces of self-stick felt, cork, plastic, foam, Velcro, and Wikki Stix.

[Illustration omitted.]

**Beginning Addition and Subtraction**

After your child has learned to read and write the numbers, he/she will begin adding and subtracting. At this point, the Braille teacher and the classroom teacher will be in very close contact. (Many teachers schedule a regular weekly meeting time in which to exchange information.) The classroom teacher will alert the Braille teacher to the new concepts and skills that will be coming up, so that the Braille teacher can teach the new Braille signs--such as plus, minus, and equals--to the blind student *before* the lesson is introduced in class. In this way, when the classroom teacher teaches that lesson, the child will be familiar with the new signs and will be able to read them.

Braille users usually learn how to set up both horizontal and vertical math problems on their own papers as soon as this kind of problem is introduced in school (usually first grade). (Print users at this point are usually only writing in the answers on their workbook pages and do not begin writing out problems on their papers until a good deal later.) The Braille teacher will teach your child an efficient method to follow in setting up math problems and a reliable method for keeping the fingers in the correct column when adding numbers with two or more digits.

As your child becomes used to setting up and working problems using the methods taught, he/she will begin to work faster and faster. The steps involved will become automatic. Soon your child will have a quick, efficient method that produces accurate, correctly spaced, neat work. Early work of this kind also prepares your Braille user for setting up problems in higher math--algebra, geometry, calculus, etc.--later in life.

**Other Beginning Math Concepts**

When your child is learning to tell time, he/she can use a tactile learning clock and raised-line clock faces on paper, which are available from the American Printing House for the Blind. You can also purchase a teaching clock from an educational supplies store and adapt it for tactile use. Braille the numbers 1 to 12 using a Braille labeler; use self-stick Velcro to mark the five-minute (hour) lines; use small snips of Wikki Stix for the other minute lines.

Blind children have no particular difficulty learning geometric and fraction concepts. These concepts, however, may be difficult for a beginner to grasp using only the raised-line drawings that appear in Brailled math books. It is better if the child can hold and examine geometric shapes and fraction pieces.

Sets of geometric shapes and various kinds of fraction kits are available from the American Printing House for the Blind and in educational supplies stores and catalogues. The fraction pieces can usually be adapted with Braille labels. If possible, supply your child with several kinds of fraction kits--fraction pies, fraction bars, etc.--so that he/she can experience fractions expressed in various ways, just as sighted children do.

**What Do Math Examples Look Like in Braille?**

As you look at your child’s math work, you will see that math examples in Braille look very much like their print counterparts and are easily recognizable.

[Illustration omitted.]

Some of the skills your child will learn as he/she begins to do math on paper will be specific to Braille:

- How to read and write the various math signs--plus, minus, times, divided by, equals, is greater than, is less than, etc.--in Braille
- How to space the numbers and signs correctly
- When to use the number sign
- How to determine when the embossing head of the Braillewriter is lined up in the correct column
- How to keep the fingers in the correct column when solving the problem

The layout and the actual figuring of the math is EXACTLY THE SAME AS IN PRINT!

[The subsections in this chapter on Writing Horizontal Problems, Writing Vertical Problems, and Long Division have been omitted. For specific instructions along with simulated Braille and print examples, see the book *Bridge to Braille*, pages 101 -105.]

**Must My Child Write Out All That Math?**

As mentioned earlier, print users in the early grades generally write their answers right onto their workbook pages. They do not begin to write out the examples on their own papers until whatever year their math book series switches to hardcover books. In contrast, Braille users are usually taught to write out the examples on their own papers starting in first grade.

**Writing Answers onto the Workbook Page**

There are several reasons why Braille teachers recommend that blind students learn how to set up the examples on their own paper right away, instead of Brailling the answers onto the Braille worksheet. If a page is removed from the Braille workbook, it needs to be trimmed evenly before it will go into the Braillewriter smoothly. Squashed Braille can also be a problem. Although it is possible to roll the Braille math page into the Braillewriter, there is usually not enough space between rows of examples for the child to Braille in the answers. The answer for one problem often ends up looking like a number to be added into the problem below! Also, it can be difficult, especially for a beginner, to align the embossing head of the Braillewriter in exactly the right position under the example to be worked. For all these reasons, Braille teachers usually recommend that the child learn how to write out the examples.

This means that the first grade Braille user will be doing work that is a little more advanced than his/her print-using classmates, but by third or fourth grade, when the print users are also writing out their examples, the blind child will be comfortable with the skill and able to do it quickly.

If the child *is* going to write in the answers on the workbook page, someone will probably have to prepare the pages for the Braillewriter by evenly trimming the left edge. For an older student who can handle a binder, someone could trim the pages, punch holes on the left, and place the pages in order in a three-ring binder for the child to use independently.

**Using an Answer Sheet**

Another way to handle math is to use an answer sheet. However, using an answer sheet also requires more advanced skills than the average first or second grader is expected to have. Using an answer sheet might work well for older students, but it has two main disadvantages for young children. First, it is difficult for a young beginner to keep track of which example he/she is on in the book, especially if only the rows, and not every example, are numbered. To work successfully with an answer sheet, the child must be taught an efficient method for keeping the place. This could be done with a small piece of Wikki Stix (the child places the piece of Wikki Stix under the example being worked) or with the Stokes place holder (a metal board is inserted behind the page the child is working on; the child places a small magnet just below the example being worked).

The second disadvantage to using an answer sheet in the early grades is even more important. In first and second grade, children are learning about the ones column, the tens column, and the hundreds column. They are just learning that the columns in the example have a relationship to the columns in the answer. Seeing these relationships is especially important when two-digit adding and subtracting is introduced and when the class learns to do addition and subtraction with “regrouping” (your child’s school might call this “trading,” “renaming,” “carrying,” “borrowing,” or some other name). Using an answer sheet does not allow the child to see these relationships. If the child is unable to see the problem as a whole because the example is on one sheet and the answer on another, it might interfere with his/her understanding of the concepts. Although writing out the whole problem might take longer at first, it does ensure that the child sees the problem as a whole.

**Dictating Answers**

While the young Braille student is still learning how to set up math examples on paper, the Braille teacher may suggest that he/she set up only three or four examples and then read the rest and dictate the answers to the teacher.

**Using the Abacus**

Another method for handling math work is to use the abacus. Sometimes the abacus is introduced as a fast way to compute without writing the whole problem out. However, since modern math teachers emphasize the *process* by which students derive an answer and not just the answer itself, they often want to see the students’ work written out. (For a full discussion of the advantages and disadvantages of the abacus, as well as for instruction in “The Paper-Compatible Abacus,” see *Handbook for Itinerant and Resource Teachers of Blind and Visually Impaired Students*.)

**The Bottom Line**

Whatever method is decided upon, it is essential that your Braille user *get the same experience with doing math* that his/her sighted classmates get. It is also extremely important that he/she *learn efficient, reliable systems* for doing the math. As the child gets older and becomes more proficient at doing all kinds of schoolwork, he/she will decide which way to do the work, sometimes choosing to write out the problems, sometimes using an answer sheet, sometimes writing in the answers on the workbook page, sometimes using an abacus, and usually using a great deal of mental math. Flexibility is important. Evaluate and reevaluate the situation frequently and teach the child to do that, too. Give the child the tools to do the work in various ways and eventually he or she will decide which method is best for the task at hand.

**Should the Workload Be Cut Down?**

Sometimes Braille teachers suggest cutting down the number of examples the blind child will be responsible for. They say this based on the idea that it takes much longer to do math work in Braille. You might hear such statements as, “We generally suggest the work be cut in half for Braille students,” or “Assign just the even numbers or every other row.” Sometimes teachers say, “If she has demonstrated that she understands the concept, she shouldn’t have to do every example.” Unfortunately, this point of view often translates into *lowered expectations* for the blind student.

Many students can demonstrate an understanding of the work long before they finish every example. But this is true for sighted children as well as blind children. Blind students need and deserve as much exposure to the work as sighted students get. If the rest of the class is expected to complete the entire assignment, the blind child should do it, too. The goal is for the blind child to participate fully and equally in class.

Braille users are just as capable as print users at getting the job done! It may at times take a Braille user longer to complete certain assignments, but in general, the Braille user can handle the normal volume of work, especially if he/she has been taught efficient methods for doing the various tasks. Think about the future--in order to hold a good job, your child will need to be able to complete the work assigned!

If a child is taking an excessive amount of time to complete assignments, of course his/her general well-being must be taken into account--time for play and relaxation is important, too. You might find that it would make sense to shorten certain assignments for a time. Look at the whole situation. Try to make a good decision. And work at getting your child up to speed!