A VI teacher inquires: My first love is braille, but I am being challenged this year, as it is my first year in over 16 years that I am co-teaching Pre-Algebra to a blind student. Next week it is factor trees; I can solve that one okay but I won't ignore any suggestions either.

Susan replies: Now that scientific calculators are required for so much in the secondary mathematics curriculum, I am happy to suggest one kind of neat use for the abacus. (Note: Although my classroom scientific calculators are not able to prime factor a number, my Scientific Notebook computer software is able to do so.) My students use the "Osterhaus" method for prime factorization on the abacus. They tend to cringe at "factor trees;" however, I know that's the way they teach it in Pre-Algebra, or at least introduce it. Eventually, however (in my book at least) they show them the "repeated division" method. The Osterhaus method is simply this repeated division method done on the abacus. I usually just show people how to do it, and it's difficult to put it all into words. Nevertheless, I'll try.

Place the whole number to be factored at the extreme right of the abacus; we'll call this the dividend. Then, beginning with 2 (the smallest prime number), check whether each prime is a factor of the dividend. If it is, place this first prime number on the extreme left of the abacus and divide the dividend by the prime. Replace the dividend with your answer (quotient) which becomes your new dividend. Continue (putting each new prime that is a factor in the column to the right of the last factor and replacing the quotient with a new dividend) until you arrive at a quotient of 1.

For example, set the whole number 420 at the extreme right of the abacus (4 in the hundreds, 2 in the tens, and 0 in the ones column). Beginning with 2, you find that it is a factor of 420. Therefore, you place 2 on the extreme left of the abacus (trillions column) and replace 420 with 210 at the extreme right of the abacus. You try 2 again and find it is a factor of 210. Place another 2 directly to the right of the first 2 (hundred billions column) and replace 210 with 105 at the extreme right. 2 is not a factor of the new dividend, so you try 3. 3 is a factor of 105. Place 3 directly to the right of the 2nd 2 (10 billions column) and replace 105 with 35 at the extreme right. Is 3 a factor of 35? No, so we try the next prime of 5. Yes, 5 is a factor of 35. Therefore, we place 5 directly to the right of the 3 on the left of the abacus (billions column) and replace 35 with 7 at the extreme right. 5 is not a factor of 7, but 7 is. Therefore, place 7 directly to the right of the five on the left of the abacus (hundred millions column) and replace 7 with 1 at the extreme right. Since your quotient is now 1, you have completed the prime factorization. Reading from left to right, your factorization would be: 2 x2 x 3 x5 x 7 or 22 x3x5x7.

You could have all kinds of variations. If the number to be prime factored is quite large, you may wish to use two abaci - one for the dividend and one for the factors. Some students may need to have a space between factors, which again might require two abaci. If the number is extremely large, the student may wish to use a calculator for the repeated divisions and an abacus (abaci) for recording the factors. My students use this prime factorization method when they need to determine the Greatest Common Factor (GCF) or Least Common Multiple (LCM) for rather large numbers.

## UAbacus

This web site provides information on how to use a Cranmer abacus for computation.  The abacus is available from the American Printing House for the Blind. The UAbacus app was developed by Dr. L. Penny Rosenblum and the staff at the Office of Instruction and Assessment at The University of Arizona. The UAbacus app is now available for free download from the iTunes App Store. Download the UAbacus Flyer (PDF 357k).

## Courses Available to Learn Abacus

The Hadley School for the Blind offers distance education courses to legally blind persons, their family members, and blindness professionals or paraprofessionals who can read and understand courses written at the high school level. "Abacus I" is one of those courses. "Abacus II" is also available. Using the abacus a person can add, subtract, multiply, and divide whole numbers and decimals.

An "Abacus II" course is available to learn to compute fractions, percents, quantities, square roots, and negative numbers.

High School math (with High School credit) courses for blind students are also available in the following areas: "Essentials of Mathematics I," "Essentials of Mathematics II," "Mathematics I - General," "Mathematics II - Pre-Algebra," "Applied Mathematics," "Algebra," "Geometry," and "Doing It the Metric Way."

by Debra Sewell, TSBVI, VH Outreach

Parents of many children with visual impairments are familiar with "talking calculators" and understand how their child can use this adaptive device to aid him/her in doing math problems. However, there is an ancient device they may not be aware of that is very important for their child to be able to use. This device is an abacus and is an adaptation of the Japanese abacus. Most of you have seen an abacus somewhere in your life, but you may never have used one. For the child with a visual impairment the abacus is comparable to the sighted child's pencil and paper, and should be considered a fundamental component of his math instruction. Just like his sighted peers, the VI student should also learn to use a calculator. Total reliance on the calculator should be avoided, however, because 1) the calculator does not allow a child to learn problem-solving skills, 2) the VI child will not have a "backup" plan when the battery goes dead. Additionally, children who are deafblind and who may not be able to hear the voice of a talking calculator, may also benefit from using an abacus.

Tactual learners may find it easier to use a device like an abacus. Some VI teachers do not teach abacus until students know their number facts to ten. In fact, the abacus can be used without knowing number facts to ten when the counting method is used.

## How to Use the Counting Method

Similar to Chisenbop (a system of using fingers for calculating), the counting method uses rote counting as beads are moved toward or away from the horizontal counting bar of an abacus.

As compared to other methods of calculating on the abacus (synthesis, direct/indirect, secrets, number partners), the counting method involves only four processes. Consequently, this method is best for students with visual and multiple impairments who would benefit from using an abacus. These students will probably learn the four processes more easily than the many steps needed to complete calculations with other methods. To be successful using the counting method, students should be capable of rote counting and have the knowledge of the concepts "one more than" and "one less than."

## Abacus Counting Method

4/5 exchange = exchanging a 5-bead for four beads set in the same column

Example: When you have four beads set and need to add one more, you set the 5-bead above the bar in the same column as you clear the four beads and count "one."

0/9 exchange = exchanging beads equaling the amount of nine for a 1-bead in the column to the immediate left

Example: When you have the amount of nine set and need to add one more, you set a 1-bead in the column to the immediate left as you clear the nine and count "one."

49/50 exchange = exchanging beads equaling the amount of 49 for a 5-bead in the same column in which the four beads are set

Example: When you have the amount of 49 set and need to add one more, you set the 5-bead in the same column in which the four beads are set as you clear the 49 and count "one."

99/100 exchange = exchanging beads equaling the amount of 99 for a 1-bead in the column to the immediate left

Example: When you have the amount of 99 set and need to add one more, you set a 1-bead in the column to the immediate left as you clear the 99 and count "one."

These exchanges are reversed for subtraction and can occur in any column on the abacus.

Reprinted with permission from TSBVI.

If you would like to know more about using an abacus, please contact Debra Sewell at (512) 206-9183 or . She has additional information on how to teach using the Counting Method and additional practice problems. You may also wish to check out the Assessment Kit she has compiled which includes an informal checklist for abacus skills.

## Posing the Question

### Three VI teachers write:

#1. I am a teacher who wonders whether it is important or appropriate to teach abacus to my blind students. It is my contention that while the abacus can be a useful teaching tool, it is not a necessary one and teaching algorithms as they are taught in the classrooms in which the children are learning (with sighted peers) is more beneficial to them than teaching a different tool - i.e. the abacus. The calculator seems inexpensive enough to be a viable, appropriate and useful alternative to the abacus with its limited capabilities. Am I wrong? I would appreciate your input as well as an indication of which of the two (calculator or abacus) is more useful to those of you who are blind or severely visually impaired.

#2. At our school, we are investigating the use of an abacus as a tool for a blind student. There are philosophical differences in the use of this item. Could you offer any insights into pros and cons of its use? Also, could you direct us to information regarding this discussion? Any assistance you might offer will be extremely helpful.

#3. I have had a request from another TVI who would like opinions regarding ABACUS? She has a fourth grade braille student who is very intelligent, and is just getting into double and triple digit multiplication and long division. She is working on her Nemeth code skills as well. What are your opinions about using abacus as a learning tool? Are there very many of you teaching Abacus, and if so what age did you start teaching it? I know it all depends on the child and their skills, but any information, comments or positive examples, negative concerns, we would like your great input. The parents really believe that the abacus is ARCHAIC, and obsolete, and feel it is a waste of time for their child to learn abacus? Any comments and opinions, and input would be greatly appreciated. Thank you very much for all your help.

## Susan replies:

I really don't like to think of this as Abacus versus Calculator. I like having all the tools I can get.

Previously, calculators were not allowed on standardized mathematics examinations even for blind students - including the TAAS (required for high school graduation here in Texas), SAT, and ACT. (The TASP still does not allow calculators, and many blind students will need to master this test before being allowed to complete their college requirements.) Calculators were also not allowed on most classroom examinations as well. Therefore, blind students were at a distinct disadvantage if they did not have an equivalent to the sighted student's pencil and paper. In my opinion, using the braillewriter to compute long computational problems is way too time intensive for the high school or college student. (I am not talking about an elementary student just learning how to perform the basic operations.) I had a student in a Pre-Algebra class many years ago when I (like the rest of the world) did not allow calculators so that they would be prepared and able to pass the standardized tests that did not allow calculator use. This student did all of her problems on the braillewriter and was staying up until 2 AM doing my homework and needing to come after school to finish tests, whereas everyone else was easily finished in a reasonable amount of time. We both decided that she needed to learn the abacus and quickly! She was extremely motivated and learned in a matter of a couple of weeks. She was then the first student to finish her homework and tests; her self-esteem increased; and math became fun. The other students wanted to know what miracle I had performed.

Now, it is recommended to use calculators in all the math classes and on most of the standardized tests. In fact, some tests "require" a scientific/graphing calculator. My students all use calculators, and I am even collaborating on finding the best way to use scientific/graphing calculators. However, I still have a definite abacus "attachment." Although everyone is using calculators, the sighted students can still use paper and pencil, if they choose or need to, when electronic power fails (be it electricity, batteries, etc.). I believe the blind student should have a fast, efficient, small, portable, non-electronic way to do a quick computation as well, if they so choose or the TASP demands it. Some of my students are surprised when even I pick up an abacus to perform a computation instead of paper and pencil. It's also non-consumable. Furthermore, I like working fractions and doing prime factorization on an abacus - not so easy on a calculator.

In secondary, students needing to learn abacus are quite often also in need of learning Nemeth Code. Ideally, they could take an abacus class during summer school and learn their basic Nemeth Code symbols while reading and answering the abacus problems. The talking calculator might be used to "check" the answers. I use the TSBVI method found in the book: Rita Livingston, Use of the Cranmer Abacus (2nd Ed.), Texas School for the Blind, Austin, Texas, 1997. See http://10.65.20.48/curriculum-a-publications/. Rita’s book also contains the Counting Method (See Using an Abacus and the Counting Method).

The abacus can be too difficult for some students however, so the individual student needs and abilities must always be your primary consideration. However, before giving up, check to see if there is a better method of calculating on the abacus than the one you are presently using. Please read Debra Sewell's comments below to see the abacus from a former elementary teacher’s viewpoint.

Following her comments, please read replies from blind users of the abacus and other vi teachers to catch their perspective as well.

## Debra Sewell, TSBVI, VH Outreach replies:

### Using an Abacus and the Counting Method

Parents of many children with visual impairments are familiar with "talking calculators" and understand how their child can use this adaptive device to aid him/her in doing math problems. However, there is an ancient device they may not be aware of that is very important for their child to be able to use. This device is an abacus and is an adaptation of the Japanese abacus. Most of you have seen an abacus somewhere in your life, but you may never have used one. For the child with a visual impairment the abacus is comparable to the sighted child's pencil and paper, and should be considered a fundamental component of his math instruction. Just like his sighted peers, the VI student should also learn to use a calculator. Total reliance on the calculator should be avoided, however, because 1) the calculator does not allow a child to learn problem-solving skills, 2) the VI child will not have a "backup" plan when the battery goes dead. Additionally, children who are deafblind and who may not be able to hear the voice of a talking calculator, may also benefit from using an abacus.

Tactual learners may find it easier to use a device like an abacus. Some VI teachers do not teach abacus until students know their number facts to ten. In fact, the abacus can be used without knowing number facts to ten when the counting method is used.

### How to Use the Counting Method

Similar to Chisenbop (a system of using fingers for calculating), the counting method uses rote counting as beads are moved toward or away from the horizontal counting bar of an abacus.

As compared to other methods of calculating on the abacus (synthesis, direct/indirect, secrets, number partners), the counting method involves only four processes. Consequently, this method is best for students with visual and multiple impairments who would benefit from using an abacus. These students will probably learn the four processes more easily than the many steps needed to complete calculations with other methods. To be successful using the counting method, students should be capable of rote counting and have the knowledge of the concepts "one more than" and "one less than."

If you would like to know more about using an abacus, please contact Debra Sewell at (512) 206-9301 or . She has additional information on how to teach using the Counting Method and additional practice problems. You may also wish to check out the Assessment Kit she has compiled which includes an informal checklist for abacus skills.

## Blind Abacus Users Thoughts

### A blind abacus user replies:

The use of both is equally important. Abacus serves as a good place holder. It can be used for fractions whereas the calculator cannot. With the abacus, the students have a better understanding of adding and subtracting where with a calculator it is just typing buttons. They don't have to do anything- they don't have to even know the steps. They have an idea of what's going on paper. The calculator can be used on tests but calculators are sometimes bigger and dependent on an external power source.

### Another blind abacus user replies:

I respond to this post from the viewpoint of a person who is 46 years old and who has always been blind. I first learned to use the Taylor Slate and type in the fourth grade and thought the abacus was a wonderful improvement for doing arithmetic. We began to learn the Cranmer Abacus in the seventh grade and I remember the feeling of fascination that it was possible to solve an arithmetic problem from left to right on the abacus just as well as it can be solved from right to left as it is on paper or via Taylor Slate. The abacus also teaches scalars in that the top beads stand for units of 5.

I have used talking calculators, computers, and the abacus and I still keep a Cranmer Abacus in my desk because it is handy for quick arithmetic or for temporarily storing telephone numbers. I would go so far as to say that the abacus is something that probably should be taught to all children because it involves several mathematical concepts and it makes doing mental arithmetic easier.

"Newsweek" magazine recently had a letter from a math teacher who was critical of the use of calculators in schools because the children grew up with no concept of numbers and how they really work. I heartily second that idea. Calculators are not bad, but students should first learn what is really happening so that they will know when to trust those electronic answers.

I would say to definitely teach the abacus and use the electronic calculators after the students have a feel for arithmetic.

For those who may not be familiar with the Taylor Slate, it was a system that made it possible for blind students to work arithmetic problems and represent the numbers with pieces of movable type on a special board that held the type in 8-sided holes which existed in rows on the slate. The advantage was that one could work problems all day and not use up any consumable materials such as paper. The pieces of type, however, frequently got spilled and higher math operations were problematic.

## VI Teachers Thoughts

### A former VI teacher replies:

When I was a VI teacher I taught abacus. I know it is being taught in our residential school now. I do not think it is archaic, I think it is a very tangible way to keep track of the various steps in more complicated math problems. A person using an abacus properly is doing more thinking than those only using a calculator, in my opinion.

It's also a quick way of recording a phone number when paper and braille writing tools, or pens are not handy, and for keeping track of purchases while shopping in the grocery store

### A new VI teacher replies:

I have limited experience with VI kids---just two years now. But, I have several other sp ed endorsements and have taught k-12 kids with many learning problems. It seems to me that the abacus is an excellent tool for developing the concepts of place value, base ten stuff, and many numerical relationships. The NFB has a good chapter in their book for vision teachers. I haven't read it yet, but understand the "paper compatible abacus" section is great. I believe the process is to use the abacus and then write the answer on the brailler.

From my own experience, it has been helpful with a first grader that is still needing manipulatives. But, the Mathline products have been more helpful when the problems would involve using the "secrets" of the abacus to find answers to easy problems that first and second graders do. After all, we don't have the luxury of tailoring all the math problems to the ones that are the easiest on the abacus. The abacus has also been great to "back up" a teenager when she has had great difficulty with concepts that were taught in grade school-----but perhaps she missed or passed over at the time. For a teen, the abacus is really just a huge pile of manipulatives that they can carry in their pocket and not be a dork! In fact, the teachers at the high School are pretty fascinated with it.

### A friend of a blind user replies:

A friend of mine (when I taught at a school for the blind a 1000 years ago) learned abacus as a child. As an adult, she chose to use it over her calculator because she could do it faster (she had residual vision such that she could operate a calculator visually).

### Another VI teacher replies:

I have a 9th grade extremely low vision student who has always been very good at math (has a Type n' Speak on which he could do calculations,) but really has enjoyed learning to use the abacus. It is his favorite activity out of the many we do (he is also learning braille) I agree with other respondents - it teaches a lot of basic math concepts, place value, etc. Also, I have heard it is a good way to quickly jot down phone numbers, etc. It is just another "tool" for the tool bag, so why not have it?

### A former VI teacher replies:

Although it has been many years since I taught the abacus, I had to enter the arena. My favorite way was the old Chisombop method. I got ahold of some of the work books for pre-abacus activities.

If you are not familiar with the Chisombop method, it was a method of finger counting where the thumb equaled '5' and the digits were (well) digits. To indicate a number a child would 'press' his/her fingers and thumbs to the table or 'lift' them to void the number. My experience was that when children had a good concept of numbers and using their fingers and thumbs for math problems they could move to the abacus easier.

I also used the finger method when introducing/reinforcing new math procedures (like division, multiplication, etc.). The students (I worked with) seemed to be able to keep track of the new math concepts easier. (Probably, due to the multi-sensory learning experience, but it was long ago, and I wasn't so sophisticated that I could label it.)

### Another VI Teacher replies:

Tell those parents to think in their own terms. Just as the pen hasn't been made obsolete for sighted folks, the pencil and eraser hasn't been replaced by the calculator. The abacus isn't hard to learn, is extremely low maintenance, and reinforces mathematical concepts in young children. Would those parents want a sighted child of theirs learning operations on a calculator only? Besides, it's a great draw for the other kids in the class, especially when they get to the sections in the math course we use (in about Gr.4) when they have historical and cross-cultural units. My kids always get to demonstrate.

### Another VI teachers replies:

I am another big fan of the abacus. I have several students who were not taught the abacus in elementary school but learned only how to do math on the Perkins. These students are severely delayed in their math skills and their math concepts because they have so much difficulty just doing the basic computation lining the numbers.

I start teaching the abacus in Kindergarten or first grade whenever the other students begin writing numbers and learning number concepts. They start right off with writing numbers on the braillewriter and the abacus.

I don't understand how anyone can NOT want to teach the abacus. It is so much more efficient and practical. The abacus can go with a student anywhere, unlike a Perkins. I use to work with elementary age students as a mobility instructor and I took the abacus and we worked on math at the store with the abacus.

### Another VI teacher responds:

I have my students use abacus from 4th grade through 6th grade and then as needed from then on. If they don't get good enough at it to use it extensively in 5th and 6th grade then it is a lost cause because the sighted kids start being allowed to use calculators beginning in 7th and therefore the blind kids do as well. If they've gotten good at abacus and used it a lot prior to that then most have enough sense to realize there are times when it is just as useful and at times more useful than the calculator.

### A VI Teacher with a Fourth Grade Student replies:

I also have a very bright fourth grade student who is learning the same things. The student has always dreaded math and the parents put much pressure on her to excel. With the frustration of the time it takes to do the work on the braille writer, I decided to try the abacus. She learned it very quickly and just loves math now! She feels very successful without having to worry about the lining up of numbers and always gets every problem correct. The parents did feel that by using the abacus she was getting the easy way out and that she needed to learn math the "normal way" as well. I did a lesson with just the parents on the abacus to really show them how it works and to emphasize that the student was not "taking the easy way out", and was actually doing all the same work just writing it down in a different way. This really helped them to understand it more and they are accepting of it now. I think it's a great tool and definitely worth teaching to both student and parents.

### A teacher replies:

I believe children should have the abacus introduced (a) as soon as their sighted peers begin doing pencil and paper math, and (b) as soon as they understand basic number facts. That is, it does not make sense to introduce abacus multiplication until (or in conjunction with) introducing the concept of it being a form of multiple adding. So a student would use it to add 6 plus 6 plus 6 to verify that three times six is eighteen and work on the times tables that way.

### Another teacher responds:

In regards to using the abacus with children before the 3rd grade, it has been my experience that students need to have some concepts firmly in place BEFORE I introduce the abacus. They need to have a clear understanding of 1:1 correspondence, the difference between ones, tens, and hundreds, and it helps if they have firmly grasped addition and subtractions facts. These concepts are more easily and thoroughly taught using manipulatives, such as Unifix cubes, before even introducing the abacus. Some children master all of these quickly, often in the first grade; some before, and most by the end of second grade. If a third grader still doesn't have these concepts, the abacus will be tough. When I start the abacus with a young child, I begin with simple counting up to 100. Of all the abacus curricula I have tried, I have found the "counting on" method developed by one of Rita Livingston's college students to be the most concrete.

## Fun Way to Use the Abacus

### A Teacher writes:

I am a Braille/mobility teacher in an elementary school. Since the beginning of this year, I have begun working with the abacus with two of my students who are in the fourth grade. They have become very proficient with addition, subtraction and multiplication using their abacus and really enjoy doing math more than when they used to compute using their Perkins Brailler. With the abacus, they compute problems faster and have an easier time erasing and starting over, if they make a mistake. I believe that using an abacus has helped them to better understand the concepts of place value and decimals.

One day a week, we have designated for playing games such as abacus Jeopardy, hangman, or Snake, all teacher made or modified games. We've even taken to playing a human race on a hopscotch mat. The two student's start off on the same square and get to move ahead if they solve the problem they draw correctly. The object of the game is to get to the last square first. Whatever the game, math and the abacus can be fun and extremely useful to blind students.