Calculators and Calculating Devices
Topic Tree | Home

Following are some items relating to calculating devices as discussed in the history of mathematics.

Contents of this Page
The Abacus
Calculators in the Classroom
The Slide Rule
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 


The Abacus

 

Counting rods were first used in ancient times to quickly perform arithmetic. The counting rods could be transported easily and made calculations accessible to many people.

The illustration above shows how the digits were expressed in counting rods. The rods were inverted for every other place, so a horizontal bar followed by a vertical bar would be 11

Some claim the origin of the abacus to be Babylonia where people drew marks in a sandbox. Sandboxes gave way to trays with lines and eventually grooved trays with balls.

Grooved trays were introduced into China by the Han Dynasty period (206 B.C - 220 A.D.) During the Ming Dynasty (1368-1636) the rod abacus was developed.

One source says the Egyptians developed rod and bead abacuses as early as 500 B.C. With the Chinese and Japanese counterparts emerging in the second century A.D.

The Chinese rod abacus, suan phan, uses a series of rods containing seven beads. The beads of each rod are divided into a group of two (heaven) and a group of five (earth) by a bar.

The Japanese rod abacus, soroban, is similar except only one bead appears on each rod above the bar, followed by five below.

There are also several web sites which allow you to try your hand at calculating on an abacus. The following two links offer Chinese and Japanese versions of an abacus. Chinese ,   Japanese

The abacus provides students with many advantages including an understanding of the base-10 number system. It also helps to promote mental math.

Many adept students of the abacus can figure problems with great speed and accuracy. In November of 1946, the abacus was positioned against the latest electric calculator in a contest of speed. The abacus won 4 out of 5 tasks, only losing in straight multiplication. The full account can be found reprinted at The Abacus

Addition, subtraction, multiplication and division are all possible on the abacus. For instructions on arithmetic functions follow these links: Addition, Subtraction, Multiplication, Division.

Contributed by Sara Collins


References.

  1. http://sddt.com/files/library/94headlines/DN94_11_24/STORY94_11_24_05.html who used the source Computer Industry Almanac, 8th Edition by Karen Petska (Juliussen) and Egil Juliussen
  2. The Abacus http://www.ee.ryerson.ca:8080/~elf/abacus/ Story reprinted from "The Japanese Abacus, Its Use and Theory", by Takashi Kojima. Charles E. Tuttle Company Inc. 1954, reprint 1987.
  3. Royce & Associates CPA firm http://www.gol.com/abacus/abacus.html
  4. Discover the Abacus http://www.users.dircon.co.uk/~damold/soroban/index.html who used the following sources: The Abacus: Its Use and Theory by Takashi Kojima. Published by Charles E Tuttle & Company. Advanced Abacus: Theory and Practice, also by Takashi Kojima. Published by Charles E Tuttle & Co. 1963.
  5. http://www.lalc.k12.ca.us/laep/smart/fti/aboverhead.html
  6. http://www.soroban.com.sg/
  7. Tomoe Soroban Company http://www.soroban.com/ and museum http://www.soroban.com/museum/index_eng.html
  8. http://mathserv.math.sfu.ca/History_of_Math/China/Beginning/Rod.html
  9. Mathematics: From the Birth of Numbers. Jan Gullberg. W. W. Norton & Company, New York 1997.
  10. A History of Mathematics. Carl B. Boyer. John Wiley & Sons, New York 1989.

Top  |  Next

Calculators in the Classroom

 

CALCULATOR: a mechanical, electromechanical, or electronic device that performs arithmetic operations automatically.

Since the first hand-held electronic calculator appeared on the market nearly 30 years ago, individuals and special interest groups have voiced extremely strong opinion, both pro and con, regarding the use of calculators in the classroom. And the debate continues today. David Gelernter, professor of computer science at Yale University, believes calculators should be totally eliminated from the classroom. He feels that allowing children to use calculators produces adults who can't do basic arithmetic, doomed to wander through life in a numeric haze. In 1997, California legislation would prohibit the use of calculators in schools prior to the sixth grade. Whereas, the state of Virginia purchased 200,000 graphing calculators to be used by all middle school and high school math students.

"According to the National Council of Teachers of Mathematics (NCTM), the use of calculators along with traditional paper-and-pencil instruction enhances the learning of basic skills." In fact, the NCTM supports "the integration of the calculator into the school mathematics program at all grade levels in class work, homework, and evaluation." (Roberts, 1991, p.51) In other words, the use of calculators should not eliminate the teaching of the basic algorithmic skills and processes of mathematics. It should be properly integrated to reinforce the basic concepts that are being taught and to aid in the application of these math processes to real-world situations. The key words here are "proper integration". This implies teacher-supervised activities relating to the mathematical concepts being learned.

One of the problems experienced with children who have used calculators from an early age without proper integration is that they have not had the opportunity to develop good "number sense". This is sometimes referred to as a "feel for numbers". The importance of estimating an answer before or after calculating is not understood. Why estimate, when the calculator will give you the exact answer? After all, a calculator is never wrong. That is virtually true, but the operator of the calculator is capable of error. And a mislocated decimal point can make quite a difference, as illustrated below.

Also we must not ignore one of the primary reasons we teach math--it trains the mind. It promotes logical and rational thinking skills and discipline. It requires using learned information to proceed to the next level of information, whether it be dealing with numbers or ideas. These types of thinking skills are imperative if students are to be able to function as thinking, intelligent, contributing members of society: the ultimate goal of education.

Many studies have been conducted and articles written regarding the use of calculators in all levels of classrooms. At least four that apply specifically to middle schools are found in Impact of Calculators on Mathematics Instruction, edited by George W. Bright, Hersholt C. Waxman and Susan E. Williams. Another source that contains articles on the positive impact and potential of calculators in the classroom is the NCTM 1992 Yearbook, edited by James T. Fey and Christian R. Hirsch. This book also contains at least three articles that look specifically at middle schools. Several of these studies support the theory that students allowed to use calculators in math classes produced higher achievement scores in problem-solving as well as basic skills. Students also had a much more positive attitude toward mathematics when allowed to use calculators. Sometimes that attitude can mean the difference between failure and success in math, especially for a student who has continually struggled with the basic skills. The question no longer seems to be whether or not to integrate the calculator into the classroom, but how to wisely use it as yet another tool to reach as many students as possible.

Contributed by Nancy Ayers


References.

  1. Bright, G. W., Waxman, H. C., & Williams, S. E. (Eds.). (1994). Impact of calculators on mathematics instruction. Lanham, MD: University Press of America.
  2. Calculator. (1997). Compton's interactive encyclopedia. The Learning Company, Inc.
  3. Fey, J. T. & Hirsch, C. R. (Eds.). (1992). Calculators in mathematics education. Reston, VA: The National Council of Teachers of Mathematics, Inc.
  4. Gelernter, D. (1998, October). Kick calculators out of class. Reader's Digest, 153, 136-7.
  5. Hunsaker, D. (1997, November 3). Ditch the calculators. Newsweek, 130, 20.
  6. Roberts, F. (1991, December). Calculators in class. Parents, 66, 51.

Top  |  Next  |  Previous

The Slide Rule

 

The slide rule is a mechanical device used for calculating the complex operations of multiplication, division, square roots, cube roots, and trigonometry, but cannot be used to complete simple addition and subtraction problems. The accuracy of the computation depends upon the size of the slide rule with the commonly used 10-inch slide rule having accuracy to 3 significant figures. Longer slide rules can have accuracy up to 4 significant figures but are not convenient to work with. The 6-inch slide rule is easy to work with but is not as accurate as the other slide rules. Computational problem solving with the slide rule is now obsolete with the introduction of the calculator.

History of the Slide Rule
The slide rule is based upon the principle of logarithms developed by John Napier (1550-1617), a Scottish mathematician. The spacing on the slide rule are marked with numbers that have been calculated from tables of numerical logarithms.

Credit is given to two separate individuals for the invention of the slide rule: Edmund Gunther (1581-1626) and William Oughtred (1574-1660), both English ministers who enjoyed the study of mathematics. Gunter was a professor of astronomy at Gresham College in London and devised "Gunter's Line" which is a logarithmic line of numbers. Oughtred designed a pair of sliding scales to be used in conjunction with one another. He also invented a circular slide rule that had two radial pointers pivoted at the center of the rule in which logarithmic distances could be added and subtracted.

Sir Isaac Newton suggested a runner for the slide rule, but none was invented until a century later. Several other distinguished mathematicians contributed to the evolution of the slide rule-Seth Partridge in 1657 invented the slide as we now know it, James Watt and Boulton in 1775 devised the "Soho" ruler and John Robertson around 1775 developed an intricate slide rule with a runner similar to the slide rules presently used.

The most common type of slide rule, Mannheim Slide Rule, was named for Lt. Amedee Mannheim (1831-1906), a French army officer from Paris, who used the slide rule to calculate artillery fire. He later became Professor of Geometry at the Ecole Polytechnique in Paris. He is noted for standardizing the slide rule and devising a cylindrical slide rule considered more accurate than his straight slide rule.

The United States interest in the Mannheim Slide Rule escalated in the later 1800's. In 1880, Washington University in St. Louis was the first college in the United States to require their engineering students to use slide rules for numerical computations. United States manufactures, Keuffel and Esser, began producing slide rules in 1887.

Parts of the Slide Rule
The slide rule consists of three parts:

  • Body-stationary part of the rule supported by clamps at each end of the rule
  • Slide-movable center section of the rule which slides through the body
  • Cursor-or hairline, the movable indicator consisting of two lined, transparent faces mounted in a frame around the body of the rule
The scales of the slide rule can be found at the extreme left and right ends of the rule consisting of: A, B, C, D, CI (C-inverted), CF (C-folded), DF (D-folded), S (Sine), T (Tangent), ST (Sine-tangent for small angles), K (Cube), and L (Log). Most slide rules will contain the A, B, C, and D scales in which calculations involving multiplication, division, proportion, squares and square roots, and problems dealing with circles and spheres. The K scale calculates cubes and cube roots. The CI scale goes in the opposite direction of the other scales and reads reciprocals.

Numerous books and instruction manuals have been written for educational purposes in learning the proper method for using the slide rule in problem solving calculations. With the introduction of the calculator, the historical applications to the advancement of calculating devices are the only practical purpose of the slide rule today.

Contributed by Judy Lasater


References.

  1. Understanding and Using the Slide Rule. (1963). Indianapolis, IN: Howard W. Sams and Bobbs-Merrill Company.
  2. Black, R. D., M'Cord M. (1938). Introducing the Slide Rule-Simplified Instructions for Beginners in the Use of the Slide Rule. Wabash, IN: Nixon Enterprises.
  3. Ellis, J.P. (1961). The Theory and Operation of the Slide Rule. New York: Dover Publications.

Top  |  Next  |  Previous