Copyright by JttUa Slisabobh Adkina 1956 AN HISTORICAL AND ANALYTICAL STUDY OF

THE TALLY, THE KNOTTED CORD, THE FINGERS, AND THE ABACUS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State U n iv e rsity

Sy

JULIA ELIZABETH ADKINS, A. B ., M. A.

The Ohio State University 1936

Approved by:

A dviser Department of Educati ACiCNOWLEDGMENT

The author is deeply indebted to Professor

Nathan lasar for his inspiration, guidance, and patience during the writing of this dissertation.

IX lâBIfi OF CONTENTS GHAFTSl Fàm 1. INTRWCTION...... 1 Pl^iflËÜaaxy Statcum t ...... 1 âtatamant of the Problem ...... 2 Sqportanee of the Problem ...... 3 Scope and Idmitationa of the S tu d y ...... 5 The Method o f the S tu d y ...... 5 BerLeir o f th e L i t e r a t u r e ...... 6 Outline of the Remainder of the Study...... 11 II. THE TâLLI ...... 14 Definition and Etymology of "Tally? *...... 14 Types of T a llies ...... 16 The Notch T a lly ...... 16 The Knot T a lly ...... 16 The Line T a l l y ...... 16 The Finger T a l l y ...... 17 The Object T a l l y ...... 17 Threaded Objects...... 17 Non-threaded O b jects ...... 17 The Word T a l l y ...... 20 Tedhniques and Matexials Used in Tallying ...... 22 The Notch T a l ly ...... 22 The Knot Tally ...... 33 The Line Tally * ...... 33 i l l TâBl£ OF CONTENTS CHAFTBEi PAGE Th# Finger Tally ...... 37 The Object T ally ...... 37 Threaded O bjects ...... 37 Non-threaded Objects ...... 37 The Word T ally ...... 43 Summary ...... 43 i n . THE KNOTTED COED...... 46 Definition of a Knotted Cord ...... 46 Purposes fo r Which the Knotted Cord Was Used ...... 46 Peoples Who Used theKnotted Cord ...... 48 Sum m ary ...... 59 The Quipu ...... 60 Introduction ...... 60 Deseripticsi of the Quipu ...... 60 Origin of the Q uipu ...... 73 Uses o f the Quipu by the I n c a s ...... 74 The Quipoeamayoes ...... 75 t Use of the Quipu by Other People ...... 76 Summary ...... 77 17. THE FINGERS ...... 78 Introduction ...... 78 The Influence o f Finger Symbolism ...... 82

i v TABLE OP OQNTprS CHAFTES zPACa Peopl«« Who D sad Finger Notation ...... 87 Writers about Finger SjnaboXiaa ...... 97 Finger Reckoning ...... 100 Suomary...... 102 V. THE ABACUS . 103 Introductii»...... 103 Definition and Etymology of **Abaeas" ...... ,...... 107 Definition and Etymology ofCoonter** .... 109 Types of Abaci ...... I l l Origin of the Abacus ...... 131 The Abacus in Egypt « ...... 136 The Abacus in Greece ...... 137 The Abacus in Rome ...... 141 The Abacus in India ...... 146 The Abacus in A r a b ia ...... 148 The Abacus in R ussia ...... 149 The Abacus in-China ...... 152 The Abacus in Japan ...... 156 The Abacus in Other European and Asiatic Countries .... !%62 The Abacus in America ...... 168 The Mhthematios of the Abacus ...... 173 The In flu en ce o f the Abacus...... 175 CHAPTER TABLE OF GCNTEMTS FAGS The dacontlnoanee of the Abacus ...... 180 axm m axy ...... 186 VI. A BRIEF HISTORICAL SURVEY OF TOE’TEACHING OF ARITHMETIC (6th century B. C. to 17th centuryA. D .) ...... 187 Introduction ...... 187 Teaching the Use of the Fingers ...... 187 Teaching the Use of the T ally ...... 189 Teaching the Use of the Quipu ...... 190 Teaching the Use of the Abacus ...... 190 Early Philosophers Who Advocated the Use of Concrete Materials in Teaching ...... 197 Summary ...... * ...... 199 V II. THE TEACHING OF ARITHMETIC IN THE UNITED STATES (Early Colonial Days to 1935) ...... 201 The Ciphering Period — from Early Colonial Days to 1821 . . 202 The Period of the Influence of Warren Colburn — Active Period: 1821 to 1857 ...... 204 The Period of the Influence of Warren Colburn — Static Period: 1857 to 1892 ...... 215 The Reflective Period — 1892to 19 1 1 ...... 220 Life A ctivities Period — 1911 to 1935 ...... 225 Summary ...... 2)5 V III. THE TEACHING OF ARITHMETIC IN THE UNITED STATES (1935 to 1 9 5 6 )...... 2)6 Textbooks on Arithmetic ...... 2)6 Vi TABUS OF CONTBITS CHAFTEE PACES A Critique of the Denriees Used ii% These Series to Indicate Place Value ...... 246 Statements of Philosophy in the Nine Séries ...... 249 Books on the Teaching of Arithmetic ...... 254 A Critique of the Books in Relation to Techniques of Teaching with Forms of Abaci ...... 262 A Criticism of the Abacus as a Teaching Device • ...... 264 Contemporary Forms o f Abaci ...... 266 The Principal Characteristics of the Present Period (1935 to 1 9 5 6 ) ...... 274 Sum m ary ...... 279 IX. SDimARY AND SUGGESTIONS...... 281 Summary ...... 281 Possible Suggestions for the Teaching of Arithmetic .... 284 Suggestions for a Sequence in the Teaching ofArithmetic . . 285 Suggestions for the Use of the Abacus in the Teaching of Arithmetic ...... 287 Recommendations for Other S tu d ie s ...... 288 APPaiDICES...... 290 BIBLIOGRAPHY...... 313

v i i LIST OF PLATES PLATE PAŒS I. Copies of Samoan War Clubs ...... 27 II. Exchequer Tally ...... 31 I I I . B eglatroa De Bamales ...... 45 17. Exniqple of the Highest Development o f the Quipu .... 63 7. Method of Tying R aots ...... 65 VI. The Strands Are Grouped by Meansof Shells orBeads • . 68 711. Finger Reckoning ...... 81 7III. Modem Finger Reckoning « ...... 95 IX. Roman A b acu s ...... 145 X, Russian Abacus ...... 151 XI. Chinese Abacus ...... 155 XU. Traditional Japanese Abacus ...... 158 XIII. Contemporary Japanese Abacus ...... 160 XI7, Medieval Counting Board ...... 165 X7. Peruvian Abacus ...... 170 X7I. Numeral Fram e ...... 207 X7II. The Abacus ...... 213 I7III. Mbntessori Advanced Fern of Abacus ...... 228 XIX. Diagram of an Abacus Used in England in the Tear 1111. • 300 XX. Symbols Used on Gerbert*s Counters ...... 302

v i i i LIST OF FIGURES m m PâGE 1-3* jtâU laa ...... 19 4* Itoekly Gàinidar ot Kio«a...... 34 5* Shop Account...... 36 6 . A Tally H older ...... 42 7 . Numerical Value o f Knots in aQuipu ...... 70 S. Okinaua C alculating D e v ic e ...... 72 9. Chinese System of Finger Notation ...... 89 10. Place Value ...... 1Q5 11-12. Medieval Counting Boards ...... 113 13. Chinese Abacus ...... 115 14# Medieval Counting Board ...... 115 15. Persian A bacus ...... 116 16. Egyptian Abacus ...... 116 17. Gerbert * s A bacus ...... 116 18-23# Modem Chinese Abacus ...... 117 24-29, Traditional Japanese Abacus ...... 119 30-34, Contemporary Japanese Abacus ...... 122 35-37, Eg^tian Abacus ...... 124 38-42, Persian A bacus ...... 125 43-46. Medieval Counting Board ...... 127 47-50, Gerbett's Abacus ...... 129 51. Greek Counting Board ...... 139 52-53, Neman Abaci ...... 143 54-55* Form of Abacus Used by Robert Records ...... 303 ix OHAPEER I

INTRODUCTION

ERELXMXNART STATEI'ÆEÜÎÎT

TJlien one opens a textbook on arithmetic that has been prepared for the first-grade level, one will find that the children are introduced to Hindu-Arabic numerals almost immediately, in fact, usually within the first ten pages. This practice tends to give teachers the impres­ sion that arithmetic can neither be taught nor understood without using Hindu-Arabic numerals. However, a study of the history of arith­ metic does not justify such a conclusion.

Arithmetic has been taught more years without the Hindu-Arabic numerals than it has been taught with them. For over two thousand years (from c, 600 B,C, to the l6th century A,D,) arithmetic for the school child was almost entirely some form of physical device. It has been only v/ithin the last four hundred fifty years that facility in the use of the Hindu-Arabic numerals has become one of the goals in the teaching of arithmetic in the western world. For centuries man has used an abacus as one of the physical devices with which he performed arithmetical computations. The origin of this device is buried in antiquity. But history reveals that the instrument was used extensively in the daily life and in the schools of many countries. The countries of western Europe ceased to use it a few hundred years ago. But Russia and some sections of the O^-ient — China, 1 2 Japan, and the East Indies — have continued using the instrument even until the present time.

A review of literature in the field of arithmetic shows that in

the United States there has recently been an increased interest in the

use of the abacus. This is perhaps due to the new philosophy inherent

in the Meaning Theory, namely, that the meaning of number is developed

by the student’s manipulating of physical devices under the skilled

guidance of the teacher.

STATEMENT OF THE PROBLEM

Some teachers are reluctant to use the "concrete” approach to the teaching of arithmetic because they associate physical devices with so- called "progressive education." They make this association, not realizing that John Dewey’s "learning to do by doing" was also advoca­ ted by Comenius (1592 - l6?0 A.D.) and by Plato (427 - 347 B.C.); not realizing that for many centuries man’s sole means of computation was with physical devices; and not realizing that, in the early schools, when arithmetic was included in the curriculum, it consisted of teach­ ing the use of the fingers and of the abacus. Other teachers are apprehensive about the use of physical devices because they believe that too much preoccupation with the concrete may make it impossible for the children to work with the abstract numerals.

The purposes of this study are: 1. to investigate man’s use through the ages of physical devices,

especially the tally, knotted cord, fingers, and abacus, as his

means of recording and communicating numbers and of computing

with numbers. 3 2 , to survey briefly the trends and practices of the schools (from

c . 600 B.C. to the present day) in the teaching of arithiitetic

in relation to the use of objective materials.

3 . to show the implications that the use of an abacus has for the

teaching of arithmetic at the present time.

IMPORTANCE OF THE PROBLEM

This study might be of value for the following reasons:

1. Present-day concepts of learning- establish that physical devices

are an integral part of good teaching and learning. In fact,

many educators believe that the "concrete" approach is the only

approach to the learning of arithmetic. Basic to an effective use

of this approach as a teaching technique is an understanding of

the technique, and a way in which one might obtain this under­

standing is by a study of the origin and development of the

technique. This study w ill be of value by bringing together in

one place an historical account of the development of the use of

physical devices in the teaching of arithmetic from 6OO B.C. to

the present time. Furthermore, the study delves into the back­

ground of these devices and portrays man*s intellectual progress

as he made use of and improved the devices.

2. Historical studies are valuable for other reasons. They serve to enlarge the teacher’s horizon of thought by enabling him to extend his experience farther into the past than his own lifetim e.

Contact with the great teachers and the great ideas of the past

might well serve to free the tradition-bound teacher from bondage, qnd to encourage him to revaluate and perhaps even to change his 4 theories and practices.

3. This study is also of value in developing a cultural appreciation

of mathematics. One finds that through the ages many poets, prose

w riters, and dramatists have made use of mathematical concepts

which are associated with physical devices,

4 . Iviany of the textbooks on arithmetic that were surveyed in this

study presented questionable concepts in relation to the princi­

ple of place value. Also, one of the books on the teaching of

arithmetic which is widely used, gave an incorrect illustration

and explanation of an early form of abacus. This study will be

of value in pointing out these errors and in presenting correct

interpretations based upon authoritative sources.

5. The survey of the textbooks on arithmetic and of the books on the

teaching of arithmetic showed that recently there had been an

increased interest in the use of manipulative materials. One of

the types of manipulative materials recommended was the abacus.

The authors of these books made suggestions as to the ways in

which this device could be used in the classroom, but their

proposals showed that they failed to realize many of the possi­

bilities that the abacus has for effective teaching. The present

study w ill be of value to the classroom teacher in suggesting

other ways in which an abacus may be used in order to make

arithmetic more meaningful to the student.

6. îiany persons believe that the approach to the development of

meaning is throu^ the use of physical devices. This study w ill 5 provide these people with suggestions for the teaching of

arithmetic,

7* Some persons think of the abacus only as a device about which

one reads for enrichment material. For them the present study

w ill be a source of historical information.

SCOBS AND LIMITATIONS OF THE STUDY

In this study a survey is made of man’s use since early times of

the most common physical devices as a means of recording numbers and

of computing with numbers. Since the arithmetic of early man was these

physical devices, they were taken into the classroom when arithmetic became a part of the curriculum. Furthermore, an examination is made of the use of these physical devices in the teaching of arithmetic since the founding of schools in ancient Greece (c. 600 B.C.).

THE METHOD OF THE STUDY

An examination and an analysis were made of a wide sampling of original and secondary sources including;

1 . research studies pertinent to the subject.

2. children’s textbooks on arithmetic published in the United States.

3 . books and articles on the history of education,

4 . books and articles on the history of mathematics.

5 . books and articles on the teaching of mathematics.

6 . books and articles on the history of the teaching of arithmetic.

7 . dictionaries, encyclopedias, and bibliographies.

8 . books on biography. 6

9. reprints of early European and Asiatic books on arithmetic.

10. vjritings of persons who have advocated the use of physical

devices in teaching and learning.

11. classical literature which contained references to mathematical co n cep ts.

12. books and articles on anthropology, philology, history, ethnology,

numismatics, classical antiquities, geography, and archaeology.

RE7IÏÏ17 Ü3T THE LITERATURE

As far as the writer of the present study could determine there was no one source which developed a general history of the four physi­ cal devices — the tally, knotted cord, fingers, and abacus. %ny articles and books on the history of mathematics have made detailed studies of some aspect of one or more of these devices, but none has been a compilation of the widespread use of each instrument, a classi­ fication of the types of each instrument, a consideration of the more important early writers about the instruments, or a historical survey of the use of these instruments in the schools.

The literature related to each instrument w ill be mentioned in the appropriate places, but a few of the more outstanding books and articles will be listed at this time. One of the books on the history of mathematics which includes material about these devices is by Smith

^D. E. Smith, History of Mathematics (Hew York: G-inn and Co., 1925), Vol. II. 7 There is no comprehensive source for infoianation about the tally,

in general, but two articles by Jenkinson^*^ include excellent mate­

rial about private and Exchequer tallies in England.

Some of the best material about early uses of the knotted cord

is found in books and articles on the history of writing, such as

those by Hoffman,^ Mason,^ Diringer,^ and M aury.A uthoritative infor­

mation about a special form of knotted cord (the quipu) is found in an Ô o article and in a book by Locke,

Information about finger notation and finger reckoning is some­ what scattered, but Richardson^® wrote an article that included

^Elilary Jenkinson,"Medieval Tallies, Public and Private," Arehaeologia. LCCI7 (1925), 289-351. 3 Hilary Jenkinson, "Exchequer Tallies," Arehaeologia. tXII (1911), 367-80.

W alter James Hoffman, The Beginnings of W riting (New York: Macmillan and Co., 1895). % illiam A. Mason, History of the Art of Vfriting (New York: Macmillian Co., 1920).

^avid Diringer, The Alphabet (New York: Philosophical Library, 1951). 7 / 'Alfred Maury, "Origine de L'Ecriture," Journal Des Savants. (Avril, 1875), 205-21.

^Leland L. Locke, "The Ancient Quipu," American Anthropologist. N.S. XIV (April-June, 1912), 325-32.

^Leland L. Locke, The Ancient Quipu or Peruvian Knot Record (New York: American Museum o f N a tu ra l H is to ry , 1923).

^^^on J. Richardson, "Digital Reckoning among the Ancients," American Mathematical Monthly. XXIII (January, 1916), 7-13. ô references to finger symbolism that are found in classical literature, 11 Chapter II of a book by Yeldham contains a translation of a chapter

from a book by Bede. Bede’s work gives present historians almost the

only knowledge of finger symbolism used in western Europe during the

Middle Ages.

Excellent sources (in English) about the abacus are by Yeldham,

Barnard,Knott,Smith and Mikami,^^ De LaCouperie,^^ and Records,^

Studies which were valuable sources for information about the

teaching of arithmetic were the following; IS Gajori discussed the teaching of all areas of mathematics from

^^Florenoe A. Yeldham, The Story of Reckoning in the Middle Ages (London: George G. Harrap and Co. L td., 1926).

^ I b i d .

P. Barnard, The Casting-Counter and the Counting Board (Oxford; Clarendon Press, I 916).

^C argill G. Knott, "The Abacus, in Its Historic and Scientific Aspects," Transactions of the Asiatic Society of Japan, XIV (1886), 18-73. E. Smith and Yoshio Mikami, History of Japanese Mathematics (Chicago: Open Court Publishigg Co., 1914J. l é Terrien De LaCouperie, "The Old Numerals, the Counting-Rods, and the Swan-Pan in China," Numismatic Chronicle. III^ (1883), 297-340.

^'^Robert Recorde, Grounde of Artes (sections from this work are given in Yeldham, op. c it.; Barnard, op. c lt.; and Robert Steele, Earliest Arithmetics in English (London: Oxford University Press, 1922), ^%’lorian Cajori, The Teaching and History of Mathematics In the United States ("Bureau of Education, Circular of Information," No. 3 OSTashington, D. C. : Government Printing Office, 189© ). 9 the elementary to the college level, beginning with the colonial

period and continuing until the time of the publication of his study

(1Ô90)* He analyzed the subject matter of the textbooks, the methods

of teaching, and the materials used. 19 Monroe, / also beginning with the colonial period, confined his study to the teaching of arithmetic, ^e reported upon the content

organization of the subject, the aim of instruction, the place of

arithmetic in the plan of education, and the methods of teaching. His

report extended until the time of publication (191?). 20 R atliff, in a study more comprehensive than that of the two previously mentioned, investigated the teaching of arithmetic from the colonial period until 1946. She analyzed the textbooks used, the methods of teaching, the philosphies advocated, and the use of objects.

Suzzallo^^ made a study of the teaching of arithmetic in the United

States, but limited the period of time to that immediately prior to

1911* The purpose of his study was to give to the educators of the day

Information relative to the methods being used in teaching mathematics in the first six grades of the American elementary school.

W alter S. Monroe, Development of Arithmetic as ^ School Subject ("U.S. Bureau of Education Bulletin," No. 10 Washington, D. C.: Government Printing Office, 19173)*

^^Lavada R atliff, "Historical Development of Methods of Arithmetic in American Elementary Schools" (unpublished Doctor of Education dissertation, University of Texas, 1946). ^^enry Suzzallo, "The Teaching of Primary Arithmetic," Teachers College Record. XII (March, 1911), 1-70* 10 Gk>rinan22 analyzed 1? series of textbooks published in the United

States during the following three periods: I 907 - 1910, 1917 - 1920,

and 1927 - 1930» Ue gave a brief background summary of the character­

istics of the teaching of arithmetic from the Colonial Period until

1931» but his principal purpose was to determine the subject matter

and methods of arithmetic in the United States as reflected in text­

books published during the periods named above.

Sm ith and Eatonanalyzed 36 books, 13 sets of boèks (2 books

from each set), and 10 series of books v/hich were published in the

United States between 1790 and 1940. Their purpose was principally to

determine the approach used in the books (inductive, deductive, or

both), the content of the books, the general order of treatment of

each topic, and the author’s philosophy relative to the use of objects

and pictures.

Two sources which contained Information about the textbooks used in early schools in the United States were :

Greenwood and Martin^^ arranged an annotated bibliography of

22s>rank Hermon Gorman, Some Pacts Concerning Changes in the Content and Methods of Arithmetic. Abstract of Ph.D. Thesis, University of Missouri, 1931» 23senry Lester Smith and M errill Thomas Eaton, An Analysis of Arithmetic Textbooks ("Bulletins 10(1942) and 19(1943T of the School Education, Indiana University" [Bloomington: Bureau of Cooperative Research and Field Service, Indiana University, 19423).

24James M. Greenwood and Arbemas Martin, "Notes on the History of American Text-books on Arithmetic," Report of the Commissioner of Education. 1897-98 (Washington, D. C.: Government Printing Office, 1899), I, 789-868. 11 textbooks used in the United States from 1861 to 1892.

Earpinski^5 catalogued mathematical works (books, pamphlets,

encyclopedias, and journal and newspaper articles) published in the

United States, Canada, and the West Indies up to the end of I 85 O, and

all of those mathematical vjorks known to have appeared in Central and

South America up to I 8 OO. The first publication he lists is one that

appeared in 1556. He also includes engraved reproductions of the title

pages of practically all of the major works included in his list,

OUTLIHS 01’ TEE REMAII3DER OF THE STUDY

Chapter II (The Tally) will present a picture of man’s beginnings

in attempting to find satisfactory ways of recording numbers. For

this purpose he used the commonplace materials that were conveniently

at hand — materials such as pebbles, sticks, and shells. At first he

matched in a one-to-one correspondence one pebble, for example, with

one of the objects he was counting. later, he learned to let one pebble symbolize one group of objects that he was counting.

As man progressed in skills and learned to make rope, he began to use this material for keeping records. Chapter III (The Knotted Cord) will show the advance in his thinking by illustrating the many varia­ tions he soon learned to use in order to increase the efficiency of this device. One group of people — the Incas of Peru — developed a

2 5 L o u 1 s Karpinskiÿ Bibliography of Mathematical Works Printed in America through 1850 (Aim Arbor: University of Michigan Press, I94Ô5T 12 highly skilled technique of keeping numerical records on knotted cords

(or quipus, as they called them). They used the principle of place

value by arranging the knots in ordered parallel rows, each roiv repre­

senting a decimal position.

Probably concurrent with the use of materials like pebbles,

sticks, and cord, for tallies, man discovered that his own fingers

could be used effectively for this purpose. Chapter IV (The Fingers)

w ill be a discussion of the practice of using the fingers for repre­

senting numbers and even for computing with numbers.

Most people in using materials like the fingers, the knotted cord,

pebbles, sticks, and shells, used them as forms of tallies. This

means that they did not use the principle of place value in represent­

ing numbers and in computing with numbers. VJhen man became intellec­

tually capable of using the principle of place value, he developed another type of device — the abacus. Chapter V (The Abacus) w ill reviev/ man's use of the abacus, including a discussion of the various forms of abaci that have been developed in the different countries of th e w o rld ,

Since schools are usually established to perpetuate the culture of a nation, or at least some aspect of the culture, and since these four devices — tally, knotted cord, fingers, and abacus — were the principal means by which man recorded numbers and computed with numbers, one finds these devices being used in the schools for the teaching of arithmetic. Chapter VI will be a survey of the use of these four devices in schools, beginning with the 6th century B,0. and including the l?th century A.D. 13 Chapters ‘VXX and VIII w ill trace the development of the teaching of arithmetic in the schools of the United States. Chapter VII will include the period from early colonial days until 1935. Chapter VIII w ill include the period from 1935 to the present time ( 1956) . The trends for this latter period were determined on the basis of a survey of textbooks on arithmetic and of books on the teaching of arithmetic that had been published in the United States during this period of tim e.

Chapter IX w ill present a summary of the study and w ill make suggestions for the use of physical devices, particularly the count­ ing frame and the abacus, in the teaching of arithmetic. CHAPTER I I

THE TAIiLT

DEFINITION AND ETÏMOIÛCÎÏ OF "TALLY”

The earliest meaning (in the English language) of the word t a l l y was:

A stick or rod of wood, usually squared, marked on one side with transverse notches representing the amount of a debt or payment. The rod being cleft lengthwise across the notches, the debtor and creditor each retained one of the halves, the agreement or tallying of which constituted legal proof of the debt, etc.

This meaning apparently was an extension of its original meaning in

Latin, for in tracing the history of the word, one finds that the root from which it is derived is the Latin word talea (cutting, rod, s t i c k ) ,

A rather interesting feature is the sim ilarity of the spelling of the equivalent word for tally (notched stick) in some of the other languages. For example, in Italian, the word is taglia; in Spanish, tara or tarja; and in French, taille.

^"Tally", Oxford English Dictionary (1933 ed.), X I, 60.

^Loc. c i t .

^D. E. Smith also connected the word with the German Zedil (number) through the root tal (D; E. Smith, History of Mathematics (kew York: Ginn and Co., 1925T7 II, 193.

14 15 There are several meanings of the word tally , but only one other

will be specifically considered in this study. This is best expressed

in one of the definitions of the verb tally; "to cause (things) to

correspond or agree.It is necessary to include this interpreta­

tion of tally, for in tracing the history of the various means by which man has kept numerical records one finds a method which was a matching process. % mtched, usually in one-to-one correspondence, that which he wished to count (like animals or days) with objects (such as sticks, rods, stones, and nuts) or with lines (or scratches) marked on the ground, on stone, or on wood. (The three-dimensional objects, such as sticks and rods w ill be referred to as tally-objects; the two-dimen­ sional marks and scratches w ill be referred to as tally-marks.)

In this study two conditions w ill be set up to determine if a method of numerical recording is to be considered as a form of tally.

These conditions are:

1. A matching in one-to-one correspondence with that which is being

counted. This matching might be that of matching one tally-object

(or tally-mark) with one object or with one group of objects,

2. An absence of the use of place value.®

^"Tally", Oxford English Dictionary (1933 ed.), X I, 6 l. ^Sorne of the uses may have been influenced by association with Latin talis (such) and talio (giving like for like) (|Loc. cit.3

See pages 104-7 for a discussion of place value. 16 T m ss OF 'PAT.T.mq

 study of the methods and materials used by man throughout the

centuries makes it possible to classify tallies according to the

following types

1. The Notch Tally.

This would refer to a type similar to that described in the first

definition (page 14), in which notches were cut in a stick of wood (or

on a piece of bamboo)* Each notch represented a unit.

2 . The Knot T a lly .

This form of tally was one in which knots were tied in a cord or

string, with simple knots representing one and more complicated knots

representing larger quantities (such as five or ten). The knot t a l l y

is discussed in more detail in Chapter III.

3. The Line Tally*

This type of tally was one in which lines were marked (or

scratched) on the ground, on stone, or on any appr<^riate material,

with each mark indicating one* Even at the present time persons make

7 So far as the author of this study could determine, there was no historical evidence of forms of tallies other than the six that are listed. However, there are possibilities of many varieties centering chiefly in the different methods man has employed for communication. Some exaü^les of these methods are: communication by sight, such as smoke-signals, semaphore, blinker- light system, colored spot lights, and the variety of effects which it is possible to produce by electrical impulses; _b* communication by sound, such as drum beats, musical pitch, tele­ graphic code, and the variety of effects which it is possible to produce by electrical impulses; communication by touch, such as the Braille System. 17 use of tally marks (HH.) in many ways, such as keeping a score for

a game, taking an inventory of supplies, or checking responses to a questionnaire.

4 . The ginger Tally.

The simplest form of finger notation was that in which the fingers were matched in one-to-one correspondence with the objects being counted. In this process the fingers could be classified as tallies (the more complicated finger notation and finger reckoning is discussed in Chapter IV).

5. The Object Tally.

Tallies of this type, which usually consisted of objects smaller than the objects that were being counted, may be classified according to two basic sub-types;

a. Threaded objects.

This form of tally was more commonly called the bead tally

and consisted chiefly of beads (occasionally beans, shells,

or berries) threaded on cords, wires or rods. Each bead

represented one object or one group of objects. The counting

frame (or numeral frame) of the present day is an example of this type of tally. The chief distinction between the bead

tally and the abacus (Chapter V) was that with the former

there was no use made of place value.

Non-threaded objects.

A tally of this form consisted of small objects (such as

pebbles, shells, nuts, wooden pegs, or slips of wood) matched

in one-to-one correspondence with that which was being counted. lÔ The earlier uses of this type of tally comprised the matching

of one tally-object with one of that which was being counted.

This technique was convenient so long as man dealt with small

quantities* But when he found it necessary to keep a record

of larger amounts, he modified the earlier technique by group­

ing the tally-objects. For example, a man might have needed

to keep a record of the number of animals he had. Suppose the

number he owned amounted t o what would be d esig nated a t th e

present time as "forty-six." He could match one pebble with

one animal until he had a total of forty-six pebbles (see

Figure 1). Then by grouping, say in tens, he had four groups

of ten each and six single pebbles (see Figure 2). This made

a much e a s ie r method f o r counting, remembering, and communi­

cating numbers.

Later, man learned to simplify his tallying even more.

After counting out a group of ten pebbles to represent ten

animals, he replaced these ten pebbles with one pebble that was of a different size (usually larger) or of a different

color. Using the example given above, he would have, as a result of this technique, four large pebbles (or four pebbles of a certain color) and six small pebbles (or six pebbles of a different color). This method had the advantage of requiring fewer pebbles and of being easier to count (see Figure 3).

Usually those tally-objects of pebbles, shells, or nuts

were placed in a heap or kept in a container, while those of oooooooooo O O OOOOOOOO oooooooooo o o ooooooo o O O O o o o

FIG

oo ooo O o ooo

O O O O O O o ooo

f ig . 2

o o o o o o FIG. 3

F ig s . 1-3 T a llie s 20 wooden pegs were inserted in holes which had been drilled in tally-boards,

6» The Word T a lly .

Rarely included as a type of tally is the use made of number

words (such as one, two, three) in the counting of objects. In this

instance each object being counted has been matched in one-to-one

correspondence with a number word, and the words themselves could be

considered as tallies.

These various types of tallies were used not only to record

numbers, but also to represent the sums of numbers, such as the total amount of money involved in a business transaction, or the total number of objects in two or more groups. People performed this operation and yet did not necessarily have words in their language to express these larger numbers which they had indicated by notching a A stick, by heaping pebbles, or by making marks. So far as the writer could determine, the most common mathematical operation performed with tallies was that of addition, with only an occasional mention of subtraction.

®Some of the American Indian tribes had numeral words as far as one billion, yet could not comprehend even moderate numbers. The only way they could form any idea of a quantity was to have the objects matched with tangible devices or with marks. Ql. R. Schoolcraft, H istorical and S tatistical Information Respecting the History. Condition. and Prospects of the Indian Tribes of the United States (!Phliadelphla; Bureau of Indian Affairs, iSgl), II, p. 20é’f T, p. 712. Volumes II, III, and IV give the names for these numerals.3 21 la a few instances people grouped by tens, such as making each

tenth cut on a stick longer than the others, or setting aside a small

yam for every hundred yams being counted. This process might well be

considered a preparatory step which eventually enabled jrwn to develop

the concept of place value.

Since the tally is usually based on a one-to-one correspondence

with the numbers being recorded, it is probably the simplest and earli­

est form of keeping numerical records. Because of this it would be

one of the instruments man would invent in an early stage of his

development. Quite possibly this could have occurred in the Neolithic

Age of man, which began in Asia and Africa about fifteen thousand

years before the Christian era. In this stage of development man was

a herdsman and accumulated numbers of animals to tend. With perhaps

a very limited numerical vocabulary, what would be more natural than

that he turn to plentiful materials, such as wood and stones, to assist

him in keeping a record of his possessions?

One of the earliest extant artistic evidences of the use of the

tally is a bas-relief^ on the temple of Seti I (c, 1350 B.C,) at

Abydos. This is a portrayal of Thot, one of the Egyptian gods, indica­

ting the length of the reign of Pharaoh by marking upon the notches

of a long frond of palm.

^Reproduced in Gaston Maspero, The Dawn of C ivilization (New York; Macmillan Co., 1922), p. 221, 22 TECHNIQUES AND mTERIALS USED IN TALLYING

In tracing the history of tallies one finds them being used for

three basic purposes; the recording of business transactions — which

might involve sums of money, total number of days worksd, or number

of cattle bought or sold; the recording of one's possessions; and the

recording of the passage of time. The techniques and materials used

in keeping these records were many and varied. Some of the more

interesting ones (classified according to type) are as follows:

1. The Notch Tally.

According to De LaCouperie, the notched stick (fu) was used by

the non-Chinese in the state of Tsu (on the banks of the Yang-tze

Kiang) in the sixth century B.C. The practice was also known in

402 Â.D. in Upper Asia. But the use of notched sticks was not

confined to the Ear East.

It was formerly greatly practiced in the West, and it still lingers in some countries. It will be sufficient for our illustrative purposes to remind our readers of the buchstaben of the Germans, the bok-stafir of the Scandinavians, the coelbren of the Welsh, • •

In 58 B.C., when the Romans invaded northern and western Europe,

they found that the Goths possessed no means of recording other

than that of using tallies of knotted cord and of notched wood.

lOTerrien De LaCouperie, "Beginnings of Writing in and around Tibet, "Journal of the Royal Asiatic Society. N.S. XVII (I 885 ) , 434*

^% illiam A. Mason, A History of the Art of Writing (New York: Macmillan Co., 1920), p. 4* 23 ç_. The Roaaas themselves probably were using tallies of notched

•wood a t an even l a t e r d a te , f o r P lin y (23 A.D. - 79 A.D.) vra-ote,

’’Willows, alders, poplars, the siler and the privet, the last

extremely useful for making tallies Cl . tesseris) , will only grow IP In places where there is water. .

The savage tribes v*.o lived in the Tibet-China area did not

reduce their language to writing until the middle of the 7th

century A.D, Before that time, when they would decide to rebel,

they would send a notched stick^^ to the ruler who represented

the Chinese government. Some type of symbol, such as a feather,

was attached to the stick, and the bearer would explain the mean­

ing of the notches and of the symbol. The notches quite often

indicated the number of soldiers who would be making the attack.

Marco Polo (1254*- 1323)» the Venetian traveler, in describing

the people of pardandan, a province in southern China, reported:

• . . They have no letters nor do writing; . . . they . . , have no dealing with the world. But I tell you that when they have to do the one with the other and wish to make their bonds or deeds of i t , which they must give and receive. they take a small piece of wood either square or round and split it in the middle, and keep, the one the one half and the other with whom he has to do the other half, a£ we here do in our way on a tally. But yet it is true that they first make two notches on i t o r th re e o r fo u r o r as many as they

^^Pliny Natural History xvi. xxxi, 76.

^^The Chinese called this a Muhk’i . The Tibetians called it a shing-tchram. ^^Terrlen De LaCouperie, op. c it., 420-1. 24 wish marking on it the amount of their dealing together. And when th e tim e i s come and th e y come to pay one an o th er th e tallies being put together agree in the little marks. And when they have paid then he who must give the money or other thing causes the half of the stick which that one had to be ,_ given back to him, and so they remain content and satisfied.^

_f. During the 13th and 14th centuries the notched tally (Kerbholz)

was used in Germny^^ for keeping accounts, and the custom did

not die out in that country nor in Austria until the 19th c e n tu ry .17

According to Marco Polo the notch tally was being used in Italy

in h is tim e and i t p ro b ab ly co n tin u ed to be common, a t l e a s t u n t i l

the middle of the l 6th century, for Tartaglia ( 1556) included a IS picture and a description of one in his book on arithmetic,

h . About 1771 the Indians at the mission of San Gabriel, Califor­

nia were using notched sticks which they had invented to record

business transactions relative to labor, money, cattle and

^^Marco Polo, The Description of the World, trans. A.C. Moule and Paul Pelliot (London; George Routledge and Sons Ltd., 1938), Book II, Part 2, p. 283 . l é Alfred Maurv ^(."Origine De L*Ecriture, ^ Journal Des Savants. (A v ril, 1875 ) f 21> made the statement that Wuttke vDde Sntstehung per SctoiftJ believed that the practice of marking numbers on tallies (called ankerben in Germany) was of Germanic origin, and that the men of the Teutonic race carried this practice to France, to England, and even to Novgorod (a town in Russia),

^"^Smith, _op, c it, , p, 194* ^%bid.. p, 195. 25 19 20 horses. * In keeping their records they jnade each tenth cut

longer, extending it entirely across the face of the stick.

About 100 years ago the natives of the Samoan Islands (South

Pacific) were still using war clubs. On the handles of the clubs

they cut a notch for each animal, including a "long-pig" (man), killed by that club (see Plate I),

J_. In 1855 notched sticks were being used by small shopkeepers in

English villages and as late as 1925 were being used in the hop- 21 gardens near Kent and Worcestershire,

An interesting use of the notched tally was that made by the Osage Tribe, In one of their religious ceremonies it was neces­

sary to sing a series of songs. La Flesche, in describing these

consecrated tally-sticks, wrote: / The songs of the ceremony proper, , , . are called Zho^-xa Wa-zhu, which, freely translated, means the songs upon which Sticks are Placed, a title that takes its name from the custom of , , , the novitiates using tally sticks for keeping a correct count when memorizing the titles and the number of the songs coming under each group , , . This stick is about 1 inch wide and as long as the lower arm of a man. Across thewidth of the stick are cut small grooves in groups to represent the number of songs in a class. These groups of marks cover both sides of the stick and a man in keeping count as he sings

^^Mason, op, c it, , pp, 31-2,

^®Walter James Hoffman, The Beginnings of W riting (New York: Macmillan and Co,, 1895), PP« 140-1,

^^Smith, op, c it, . p, 195• 26

Plate I. Copies of Samoan War Clubs.

The tv/o clubs pictured are copies of war clubs owned by Mr. and Mrs, Alvin Gibbs of Columbus, Ohio, While on the Samoan Islands in 1927, the Gibbs obtained the clubs from a native who said that they were copies of clubs in his family. The notches on the handles indicated the number of animals killed by members of his family with that particular club. a.7 2S begins to count from the lower end of the stick and proceeds upwards toward the top. When he reaches the top he turns the stick over endwise and continues his upward count from the end nearest to him.

These tally-sticks and the way they were used resemble a rosary

as used by the Catholics, by the Mohammedans, and by the Buddhists

o f Burma.

_1 , As late as 1951 the Khas of Indo-China and the Bangala of the

Upper Congo River in Africa were keeping their accounts by means

o f notched bamboo rod s and s t i c k s .

m. Many of the more primitive tribes used notched sticks as a

means of measuring the passage of time. Some of these were: the

natives of the Fiji Islands with their moon sticks (used to tally

ttioons, which early man tallied as years); the natives of Guiana

having the white man cut notches in a stick to indicate the number

of days until "Crissimus”; Indian tribes in the United States with

sticks notched to indicate nights (for days were counted by

nights), or even months and years; the natives of the Solomon

Islands in the Pacific notching a stick at the appearance of the

new moon; and the natives in New Guinea counting the months by

cutting notches in trees.

Francis La Flesche, The Osage Tribe: The Rite of Vigil ("Bureau of American Ethnology, annual report," No. 39 (Washington, D. C.: Government Printing Office, 1925/)» P» 77*

^^Davld Diringer, The Alphabet (New York: Philosophical Library, 19 5 1 ). p . 28 . 29 n. One of the jnost unusual uses of a notched stick as a calendar

was th e Clog(g) Alm anack also known a s Log Almanac, Runic

Calendar, Rune-stave, Rune-staff, Rimstav, aM Primstav. This

was a square block of wood, brass, or bone about eight inches

long, notched with special designs to indicate Sundays and the

church festival days. These calendars were tised by the Romany;

but it is believed that the Scandinavians invented this form, of

calendar and brought the idea to England when they invaded the

country in the 11th century. The English continued the use of

the Clog Almanac until the end of the l?th century, and the Danes

were using this form of calendar as late as the 1900’s.

Omitted from the previous list was the use of the notch tally in

the English Exchequer (see Plate II). After 1100 the notched stick

became the recognized form of receipt for the payments into the Royal

Treasury and continued as such until 1826. Charles Dickens, the well-

known a u th o r, made a speech in 1855 in which he derided th e government

for using tallies when more modem materials were available. (See

Appendix I).

The tally as used by the Royal Treasury merits a separate discussion

for it reached a highly developed state as an official instrument cut

strictly according to certain rules — rules which governed the

follow in g:

^4àn engraving of one is given in William Camden, Britannia, trans. from l6o? edition (4 vols.; London: J. Nichols and Son, 1806), II, facing p.499. 30

Plate II. Eïchequer Tally

From Hilary Jeakinson, "Exchequer T allies," Archaeologia.

IXLI (1911), Plate L, Fig. 1. Permission for use granted by Society of Antiquaries of London, 'wwm

*3«

Fig. I. Exchequer Tally: stock and foil; nineteenth century. About \

Cii 32 (1) the size of the tally; (2) the use of stook^^ and foil^^ and their relative size; (3) the way in which the tally was cut half through and (4) at the right end; (5) the size and (6) the position of the notches; (7) the way it [the infor­ mation] was written on the face and (8) on the edge; (9) the form of wording employed, both on face and edge; and (lO) the way in which i t was s p l i t ,

As mentioned in (5) above, definite rules were made which

determined the sizes of the notches, the size varying according to the amounts of money involved. From the top of the stick, where thousands

of pounds were represented by cuts the thickness of the palm of the

hand, the notches decreased in size to the lower part of the stick

where a penny was represented by a single cut in which no wood was

removed.

After being notched the stick was divided into two parts of un­

equal size (see Plate 11) by having a cut made half through its thick­

ness at a distance from one end which was usually a quarter or a third

of the entire length. Then the tally was split longitudinally from

the other end down to this point.

In contrast with the Exchequer tallies, the use and the notching

of the private tallies were not systematized to the same degree. Tliey

were chiefly cut according to an agreement between the interested

parties. Quite often the meaning of the notches, that is, whether the

^^The larger part which was given to the payer. ^^The part kept by the Exchequer.

^"^Hilary ^enkinson, "Medieval Tallies, Public and Private,” Archaeologia. LXXCV (1925), 320. 33 transaction involved money, cattle, or com, was made clear by written

Information upon the tally.

2. The Knot Tally

Illustrations of this form of tally are given in Chapter III,

3i The Line Tally

a. A native invention used by the Kiowa Indians (United States)

in order to help the uneducated members of the tribe "keep track

of Sunday" is shown in Figure U. Six straight parallel lines were

marked on a convenient surface to represent the week-days. The

seventh line (to represent Sunday) had an eagle feather drawn at

the top. The Sundays on which the Peyote ceremony was to be en- 28 acted had a small "peyote button" drawn above the eagle feather, 29 b. Adair, who lived among the Indians of southeastern United

States (c, 1761 - 1768), wrote of the method used by the Chickasaw

in figuring merCahtile transactions on the ground. The system

was called yaka-ne tàlapha, or "scoring on the ground," They

made a single mark for each unit and a cross for ten, and could

readily add together the tens in order to determine the total,

c. The Passamaquoddy division of the Abnaki tribe (from Maine)

kept their shop accounts by a method which was a combination of

28 J, D. Leechman and M, H, Harrington, String Records of the North­ west (New York: K&isetaa of the American Indimi, 19^1), 6^3^, 29 James Adair, History of the American Indians, ed, S, C, Williams from 1775 edition (Johnson City, Tenn, : Johnson City Press, 1930), p, 81, F ig . 4 Weekly Calendar of Kiowa J. D. Leechinan and M. B. Harrington, String Records of the Northwest (New York: Museùà of the American Indian, 1921), p. 63. Permission for use granted by the Nuseum of the American Indian, Heye Foundation, New York, New York. 35 tallies with ideographic writing. An exainple and an explanation of one shop account is as follows (see iFigure 5):

A deer hunter brings 3 deerskins, for which he is allowed $2 each, making #6; 30 pounds of venison, at 10 cents per pound, making #3. In payment thereof he purchases 3 pounds of powder, at 40 c e n ts per pound; 5 pounds of pork, at 10 cents per pound; and 2 gallons of molasses, at 50ç! per gallon. The debit foots ■f3»30» according to the ^ndian account, but it seems on calculation to be 30 c e n ts in excess, an overcharge, showing the advance in civilization of the Passamaquoddy trader.

The following explanation w ill serve to make intelligible the characters employed, . . . The hunter is shown as the first character in line _a, and that he is a deer-hunter is . . . indicated by the figure of a deer at which he is shooting. The three skins referred to are shown stretched upon frames in line _b, the total number being also indicated by three vertical strokes, between which and the drying frames are two circles, each with a line across it, to denote dollars, the total sum of #6 being the last group of dollar maries on line

The 30 pounds of venison are represented in line the three crosses signifying 30, the T-shaped character designating a balance scale, synonymous with pound, while the venison is indicated by the drawing of the hind quarter or ham. The price is given by uniting the X, or numeral, and the T, or pound mark, making a total of $3 a-s completing the line _c.

The line _d refers to the purchase of 3 pounds of powder, as expressed by the three strokes, the T, or scale for pound, and the powder horn, the price of which is four Xs or 40 cents per pound, or T; and 3 pounds of powder, the next three vertical strokes succeeded by a number of spots to indicate grains of povfder, which is noted as being 10 cents per pound, indicated by the cross and T, respectively. The next item, shown on line charges for 5 pounds of pork, the latter being indicated by the outline «Sf a pig, the price being indicated by the X or 10, and T, scale or pound; then two short lines preceding one small oblong square, or quart measure, indicates that 2 quarts of molasses, shown by the black spot, cost 5 crosses, or $0 cents per measure, the sum of the whole of the purchase being indicated by three rings with stems and three crosses, equivalent to # 3 »30.^®

^^Garrick îfellery, Picture-W riting of the American Indians ("U.S. Bureau of American Ethnology, annual report," No. 10 U^/ashington, D.C.: Government Printing Office, 18933), PP» 259-60. 36

5 t c -R

0 o

3 -s.^ 4" ■0 r c E ^ Ar

1 ^ :k co en V o-n . t o L

1

H

j w 3 ■ c d U Ü) 37 4* The F inger T ally

This type of tally will be discussed In Chapter XV, 5* The Object Tally

a* Threaded Objects;

(1). The Apache scouts (tribe of American Indians in the

southwest) kept a record of the time of their absence on a

campaign by using colored beads threaded on a string. Six white

beads were used to represent the days of the week, and one black

(or a colored bead) to represent Sunday. (2). The tribes who lived at the southern corner of Lake Nyassa

(Africa) used pieces of wood threaded on a string to indicate the days of the month that had passed.

Non-threaded objects:

(1). From the writings of Denig, one might infer that some of

the Indians in the United States could perform the mathematical

operations of addition, subtraction, and multiplication by using

a combination of their fingers and of sticks. He said:

They cannot multiply or subtract uneven sums without the aid of small sticks or some other mark. Thus to add 40 to 60 would be done by the fingers, shutting down one for each succeeding ten, naming 70, 80, 90, 100. But to add 37 to 94 would require some time; most Indians would count

^ John G, Bourke, The Medicine-Men of the Anache (*'U. S, Bureau of American Ethnology, annual repbrt,” No, 9 CWashington, D. C.: Government Printing Office, 1892]), p. $ 62.

32h . s . Stannus, "Notes on Some Tribes of British Central Africa," Joumal. of the Royal ^tothropologieal Institute of Great Britain and j-reland. 30711910), 288. 38 37 small sticks and beginning with 94, lay one down for each succeeding number, naming the same until a ll were counted. Now te ll them to add 76 to 47 and subtract 28, In addition to the first process, and counting the whole number of sticks, he would withdraw 28 and recount the rem ainder,

(2 ). Some of the Indian tribes of North America used bundles

of sticks to indicate to each other the day on which to attack the

white man. For example, once when the Natchez and the Ohocktaw

wished to attack the French in Louisiana, each tribe received a

bundle of sticks. One stick was to be withdrawn each day; when

the last stick was reached then they knew that the appointed time had arrived.34

(3 )' The Negroes in Africa used bundles of straws and twigs to

show how many women, c h ild re n , and cows had been c a rrie d o ff by

the slave traders. Since, of the three, the cows were the most

valuable property, they were denoted by the longest straws,3^

(4). To facilitate counting, the Tahitians (South Pacific)

tied strips of coco-palm leaf into bundles,3^

33Edwin T. Denig, Indian Tribes of the Upper Missouri (”U. S, Bureau of American Ethnology, annual report" No. 46.[Washington, D. C.: Government Printing Office, 19303), p. 420,.

34Le Page du Pratz, Histore de la Louisiane (Paris: Lambert, 1758 ) , I I I , 241. 35j'piedrich Ratzel, The History of Mankind (New York: Macmillan Co., 1897), II, 328 .

3^Ratzel, op. c i t . . I , 192. 39 (5)« To keep records of one's possessions, the Baganda (who

live by Lake Victoria Nyanza in Africa) used bundles of twigs or

reeds which were three or four inches long.3?

(6), The pygmy Hegritos of the Hiilippines used rice or small

stones in order to perform addition.

(7)* Cotsworth described the 5-bundles form of calendar as

probably the earliest type of record used by our remote ancestors

to count the days. From each bundle (containing 30 sticks) a

stick was drawn each morning to denote the passing day. He

believed that this type of calendar was "the form most natural ly

used by the Bible patriarches, from Noah to Abraham and Isaac. The North American Indians also used this type of calendar, and

as late as 1922 the Red Indians of Northwest Canada were using it

s e c r e t ly .

(8). The Nahyssan tribe of South Carolina used pebbles and

bundles of short reeds to help them in counting. They also piled

up heaps of stones to indicate the number of persons killed in

37j. Hoscoe, "Further Notes on the Manners and Customs of the Baganda," Journal of the Anthropological Institute of Great Britain and Ireland. IDQΠ1T 1902) . 71. ' ~

3 ^ , A. Reed, Negritos of Zambales ("Ethnological Survey Publi­ cations," Vol. II Part I tManila: Bureau of Public Printing, 1904] ) , p . 64. ^^Moses B, Cotsworth, "The Evolution of Calendars," reprinted from the Bulletin of the Pan American Union (June, 1922), 4» 40 battle or the number of emigrants to some distant region.40

(9). The Tupi tribe of South Brazil used the word akayu

(cashew-tree) to mean year because the tree blossoms once a

year. Thus each time the tree blossomed they reckoned their

age by laying aside one stone, keeping the stones in a small basket reserved for this purpose.4^

(10). The New Zealanders recorded the passage of time by add­

ing every month a little piece of wood or a small stone to a h e ap .42

(11). Codrington, writing in I 891 , told that the natives on

Florida j^sland (an island in the western Pacific) used stones and

canarium shells to help them in counting. When they had a feast

some one went around with a basket, and each person present

dropped some small object into it. By counting these objects they knew how many th ey had e n te rta in e d .43

40james Mooney, The Siouan Tribes of the East ("IT. S, Bureau of American Ethnology," Bulletin 22 LWashtngton, D. G.: Government Printing Office, 1894]|), p. 32.

41oharles de Rochefort, Histoire Naturelle et Morale des Isles Antilles (Rotterdam: Reinier Leers, I 68 I ) , p . 73» 42i^. P. Nilsson, Primitive Time-Reckoning (Oxford: University P re s s , 1920) , p . 321.

43e , H. Codrington, The ^lanesians (Oxford: Clarendon Press, 1891), p. 353. Ui (12), Some groups of people inserted wooden pegs into holes

which had been drilled in a tal]y-board.

(a), Mtesa (a ruler in Africa) used a board of this type

in order to keep a record of the number of units of

force in his arnqr.^

(b). The Creek tribe (Indians in the United States) kept a

record of the days of the week by inserting pegs in a

b o ard .T h e Arawak tribe (fbrench Guiana^ South

America) kept a record of the days in a sim il^ manner

except that they used sticks [adda-(ek)kishihi: wood

signs 1 in which holes had been drilled.

(c). The North American Indians played many types of games. Oihs of the ways of keeping score or of keeping a record

of the number of throws was by using a board in which

holes had been drilled. Usually the boards were

arranged in four divisions, consisting of ten places ) *7 each (see Figure 6),

^Ratael, op. cit., II, 328.

^^Schoolcraft, o£. c it.. I, 273.

E. Roth, ^ Introductoy Stu^ of the ^ts. Crafts, and Customs of the Guiana Inmans (*'tj. 3T"Bureau""^ American kthnolo^, annual report," No. 3b LWashin^on, D. G. : Government Printing Office, 19214.']), p . 719. U7stewart Culin, Games of the North American Indians ("U. S. Bureau of American Ethnology, annual report, ” No. 21; [Washington, D. G.: Government Printing Office, 1907]), p. U5. A2

oooooooooooooo

Fig. 6 A Tally Holder Culin, op. Pit.. p. 20A. Permission for use granted by Bureau of American Ethnology, A3 6 . The Word T a lly

T/ifhen the natives on Florida Island counted yams they used a

process which might be classified as a combination of word tally and

object tally. Three men worked together. Two of the men would count

five yams each, making a total of ten. As each ten was counted they

called out "one", "two", and so on; the "one" indicated one ten, the

"two" indicated two tens, etc. For the first "nine" they did not

record the number of tens by tangible means, but only by voice. For

each "ten" they called, indicating ten tens or one hundred, the third

man put dovm a small yam as a tally.^^

SIB'ήARY

As indicated by the preceding listings and discussion, the use of the tally v/as fairly universal from the time of primitive man until the fourteenth century, and in a few cases until the twentieth century.

This device served as a valuable instrument in enabling primitive and illiterate persons — with the exception of the use of tallies in the

English Exchequer — to keep numerical records and to perform simple computations of addition and subtraction. These records were used even by those people who had no words in their language to express the quantities that were being indicated by the tallies.

^^Oulin, op. c it., p. A5« 44

Plate III. Registros de Ramaies

Cadmus was considered by the Greeks to be the inventor of letters. Elysium, in Greek religion, was the happy otherworld for those favored by the gods.

From Leland Locke, The Ancient Quipu or Peruvian Knot Record (New York: American Museum of Natural History, 1923)» on the page preceding the Foreword. Permission for use granted by the American Museum of Natural History. HEGISTROS DE RAMALES

l.rtfe rn A p o liiÿ flir it, t7 ''h "If Cailiuus is in làlysiuin, tlie iiivt'ntor of knot-writing must Lave some place of lionor in tlie land ol souls; and if the spirits in that dreamy country are capable of satisfaction from what passes in this grosser world, some feather- cinctured sage may be rejoicing in the chain of accidents which at this distant period brings his creations once more before the thoughts of living men."—G e x . T . P, Thompsom. 'A CHAPTER I I I

THE KNOTTED CORD (A Form of Tally)

DEFINITION OF A KNOTTED CCRD

The knotted cord is, as the name Implies, a cord, string, or

leather thong in which knots are tied in order to record numbers. This

recording was done in different ways: (1) in some cases the person

tied a knot in the cord as each object was counted; (2) in other cases,

particularly in relation to checking the number of days until a special

event, the required number of knots was tied in the cord first; then as

each day passed, one knot was untied (or cut off), until, at the

untying of the last knot, the person knew that the designated day had a rriv e d .

As mentioned in Chapter II, the simple form of knotted cord is a type of tally and was used to record numbers and to add together quan­ tities from two or more groups. Février said that subtraction was also performed on this device by the natives on Okinawa (an island between the East China Sea and the Pacific)

PURPOSES FOR WHICH THE KNOTTED COED WAS USED

Knotted cords were used for the recording of business transac­ tions, of one’s possessions, and of the passage of time. This device

^James G. Février, Histoire de L’Ecriture (Paris: Payot, 1948), p • 21. 46 h i was also used for keeping records in the taking of census and in the

collecting of taxes, and for keeping biographical and genealogical re c o rd s.

This device was used in many parts of the world and "may be said

to be indigenous to all lands in which the art of spinning, weaving, 2 and dyeing have been cultivated." Just as the use of tallies might possibly have originated in the Neolithic Age of man, so it is be­

lieved that weaving, too, began in this a g e . 3 a crude form of rope

(plaited thongs) was used in a much earlier time.

An interesting variation of a mathematical use made of a knotted U rope was that of the rope-stretchers (harpedonaptae) of Egypt. Accord­

ing to some w r i t e r s , 3 Egyptian surveyors of the fourth century B.C.

made use of a special application of the converse of tie Pythagorean

Theorem. By using a rope knotted in segments whose ratios were 3 : 4 : 5 ,

2 Leland Locke, "The Ancient Quipu," American Anthropologist. N,S* XIY, (April-June, 1912), 325-32.

^"Eope and Ropemaking," Encyclopedia Britannica (1953 éd.), XIX, 545. . ^ 4possibly analogous to the Hindu sulvasutra and to the Semitic mashopha Csolomon Gandz, "On Three Interesting Terms Relating to Area," American kiathematical Monthly. XXXIV (February, 1927), 82.^

%ot all writers are in agreement on this question. For addition­ al references see: Titus Flavius Clement, Stromata, tians. William Wilson (Edinburgh: T. and T. Clark, 186?), p. 397; T. Eric Peet, The Rhind Ivlathematical Papyrus (Liverpool: University Press, 1923), pp. 31-2; Thomas Heath, A History of Greek Mathematics (Oxford: Clarendon Press, 1921), I, p. 122; Solomon Gandz, "Die Harpedonapten Oder 8eilspanner und Seilknupfer," Quelien und Studien zur Geschichte der hlathematik ( 1930), 256- 60, 4 8 they were able to lay off right angles.^ Some historians believed

that this practice was followed in early times by people in Palestine,*^ India, ft and probably China, ft

EEOPIES mo USED THE KHOTTED CORD

A few of the peoples who have used or are using knotted cords (for the purposes mentioned on page 46) were the following:

1. The ancient Egyptians, who lived long before the rope stretchers, were probably familiar with knotted cords. This seems evident from the fact that one of the hieroglyphic symbols consisted of the knotted and looped cord itself,^ Since their hieroglyphic system of writing was already perfected by 3400 one might assume that they had been using this device prior to that date,

2. The Chinese used knotted cords, but it is rather difficult to determine when the practice began. Listed below is the information as given by several authors:

^ era Sanford; A Short History of Mathematics (New York: Houghton Mifflin Co,, 1930), p. 28é.

^Solomon Gandz, "Studies in Histoiy of Mathematics from Hebrew and Arabic Sources," reprinted from the Hebrew Union College Annual. 71 (1929), 275 . %eath, loo, cit. ^Walter James Hoffman, The Beginnings of Writing (New York: Mac­ millan and Co., 1895)» pp. 40, I 40, ^^"Hieroglyphic", Columbia Encyclopedia (1950 ed.), 894 .

"Egypt", Columbia Encyclopedia (1950 ed.), 596. 49 &. De LaCouperie said that prior to the movement of the ancient

Chinese (the Bak tribes) into western China, non-Chinese aborigi­

nal tribes lived near the sea coast of China, These aborigines

used knotted cords.

Chinese legends stated that the Chinese were using this device b efo re 2800 B.C.^^

_c. According to De laCouperie, the oldest statement about knotted

cords is found in the appendix of the Chinese historical work

entitled Yi-King. The statement, attributed to Confucius^ (550 -

47® B.C.), is: "In the highest antiquity, governiiient was carried

on successfully by the use of knotted cords (to preserve the

memory of things). In subsequent ages the sages substituted for these written characters and bonds,"^5 This statement implied

that prior to the invention of writing the Chinese used the

^^Terrien De IjaCpuperie, "The Beginnings of W riting in and around Tibet," Journal of the Royal Asiatic Society, N,S, XVII (I 8 S5) , 422- 423,425. ^3c. T. Gardner "On the Chinese Race," Journal of the Ethnological Society of London. II (I 870 ) , 5»

14Terrien De LaCouperie did not believe th is was the work of Confucius, but that it was written at a later date (De ^aCouperie, op. c i t . . 426).

^^Yi-Xing. the Great Appendix, Section II, Chapter II, p. 313» The sixty-four combinations of strokes in which the Yi-Eing was written were devised by the first Chinese sage, Ru Hsi, about 3322 B.C. Cz.D. Sung, The Syn&ols of Yl-King (Shanghai: China Modern Education Co., 1934)» p . i'x .5 These strokes consisted of unbroken and broken lines, and some authors believed that the symbols of Eu Hsi were pictorial representations of the knotted strings (Gardner, op. cit., g). 50 knotted cord. As yet the date for the invention of Chinese ideo­

grams is unknown. But "The oldest written records date from the

Shang dynasty (o. 1523 - C, 102? B.C.), , , . the elaborate system

of notation used even then argues an earlier origin,Again De

LaCouperie disagreed, for he maintained (as mentioned in that

the noneGhinese aborigines were the ones who used knotted cords,^7

_d. Lao-tze (born c. 604 B.CP.), referring to the dévice in his Tao-

Teh-King, ° a famous classic, said "Induce people to return to

[the old custom of] knotted cords^9 and to use them [in the place

o f w r i t i n g ] , "20 would indicate that the method had been

commonly used and had already been replaced by another method, A e_. In 225 B.C, Oh*eng Kiang Chen wrote about knotted cords being

used for keeping accounts.21

l6ifohinese Literature," Columbia Encyclopedia (1950 ed.), 3S2 , ^?Terrien De LaCouperie, jo^. c it. , 426,

l^some authors doubt the existence of ^o-tze, and also believe that the text of this book must have been composed several centuriei after his supposed lifetim e ["Lao-tze," Columbia Encyclopedia. (1950 ed,), 1095^' But Carus said, "As to the authenticity of the Tao-Teh- Eing and the historical reality of Lao-Tze’s life, there can be no doubt [Lao-Tze, Tao-Teh-Eing, trans. Paul Carus (Chicago: Open Court Publishing Co., l89ôT,' p. 6.]

19chieh-shing.

^^Lao-Tze, op. c i t . , p. 137»

21d , E. Smith, History of Mathematics (Hew York: G-inn and Co., 1925) , I , pp. 139, 552, 51 3. Gandz gave an interpretation of a few passages from the Bible which

he believed referred to the use of the knotted cord in the pre-Chris­

tian era. Some of these were;

In one of the books ascribed to Moses he related a conversation

Judai?^ had w ith Tamar; "He s a id , *¥Jhat pledge s h a ll I give you?* She replied, *Your signet and your cord, and your staff that is

in your hand'"^^ Gandz thought that the cord mentioned in this verse was a type of Quipu note book, or register, that Judah was

carrying with him. He was on his way to Timnath to see his sheep-

shearers, and it would be only natural to assume that he might be planning to settle the yearly accounts. For this purpose he

would take along his account books (the knotted cord)

_b. Writing about 6^0 B.C., Jeremiah said, "Can a maid forget her

ornaments, or a bride her attii*e?"^^ The Hebrew word for attire

(qishshurim) literally meant "knots" or "Imotted strings."

Gandz^^ believed that this verse referred to the bride using a

knotted cord to count the days until her wedding.

29 A son of Jacob.

^3(3enesis 38:lSa (Revised Standard Version).

^^Solomon Gandz, "The Knot in Hebrew Literature, or from the Knot to the Alphabet," Isis, XIV (lÆay, 1930), 200-10.

25Jeremiah 2 ;32 .

26golomon Gandz, op. c it. . 204-5 52 ç_. The prophet Ezekiel (c. 592 B.C.) described a vision of the

ideal temple which appeared to him, A part of this vision was;

"... behold, there was a man, whose appearance was like the

appearance of brass, with a line of flax in his hand, and a 27 measuring reed; ..." In Gandz’ opinion the line of flax was

the "blank note book" upon ifldiich the architect would record his OQ data as he measured his temple.

4« Herodotus (4-Ô4-? - 425? B.C.), the Greek Historian, told of Darius

{558 ? - 486 B.C.), King of Persia, during one of his invasions giving

instructions to the lonians:

Having so said, the King took a leathern thong, and tying sixty knots in it, called together the Ionian tyrants, and spoke th u s t o them: — "Men o f Io n ia , my form er commands to you concerning the bridge are now withdrawn. See, here is a thong; take it, and observe zqy bidding with respect to it. From the time that I leave you to march forward into Scythia, untie every day one of the knots. If I do not return before the last day to which the knots will hold out, then leave your station, and sail to your several homes."^9

5. From the writings of Pliny (23 A.D. - 79 A.D.) we learn that the

Romans used th e k n o tted cord:

. . . Fabius Pictor records in his Annals that when a Roman garrison was besieged by the Ligurians a swallow taken from her nestlings was brought to him for him to indicate by knots made in a thread tied to its foot how many days later help would ai'rive and a sortie must be made. 30

27Ezekiel 40:3. ^^Solomon Gandz, op. c it. , 210.

^^erodotus Melpomene iv. 98 . 30piiny Natural History x. xxxiv. 53 6 . In the middle of the 12th century the Marquesans (Marquesas Islands

in th e South P a c ific ) used a mnemonic device (ta*o m ata) made of se n n it

(plaited straw, grass, bark or palm leaves) with a knot for each

generation in a lineage. This was similar to the genealogical stick

(rakau whakapapa). with knobs to represent the generations, which the Maoris (New Zealand) used. 7. At the ceremonial for settling the preliminaries of a Khond mar­

riage (southern India), a knotted string was given to the siridih* pa

gâtâru (searchers for the bride), and a similar string kept by the

g irl's family. The reckoning of the date of the betrothal ceremony

was kept by undoing a knot in the string every morning.^^

8 . De Goguet, writing in 1775, told about the Negroes on the coast of

Juida who "know nothing of the art of writing, and yet they can calcu­

late the largest sums with great facility, by means of cords and knots

which have their own signification.

9. Von Humboldt^^ in telling of his travels (1799 - l80i^) related that

^^Sir Peter Buck, The Coming of the I>Iaori (Wellington: Whit combe and Tombs Ltd., 1952), p. 26.

32i,eland Locke, The Ancient Q,uipu (New York: American Museum of Natural History, 1923), p. 62", citing E. Thurston, Ethnographic Notes in Southern In d ia .

53Antoine Yves De Goguet, The Origin of Laws. A rt. and Sciences, trans. Robert Henry (Edinburgh: George Robinson and Alexander Donald­ son, 1775), I. p. 224. 3AAlexander Von Humboldt, Personal Narrative of Travels, trans. Thomasina Ross (London: Henry G. Bohn, i8'53), III, p. 88 . 54 knotted cords were found in Canada, in the plains of Guiana (South

America), and in Central Asia,

10, In 1838 the Hawaiiens were s till using a so-called ''revenue book"

which consisted of a rope four hundred fathoDis long (one fathom = six

feet) divided into sections corresponding to the various districts of

the country. The tax collectors kept a record of the taxes to be paid

in dogs, hogs, and other commodities by tying loops, knots, and tufts

of various sizes and colors to lines of cordage. The weak point in the

system was that each division collector had to "go along” with the 35 record to explain and interpret it!

11, The Paloni Indians of California each year selected a certain

number of their tribe to visit the settlement of San Gabriel to sell

native blankets. Each Indian who was sending goods gave a salesman

two cords made of twisted hair or wool. On one of these was tied a knot for every "real" received, and on the other a knot for each blanket sold. Vifhen the sum reached ten "reals" (one dollar) a double knot was made.^^

12, Other peoples who have used knotted cords are tribes in Africa^*^^^

35w, A, Mason, History of the Art of Writing (Hew York: Macmillan C o,, 1920), p. 35.

3%offman, op. cit. . p. 137.

3?David Diringer, The Alphabet (Hew York; Philosophical library, 1951). P* 26, 3%idward B. Tylor, Early History of Mankind (New York: Henry Holt and Co,, 18?8) , p, 155. 55 the Bratyki and Buriats of Siberia|^ and the Japanese.

13. Some of the primitive tribes who were still using the knotted cctrd

in 1951 were the Li of Hainan (an island off the coast of China), the

Sonthals of Bengal (India and Pakistan), and some tribes of the

Japanese Byukyu Islands (between Formosa and Japan)

14. In 1872 the Santals in the wilder parts of Santal Parganas (India)

used knots in four colors of cords in order to take the census. Black

signified an adult man; red, an adult woman; white, a boy; and yellow, a g irl. "The census was taken by the headmen, who, being unable to w rite, simply followed the popular method of keeping a numerical re c o rd .

15 . Many primitive tribes reckoned time by using knotted cords. A few of these were:

_a. The Nahyssan Indians of South Carolina used strings of leather

with knots of various colors. This system proved so convenient

in dealing with the Indians that a South Carolina governor

adopted it for that purpose.

When the Miwok tribe of California decided to hold a dance^

QQ / ■'^'Alfred Maury, "Origine De L'Ecriture," Journal Des Savants. (Avril, 1875), 216-7. ^%iringer, op. cit., p. 26.

^^Smith, op. cit. , II, p. 195. James Mooney, The Slouan Tribes of the East ("H. 8. Bureau of Ethnology," Bulletin 22 [Washington, D. C., Government Printing Office, 1894]), p. 32. 56 the chief would send messengers to the neighboring rancherias,

each bearing a string on which a certain number of knots was

tied. Each morning the chief of the invited tribe would untie

one knot, then when the last one was reached, the group set out

for the dance.

jo. The Negritos of the Philippines, who are pygmies, counted

days ahead by tying knots in a string of bejuco (the cane or

rattan palm), then cut off one of the knots each day.

jd. The Melanesians (who live on Islands in the Western Pacific)

when inviting guests sent each person a cycas frond. The

appointed number of days until the special event was marked by the pinching off or turning down of a leaflet as each day p a sse d .45

_e. Knotted strings of pandanus- leaf (swordlike leaf of screw-

pine family) or coconut fibre served the Polynesians and the

Micronesians for the reckoning of time, and many of their chiefs

wore them around their necks for that purpose.4^

43stephen Powers, Tribes of California ("Contributions to North American Ethnology, U. S. Geographical and Geological Survey of the Rocky Mountain Region," Vol. IIlDWashington, D. 0.: Government Printing Office, 18773), p. 352. 44w. A. Reed, Negritos of Zambales ("Ethnological Survey Publi­ cations," Vol. II, Part 1 [Washington, D. G. : Government Printing Office, 1904]), PR" 13, 64.

45r . h . Oodrington, The Melanesians (Oxford: Clarendon Press, 1891), p . 272 .

4^r4edrich Ratzel, The History of Mankind (New York: Macmillan Co, 1896), I, p. 199. 57 16, The Indians of the Northwest (United States and Canada) used

knotted strings to keep a biographical record of a person's life. This

record was usually started by the mother at the birth of the child and

served not only to record the child's age but to record any unusual

events in his life. The record consisted of a single string with

simple knots arranged in groups. Usually each knot represented a day

and each group of knots a week. Markers (representing special events)

were attached or wrapped around some of the knots; these markers gener­

ally consisted of beads, rags, bark, string, hair, and sinew. One

record Ibund near M erritt, British Columbia was a large ball of string about 180 feet in length tied into more than 6600 sim ple k n ots. 47

Some other ways in which the knotted cord has been used are as a; 1 . record of the results of the h a rv ests, 4®

2 . notice from the tax-collector of the tribute due,4^

3» tax-collector's "account book" — this was used in Palestine in

the 2nd century A.D,4®* 49

4 * receipt from the tax collector,

5» genealogical record — the father made a knot at the birth of each

child and untied the knot if the child died. When a son married.

47j. D. Leechman and M, R, Harrington, String Records of the Northwest (New York; Museum of the American Indian, 1921 ), pp. 5-9»

4%eVrier, op, cit, . p, 22,

49golomon Gandz, op, cit. . p, 210,

^Opevrier, loo, cit. 58 an additional knot was added for his wife. The woven body of the

record was called too.

6. record of deaths — in ancient times on the island of Hiva (South

Pacific), a priest was delegated to register each death by tying a

knot on one of the strings attached to a post.

A few of the tribes that used both the notched stick and the knot­

ted cord were: (a) the Indians on the Island of Titicaca (an island on Lake Titicaca which is between Peru and Bolivia); (b) tribes in

T ib et ; » 54 ( q ) the aborigines (non-Chinese) who lived on the seacoast o f China(d) the Bangala of the Upper Congo River (Africa)(e) the Tang Tung who lived north of Tibet(f) the Sonthâlà of Bengal

(India and Pakistan)(g) natives of the Solomon Islands (Pacific

5^Ralph Linton, The Material Culture of the Marquesas Islands ("Memoirs of the Bernice Kiuahi Bishop Museum," Vol. VIII, No. 5 (konolula, Hawaii: Bishop Museum Press, I923J), p. 443*

^^Adolph Bandelier, The Islands of Titicaca and Koati (New York: Hispanic Society of America, 1910), p. 89.

W. Bushell, "Early History of Tibet," Journal of the Roval Asiatic Society. N.S. XII (I 88 O), 440,473.

^^Diringer, op. cit.. p. 26.

^^Terrien De LaCouperie, op. c it. , 422-3.

^^iringer, op. cit., p. 28.

57s. W. Bushell, cit., 527.

5%. G-. Man, Sonthalia and the Sonthals (Calcutta: George Wyman and Co., 186?), p. 42. 59 Ocean) ^^and (b) the Indians In the Southeastern part of the United States.

SUmÂEY

The simple form of knotted cord was used extensively in the early

civilizations of Egypt, China, Persia, Rome, America, and in many

other countries. This device was also used among primitive groups,

probably beginning in the Neolithic Age and continuing even until the

present day.

Some of the advantages of using knotted cords were: they were

usually small and light in weight, therefore easy to carry; they

provided a permanent record of numerical information; the mathematical

operations of addition and subtraction could be performed upon them; and they could indicate not only the numerical record, but by vary­

ing the colors of the cords or by tying other materials into the knot, could convey the meaning of what the numbers indicated.

5?m, p . Nilsson, Primitive Time-Reckoning (Oxford: University P re s s , 1920) , p. 151. ^^C odrington, op. c i t . . p . 353» ^Ijàmes Adair, Histoiy of the American Indians ed. S. 0. Williams from 1775 edition (Johnson City, Tennessee: Johnson City P re ss , 1930), p* 79 . THE QUIPU

INTRODUCTION

The knotted cord reached its highest state of development in the

quipu^^*^^ which the Incas were using at the time the Spaniards in­

vaded Peru in the sixteenth century (c. 1527).^^ Before the Spaniards

came " . . they of ^eru had no kind of writing, either letters, char­ acters, ciphers, or figures, . . ."6$,66 Lacking written symbols, the

Incas used the quipu as an instrument for keeping numerical records.

DESCRIPTION OE TEÎE QUIPU

The quipu basically consisted of a main cord from which hung a

number of pendent cords in a manner similar to a fringe. The main

Q,uipu was a word of the Quichua language (the language spoken by the Incas) which meant "knot."

The natives of the Carib nations (originally extending from the Virgin Islands east of Porto Rico to the mouth of the Amazon) called this device the cordonclllos con nudos (cords with knots). The Mexican name was nepohualtzitzin. In Chili, where the quipu was introduced in the fifteenth century, it was known as pron (knot), ^^A lfred liauxy (op. c it.. p. 217) said that Wuttke Fuie Sntstehung der Sohriftl established that the quipu was used in South -Ajnerica before the beginning of the 15th century.

^^Father Joseph De Acosta, The Natural and Moral History of the Indies (London: Hakluyt:: Society, 1880), II p. ^05.

Tradition says that in the reign of Huanacauri Pirua, the third of the old kings of Peru in the list of Montesinas, the use of letters was known and the art of writing on plantain leaves, and that the eighty-first king, Tupaco Gouri Pachacuti (c. 1300 A.D. ), prohibited the use of plantain parchment and introduced knotted strings" (Locke, op. c i t . . p. 9û).

60 6l cord varied in length from a few inches to a yard or more and was

usually thicker than the pendent cords. The latter varied in number

from one or two to a hundred or more. Knots were tied in the pendent

cords to represent numbers (see Plate IT).

By varying certain features of the quipu it was possible to record mathematical facts pertaining to many different situations. Some of these features that were varied were the following:

1. The color of the cord.

In relation to this Mead wrote:

These cords, hanging from the main cord, are usually of several different colors. . . . The contrivance is simplicity itself as a supposed case w ill show, . . imagine that the owner of a large number of llamas goes from home, for a lengthy stay, leaving a shepherd in charge. In keeping account of the changes in the flock the shepherd may have the blue cord represent the old male llamas, and the red, the old females. The increase and loss in the flo ck may be shown by other c o lo rs, . . . 2. The character of the knots.

In some cases they used single overhand ties with each knot re­ presenting a unit. In other cases they used a long knot formed by wrapping a series of loops before drawing the cord tight (see Plate V).

The Zunis (Indians from western New Mexico) represented numbers on knotted cords in a different manner. Cushing, in describing their system, wrote:

The simplest knot known to the Zunis was spoken of as the "finger-knot" . . . , because tied by a twirl of the forefinger

^'^Charles Mead, Old Civilizations of Inca ^and (New ^ork: American Museum Press, 1924), p. 9Ô. 62

Plate IV* Example of the Highest

Development of the Quipu.

Prom Locke, op. cit. , frontispiece. Permission for use granted by The American Museum of Natural History Front tftptfrr

E x !U1H»Io o f Iho Di*vc*loj»nn'iit o f flip Q uipu. Q u ip u Xo. I, from ( luiiii :iy. 64

Plate V. Method of Tying Knots

From Locke, op. cit. , p. 13. Permission for use granted by The American Museum of Natural H istory. 8

Fig. 1. Metliod of Tying Knots. 66

alone, as the same kind of knot is now tied by seamstresses. ^ Because tied with a fiiiger, it was not only known as a finger- knot, but came naturally to mean one. . . . The finger-knot, given an additional tw irl by means of the thumb and forefinger together (as used in joining two threads by expert weavers and spinners), befame, with the Zunis, the thumb-knot. . . , Now the thumb-knot, of course, meant five, and this with a single finger-knot before or above it meant four. . . , and with one, two, or three finger-knots after or below it, meant six. , . . seven, and eight.

So also either the knot which required two hands for the tying of it, or the double thumb-knot in one. . . , came to mean "the two hands," and so ten. placing a single finger- knot before or above this double thumb-knot. . . nine was signified, whilst a double thumb-knot followed by a single finger-knot. . . meant eleven, and two double thumb-knots followed by a single finger-knot. . . meant twenty-one ....

It may readily be inferred that a people given to weaving, netting, feather-tv/ining, and embroidery so much as were the ancient Zunis and Peruvians would hit most naturally upon this method of recording.®®

3. The use of shells or beads to group the cords:

By grouping they were able to determine which records referred to

certain villages. Occasionally, when several pendent cords were

grouped together, an additional top cord was used on which to

record the sum total of all the cords in the group, (see Plate

VI).

4. The use of supplementary cords.

The supplementary cords were attached to the pendent cords to

serve as exceptions to the chief record. For example, if in a

census record the pendent string recorded the number of married

men of a certain age, the supplementary cord gave the number of

^%rank Hamilton Cushing, "Manual Concepts," American Anthro­ pologist. V (October, 1892), 300-2. 67

Plate VI. The Strands Are grouped by Means

of Shells or Beads.

Locke, op. cit. , Plate XUII Permission for use granted by The American Museum of Natural History ri,»TK xi.iii

Q llijm N o. XÔ. I Iin S tru iu ls urn (jnm iw d liy M nuiis of Hlinlls o r Mn:ii|>. ûC 69 widowers of the same age (Figure 7. The short cord, extending

from the pendent cord which represents 598, is a supplementary c o r d .)

The position of the knots in relation to the main cord.

The knots representing hundreds were tied in a group nearest

the main cord; in a separate group next below the hundreds’.

group, the knots representing the tens were tied; and next

below the tens’ group, the knots representing the units were

tied. There were never more than nine knots in a group, and

all the knots representing the same order (such as hundreds,

tens, and units) were tied in a horizontal line (parallel to

the main cord) across the pendent strings (see Figure 7)*

Nordenskiold^^ believed that the arrangement of the knots

according to the decimal system (as described above) was

entirely original and undoubtedly invented by the Incas,

Smith agreed somewhat whai he wrote:

. . . It would seem, therefore, that we have here the earliest known decimal notation of the Western World, at any rate the earliest that admitted of easy transportation. For the latter Cthose who work in the domain of the history of mathematics^, the quipu forms a chapter in the extensive history of the abacus .... fsee Chapter ?]*'

69Erland Nordenskiold, "The American Indian as an Inventor," Journal of the Royal Anthropological I n s t i t u t e of Great Britain and Ireland. LÊrTl929)~, 283.

70d , e . Smith in Leland Locke, "The Ancient Quipu," Ameriean Anthropologist, U.S. XIV (April-June, 1912), 325. JQ_

I I O I

I o lO I o I o I o I O

I o o

|00 O

top loo I oo ( oo 1 oo loo loo . loo L loo I loo

r to lo lO l o lo lO lo lo I O ; lo • lo t t o > t o f* to kt to lo % |0

48>

Pig. 7 Numerical Value of Knots in a Qui pu. Adapted from Locke, op. c it.. opposite p. l6. Permission for use granted by the American Museum of Natural History, 71 FeTBiéri- described another way in which the idea of place value was used in recording on kmtted cords. The technique was used by natives in certain mountainous districts of Okinawa (South Pacific

They used four cords, corresponding respectively to units, tens, hundreds, and thousands. With this arrangement they could not only record numbers, but could also perform addition and subtraction by adding or cutting off knots. When the number of knots on a certain cord reached or passed ten, they cut off these ten knots and made a knot on the cord of the next higher order. This latter arrangement of the cords to indicate place value is similar to the method used in many of the abaci as well as in the common decimal notation using

Hindu-iirabic numerals. Because of this use of the principle of place 72 71 value, the quipu might well be designated as a knot-abacus. *

An e n tir e ly d iffe r e n t c a lc u la tin g technique used on th e Okinawa

Islands was described by Simon. This method was also based upon the principle of place value. Simon wrote:

An interesting . . .calculating machine . . . used . . . on Okinawa as la t e as 30 t o 1^0 years ago . . . (jjas] a tassel of rush straw Ipee F igu i’e 8] . . . (made]from a chosen number o f fringes which would be worn in a girdle . . . Its application is as follows: . . . designate the spaces between the five fingers on the hand as A B 0 D, beginning with the thumb space A.

71j'evrier, op. cit. , p. 21.

72solomon Gandz, op. c it., 205 -6

73solomon Gandz, (op. c i t . , 206-7î showed that traces of the knot abacus are still to be found in some of the ancient languages. His interpretation was different from that of most writers. /

F ig . 8 , Okinawa Calculating Device

Von Idmund Simon, "Uber Knotenaohriften und Ahnliche Knotenschnüre der Riukin - Inseln,” Asia Ifejor. I (1924), 659. Permission for use granted by Percy Lund, Humphries and Co, Ltd. 73 For . . . calculation one assigned thousands to space A, hundreds to space B, tens to C, and units to D. To express th e number 1253 . . . lay one fringe of , , , rush tassel . . , into space A, two fringes into space B, five in C, and three into D . . . to add, e.g., . . , 372 . . . add two fringes into space D, seven fringes in C, and three to B . . , since this would crowd space 0 beyond ten fringes, . . . remove ten from space 0 and add one fringe to space B with the result of 1625, 'A

ORIGIN OF THE QBIHJ

As yet no one knows where the Peruvian quipu originated. De

LaCouperie advanced the theory that the custom of the knotted cord

reached the new world through the Polynesians On the other hand

Heyerdahl believed that a race of people who preceded the Inca Indians in Peru migrated to the Polynesian Islands and brought with them many skills. The learned men on the islands, with the aid of a "complicated system of knots on twisted s t r i n g s , "^6 could enumerate the names of all the island’s chiefs back to the time when the islands were first peo­ pled, By allowing approximately twenty-five years for each generation, scientists have estimated that the first settlers arrived about y!,00 -

500 A.D. Brown seem ingly agreed w ith H eyerdahl when he w rote:

There is one more feature of Peruvian culture that is widespread in Polynesia and Micronesia; it is the Inca Quinu.

7Avon Edmund Simon, "Tiber Khotenschriften and Ahnliche Knot­ enschnüre der Eiukiu-Inseln,” Asia Major. I (192^), 659. I This section was translated for the writer of the present study by T. F. Kuechle of Columbus, Ohio.]

75Terrien De LaCouperie, op. c it. « p. A29,

7^hor Heyerdahl, Kon-Tiki (New York: Rand McNally and Col, 1951) , p. 21. 74 or system of knotted cords for remembering facts and especially numbers • . . none had such wide use as th e mnemonic system of knotted cords which in Maori was called tauponapona , , , .77 A third point of view was expressed by Tylor:

This Tthe knotted cord! is so simple a deride that it may for all we know have been invented again and again, and its appear­ ance in several countries does not necessarily prove it to have been transmitted from one country to another.'®

USES OF TEE qUIPU BT THE IHCAS

The quipu was used for keeping more complicated numerical records

and for a greater variety of purposes than the other types of tallies.

These records were essential because of the type of society the Incas

had. Everything in the Empire belonged to the Inca (the ruler).

This necessitated the keeping of records pertaining to the collection

of revenue and to the quantity of materials distributed to individual members of th e so c ie ty . Mead re la te d th ese p ra c tic e s :

At the time of shearing, the wool was all put in public magazines and . . . dealt out by officers appointed for that service, to the women of each family to spin and weave into clothing . . . Accounts of these things were accurately kept by means of the quipu, . . . The quipu was also useful in connection with the collection of revenue.79

Additional governmental uses made of the quipu were the recording of (1) the census (distinguishing between taxpayers, aged, invalids.

7 7 j, M. Brown, The Riddle o f the P a c ific (London: T. F ish er Unwin L td ., 1924), p. 264.

7^Tylor, 0£ . c i t . . p. 155-

7%ead, op. cit. , p. 59* 75 women» and children), (2) lists of arms and troops, (3) inventories

of the contents of storehouses, and (^) the number of wild game animals killed.

Individuals made use of the quipu in order to keep records of

their possessions (numbers of animals, quantity of grain), particu­

larly if they were the owners of a larger number of animls and had

someone tend,lag their flocks. Those who were wealthy enough to have

vassals working on their lands and taking care of their property used

the quipu to determine the tribute their vassals owed them.

THE Q,UIP0CA1/IA.Y0CS

The numerical records kept by the government were so complicated that it was necessary to provide special training in order for persons to be qualified to assume this responsibility. The persons so trained and placed in this official position were called quipocaraayocs. Every village, no matter how small, had at least four quipocamayocs, and the So number might be as high as twenty or thirty.

About 1200 A.D. Inca Rocca established schools (yacha huasi) at

Cuzco where youths were trained and instructed as Amautas (learned men responsible for remembering the history and traditions of the tribe)

^Oynea Garcilasso De la Vega, Royal Commentaries of the Yncas. trans. Clements R. Markham (London: Hakluyt t’- Society, I869 ), II, p. 1 2 3 . 76 81 and as quipocaiuayocs, Another school, a type of university, was

established at Eacarictampu where, as a part of the curriculum, the boys were tau g h t how to use the q u ip u .^

USE OF THE QUIPU BT OTHER lEOPLE

According to Bennett there were other tribes in the Andean region who used the quipu. wrote:

The Araucanians used a quipu for keeping accounts of livestock, for recording events, for keeping track of the number of days of work done, and for indicating the number of days which would elapse before the warriors should assemble. The Pehuenche are said to have sent quipus to their enemies in which the knots in d ic a te d th e amount of blood money demanded fo r some offense . . . The Popayan and the Cooonuco used cords with colored knots to record events or time, and the Choco kept a count of days and months and a record of amounts of certain objects on quipus. Outside of the Andean region, knotted strings are used only in the Guianas, for keeping track of future dates. A knot is tied for each day intervening before the date set. A knot is untied each day, the last one representing the date of the c e le b ra tio n .^3

The Chinese and Japanese used knotted cords (see pages 48 , 49»

50, 54» 55)» Most writers considered the type they used as of a simple form, but Matiry®^ compared their systems quite favorably with the one used by the Incas, He said that the ancient Chinese and

®%ir Clements Markham, The Incas of ^eru (London; Smith, Elder and Co., 1912), p. 142. ^^Fernando Montesinos, Mej^rias Antiques Historeales Del Peru. trans. Philip Ainsworth Means (London: Hakluyt- Society, 1920), p. 64.

^Ven#ell C. Bennett, Mnemonic and Recording Devices ("Handbook of South American Indians, Bureau of American Ethnology,” Bulletin 143» Vol. V Oîîashington, D. C.: Government Printing Office, 1949J), p. 619.

^^Alfred Maury, op. c it. , p. 2l6. 77 Japanese (as well as certain tribes of the Pacific Ocean) used a

combination of strings of different colors in order to designate

certain things. However, Maury did not indicate that their use in

these countries included the principle of place value,

SUMMAET

The Incas of Peru seem to have devised a more complex record system with knotted cords (or Quipus) than probably any other group of people. By varying the structure and the color of the knotted cords and by using other materials in combination with them they were able to determine not only the numbers that the knots represented, but also the object or information to which the number referred. CHAPTER IV

THE FIMJERS (A Form o f T a lly )

INTRODUCTION

Early in man's intellectual development he learned the advantages

of using his fingers as a means of representing and communicating nu­

merical quantities and of performing mathematical computation. The use

of the fingers for these purposes may be said to have progressed

through three basic stages:

1. The simplest and most primitive stage was the one of using the

fingers as tallies.. This stage is comparable to a method used by

children at the present time, which is commonly called "counting on the fingers." Usually one matches in one-to-one correspondence one finger with one (or one group) of that which is being counted. By this process not only is one able to represent numerical quantities, but may also perform the operations of addition and subtraction.

2. The second stage was a more advanced and complicated technique for representing numerical quantities. This stage of development is discussed more fu lly on page 79»

3. The third stage was that of actual mathematical computation, generally called finger reckoning. This is briefly described on pages 100-1.

Finger notation of the second stage was "a method of representing numbers by the position of the fingers or hands, analogous to the 78 79 digital language of deaf m utes.For example, in a system^ used in

th e 13th century, the operator held up his hands so that the fingers were erect, the palms facing outwards. The gestures were as follov^s

(see Plate 711):

(a) On the left hand:

f o r 1 , h a l f - c l o s e th e 5th finger only f o r 2 , h a l f - c l o s e th e 4th and 5th fingers only f o r 3 , half-close the 3rd , 4 th and 5th fingers only f o r 4 » half-close the 3rd and 4 th fingers only f o r 5 > half-close the 3rd finger only fo r 6 , h a l f - c l o s e th e 4th finger only f o r 7 , close the 5th finger only f o r 8 , close the 4th and 5th fingers only f o r 9 » close the 3r d , 4th and 5th fingers only

(b) The same operations on the r i^ t hand gave the thousands,

from 1000 t o 9OOO.

(c) On the left hand;

fo r 1 0 , apply the tip of the forefinger to the bottom of the thumb, so that the resulting figure resembles 6 . f o r 2 0 , the forefinger is straight and is separated by the thumb from the remaining fingers, vfhich are slightly b en t f o r 3 0 , join the tips of the forefinger and thumb f o r 4 0 , place the thumb behind (on the knuckle of) the fore­ f in g e r f o r 50, place the thumb in front (on the ball), of the fore­ f in g e r fo r 6 0 , place the thumb as for 50 and bend the forefinger over it, so as to touch the ball of the thumb f o r 7 0 , rest the forefinger on the tip of the thumb for 80, lay the thumb on the palm, bend the forefinger close over the first joint of the thumb and slightly bend the remaining fingers f o r 9 0» close the forefinger only as completely as possible.

^D. E. Smith, "Finger Notation," Encyclopedia Americana (1948 ed.), X I, 2 2 1 . ^Nicholaus Ehabdas of Smyrna, about 1341» wrote E pistles on Arith- metic in which he described this system. He gave the notation as it had probably been used in Asia Minor for many centuries. 80

Plate 711. Pinger Reckoning

Prom. L o u is K a r p in s k i, The H is t o r y o f A r ith m e tic (Hew York;, Rand i'lcNally and Co., 1925), p. 2% Permission for use granted by Mrs. L. G. Harpinski. X ^

5 0 ^

(

.

FISIJKR KHCKUNIM'i This illustration is from the works of Xoviomagus (Bronkhurst), Df Numéris (Cologne. 1Ô44). Similar illustrations appear in some editions of Recorde’s arithmetic and in Paciuolo’s great Italian treatise of 1494. The Venerable Bede wrote, probably early in the eighth century A .D .. a treatise on the subject, explaining this system. Hundreds on the right hand follow the tens on the left, and thousands arc like/units on the left. 82 (d) The same operations on the right hand gave the hundreds, from 100 to 900,^

THE IKFLTIENGE OF FINGER SY1ÆB0LISM

The plan of indicating numbers by the fingers of one or both hands

is so natural that it was most likely used by early man. Historians

agree that probably all peoples used finger notation at some time in

their development. But there is disagreement as to whether the fingers

were the first physical device to which man turned when he needed a

means of cojjimunieating numerical ideas and of performing certain mathe­

matical operations. The argument pivots on the origin of numeral words in the various languages of the world.

Grimm’s^ axiom that "all numeral words arise from names of the fingers of the handslends support to those who advance the theory that the fingers were the first physical device. Knott seemingly agreed with the idea when he said, "The general theory that the names of the numerals in all languages are connected with the peculiarities of the hand is as highly probable as it is difficult of proof.

^James Gow, Short History of Greek Mathematics (Nevj York: Hafner Publishing Co., inc., 1923)» p. 25. ^Jakob Ludwig Karl Grimm (1785 -I 863 ) was a German philologist, but is probably better known for his collection and publication of German folk tales, commonly knovm as Grimm's Fairy Tales.

^Levi L. Gonant, "Primitive Number-Systems," Annual Report of the Smithsonian Institution. 1892, p. 58 A.

^Cargill G, Knott, "The Abacus in Its Historic and Scientific Aspects." Transactions of the Asiatic Society of Japan. XIV ( 1913)» 35* 83 Koelle also upheld the theory when he wrote:

. . . in the Tartar languages the Numerals were originally not designations for the abstract idea of nuDieric order, but ex­ pressions, still capable of being understood, to mark certain peculiarities of the fingers, with whose help people used to c o u n t.7

Tylor agreed with the theory as far as the words used to express five and above are concerned. He said:

. . . among many and d is ta n t t r ib e s , men w anting to express 5 in words called it simply by their name for the hand which they held up to denote it, that in like manner they said two hands or half ^ man to denote 10, that the word foot carried on the reckoning up to 15, and to 20, which they described in words as in gesture by the hands and feet together, or as one man, and that lastly, by various expressions referring directly to the gestures of counting on the fingers and toes, they gave names to these and intermediate numerals.®

But Tylor did not agree with Grimm in regard to the names that the

more primitive races gave to the numerals 1 to 4. He (Tylor)stated:

The original meaning . . . is obscure. They may have been named from comparison with objects, . . . but any concrete meaning we may guess them t o have once had seems now by modification and mutilation to have passed out of knowledge.9

Du Pasquier^® admitted that it was difficult and perhaps impossible at times to discover the origin of numeral words, but he nevertheless stated as a probable theory that the names for the numbers less than

7s. W. Koelle, "Etymology of the Turkish Numerals," Journal of the Royal Asiatic Society. N.S. Z7I (1884)» 142. %dward B. Tylor, Primitive Culture (New York: Henry Holt and Co., 1889), pp. 2U6-7. 9lbid. . p. 271. ^^Louis-Gustave DuPasquier, Développement de la Notion de Nombre (Paris: Attinger Erères, ^iteurs, 1921), pi 47. 84 five had, for the most part, a concrete origin. In addition he set up,

for these numbers, a symbolic formula:

The name of a number = The name of the co n crete o b jec t from which the Idea originated.

There are other writers who believe that early man began using

two or three objects (such as pebbles, shells, sticks) to assist him

in his counting, and it was not until he reached the stage of develop­ ment in which he was able to count to five that he began using his fingers. According to these w riters, in many primitive languages the words for the first four numerals are associated with concrete objects.

Some examples are:

1, In the Malay and Aztec tongues the number words for "one", "two", and so on, are the same as the words for "one stone, two stone" and so on.^^

2, The natives in the South Pacific use "one fruit, two fruits, three fruits", where those who speak English would use one, two, or three,

3, Similarly, the Javans use "one grain, two grains", and so on, Gow w rote:

, , . men did not arrive at this use of the fingers till they had already made some little progress in calculation without them. That this is the true history of the art of counting is evident if we consider the following facts in order. First, there is hardly any language in the world in which the first three or four numerals bear, on the face of them any reference to the fingers. Secondly, there are many savage languages in which these numerals are obviously taken (not from the fingers

^^Frederick Eby and C, F. Arrowood, The History and Philosophy of Education (New York: Prentice-Hall, Inc., 1942), p. 54» 85 but) from small symmetrical groups of common objects. Thus «two” is, among the Chinese ^ and ceul. which also mean "ears”: In T hibet pakaha "wing”; In Hottentot t*Koam "hand”: and so also among the Javanese, Samoyeds, Sioux and other peoples . . . Thirdly, there are also many savages who, having only a very few low numerals, count to much higher numbers dumbly by means of the fingers.12

Peacock was not interested so much in which man used first -

fingers or small objects - as he was in providing evidence that would support the following theory:

. . . an examination of the structure of numerical language w ill in many cases more completely establish Cthe following general proposition] ; which is, that amongst all nations practical methods of numeration have preceded the formation of numerical languages .^3

An effect even more important than the one upon the number-words

of many languages, was the effect that finger notation has had upon

the bases of number systems. Conant said:

With the exception of a small number of isolated cases . . . it may be laid down as a universal law that every language con­ taining a number system extending beyond 5 reveals the use of one of the three numbers, 5» 10, or 20, as the base, of that system . . . by far the greatest number of uncivilized people perform their reckoning by tens; and . . . with five or six exceptions, all civilized peoples have done the same.14

As long ago as the third century, B.C. the Greeks were wondering as to why ten had been chosen as a base for so many number systems:

Why do all men, both foreign and Greek count in tens, and not in any other numbers? . . . For as this is invariably done it

^^Gow, op. c i t . , pp. 6 -7 .

13George Peacock, "Arithmetic,” Encyclopaedia Metropolitana (1845 e d . ) . I , 371. l^Conant, o£. c it. , pp. 591» 594. 86 cannot be due to chance; for what is invariable and occurs in every case cannot be due to a chance arrangement but must be due to nature , , . is it because there are nine traveling (heavenly) bodies? . . . Or is it because all men have ten fingers? Having then counters of a natural number, they number all other quantities by this number,^5

The latter explanation as given by A ristotle’s followers is the

one accepted at the present time. One might say that decimal arith­

m e tic i s b a sed on human anatom y.

Another possible influence of finger notation is shown in early

numerical notation. Many systems used vertical strokes to symbolize

the first few numbers. Some writers believed that these strokes

represented the fingers. A few systems in which this technique was

used were the Egyptian hieroglyphics,^^ the first numeral forms of the

G r e e k s , and the Roman numerals.

A most interesting (but highly improbable) influence that only

one author (as far as the writer could determine) suggested, was the

one upon the origin of zero. Job of Edessa (c. ?60 A .D . - c . 835

A.D.), in explaining how early mathematicians counted on their fingers, wrote;

l^Aristotle Problems xv. 3. (Aristotle is not considered to be the author of the Problems, but the work is undoubtedly the product of the Peripatetic School — A ristotle's School.)

I6w. w. Rouse Ball, Short Account of the History of Ma them tics (New York: Lîacmillan Co., 1901 ), p. 130.

E. Smith, History of Mathematics (New York: Ginn and Co., 1925), II, p. 47. iQ ibid.. p. 55. 87 Lwhen they reached3 the ntuuber nine [vfhioh would be the fore­ finger ] . . . the movement of mounting-up stops . . . It is after this that • « . an addition begins towards the number ten, as if the number nine wished to maké a kind of link, , . . and so that it might be linked, it reverted towards the number one [the little finger of the other hand] in a circular way. The movement of numbering is thus completed in a kind of cycle. It is for this reason that the ancients invented, as a first sign for this number (ten) the (empty) space between the forefinger and the thumb, formed in a circular w a y . ^9

EEOPLES m o USED FINGER NOTATION

As previously mentioned practically every group of people at some

stage of intellectual development used fingers (occasionally the toes

also) to aid in counting and in computation. In tracing the history

of finger notation one finds the following (finger reckoning w ill be

discussed in more detail in a later section):

1. Ifeny authors believed that Solomon (tenth century B.C.) was refer­ r in g to f in g e r n o ta tio n in Proverbs 3 :l6 when he w ro te, **Length o f days is in her jjVisdom^s] right hand; ..."

2. The practice was common among the ancient Egyptians, Babylonians,

Greeks, and Romans. Their systems were so developed that they could express numbers from 1 to 10,000, and sometimes even larger numbers

3. The Chinese devised a system 21 that was superior to that of the

^Job of Edessa, Book of Treasures, trans. A. Mingana ( Cambridge : W. Heffer and Sons Limited, 1935)^ p. 262. CEncyclopedia of Philo­ sophical and Natural Sciences as taught in Baghdad about A.D. 81?J.

^®Leon J. Richardson, "Digital Reckoning among the Ancients," American Mathematical Monthly, XXIII (January, 1916), 7-13»

2^Sir John Leslie, Philosophy of Arithmetic (Edinburgh: William and Charles Tait, 1820), p. 22^. 88 Romans. Based on the fact that every finger has three joints, and by

using three sides of each finger they were able to rex^resent nine dif­

ferent numbers on each .finger. For example, the little finger signi­

fied units. Using the thumb-nail of the other hand to touch the finger

at the joints, they began at the base on the external side of the

finger to represent 1. Passing up that side they touched the joints

in succession to represent 2 and 3 (see Figure 9)* Then starting at

the top joint in the middle of the finger they passed down the middle

touching the joints to represent 4, 5» and 6. The base of the other side of the finger represented 7, and passing up that side the succes­ sive joints represented 8 and 9»

In a similar manner numbers were indicated on the other fingers.

The next finger signified tens, the middle finger hundreds. the index 22 finger thousands, and the thumb hundred thousands. With this combi­ nation of positions it was possible to advance by signs as far as a m illio n ,

U» The classical literature of Greece and Rome had numerous references to this practice of counting on the fingers. Some representative passages^3 are the following:

a^. The oldest passage is from a play by Aristophanes

(Athenian dramatist) which was produced in ^22 B.O.:

^^Since the Chinese used the principle of place value, this system of finger notation could also be classified as a finger abacus (See definition of abacus on page 10?)• 23see Appendix II for additional references. Fig, 9 Chinese System of Finger Notation 90 Bdelycleon: Then listen iny own little pet Papa, and smooth your brow from its frown again. And not w ith pebbles p re c is e ly ranged, but roughly thus on your fingers count The tribute paid by the subject States, and just consider its whole a m o u n t ;

_b. Suetonius (Roman biographer and historian, 2nd century A.D.)

described the behavior of the emperor at gladiatorial shows:

Now there was no form of entertainment at which he was more familiar and free, even thrusting out his left hand,^ as the commons did, and counting aloud on his fingers the gold pieces which were paid to the victors

G_. Quintilian (Roman rhetorician of the 1st century A.D.), in

discussing the education of an orator, said:

Geometry has two divisions; one is concerned with numbers, the other with figures. Now knowledge of the former is a necessity not merely to the orator, but to any one who has had even an elementary education. Such knovjledge is fre­ quently required in actual cases, in which a speaker is regarded as deficient in education, I will not say if he hesitates in making a calculation, but even if he con­ tradicts the calculation which he states in words by making an uncertain or inappropriate gesture with his fin g e r s . ^7 / 5. Firdausi (935 A.D. - 1020 A.D.), national poet of Iran, in the cel- ebrated satire upon Shah/ Mahmud / made an allusion to finger counting. ° O ft

^^Aristophanes The Wasps 655-657.

25au undignified performance for an emperor since he was supposed to have kept his hand covered with his toga.

^^Suetonius The Deified Claudius v.xxi. Quintilian Institutio Or at or ia i. x. 35»

H. Palmer, "Explanation of a Difficult Passage in Firdausi," Journal of Philology. II (1869), 247-52. 91 6. A 13th century Spanish codex contained a Byzantine-style illustra­ tion of finger reckoning.^9

7« In the time of Fibonacci (12 0 2 ) finger symbols were used especially as an aid in remembering certain numbers in division.30

8. One of the requirements for a master’s degree at Prague (in 138 ^)

and at Vienna (in I 389 ) was to attend lectures on finger reckoning.

9. At medieval fairs in the Orient, and continuing until the present

time, an international system of finger signs was used in order to

facilitate trade. Another advantage of its use was that the dragoman

(an interpreter) could use it secretly when he wished to have one

price with the seller and another with his master.The dragoman and

the seller made this secret bargain by the position of their fingers while appearing merely to be shaking hands.

10. The absence of inexpensive writing material (cheap paper being a recent invention) and the lack of public education systems caused the method of digital computation to continue to be used in England as late as the 17th century.

11. Tylor^^ discussed the languages of several groups of people whose

29vf, T. Sedgwick and H. W. Tyler, Short History of Science (New York: Macmillan Co., 1939)» P* 19Ô.

3% m ith, op. c i t . , p. 202.

31sedgyjick and Tyler, op. c it. , p. 213.

5^Gow, op. c i t . . p. 27 . ^% ilary Jenkins on. The Later Court Hands in England (Cambridge: University Press, 1977/, p. 95* ^^Tylor, 0£. c it. , pp. 247-251» 271. 92 numeral-words revealed a background of finger counting. These were:

a.. South American tribes such as the Tamanacs, Cayriri, Tupi,

Abipone, Carib, and the Muyscas.

the Aztecs of Mexico,

2' the Greenlanders•

à, the Tasmanians (South of Australia)* e^. West Australian tribes,

_f, the Melanesians (South Pacific).

the Polynesians (Central Pacific).

_h, the West Africans,

the Zulu (South Africa),

the Eskimos •

12. The Konkau tribe (California) celebrated their New Year's Day by

having a dance. This was the occasion for settling their accounts and

wiping out all old debts. The members of the tribe not presently

engaged in the dance squatted around the fire in twos and reckoned

their accounts on their f i n g e r s , ^5

13. The natives on the Banks' Islands (Western Pacific) used a curious variation of finger signs. The number of fingers turned down were the ones to be noticed, not those that stood up.^^ 14. The Kaliana (Indian tribe in South America) had but one numeral

^^Stephen Powers, Tribes of California ("Contributions to North American Ethnology. U* S, Geographical and Geological Survey of the Rocky Mountain Region,” Vol. I ll [Washington, D, C. : Government Printing Office, 1877]), p. 438. 3 ^ , H, Codrington, The Melanesians (Oxford: Clarendon Press, 1891), p . 353. 93 word, xaeyalcan. which th e y rep eated as th ey counted on t h e i r fin g e rs and then on their teeth.

15. The Aymar a (South American Indian tribe) not only counted their fillers but also counted the spaces between them,^^ 16. A more modern use of finger notation is the one that was being used as late as I 925 on the floor of the Chicago Board of Trade.

Plate VIII shows the position of the fingers to indicate various prices, Karpinski described the procedure as follows;

%ice is indicated always with reference to the last sale, by the hand held with palm toward the buyer and with palm outward for a broker trying to sell. A broker wishing to buy $000 bushels of wheat at IO 6 5/8 holds his five fingers outstretched, palm in. Another broker nods acceptance: the buyer indicates 5000 by a single finger held vertically.

17 » Other peoples who used finger notation were:

a_. the natives of the F iji Islands (Southwest Pacific Ocean) b. the Arabians.^^

^"^Walter Edmund Roth, An Introductory Study of the A rts. Crafts. and Customs of the Guiana Indiana ("United States Bureau of American Ethnology, annual report," Vol. XXXVEII [Washington, B.C.: Government Printing Office, 192A]), p. 719» 38vjendell C. B ennett, Mnemonic and Recording Devices ("Handbook of South American Indians," Bulletin 143, Vol. V [Washington, D.C.: Government Printing Office, 1949]), p. 614. 39Karpinski, op. cit., p. 25.

AOMoses B. Cotsworth, "The Evolution of Calendars and How to Improve Them." Bulletin of the Pan American Union. (June, 1922), p . 2 .

^^Karpinski, 0£. cit.. p. 23. 94

Plate VIII. Modem Finger Reckoning.

From Karpinski, op. pit. . p. 25. Permission for use granted by Mrs. L. C. Karpinski. Even C e n t

$ Cent /o 96 c. the Turks.4':

_d. the Japanese.^

Richardson^^ stated that no trace of finger computation (of the

type discussed in this study) had been found among the Hindus. This

statement may be true if the word "Hindus" is accepted in its limited

meaning and not considered as including all the people of India. In

early Buddhist literature there were references to finger notation:

In the art of calculation by using the joints of the fingers as signs or m a r k s , ^ 4 i n the art of arithmetic pure and s i m p l e , 4 5 in the art of estimating the probably yield of growing c r o p s , 46 and in the art of writing, 0 King, the beginner is clumsy. 47

The above quotation is from The Questions of King Milinda

(Milindapanha) which w^s w ritten in Northern India about the beginning

of the Christian era. It is the only work composed among the Northern

Buddhists which is regarded with reverence by the Orthodox Buddhists

of the southern s c h o o ls . 4® In discussing the Milinapanha and other

42iùiott, loc. cit.

43Richardson, loc. cit. 44”Muddâ . . . 'the finger-ring art, so called from seizing on the joints of the fingers, and using them as signs.*" 45«Gamna . . . 'the art of unbroken counting,' . . . probably mftariH arithmetic without the aids involved in the last phrase. We have here in that case an interesting peep into the progress of arithm etical knowledge. When our author wrote, the old way of count­ ing on the fingers was s till in vogue, but the modern system was coming into general use."

4^«Sank^, literally 'calculation.'"

4 7 ?. Max Muller (ed. ), "The Questions of King Milinda," The Sacred Books of the East. Vol. ZXXV (Oxford: Clarendon Press, 1390), pp. 91-2. 4%uller, op.. cit.. pp. xi - xii. 97 Buddtiist literature, Datta said:

Thus as early as the fifth or the sixth century before the Christian era the Indians were accustomed to distinguish between three classes of arithmetic; (1) mental arithmetic Cgananal, pure and simple; (2) arithmetic with the use of the fingers [ mudra or mudd^j; and (3) higher arithmetic in general [ sankhyana or sankha] .... It is also noteworthy that the above classification of arithmetic is met with in B uddhist w ritin g s , nob in o th e r In d ian literatures. 49

limiTSES ABOUT î'INGER SYMBOLISM

A system so widely practised throughout the world was certain to

become a topic in many books. The earliestw ritten instructions on finger symbolism that are available today, the Romana Comnutatis. were written in the seventh century A.D. This was a short pamphlet whose chief value was that it served as a source of knovjledge for the better known writings of the Venerable Bede (English scholar, historian, and theologian who was born c. 6?3 A.D. and died 735 A.D. ). A part of the f i r s t c h ap ter of B ede's De Temporum R atione^^ (725 A .D .) gives present historians almost the only knowledge of the finger reckoning or symbolism used in western Europe during the Middle Ages.

Next to Bede's writings the best source of information about

^■^Bibhutibuhusan Datta, "The Science of Calculation by the Board," American Mathematical Monthly, XXXV (December, 1928), 524-5*

^^Charles W. Jones, Bedae Pseudepigrapha: Scientific Writings Falsely Attributed to Bede (Ithaca, N. Y. : Cornell University Press, 1939), p. 54.

^^Ohapter II of Florence A. Yeldham, The Story of Reckoning in the Middle Ages (London; George G, Harrap and Go. Ltd., 1926) included a translation of this chapter. 98 finger notation and computation is that of Nicholas Rhabdas,^^ a Greek

w riter from Smyrna. About 1341 A.D., while in Constantinople, he

wrote Epistles on Arithmetic. I n these letters he gave an account

of the digital notation as it had probably been used in Asia Minor

f o r many c e n tu rie s (see page 79 ).

A third man whose work was important in this area was a Bavarian, Johannes Aventinus (Johann Thurnmayer), who was bom in 1477 and died

in 1534. His book contains the most complete explanation extant. The

title is Abacus atique vetustissima veterum latinorum -per digit os manusq numerandi . . . coseutudo. Ex beda ou picturls et imaginabus (published at Nürnberg in 1$22).

The subject of finger symbolism^^’^^ was also treated in the books on arithmetic written by the following men:

1 . Ehabanus lïarus.57 In his book an elaborate explanation was given

^^Also called Rhabda or Artabascda,

53lnco rpor at ed in the work of Nicolaus Gauss inus. De eloquent ia sacra e t humane (P a ris, I 636), IX, Ch. 8 , pp. 565- 8 .

54j’or the Greek text with French translation consult Paul Tannery, "Notice sur les Deux Lettres Arithmétiques de Nicolas Rhabdas," Notices et Extraits des Manuscrits de la Bibliothèque Nationale. Part I XXXII (1886). 142-243. 55j>or additional references see the footnote on p. 200 of Smith, op. c it., II, and D. E, Smith, Rara Arithmetioa (Boston: Ginn and Co., 190877" ^^Another notable account is contained in the Persian and Arabic lexicon of Ghiyés ud din Mohammed of Rampore (translated into English by E. H. Palmer). 57On Reckoning, issued in 820 A.D. 99 o f f in g e r reckoning a f t e r th e old Rojnan p la n .^ ^

2. Luca Bacioli^^ (bora in Tuscany c. 1M5; died o. I 5I 4 ). This book

was a summary of all the knowledge of his day on arithmetic, algebra,

and trigonometry, and was the first printed work to illustrate the finger symbolism of number,

3 . Moss en Juan Andrews.

U» Johann Noviomagus^^ (Jan B ronckhorst Neomagus) born a t Nijmvegen in

1494 and died in 1570. His book set forth the finger notation as ezplained in the works of Bede.

5 . Roberts Recorde^^*^^ (born in England c, 15IO; d ied 1558 ).

6. Joannes Pierius Valerianus Bellunensis.^4 7 . Juan Perez de Moya^^ (born in San Stefano in the Sierra Morena in the first third of the sixteenth century). This was the most

^^llwood P. Cubberley, History of Education (New York: Houghton M ifflin Co., 1920) , p. 158 .

^9guma de Arithmetics Geometria Pro-portioni e_ Proportionalita. V enice, 1494»

^Qsumario breve de la pratica de la Arithmetioa. Valencia, 1515.

^^De N um éris. L ib ri I I , P a r is , 1539.

Grounde of Artes. London, c. 1542.

^^In Robert Steele, Earliest Arithmetics in English (London: Oxford University Press, 1922) , on piages 66-9 is reprinted a portion from the 1543 edition of Records’s arithmetic (The arte of nombrynge by the hande).

^^Hieroglyphlca, p. 454 (Frankfort a.M., I 556)

^SArithmetica Hractica, p. 627 (Salamanca, I 562). 100 noteworthy book on mathematics published in Spain in the l6th cen tu ry ,

FINŒR EECKOÎÎING

According to Records, whose book was mentioned above, the fingers were used for notation only and not for computation. He wrote:

And as for Addition, Subtraction, Mu].tiplioation, and Division (which yet were neuer taught by any man as farre as I do knowe) I wyll enstruct you after the treatyse of fractions, ' But most authors disagreed with this point of view and either stated that the reckoning was done on the fingers or gave the rules for the operations.

Addition would not be difficult to perform as long as the numbers did not become too large and complicated. When a series of numbers was to be added, the fingers indicated the first number, then the sedond number was added mentally and the fingers changed to a position to indicate the sum. This procedure continued until the final sum was obtained. Subtraction was a process similar to addition, except that the two numbers were subtracted mentally. In both cases just the sight of the fingers was an aid to the operation.

M ultiplication was the most important operation for which the fingers were used. By this method a person had only to know the mul­ tiplication table as far as $ x $ and could avoid learning the table

^^Smith, Rara Arithmetics, p, 309. ^^Steele, op., c it. , p. 63, citing Records, Grounds of Artes. p. 137. 101

as far as 10 x 10, Eves described a method of m ultiplication;

, . « to multiply 7 by 9» raise 7-5=2 fingers on one hand and 9-5=4 fingers on the other hand. Now add the raised f in g e r s , 2/4=6, for the tens digit of the product, and multiply the closed fingers, 3 x 1 - 3 for the units digit of the product, giving the result 63

This complementary plan of m ultiplication was found in many l6th

century books, and is s till used among peasants in certain parts of

Russia and Poland, Smith^^ described a variant of this method which

until recently was used in certain towns in Russia.

In relation to the use of the fingers as a means of m ultiplica­

tion, Richardson related an interesting situation which s till

existed as late as I 9 1 6. He wrote:

Dacia, which in ancient geography corresponds in the main to modern Roumania and Transylvania, became a Roman province under Trajan. It was in fact governed from Rome from about 101 to 256 A.D. , , , The Roman occupation, though comparatively short, may still be traced in , . . the language . . . of the Roumanians.

Interestingly enou^, the Walachian peasants, who dwell in southern Roumania, have preserved an old method of m ultiplication on the fingers. How old the custom is, or whence derived, no one can tell. It can hardly be of local origin, for it is not absolutely unique. It may hark back to oriental or Greek sources, for an algorism manuscript of about 1200 A.D., now preserved at Heidelberg, contains something sim ilar . . . Again it may be of Roman origin. The last theory is supported by the fact that sim ilar usages, nrobably of Roman origin, have been noted among Prench peasants?^ . . .

^®Howard Eves, Introduction to the History of ^Mathematics (New York: Rinehart and. Go.Inc.V 1933), p. 24.

^^Smith, History of Mathematics, II, p. 120.

7°Leon J. Richardson, or£., oit,. 11. On pp. 11-3 he explains the process discussed in the above section. 102 smajAKT

The fingers were perhaps the earliest and most widely used form of tally. They have had marked effects upon the number-words of many languages and upon the bases of the majority of number systems.

Most people have used the fingers as a form of tally, but the

Chinese are credited with developing a form of finger notation that was based upon the principle of place value. CHAPTER V THE ABACUS

INTRODUCTION As nas pointed out in Chapter 11^ the tally served as a very convenient device for man to use in recording and communicating numerical quantities and in performing simple arithmetical computa­ tions. The earliest type of tally was undoubtedly one in ehich there was a one-to-one correspondence between one of the form of tally being used (such as one pebble, one mark, or

C o MTAtMEÏ2.t3

e . a. O O O TEW6, UNITt>

HUNPCE06 TEW5> u w irt.

4 .

>• O o o UKIITt? T6Nλ HUMOEeO^.

UN»Tt> T&Nt> HUNOeEPb

C. UNITt? o

T&M5> o TENf? o H U N D eEO f» HVJK)D12E.Pi>

d, HUMOC&oi? O h . HUNDREDS,

o TEN t,

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Fig. 10. Place Value 106 ti#* upon the primcdfle of place vaLie. On page 15, the atatemeat iraa made that @ae of the conditions set up for classifying a recording and computational process as a tallying process is the absence of the use of the principle of place value. In contrast, the basic condition set up for a device to be classed as an abacus is that the principle of place value be employed. In order better to understand shat device is an abacus, one should also know shat is not an abacus. Two examples of the latter are: 1. Assume that one uses counters of three different sizes; the smallest represents units, the medium-sized represents tens, and the largest represents hundreds. Then, no matter shat arrangement is used, a definite number of each type of counter would represent a definite number. For example, in each of the illustrations given below the counters represent 213: V o’ o o o o O O o o t o o

2. Assume that one uses counters of three different colors; yellow represents units, green represents tens, and red represents hundreds. Again, no matter shat arrangement is used, a definite number o f each type o f counter represents a d e fin ite number. In each of the illustrations below, the counters represent 213% ® 5E.

® ® <2> © © © © @ ® © ^ ® (g) (E)

The devices illustrated above are not abaci because the counters representing units, tens, and hundreds are not arranged in that order proceeding from right to left, or from left to right, or from top to bottom, and so on.

DEFINITION AND ETYMOLOGY OF "ABACUS"

In this study an abacus w ill be defined as a device with which counters are used to represent numbers and to perform the simple arithmetical operations, based upon the principle of place value; the true value of any counter, or of any number of counters, w ill depend upon the column (or row) occupied in relation to the unit column (or row ). The original forai of abacus (the dust abacus) is believed to have developed from the technique (described on page 104) of drawing lines upon the ground to separate the units', tens', and hundreds' groups of pebbles. From this form evolved those having more intri­ cate structures. In explaining the meaning of the term "abacus,"

Smith classified abaci as three standard types — the ancient dust board, the table with loose counters, 2 and the table with counters

E. Smith, History of Mathematics (New York: Ginn and Go., 1925), H , p. 157. 2 See pages 109-11 for a discussion of counters. 108 fastened to the lines. But limiting the meaning of abacus to these three forms is not entirely satisfactory since these forms do not include a ll types of abaei^ Several etymologies of* the word abacus have been given. Two of the more commonly accepted ones Are that (1) the word was adopted

> from the Latin, abacus . which was formed on the Oreek, ( a boaz^ or slab)^; and (2) the word refers to a Semitic word, abq (d u st).4

The original meaning of "abacus" as used in the English language (1337 A.D.) was* A board or tray strewn with sand, for the delinea­ tion of figures, geometrical diagrams, etc.^ Equivalent words for abacus in some of the other languages are: Chinese: swan-pan, suan pan, swan-p'an, swanyan Southern China* soo-»pan Calcutta (used by Chinese): swinbon literary name: chou-p'an (ball table or pearl table) Japanese* soroban Russian* Tchotu, tschotu, s'choty Turklaht coulba Armenian* thoreb Common Sanstoit tenu* path!

3wgbacus." Oxford EmgÜsh Dictionahv fl933 ed.) I , 5* Asmithj op. c i t.. g . p. 1 5 6 * ^"Abacusj" Oxford English Dictionary (1933 ed.) I, 5 . 109 French» abaque boulier compteur (more commonly ueed in French schools) Spanish» abaeo Ita lia n » abbaco Gezman: rechenbrebt

DEFINITION AND ETIMOLOQÏ OF nCOUNTER» In the Oxford English Dictionary counter is defined as "a round piece of metal, ivory, or other material, formerly used in performing arithmetical operations.**^ In this study a counter (a) w ill not be limited in shape; (b) will be extended to include knots on a cord, fingers, and marks or scratches; and (c) will be discussed as it is used in recording numbers as well as in the performing of arithmeti­ cal operations. Some of the materials that various groups of people have used for counters were pebbles, ivory, metals (such as bronze, copper, brass, and lead), colored glass, small silver coins, gold coins, wooden beads, hozn, baked clay, and bone. The first pieces of metal used for computing counters (or jettons) during the middle ages were round, flat, well polished, and quite plain. During the 14th century the French originated the idea of en- gzmving the counters with flow ers, leaves, and other ornaments.^

f"Counter,** Oxford English Dictionary (1933 ed.), II, 1058. ^Thomas Snelling, A View of the Origin. Nature, and the Use of Jetons or Counters (London» [no publisher J , 17^9), p. 4. 110 About the close of the 15th century, in the Low Countries, the practice began of giving jettons as New Year's gifts to government officials. These jettons were of silver and copper. Prance imitated this custom and in 1769 was still giving gold jettons to the King, to the Royal Family, to the Princes of the Blood, and to many high officials in the govezrunent. A One author, Bubnov, claimed that the Greeks were familiar with the idea of zero in the form of a blank stone or counter for reckoning. In addition, he stated that when the abacus made its appearance in western Euix>pe in the 10th century: The counters of the abacus were strung on a cord when not in use so that they would not be lost and thus each counter, whether blank or with a symbol thereon, had a hole in it. These blank counters with holes in the centers were put in blank columns (for convenience, not of necessity) and from them the zero was derived, being simply the written arithmetical sign for this blank counter, or rather the hole.° Vihen tracing the derivation of the word "counter," one finds that

it was adopted from Anglo-French counteour or countour. Equivalent words for counter in some of the other languages are: Latin: aera (bronzes) calculi (pebbles, marbles) abaculi projectilia (pro, forward / .jacere. to throw) denarii supputarii (computing pennies) calculi supputatorii tesserae

% arriett Pratt Lattin, "The Origin of our Present System of Notation according to the Theories of Nicholas Bubnov," Isis. (1933), 131-94. 9lbid. XIX, 133. lOiiGounter," Oxford English Dictionary (1933 e d .) , I I , 1057. I l l French: jetons (jéter, to throw) (variations in spelling: gects, jectoirs, gietons, jettons, gectz, getoers, getoirs, gettoirs, gettenrs, jectoer, jetoir, giets, and gitones)

Chinese: (literary name) chou (pearls) tse (son, child, grain)

German: zahlpfennig (number penny) tand (reckoning penny) rechenpfennig (reckoning penny) raitpfennig

Japanese : tame (ball)

G reek: t a l i

Dutch: telmarck (count-marker) ^ legpenning (laid-penny) leggelt (laidScoin) werpgeld (cast money or thrown money)

Spanish: contadore g ito n

TYPES OF ABACI

Many forms of abaci^ have been used by man throughout the cen­ turies, and it is possible to catalogue these forms on different bases.

In this study the classifications w ill be made according to the follow­ ing categories: (l) materials, and (2) structure.

On the basis of the materials used in their construction, the better-known types were the following:

1. Sand or dust abacus.

a^. Lines drawn in sand or dust on the ground.

b. Lines drawn in saœd or dust sprinkled on a board or table.

11 See Chapter VIII for a discussion of some contemporary forms o f a b a c i. 112 2, Cloth abacas (Reckoning cloths used with loose counters). 3» Knot abacus. 4* Finger abacus. 5. Paper^2 abacus (tableaux a colonnes). 6. Rod-and-bead abacus. 7. Groove-and-button abacus. Ô. Wax-tablet abacus (used with stylus). 9. Marble abacus (used with loose counters). 10. Wood abacus (used with loose counters). On a basis of structure, abaci may be classified according to : 1. The method of indicating places. This was done by: a. Lines (horizontal, vertical) Grooves, (horizontal, vertical). Rods (horizontal, vertical). Spaces (horizontal, vertical). b. Combination of lin es and spaces (horizontal) Combination of lin e s , spaces, and lyers^^ (horizontal). Neither lines nor spaces (l«yers used only) [ see Figure 12.] Neither lines nor Lyers (no guides used to help the eye).

^"^he word paper is used in its extended meaning to include parchment, vellum, and similar materials. ^^Stationary jettons (see Figure 11) used to guide the eye. These were often called the Tree of Numeration [ F. P. Barnard, The Casting-Counter and the Ccuntiha Bohird (Oxford: Clarendon Press, 1916), pp. 237, ^ -7 1 , and PlatelBEF] . 1 1 3 uveKV

Fig. 11. From F. P. Barnard, The Casting-Counter and the Counting-Board (Oxford: Clarendon Press, 1916), p. 2^7. Illustrating Awdeley's technique of using lyers in combination with lines and spaces. Permission for use granted by Oxford University Press, Inc,

m

a n y cx>

Fig. 12. The board shows f 3647 13 s. 9 d. 1 shilling(s) = 12 pence (d)j /-I * 20 8. From lean Trenchant, L|Arithmétique de lean Trenchant . . .avec l'A rt de Calculer aux Gettdns(Lyon: Claude Rigaud et Claude Obert, 1631)> p.3^9. Illustrating Trenchant's technique of using lyers without lines. Trenchant used overlapping jettons.

Figs. 11-12. Medieval Counting Boards. 114

2. The arrangement and number of counters: a. Limited number of counters on each rod or groove. (1 ). Four beads cm fir s t and fourth rod and ten beads on each of the other rods. (2 ). Five beads (one bead in upper groove and four beloir); additional grooves with beads representing fractions. (3 ). Five beads (one bead above partitioning bar and four below). (4 ). Six beads (one bead above partitioning bar and fiv e below) . (3 ). Seven beads (two beads above partitioning bar and fiv e below) . (6) • Ntee beads. (7). Ten beads. b. Unlimited number of counters on each line or space. In general, the materials used in making abaci had little or no effect upon the techniques used in recording numerical quantities or in performing simple computations upon the abaci. The chief factor in detezmining recording and operational techniques was the structure of the abaci — that is, (1) the method of indicating the places (un its, ten s, hundreds) o f the number system; and (2) the arrangement and number of counters. 115 Recording Technlau# ^ tha reebraing of a namerlcal quantity. (Each abacus mdll show 5,073) As far as the recording of a numerical quantity is concerned^ there are fundamentally three different techniques. On this basis the historical forms of abaci may be classified as follows: 1. a. Those abaci having a limited number of counters and a parti­ tioning bar or qpace^^ (see Figure 13 — although the diagram is that of the Chinese abacus, these techniques also apply to the Roman abacus (see Blate IX), and to the traditional and the contenporary forms of the Japanese soroban. b. Those abaci having an unlimited number o f counters and using a combination of lines and spaces (Figure 1U — the medieval _JïounyngLbo§rd).„ $060 ÇOO ÇO ^ Sf»c«p

^On Abaci having partitioning bars (e.g., the Chinese or Japan­ ese abacus) or separating spaces (e.g., the Roman abacus)only the counters near the bar or^pace are the counters to be considered in detenuining the number represented. This position is called the indieated-number position. ^^Qn abaci having divided vertical rods the atoclgpile position is the position at the top and the bottom of the frame. 116 2. Those abaci having a limited number of counters and either nine or ten beads (see Figure 15 — the diagram shows the ten-bead abacus, but the same technique would apply to the nine-bead abacus.) — Persian abacus, b. Those abaci having an unlimited number of counters and using lines or spaces only (see Figure 16) — Egyptian abacus.

’HiflUMnis fans unii's

tens

lb hum be f poii p o S ( ‘ fion

Fig. 15 Fig. 16 3. Those abaci having an unlimited number of counters on which 17 numerals were indicated (Figure 17) — Gerbert's abacus.

f e n s unifi !■

Q> G>

Fig. 17

^^On abaci having horizontal rods the stoclgile position is the position on the right side of the frame. 17See footnote 7 on page 298 • 117 Operational Technique the addition^^ of two numbers.

EXAMPLE: 62 / 53. As far as the addition of numbers is concerned there are funda­ mentally seven different operational techniques. On this basis the historical forms of abaci may be classified as follows: 1. Those abaci having two beads above the partitioning bar and five beads below (modem Chinese abacus).

Fig. 13. Indicate the Fig. 19. Indicate the second first number (62) number (53) by pushing down the second bead on the tens' rod above the bar, and pushing up three beads on the units' rod below the bar. This operation results in the positions shown in the above diagram.

18. Since the primary puipose of this description is to show that computation on an abacus is determined to a great degree by the struc­ ture, only the operation of addition as performed on the different types of abaci will be analyzed in detail. [jSome methods described in the present study would probably not be used by professional or eaperb calculators.3 The other simple arithmetical operations (sub­ traction, multiplication, and division) could be analyzed in a similar manner. For sources of descriptions of those other operations on the different forms of abaci see footnotes 7$(Roman abacus^ 110(suan pan), U2(soroban) in Chapter V, and Appendix III (medieval counting board). lid

-- T

Fig. 20. Since there are Fig. 21 now 5 beads on the units* rod below the bar, these are pushed down to the stockpile position an# ex­ changed for one bead above the bar on the units' rod. The resu lt is shown in Figure 21.

-1

Fig. 22. Since there are Fig. 23. The final re­ two beads above the bar on su lt i s one bead below the tens' rod, these are the bar on the hundreds' pushed up to the stoclg>ile rod, one bead below the position and exchanged for bar on the tens' rod, and one bead below the bar on one bead above the bar the hundreds' rod. The on the units' rod. This resulting positions are arrangemwt indicates 1 1 5 . shown in Figure 23. 1 1 9 2. Tbos« abaci having on* bead above the partitioning bar and five beads below the partitioning bar (traditional Japanese abacus).

Fig. 2 4 . Indicate the f ir s t number (62)

Fig. 2 5 . Indicate the 3 units of the second number (53) by puablng up from the stockpile three beads below the bar on the u n its' red. The result is shovm in the above diagram. 120 ±

-L

Fig. 26. Since there is only one five-bead on the tens' rod and this one has been used in representing the first number (62), the five tens of the second number (53) can be added by using the complementary method.That is, add 100 and subtract 50. This is done by bringing up from the stockpile one bead below the bar on the hundreds' rod, and pushing up into the stockpile one bead above the bar on the tens' rod. The result is shown in Figure 2?.

Fig. 27. ^^viTith abaci having five beads below the bar, a method other than the comp lament ary method is theoretically possible. But this other method is rarely used. To use this method proceed as follows from the positions shown in Figure 25: 1. Push up ffom the stockpile on the tens' rod the four beads in the stockpile, counting ten, twenty, thirty, forty. 2. Now, there are five beads below the bar on the tens' rod along with the one five-bead above the bar (on the tens' rod). This combination totals one hundred, so the counting is interrupted while an exchange is made. These six beads on the tens' rod (the five-unit beads and the one five-bead) are pushed back into the stockpile position and replaced by one bead below the bar on the hundreds' rod* 3. The counting is resumed by bringing up one bead from the stockpile on the tens' rod below the bar. This makes a total of fifty and the result is the same as Figure 2?. (The five unit-beads on th,ft^ units' rod are exchanged for the one five-bead on the units' rod, as was designated for Figure 28.) 121

-- T

:r T

Flg. 2d. Sixica there are five beads below the bar OQ the units' rod, push these beads back to the stoclQ>lle position and replace them by pushing down one bead above the bar on the units' rod. The result is ahom In Figure 29.

Fig. 29* The result is one bsad below the bar on the hundreds' rod, one bead below the bar on the tens' rod, and one bead above the bar on the u n its' rod. 122 3* Thos» abaci having one bead above the partitioning bar (contem- porary toxm. of Japanese abacus) or space (Roman abacus) and four beads below the partitioning bar or space.

Fig. 30. Indicate the first number (62).

- - T

— T

Fig. 31* Neither part of the second number (53) can be in­ dicated simultaneously with the first number (62). (There are only two beads in the stockpile on the units' rod below the bar, and no beads in the stockpile above the bar on the tens' rod.) Therefore, a coiiplementfiry method must be used. To add the 3 units, add 5 and subtract 2. This is done by bringing down to the bar one bead above the bar on the units' red, and pushing back into the stockpile the two beads below the bar on the units' rod. The result is shown in Figure 32.

50 The word "space" as used here refers to the space separating the unit beads from the five-beads, the ten-beads from the fifty- beads, and so on. 123

Fig. 32.

- ±

Fig. 33» To add the 5 tons (of the 53), again use the conplemenbary- method. That is, to add 50, add 100 and subtract 50. This is done by bringing up to the bar one head below the bar on the hundreds' rod, and pushing back into the stockpile position one bead above the bar on the tens' rod. The result is shown in Figure 34. L24

Fig, 34* The result le one bead below the bar on the hundreds' rod, one bead below the bar on the tens' rod, and one bead above the bar on the units' rod, 4* Those abaci having an unlimited number of counters and using lines or spaces only, lluxlvftit fntts,

Fig* 3^* Indicate the first number (62). \ I j few* unifi |

Fig, 36, Indicate the second number (53) simultaneously with the first number, (This is possible because of the unlimited number of counters). 125

Flg. 37. Vlh«n tho counters ozx both the tens' and units' rods are pushed together one notes that there are 11 counters on the tens' rod. Remove 10 of these counters and replace them with one counter on the hundreds' rod. The result Is one counter on the hundreds' rod, one counter on the tens' rod, and five counters on the units' rod. N0T2: With this form of abacus there is no need for the use of the cony^lementaxy method except as a short­ cut. The two numbers may be Indicated simultaneously, and by being able to use the principle of exchange one has a simple and easy technique for adding numbers. 5. These abaci having nine or ten beads. (For the Illustrations the ten-bead abacus will be used, but the techniqu^ea w also app]^ to the nliie-bead abacus.)

------imiiiHi

MP«II IMI

uni ^5 J| 11 llll 1 III Flg. 38. Indicate the first number (62).

y 21q|j abaci having nine or ten beads horizontal rqds, only the coAters against thé left side of the frame are the counters to be considered in determining the number represented. 126

+e»»5

39. The 3 units of the second number can be indicated by pushing from the stockpile three beads the units' rod. The result is shown in Figure 40. kvn^re(i>

f e n %

w r >4 I

Fig. 40. Since there are only four beads In the stockpile on the tens' rodr, the five tens (of the 53) cannot be indicated directly. The cqn^lementary mebhod^^ can be used (see Figure 41).

^With abaci having ten beads. Another method is theoretically possible. Proceeding from the positions as shown in Figure 40, the technique would be as follows: 1. Move to the left the four beads in the stockpile on the tens' rod, counting ten, twenty, thirty, forty. 2. Now, there are ten beads against the left' side of the frame on the tens' red, and the counting is interrupted while an exchange is made. The t^ beads on^the tens ' red are pushed to the stockpile poiiti^ and replaced by pusHing to the left (from the stedqdLle) èhe bead on the hundreds' rdii 3* The counting is réiëümed bÿ pushing to the left one bead (from the stockpile) on the tens' red. This makes a total of fifty, and the result is the same as shown in Figpre 42, 127 irtAs h Ilf winhwn

te n s H im ten s

o n • t S uni ts

Flg. 41. To add 50 by tha Fig. 42. The result is one bead complomenbaxy method mean# on the hundreds' rod, one bead to add 100 and subtract 50. on the tens' rod, and five beads This is dcme by pushing from on the units' rod. the stockpile one bead on the hundreds' rod, and returning to the stockpile five beads on the tens' rod. The result is shown in Figure 42. 6. Those abaci having an unlimited number of counters and using a combination of lines and spaces (medieval counting board).

C 4 L # X X y y X X

Fig. 43. Indicate the first Fig. 44. In column XI Indicate the number (62) in Column 1. second number (53) simultaneously with the first number. Since there are five counters on the units' line, these are removed and replaced by one counter in the fives' space, resulting in the positions shown in Figure 45« 128

C L X V

Flg. 45. 3inc9 there ara two counters in the fifties' space, these are removed and repla#0d by one counter on the hun­ dreds' line, resulting in the positions ahom in Figure 4 6 .

C L K V X

Fig. 4 6 . The result is one counter on the hundreds' line, one counter on the tens' line, and one counter in the fives' space. 129 7* Those abaci having an unlimited number of counters on which numerals^) were indicated (Gerbert's Abacus). .1 fans on»'/*

© ©

Fig. 47« Indicate the first number (62).

fen S u rtits

© ©

© (D

Fig. 48. Indicate the second number (53) simultaneously with the first number. The operator must know his addition combinations, namely that 2 /3 " 5# The counters in the units' column having the symbols "2" and "3" engraved upon them are removed and replaced in the same column by a counter having a "5" engraved upon it, resulting in the positions shown in Fignre 49.

^^he numerals engraved i%)on the counters used by Gerbert were the gobir numerals and were not exactly like the Hindu-Arabic numerals, (see Appendix III). Florence A. Yeldham, The Storv of Reckoning in the Middle Ages (London: George G. Harrap and Co. Ltd., 19267% p. 37 illustrated these numerals, [see Plate JOC ] . 130

fens u*|i ts

© < D ©

Fig. 49. Again tha addition combinations must be know, namely^ that 6 tens / 5 tens " 11 tens m 1 hundred / 1 ten. This is indicated by re­ molding the two counters in the tens' column having the symbols "6" and "5" engraved upon them, and replacing. them by one counter in the hundreds' column having a "I" engraved upon it, and one counter in the tens' column having a "1" engraved upon it, resulting in the positions shown in Figure 50.

fenS uyilts

o o a>

Fig. 50. The result is one counter in each column — the counters in the hundreds' and the tens' column each having a "I" engraved upon it, and the counter in the units' column having a "5" engraved upon it. 131 ORIGIN OF THE ABACUS Since the duet abacus Is of more primitive construction than the other abaci, it was probably the earliest fom of abacus used. Defi­ nite infoxmation as to the origin of this type, or of the counter abacus, has been lost in antiquity. Some historians entirely ignore the question of its origin; some admit that the origin is unknomn; a few have made an issue of the question and have attempted to advance theories as to the group of people to whom credit should be given for having invented the abacus. Those who have attempted to discover the origin usually sought this information by making one or more of the following studies: 1. Development of the symbols used for the number systems of various countries, particularly concentrating on the Hindu-Arabic numerals, and more specifically endeavoring to detexnixrp the origin of the present-day symbol for zero. 2. Derivation of the number words for the symbols located in a study of the type described in 1. 3. Analysis of the mathematical works of ancient writers to determine if their explanations of mathematical operations implied the use of an abacus. The most commonly advanced theories about the origin of the abacus are as follows: 1. Pythagoras (or the Pythagoreans) invented the abacus. 132 Boîithius,^ in his work Quae Fertur Geometria.^^ stated that Pythagoras invented a tableau a colonnes^^ $giich was named in his honor, mensa Pvthagorea. The27 device was later called an abacus.

^^Boethiua (bom at Rome c. 460, died $24). His works on geo­ metry and arithmetic were edited by Godofredus Friedlein, Leipzig, 1667 (see pp. 395-399). 25 Many historians believed that this geometry attributed to Bo^hius was spurious, ^^From M. F. Woepcke, "Mémoire sur la Propagation des chiffres Indiens," Journal Asiatique. I 5 (JanvieBÿ-Février, I863 ), 33n.

% Tableau a colonnes "Le tableau à colonnes foumait donc un moyen d^ecrire tous les nombres, quelque grands qu'ils soint, au moyen de neuf chiffres, en donnant à ceux-ci des valeurs différentes, selon leur position, et sans faire usage du zero [The tableau a colonnes thus furnished a way of writing all the numbers, however large they may be, by means of nine numerals, by giving to them different values, according to their posit ion .and without, u sing zero.] " 5 5

Ô 4 7 (a 1 4 6 4 The tableau a colonnes shows the numbers: 305 84009076 10k0084000 Woepcke illustrated the tableau a colonnes in this form,but if Pythagoras had been the inventor he would have used one of the Greek fozms of numerals — either the "Herodianic" or the alphabetic system. ^7lbid. 38. 133 Brooks Inyslled that Pythagoras had Invented an abacus, or at least that one nas named j£er .him, when he wrote: • . . it is supposed that the Neo-Pythagoreans,^ who were probably the fijrat teachers of ciphering among the Greeks and Romans, became acquainted with the Indian figures, and adapted them to the P;^hagorean âbacus, and that Boethius, or his continuator, made these figures generally known in Europe by means o f mathematical handbooks,^”. . . 2. The Chaldeans^ invented the abacus. lamblichus,^^ in his Be Vita Pythagoras (chapter V), stated^

(c. 3 2 5 ) that Pythagoras introduced the abacus into Greece, and hinted that ha may have brought his knowledge of the instrument from Babylon. Radulphe, 34,35 Qf Laon, writing about 1125 declared that

^{foup (2nd and 1st century B.C., centering at Alexandria) Wiich atteaç>ted to revive the doctrines of Pythagoras (c.5B2 - C.5 0 ? B.C.) ^^Edward Brooks, Philosophy of Arithmetic (Lancaster, Penn.: Normal Publishing Co., 1880), p. 29. 3^Chaldea was the #outhemmost portion of the valley of the Tigris and the Euphrates. Sometimes it is extended to include Baby­ lonia and thus comprise a ll of southern Mesopotamia.

3^A1so spelled Jamblichus, Syrian philosopher (died c.330). 3%atin text of statement given in Smith, o£. cit. . p. 160 n. An English translation (Life of Pythagoras) has been made by Thomas Taylor. 33gmith used Babylon and Chaldea synonymously.

34a 1so known as Raoul (died c. 1131). 33His arithmetic, which is almost wholly an explanation of abacus reckoning, has been transcribed and published with an introduction by Nagl (Der arithmetische Tractat des Rudolph von Laon. Abhandlungen Zur Geschichte der Mathematischen Wissenschaften. ?, 05-133)• 134 the Greeks obtained the names for the units used by the Neophythagoreane^ with the abacus, from Chaldea, Bayley believed that "this assertion

is in a great measure corroborated by facts . . and further added .

that "Pythagoras, who is reported to have been carried as a prisoner into Babylon by Cambyses, and who spent a long captivity there, may well have learned his arithmetic and the use of the abacus in ttmt 37 country," Woepcke, another well-known authority, apparently agreed with the theory as presented by Radulphe, and gave the Latin text from 38 his work.-^ 39 3. The abacus is of Semitic origin.

Knott stated:

The Chinese write in vertical columns from above downwards; and if they ever are compelled to write in a horizontal line they work from right to left. Now the j^bacus is worked from left to right, a fact which tends to prove incidentally that the Abacus is not indigenous to China. The sim ilarity between the numerals as written and the Abacus indications of the same would not be so striking to the Chinaman as to the Aryan or Semite, since those

36 Sir E. Clisre Bayley, "On the Genealogy of Modern Numerals," Journal of the Royal Asiatic Society, N.S. XIV (1882), 353»

^^Bayley, og. c it. . N.S. XV (1883), 9.

^%foepcke, og. cit., 40-49n. 39 "The Semitic peoples occupied the territory represented by the great Arabian peninsula and pushing back between the eastern end of the Mediterranean and the table-lands of Persia to the Taurus Mountains in the North. In this territory there lived in Bible times a group of peoples speaking Semitic languages and more or less closely related by blood. These people were the Arabs, Babylonians, Assyrians, Arameans or Syrians, Phenicians, Moabites, Ammohites, Edomites, Canaanites (or Amorites), and Hebrews . . . They were in constant contact one with another and developed a common type of civilization." ["Semitic Re­ ligion," New Standard Bible Dictionary (3rd ed.), 81?.] 135 wrote in horizontal lines. Now so far as evidence goes^ our numeral systems all passed to the races of Aiyan origin through the Semitic peoples, who generally wrote from right to left. A# will be seen in the subsequent part of the present paper, the Semite named his numbers by beginning with the unit or smallest denomination. Thus in Arabic it is five and twenty and one hundred, instead of one hundred and twenty-five. But in writing doTim a number he would write it as he named; and as he both, so to speak, wrote and named it backwards, the result would appear as it is on the Abacus, -125. Now the early Indian spoke like the Arab, but wrote from left to right ; while the Chinese always spoke as we do now but tended to write from right to left. Hence if the Abacus had been an Indian or Chinese invention, the columns would probably have gone the reverse way, with the units to the left, so that one hundred and twenty^five would have appeared as 521. This argument of course cannot be urged in the face of evidence to the contrary; for we know that in ancient days both modes o f w ritin g were in use by th e same p eo p le. In some in ­ scriptions ihdoed the w riter has turned backward along the next line, plough"^mah?^like. S till as the Chinese write in vertical columns, so the>Semitic peoples generally write from right to left and the Aryan from left to right. Hence, unless there were definite evidence to the contrary we should be inclined to regard the Abacus as not being primarily an Aryan invention, but more probably introduced to the Aryan races through the.Semitic p e o p le s .^

Smith^ agreed with Knott but said in addition, "There is some

reason for believing that this form [the counter abacus^ of the abacus

originated in India, Maaopotamia, or Egypt,

^^Cargill G. Knott, "The Abacus> in its Historic and Scientific Aspects," Transactions of the Asiatic Society of Japan. XCV (1886), '3 3 - 4 i, ' ^Smith. op. cit. , p. 157. ^Ibid., p. 159. 136

4. Tha abacus was invented independently in different parts of the

ancient world.

SedgiDfick and Tyler,Ba]^,^ and Gandz^^ were advocates of the

theory that credit should not be given to any one country for the

invention of the abacus. Many ancient peoples invented it independent­

ly as a step in their intellectual development.

THE ABACUS IN EGYPT

Evidence that the Egyptians had an abacus is based on the writings 46 of Herodotus. He declared that the Egyptians used one as early as

461 B. C. In his history he stated:

When they [the Egyptians] write or calculate, instead of going, like the Greeks, from left to right, they move their hand from right to left; and they insist, notwithstanding, that it is they Wio go to the right, and the Greeks who go to the left.47 '

T. Sedgwick and H. W. Tyler, A Short History of Science (New York: Macmillan Co., 1939)> p. 27.~

W. Bouse B all, A Short Account of the History of Mathematics (New York: Ifecm illan Co., I 90I), p . 127.

^^Solomon Gandz, review of B. Datta and A. N. Singh's History of Hindu Mathematics; A Source Book. Part I. in Isis. XXV (1936), 4 8 4 .

^Greek Historian of the 5th century By' C.

^^Herodotus The History of Herodotus/i i « 36. 137 No specimens of an Egyptian abacus have been found. None of the wall pictures yet discovered give any evidence of an abacus, but many disks have been found which might have been used as counters. 48 Gow implied that the Egyptians and Chinese invented independently the counter-and-line form of abacus and imparted it to the nei^bor- ing nations. Mahaffy^^ thought probably the Greeks derived their abacus from Egypt.

THE ABACUS H'î GREECE

Historians are agreed that the Greeks had forms of abaci, but are not agreed as to their origin. As mentioned on pages 131-3 , some writers believed that Pythagoras (or the Pythagoreans) invented the abacus, while others believed that Pythagoras Introduced the abacus into Greece, having obtained the idea from the Chaldeans.

Evidence of the use of the Abacus in ancient Greece exists in the form of references in literature,an e x is tin g specimen,and 52 classical antiquities* According to available information, there were basically three forms of abaci used:

^James Gow* Short History of Greek Mathematics (New York: Hafher Publishing Co*, Inc*, 19237, p* 28.

P. Mahaffy, Old Greek Education (New York: Hajper and Brothers, 1882), p. $6.

^^See Appendix IV.

^^he Salamis abacus (see page 138 ) *

^^he Darius Vase (see page 141 ). 138 1. The dust abacus.53

This fom of abacus was usually a board on vdiich dust or sand was sprinkled. A stylus was used for marking in the dust. 2. The counting Board.

Cubberley,^^ Mahaffy,^^ and Freeman^^ described this form of abacus (see Figure 51). It consisted of several grooves in which pebbles of different sizes or colors were used. Pebbles placed in the division at the left side of the table had their value multiplied by f i v e .

Yeldham,interpreting Herodotus* remarks, illustrated the

Greek abacus as being arranged with the pebbles in vertical lines.

The highest order was on the left side of the table.

3* The Salamis Abacus.

A marble table found (c. 1846) on the island of Salamis^® by

^^Smith, o£. c it., p. 158.

54Bllwood P. Cubberloy, History of Education (New York: Houghton M ifflin Co., 1920), p. 2?.

55iiahaffy, o£. c it. . p . 5 6 .

J. Freeman, Schools of Hellas (London: Macmillan and Co. L td ., 1932) , p . 10 4.

57yeldham, o£. c it. . pp. 25- 6 .

^^An island east of Greece in the Gulf of Aegina, southwest of A thens. 139

THOU

ooo

Hume

oo T6 Nt)

O O

UMiTt)

O oooo

Fig. 51. Greek Counting Board (The board shows 15,269). Ellwobd P. Gubberley, The History of Education (Boston; Houghton Miff!In Publishing Co., 1920), p.2?. Permission for use granted by Houghton M ifflin Co. lito

Rangabe is believed to be a Greek a b a c u s , I t i s now i n th e E pi- grapMcal Museum at Athens, Descriptions and illustrations of this abacus were given by the following w riters: Sm ith,^ Heath,Gow,^^ and H arper,Steele^ gave a good description in his book.

59 There has been some disagreement as to the purpose for which the Greeks designed this table, Hangabe analyzed the structure of the stone along with passages from classical literature, and arrived at the conclusion that the table was designed as one on which to play certain games, Letronne did not make an analysis but simply stated that he believed the stone to be a Greek abacus of a period prior to the time of Euclid, and as such it would be the oldest one known, Vincent compared the stone with the Roman abacus (the form on which the buttons slide in grooves) and reached the conclusion that the table had a double usage - - the primary one being that of an abacus and the secondary one that of a game table CThe articles ^ by the above-mentioned men are: A, R. Rahgabe, “Lettre de M, Rangabe a M, Letronne,” Revue Archéologique, III (181:6), 293-301:5 Letronne, "Note sur L'& helle kum(^riqued'un Abacus Athénien, et sur la Division de L'Obole Attique, ” Revue Archéologique, III (181:6), 30^-8; A, J, H, Vincent, "Lettre a M, letronne sur un Abacus Athénien, " Revue Archéologique, III (181:6), 1:01-5J, 60 Sm ith, 0£, cit,, pp. 162-k,

61 Thomas A, Heath, History of Greek Mathematics (Oxford: Clarendon Press, 1921 ), Ï, ' pp. 1:9-51.

62 Gow, 0£, c it., pp. 33-5.

63 "Abacus," Harper's Dictionary of Classical Literature and Antiquities (l897 ed.),' 2 -3 .

6U Robert Steele, Earliest Arithmetics in English (London: Oxford University Press, 19^i2), pp. v ii-v iii. 141 One of the pictures on the Darius vase^^*^^ is believed to por­

tray a man sitting at a table using an abacus of the same form as the

Salamis abacus. This picture referred to the Persian wars (c. 500

B.C.)during the time of Darius, The vase was found in 1S51.

THE ABACUS IN ROME

Evidence of Roman abaci exist in the form of references in

literature,an existing specimen,^® and classical antiquities.^^ 70 According to Latin writers there were three forms of abaci used in e a r ly Rome:

1. The sand board or the wax tablet.

A stylus was used for marking upon this form of abacus.

2. A marked table for counters.

65 Found at Cano sa (Canusium), a rural city in Southern Italy.

^descriptions are given by: Heath, o£. c it. . pp. 48-9; D. E. Smith and Jekuthiel Ginsburg, Numbers and Numerals (Washington, D.C.: National Council of Teachers of Mathematics, 193?), p. 13; Smith.op.cit p p . 161-2.

^^See Appendix V.

^% ireher Museum at Rome (See Plate IX).

^^(a) Figure of a youth with a calculating board on a grave- r e l i e f in th e C a p ito lin e Museum i n Rome; (b) An E tru scan (p e rta in in g to Etruria, an ancient country in central Italy) gem engraved with a youth working at a calculating board. CBoth (a) and (b) are pic­ tured in Theodor Schreiber: Atlas of Classical Antiquities (London: Macmillan and Co*, 1895), pp. 115-116;,175.3 70 Sm ith, c it. . pp. 165-8. 142 According to 3mith^^ and Leslie,Rome acquired the line abacus

from Greece. The early fom of this type of abacus is shown in Fig­

u re 52. This abacus could be made with as many columns as one pleased, and pebbles (calculi) were usually used as counters.

Later (probably in the 2nd century) the Romans developed the

idea of dividing each line so that each counter above the numbered p a r titio n (when moved down) had fiv e tim es th e v alu e of each counter 73 (on th e same lin e ) below th e numbered p a r titio n (when moved u p ).

T his form i s shown in Figure 53. Pebbles were originally used as counters, but as people acquired wealth they began to use disks of ivory or small silver coins,

3, A grooved table with beads.

This fom of abacus (see Plate IX) was a relatively late in­ vention of the Romans, It was a board^^fYS marked off with grooves

(along which balls, counters, or buttons could be moved) representing

S, Smithy Mathematics (Boston: Marshall Jones Co,, 1923), p , 28,

^^Sir John Leslie, Philosophy of Arithmetic (Edinburgh; William and Charles Tait, 1820), p. 94.

^^Smith, Loc, c it,

^^he abacus was made of wood, of brass, or even of silver (Leslie, og, c it, . p , 95)*

75piorian Gajori £ History of Elementary Mat hematic s (New York: Macmillan Co.^ 1897), pp, 38-40J3 gives a brief e:iqplanation of the method, for performing the four basic operations upon this form of abacus. 143

M C K I 0 0 0 0 0 0 0 0 0 0 0 0

Fig. 5 2 . Tha board shows 362I. Louis Charles Karpinski. The History of Arithmetic (Now York: Rand McNally and Co., 1925), p. 2 6 . Peraiission for use granted by Mrs. L. C. K arp in sk i.

# # e

• e # e M c X Ï c X I # e # e # e e # # # e A #

# e e e e e • # e # # # • e e e Pig, 53» The board shows 8,760,254* Cubbarley, op. c it. . p. 65 Permission for use granted by Houghton M ifflin Go, Figs. 52-53* Roman Abaci 144

Plate IX. Boman Abacus

David Xugeae Smith aM Jekuthiel Ginsburgj Numbers and Numerals (Washington^ D.C.: National Council of Teachers of Mathematics, 1937), p. 25.

Permission for use granted by the National Council of Teachers of Mathematics. Ancient bronze abacus used by the Ronians 146

tha several orders of numbers. Just as in the abacus of Figure 53,

each counter in the upper groove (when pushed down) had a value five

times that of each counter in the groove immediately below (when pushed up). The counters in the two extreme right sections had fractional values, as marked.

THE ABACUS IN INDIA

A very controversial subject is that of the existence of the abacus in ancient India. Some of the writers who maintained that In­ dia did have an abacus were: Fleet, 7^ K nott,77 Bayley,7^ R o d e t , 79 lyer,®^'®^ Clark,Smith,and Gandz.®^*^^ The latter cited in

7^J. F. Fleet, "The Use of the Abacus in India," Jourml of the Royal Asiatic Society of Great Britain and Ireland. XT.TTT (igiTy, 518 - 21.

'^’^Knott, O P . c it. . XIV, 18-73.

"^^Bayley, op. c it., N.S. XIV, 353i N.5. XV, 9, 10, 30, 38, 66. *7Q / / ^ Leon Rodet, "Sur^la veritable signification de la Notation Numérique ^ventee par Aryabhata," Journal Asiatique. XVI« (Octobre- Novembre-Decembre, 1880), 465. '

®*^R. Venkachalam Iyer, "Pâtiganita and the Hindu Abacus," The ifethematics Student « XVIII (Septamber-December, 1950), 79-82.

®^R. Venkachalam Iyer, "The Hindu Abacus," Scripts Mathematics, XX (March-June, 1954), 58-63.

E. Clark, "Hindu-Arabic Numerals," Indian Studies in Honor of Charles Rockwell Lanman (Cambridge: Harvard University Press, 1929), pp. 228-9. ^%mith. History of Mathematics. II, p. 158.

®^Solomon Gandz, "Did th e Arabs Know the A bacus," American Mathematical Monthly. XXXIV (June-July, 1927), 308n.

®^Solomon Gandz, review of B. D atta and A.N. Singh * s History of Hindu Mathematics in Isis. XXV (1936), 484. 147 his article that the following historians (in addition to most of

the ones already named) believed that tha abacus existed in India;

Humboldt, Chasles, Woepcke, Kaipinski, Taylor, and Burnell.

The type of abacus that the above-mentioned writers believed

S the ancient Indians had was either the dust-board or the tableau a

colonnes. The Hindus called their abacus either pati (board) or

dhuli (dust).

Kaye^^'^ thought the abacus was of comparatively recent intro­

duction into India. In his article he showed the weaknesses in the

arguments of Bayley, Rodet, Woepcke, Burnell, and Taylor.

Datta was one of the smaller number of w riters 'vriio believed

that ". . . if an abacus,was ever in use among the learned men of

India, it was discarded long ago.Gandz also cited Cantor as one

of the historians opposed to the theory that the abacus existed in

In d ia .

R. Kayo "The Use of the Abacus in Ancient India," Journal and Proceedings of the Asiatic 5ociety of Bengal. IV (June, I 9OÔ), 293- 7 . 87 . . often the arguments of Kayo, whose absolute unrelia­ bility on these questions Ccontributions of the Hindus 1 has been de­ monstrated ..." Dtouis Karpinski, "Is There Progress in Mathematical Discovery and Did the Greeks Have Analytic Geometry?," Isis. XXVII (May, 1937 ) , 46.] ®%ibhutibuhusan Datta, "Early Literary Evidence of the Use of the Zero in India," American Mathematical Monthly. XXXIII (November, 1926) , 450. See also Datta, "The Science of Calculation by the Board," American Mathematical Monthly. XXXV (December, 1928), 520-9. 14S THE ABACUS IN ARABIA

Arabia is another country about which there is a controversy

as to whether the abacus was used in early times. Not as many writers

entered into the discussion as did in relation to the abacus in India,

A few of those who believed Arabia had a form of abacus were Karpinski, Weissenborn, Woepcke,Smith,and Gandz^91,92,93 D a tta w ro te

an article in which he refuted Gandz's claims that the Arabs had an

abacus. Treutlein, Hankel, and Cantor also opposed the idea of an

early Arabian abacus.

The types of abaci, as described by the writers who believed the '

Arabs had an abacus, were; (1) a fom. of abacus which had ten beads on

each line, and (2) the dust board. The argument for the latter form of — 95 abacus was based on a history of the ghubar^ numerals. The Arabs used two terms in referring to their abacus, namely takht (board) and ghubar

(d u s t) .

go ^Woepcke, c it ., I ^ , 58. 90 Sm ith, c i t ., pp. 174-5 • 91 Solomon Gandz, "Did the Arabs Know the Abacus," American Mathe­ m atical Monthly. X3CXIV (June-July, 1927), 308-16. 92 _ Solomon Gandz, "The Origin of the Ghubar Numerals, of the Arab­ ian Abacus and the A rticuli," Isis, XVI (1931), 393-424.

93solomon Gandz, Review of B. Datta and A. N. Singh's History of Hindu Mathematics in Isis. XXV (1936), 484»

94gj^bhutibuhasan Datta, "The Science of Calculation by the Board," American Mathematical Monthly. XXXV (December, 1928), 526-9. ' 95 ' ' ' An_Arabic trw islation of the Latin word (pulvis) for abacus. Both ghubar and pulvis originally meant "dust, dust board." 149 THE ABACUS IN HUSSIA There ie umeertaimty as to the origin of the Buasian abacus. Baylej attributed the Introduction of the instrument to the Mongols mho conquered Russia at the end of the middle ages. According to 97 SSlth,^ the Russians themselves spoke of their abacus as a Chinese abacus; but because of the general form he believed that it came from Central Asia. 98 Gon* in describing the abacus, said that each wire carried ten balls of different colors, some being white and some black. Smith^^ pictured the abacus (see Plate X) as having ten horlaontal wires, the first and fourth wire having four bails each, and the remainder of the wireit having ten balls each. The Russians have continued using the abacus in business establishments even until the present time. Recent photographs^^ indicate that the form described by Gow is the one being used today. THE ABACUS IN CHINA The dates given for the time that the Chinese began using the abacus range over a period of approximately four thousand years.

^^!Sayley, gg. c it. . N. S. XIV, 369n. 97 Smith, git., pp. 169, 175-6. ^Gow, gg. gl^., p. 31. 99 Smith, gg. cit. . p. 176. 100 See illustration in the article: "Moscow for the Tourist," Time. UVX (November 28, 1955), 52-61. 150

Plate X. Russian Abacus

Reprinted from David Eugene Smith, H istory o f Mathematics» Volume I I » Perm ission fo r use granted by Ginn and Company. i ^ l

100,000

10,000

1,000

lOO

># #hWy. I ruble i w i ruble

lo kopeks

I kopek

J kopek

RUSSIAN ABACUS 152 Th# table below eh 507 4. Di 103 Oe LaCoupj^e'*^'^.104 J- 12th eentury A. D. Sdth®^ 107 Tan N am e^ io@ | Spdth and tfikaai (citing Hal Wen-tlng)*~^ j 14th century A. D. V iL09 Vissiere 14th century A. D. ^®^errien Do LaGouperie, "The Old Numerals, the Countlng-Bods, and the Sean^Pan in China," UamAm^Mç chronicle, HI* (1883), 299-300, Citing Martihi, Sinioae Historiae Peèas^rina. I. Munich, 1658. ^^honas Cheng, Abacus Arithmetic in ChinA ("at. Bohaventure Science Studies," 17), 1936, p p . 20-9. citing Thesaurus of Chinese. Coomerdial Press, Shanghai [Story of the abacus first appeared in the Jan Man Chew Pih Juh j- ^% em adette Lamey, "How the Middle Ages Counted," School Science and MathematicsmKM oveeber. 193Ü. 919-30. (First defi­ nite description given in 1174 in Pan chu tsik .l ^®^e LaCouperie, gg. c it.. lU ^, 297-340. ^^Karpinski, History of Arithmetic. P. 29. ^Smith, SB. sit., p. 168. 107 'Addison Van Name> "On the Abacus of China and Japan," Journal of the Americen Oriental Society. X (May. 1875), proceedüngs, pp. cx- cx ii. (The Samnd daiaen. a Japanese treatise on mathematics published in 18% gave the middle of the 14th century as the time that the swan p'an was introduced into China). B. Smith and Toshio Mikawl, History ^f Japanese Mathematics (Chicagot Open Court Publishing Co., 1914), p. 29 LMei Wen-tiag(lo33- 1721) . Ku-auan-K*i-K»ao. thou^t probably the suan-pan began to be used abodt 13840 ^^A. 'i7#aiere, "Bech^bes sur l*Origine de I'Abagie, Chinois et sur sa D erision des Anciennes Fiches a Calcul, " BulletiÉ de QeegM- phio H istorié et BescriPtiVe. VU (1892) . 54-80. 1^3 \ Of th e -works c ite d above, th e ones by De LaCotçjerie and V issie re

seemed the most scholarly and detailed studies of this topic. On a basis of their studies, one may say that the suan-pan was introduced

into China in the 12th or in the lUth century. 110 Each rod of the suan-pan (see Plate XI) holds seven beads, five of them below the partitioning bar and two above. Each of the two beads above the bar (when pushed down) has a value five times that of each bead (on the same rod) belox-r the bar (when pushed up). Any rod of the abacus could be chosen as the units' rod. Then the value of each bead on the rods to the right of the units' rod would represent 1 * 2 Q one tenth, one hundredth, one thousandth (that is, 10” , lO” , 10”-^, respectively). Similarly, the value of each bead on the rods to the left of -the units' rod would represent tens, hundreds, thousands (that is, 10^, 10^, 10^, respectively).

The Chinese have continued using the abacus to the present time,

"Bead Arithmetic" is a required subject in the primary schools of

China, In almost e-very home one finds an abacus. Clerks in stores and business establishments prefer an abacus to a modern computing machine, even bringing the suan-pan to the United States to use in their places of business.

General directions for performing the simple arithmetical operations upon the suanpan are given in: Yi-Yuh Yen, "The Chinese Abacus," ]%thematio8 Teacher, XL (December, 19^0), ii.02-Uj Keychow Tin, %e t^Vmdamental Ùpérations in Bead Arithmetic (Manila, P,I, : Depar-taient of Agriculture and Natwal Resources, 1920); W,D, Loy, "How to Use an Abacus," Popular Science Monthly, CLIII (August, 19i|.8), 86-9; and Cheng, op^. c it,, ZO-9. 154

Plata XX. Chinese Abacus

Reprinted frcm David Eugene Smith, History of Mathematics. Volume I I . Pezmission for use granted by Ginn and Coogpany. m

MODERN CHINESE ABACUS The suan-pan, known to have been used as early as the 12th century 156

THE ABACUS IN JAPAX Host «rlters »r# agreed that the Japanese reeeired the abacas frem China dhiing the sixteenth century or at the beginning of the seventeenth century. The name soroban is believed to be a mispronanciation of the Chinese eord Seanvan. - 113 114 The soroban is similar in eonstraction to the seen pan, mith the exception that on the former there is one bead above the bar ehile on the latter there are two beads above the bar (Plate XII). The Japanese are noted for their poetic manner of expressing thesuselves. 115 This is rather interestingly shoen in a book written by Toshino. He called the section above the bar the "upper part or Heaven," and the section below the bar the "lower part or Earth." In explaining how to represent a number on the abacus he said, "To designate 6 a 3 .1 4 * counter o f Heaven and one o f Earth are used. The Jeqpanese, in a

^4)e LaCouperie, o£. cit.. UI^, 301. 112 Smith and lUkami, History of Japanese pp. 31-44. This is a very good discussion of the soroban. — ^ ^ o t t , igB» c i t . . XIV, 45. OMitedporary form of the soroban has one bead above the partitioning bar and four beads below (see Plate XIII). A book iMeh discusses this form of abacus is by Takashi Kojima, The Japanese Abacus (Rutland. Terment % Charles E. T u ttle Go. 4 1954.) 115 / J. Toshino, The Japanese Abacus Rvplaiaed (Tokyo, Japan* Kyo Bun Kwan, 1937).

p . 3 . 157

Plata XXI. Traditional Japaneso Abacus

From the collection of Professor Nathan Lazar, Ohio State University, I.W: 159

Plate XIII. Contemporary Japanese Abacus

This abacus is in the collection of Professor Nathan Lazar, Ohio State University, ...... / ^ o 161 manner sim ilar to the Chinese, also used their abacus for decimàl f r a c tio n s .

Several writers have described and explained the soroban, but one of the best sources of information written in English is the one 117 by Knott. He gave a description, illustration, and explanation of the following operations: addition, subtraction, m ultiplication, division, extraction of square root, and extraction of cube root.

Many persons think of the abacus as a toy or as a primitive mode of reckoning. The following incident had results which would contra­ dict both of these concepts:

Soon after World War II a unique contestwas held in Tokyo.

Kiyoshi Matsuzaki (a clerk in the Japanese Communications Ministry) u sin g a s o r o b a n , ^^9 and Private Thomas Wood (a clerk in Army Finance) using an electric calculator, worked arithmetic problems to detemine which could be the first to cojaç)late each type of problem. The one using the abacus finished first in the following parts of the contest: the addition, subtraction, division, and composite problems. The

American soldier was first with the multiplication problem. The man using the abacus also made fewer mistakes.

P-'^Knott, o£. cU ., XIV, 46- 6 9.

^^Addition on the Abacus-Japanese Style," Time, XLVIII (Novem­ ber 25, 1946) , 35. 119 ■ The content ora ry form of soroban was used (The fom having one bead above the partition and four beads below the partition. ) 162 THE ABACUS IN OTHER EUROPEAN AND ASIATIC COUNTRIES

The abacus, in various forms, was used in other countries of

Europe and Asia. In Persia,^^^*^^^'^^^ Armenia, and Turkey^^^ there was in use a type similar to the Russian abacus ---- horizontal wires with ten beads. The Koreans^^ used both the suan pan and the soroban. In the following countries the form of abacus was a type that made use of counters with lines, probably horizontal: the Catho­ lic Low Countries, Germany, Italy, the countries of the Iberian Peninsula, Scandinavia,Poland,and 131 Austria. The Assyrians used an abacus, but the writer of this

Karpinski, o£. c it., p. 2^. 121 Smith and Mikarai, 0£. cit., p. 19, 122 Smith, History of Mathematics, II, p. 17U. 123 Loc. c i t . 12li. a n ith , 0£. cit., pp. 171-U. 125 Low region on the North Sea, modern Netherlands, Belgium, and Luxembourg. 126 Barnard, 0£. cit., pp. 25,90. 127 Smith, o£, c it., pp. 185, 190-1, 128 Barnard, 0£, c it., pp. 25, 105.

^^The southwest part of Europe, comprising Spain and Portugal, ^^^Bamard, 0£, cit., p. 25. 131 Smith, 0£. cit., p. 185. 163 stïKîy was unable to determine the form. The Hebrews also had an abacus,

Sometime between the beginning of the Middle Ages and the 13th

century the more popular form of abacus in Western Europe changed

from one which used the vertical lines to one which used the horizon­

tal lines and the spaces between the lines (see Plate XIV), France 133 and England both used this form of abacus.

The first description of counter reckoning printed in France

appeared about 1^00,^^^ Literary evidence indicated that this form

of computation was s till common in the 17th century. For example, in

Malade Imaginaire, Moli^re^^^ (1622-1673) described the activity on stage at the opening of a scene as follows:

Act I, Scene I,

Argan, seated alone in his room, a table in front of him, adding up his apothecary’s bills with counters; . , ',

132 Solomon Gandz, "Did the Ara.bs Know the Abacus?" American Mathematical MonthUy, XXXIV (Ju n e-Ju ly 1927)» 3l5h, 133 See footnote 8 in Appendix III for sources of books which give descriptions of the recording of numbers and of the performing of operations upon the medieval counting board, 13U Smith, o£, c it,, p. 191. 135 French actor and dramatist (Pseudonym of Jean Baptiste P o q u e lin ), 164

Plata XIV. Medieval Counting Board. In the background presides the genius of numbering, Typus Arithmetica^e; she bears a book in either hand and the Arabic sg^ols are shown upon her skirt. On the left Boethius is ciphering with Arabic numerals. On the right Pythagoras manipulates an abacus. (Yeldham, o£. cit.. frontispiece)

From F. P. Barnard, The Casting-Counter and the Counting-Beard « Plate XLVII. Pennission for use granted by Oxford University Press, Inc. PLATE XLVII

v :

REISCH’S.... ————' ' -— — - — —/_ , . . . philosophic, ^ : 1(01 V 166 The ïVench continued using the oounter-reckoning until the time of the Revolution (1789-179^)*^^^ Since writing was not as common among women as among the men, the women used this form of reckoning long 137 after the men had adopted the algorism. Le Gendre wrote in his arithmetic of 1792, "Cette manière de calculer est plus pratiquée par les femmes que par les hommes &his method of calculation is practiced more by women than by men.1

Haskins^^^ b eliev ed th a t in E n^and th e abacus was known to members of the curia^® during the reign of William the Conqueror

(IO66-IO87 ). Since Lorraine^^ was the chief center for the study of the abacus, probably the introduction into England came from that direction.

The English Exchequer used an abacus on which transverse and vertical lines were drawn to form a checkered surface. Coins or loose

136 Barnard, op, c it,, p. 8^, 137 ~ The a r t of c a lc u la tin g by means o f nine fig u re s and zero, 138 îVancois Le Gendre, L 'Arithmétique en sa Perfection, , , , contenant , , , un Traite de Arithmétique aux Jetons (limoges : Î4rtial Barbou, I78 I ) , p , ll 97 . 139 C, H, Haskins, "The Abacus and the King's Curia," English Historical Review, XXVII (January, 1912), IO6 . HJ5 The K ing's Court, l i a Medieval kingdom. West Central Europe, the valleys of Rhine, Meuse, and Moselle, 167 counters were used with this instrument. The Exchequer continued using the abacus until the l6th century.

References in literature help to determine the period of time during which counter-reckoning was commonly used. In The I&lleres

Tfllig (lines 22-25), Chaucor^^ gave the following description:

His almugest, and bookes gret and smale. His astrylabe, longrog to his art. His augiym stoones%l43 leyen faire apart On schelves couched at his beddes heed.

The writings of Shakespeare^^ inplied that by his time counter­ reckoning was used only by the uneducated. In order to portray a character as one to be treated with derision he used the following means:

The Winter's Tale, Act IV. Scene III

Clown: I cannot do't without c o m p t e r s , ^^5

Othello, Act I, Scene I. Line 31,

By debitor and creditor — this counter-caster.

The casting-counter disappeared sooner in England than in most

îkiropean countries. By the middle of the l?th century this form of computation had become p r a c tic a lly e x tin c t,

^^Geoffrey Chaucer (c, 1340-1400), English poet. 3^3Qounters. ^^*^illiam Shakespeare (I 564-I 6I 6), English poet and dramatist.

Pieces of metal used in making calculations,

^^Bamard, op, cit.. p. 2pS 1 6 8 THE ABACUS IH AMERICA Chapter III Included a discussion of the quipu that the Incas of Peru used for the recording of numbers and for the simpler compu­ tations. In summarizing conclusions about the use of the quipus,

Locke «ays, "The quipu vms not adapted to calculation. For this pur­ pose small pebbles and grains of maize were used. One of the best sources of information about the instrument the Incas probably used for calculations, before recording the results on the quipu, is a manuscript discovered in the Royal Library at Copen­ hagen. The manuscript (entitled El primer nueva coronica buen gobiemo) was written by D. Felipe Huaman Foma de Ayala (c. 1583-1613). This manuscript contained a large number of pen-and-ink sketches, and it was a detail of one of these sketches (page 360) that gave evi­ dence of a form of abacus (see Plate XV). In relation to this detail in the lower left hand comer of the sketch, Wassœ said* ....I am of the opinion that it represents an abacus by the aid of which the Indians were able to work out computation, the re­ sults of which were subsequently recorded by knots on the cords of the quipu8. I believe that an abacus of this kind consisted of 4 by 5 squares, with 5, 3* 2, and 1 holes, respectively. In computing, pebbles, beans, seeds, or similar objects were put into the holes, so that in this way different numbers could be marked.

^^Leland Locke, The Ancient Quipu or Peruvian Knot Record (New York: American Museum of Natural History, 1923), p. 32. 1 y A ''y ..... Henry Wassen, The Ancient Peruvian Abacus ("Comparative Ethnographical Studies," IX.), pp. 196-7. 169

Plate XV. Peruvian Abacus.

From Leland Locke, "The Ancient Peruvian Abacus," Scripta Mathematica. I (September, 1932), 38. Permission for use granted by Scripta Mathematica. 5 2 g

2

N Ç> 171 De Acosta, who visited Peru in 1571 and remained for fifteen

years, later wrote:

But it seemes a kinde of witchcraft, to sea an other kinde of quippos, which they make of graines of Mays, for to cast vp a hard account, wherein a good Arithmetitian would be troubled with his penne to make a division; to see how much every one must contribute: they do drawe so many graines from one side, and adds so many to another with a thousand other inventions. These Indians will take their graines, and place five of one side, three of another, and eight of another, and will change one graines of one side, and three of another. So as they finish a certaine account, without erring in any poynt: and they sooner submitte themselves to reason by these qUippos,,,g what every one ought to pay, then we can do with the penne.

In 1601 - 1615 Herrera Tordesillas published a book in which he

apparently referred to this method of calculation:

. . .les grains de mais leur servent aussi d’ordinaire pour des arrangements de comptes fort difficiles, donnant à chacun la part qui lui revient (7. . .the grains of maize ordinarily serve them also for the more difficult computations, giving to each one the part which is due himj

The form of abacus used by the Incas was different from any

known forms used in European and Asiatic countries. Because of this, / 151 Vfassen and Locke both concluded that this type of abacus was of

Inca origin.

IA.9 Father Joseph De Acosta, The Natural and Moral History of the Indies (London: Hakluyt Society, 1880),11, pp. 407-8. 150 Locke, The Ancient Quipu or Peruvian Knot Record, pp. 39-40, citing Antonio De Herrera Tordesillas. Historia General de los Hechos de los Castellanos en las Islas i Tierra Firme del Mar Oceano, pp.v, v i. 2.

^^Leland Locke, ’’The Ancient Peruvian Abacus," Scripta Mathe­ matica . I (September, 1932), 43. 172

Smlth^^^ and Hooper believed that the Lfeiyas^53 had used an abacus

in early times. Smith's evidence was based on the pebble-and-rod

form of numerals they used. Hooper reasoned as follows:

They £the Mayas] even had a symbol for zero, which suggests that they must have had some kind of abacus, or counting frame, for, as we shall see, the work done by the symbol 0 i s to in ­ dicate that there are no counters or beads on that particular wire or line of the counting f r a m e ,^54

The descendants of the Quiches^55 used a form of quipu which

they probably modified themselves. They pierced beans and hung them

on different colored strings, each color representing one of the

column places used in decimal arithmetic. A green string signified

thousands; a red one, hundreds; a yellow one, tens; and a white one,

u n its .

A form of abacus of modem derivation (used in the more remote

sections of Peru and Bolivia) was called a "chimpu,"457^15^ it con­

sisted of pierced beads or shells strung on strings. The number of

^^^Smith, op. c it,, p. 4 5 . I53xribe of Indians which inhabited Yucatan [Peninsula, (south­ west Mexico, B ritish Honduras, and North Guatemala) separating Gulf of Mexico from Caribbean Sea] . T54&lfred Hooper, Ifekers of Mathematics (New York: Random House, 1943), pp. 6-7. l55Tribe of American Indians. l56Qai.rick Mallery, Picture-Writing of the American Indians ("Bureau of Ethnology, annual report," No. lo) [Washington, B.C.: Government Printing Office, 1893]), p. 225. 457î|fendell C. Bennett, Mnemonic and Recording Devieea ("Hand­ book of South American Indians, Bureau of American Ethnology," Bulletin 143> V CWashington,D.C.: Government Printing Office, 1949]), p . 614. 15% ,t , Hamy,”Le Chimpu," La Nature.X3Œ (December 3, 1892), 11. 173 strings threaded through each bead indicated the order of units, tans, hundreds, and so on. For example, to represent the number five hundred sixty-eight: one string would be threaded through eight beads, two strings through six beads, and three strings through five beads.

THE MATHEMATICS OF THE ABACUS

Most of the known^59 forms of historical abaci were so designed that the ratio of the values represented by the columns or rows was constant (such abaci will be called "constant-ratio” abaci). This meant that two basic requirements were met: ( 1) the relative position of the columns or rows upon the abacus was always ordered in the same way — from right to left, or from left to right, or from top to bottom, and so ou (see Figures lOe to lOh, page 10$); and (2) the ratio between the value of any one counter of a column (or row) and the value of any one counter on an adjacent column (or row) was constant (with most abaci this ratio was 10:1 or 1 : 10, depending upon the direction in vrtiich the operator was proceeding). Under these conditions tan units in a given column (or row) are equal to one unit in the next higher column (or row). For example, if an abacus were of the form listed first in ( 1) above — that is the value of each counter on the columns increased in value from right to left — the value of each counter on the columns, proceeding from right to left, would be: 10^ , 10^, 10^ , 10^, and so on.

1 5 rt .The discussion of the Peruvian abacus is not included because at the present time the explanation that has been given is one of conjecture only. 174 Some of the forma of abaci (such as the common form of the

medieval counting-board, the ’’whole-number part" of the Roman abacus,

the Chinese suannan. and the Japanese soroban) may give the impres­

sion of not being constant-ratio abaci. But in spite of the presence

of counters having values of 5 , 50, 500, and so cn, as well as

counters having values of 1 , 10, 100, and so: on, these abaci really

belong to the same type. In the "whole-number part" of the Roman

abacus, in the suanpan, and in the soroban, the partitioning bar

(or space) divides the counters on the columns into groups of five

and two (or five-and-one or four-and-one) for the purpose of conven­

ience — that is, in order to have an arrangement that the eye can

"take in" more easily. Similarly, on the medieval counting-board the

value of each counter in the spaces represented $, , $(>,500, and so on

(in order from bottom to top) ; the value of each counter on the lines

represented 1, 10, 100, and so on (in order from bottom to top). In

spite of this seeming variation in the ratio of the values of the counters, this form of abacus was a constant-ratio abacus (1:10 when taken in order from bott

Three exceptions to the constant-ratio abaci were: (1) one form of the Russian abacus (see Plate X — the form having the beads arranged. 4, 10, 10, 4, 10, 10 , , , in order from bottom to top);

(2) one of the foms of the c ounting-board used in medieval England

(see Figure 12) the lyers were arranged to represent pence (d). 175 shilling (s), £ 1, 10, 4100, lOOOj and (3) the e^reiue ri^ t section

of the Roman abacus on which fractions were represented. The first

two forms of abaci named (the Russian abacus and the medieval counting

board) were arranged for convenience in order to match the money

system of the day. The right section of theRoaian abacus was arranged

to match the Roman money system and system of weights.

Although historical abaci have been based almost exclusively on

a power base of ten, it is possible for an abacus to be structured

on any number as a base. Leslie (see page 20U) proposed using the

abacus to operate with number systems on the binary scale, the tem ary

scale . . . and the denary scale. Hogben^^*^ gave a brief discussion

of the abacus of the one-armed man (an abacus based on 5 ) and of a

two-fingered man (an abacus based on 2).

INRLUEKCE OF THE ABACUS

The abacus, while in use during the centuries, exerted an in­

fluence that in some cases is in evidence even at the present time.

Some of these effects were:

1. The columns in account-books.

Tylor pointed out that "the columns in our account-books for i . s.d. or cwts. qrs. lbs. . are surviving representatives of the

old method of the abacus.

^^^Lancelot Hogben, Mathematics for the M illion (New York: W. W. Norton and Co:, Inc., 1937), 292-3w

B. Tylor, Anthropology (New York: D. Appleton and Co., 1913), p. 315. 176 2 . The system of separating large numbers into groups of three by

means of a comma.

On the abacus of the Middle Ages the thousands' and millions'

lines were each marked by a cross in order to aid the eye in

locating those lines easily. Influenced by these crosses,

medieval writers developed different techniques (with dots and

bars) for grouping the Hindu-Arabic numerals, until finally the

method used today evolved,On the other hand, Bayley^^^

believed that it was the grouping of the columns of the abacus

(Gerbart's abacus — see Appendix III) that gave rise to the

modem method of the grouping of numbers. But in either case,

the system of grouping developed from techniques used with the

abacus,

3* The spread of the Arabic numerals,

Wright stated:

It is in connection with the abacus that the Arabic numerals f i r s t became known. In drawings of th e abacus, in m edieval manuscript treatises explaining its use, they appear at the heads of the columns. They ware probably derived from the gobar ^°4 numerals used on the dust abacus , , , Next, these characters were inscribed upon counters, . , Their intro­ duction is traditionally credited to Boethius (c.A.B. 473- 524), and their use on counters to Gerbert . , ,1°5

I62sjnith, op, cit,. II, pp, 36-7. ^^% ayley. Op, c i t . , XV, 65, ^^^Solomon Gandz ["The Origin of the Ghubar Numerals, or the Arabian Abacus and the A rticuli," Isis. XVI (1931)> 393-424} be­ lieved that ghub^ (or gôbâr) was the Arabic word for abacus. ^^^G. G, N eill Wright, Writing of Arabic Numerals (London: University of London Press Ltd., 195^> pp. I 26- 7 , 177 4* The concept of place value and zero (though not the symbol for

zerê) in the Hindu-Arable number system,

Bayley wrote:

the semi-savage, who counted upon his fingers and recorded the results of his calculations in rows of mere scratches upon the sand, gave the first hint of the abacus. So the rude numeral signs composed of groups of single lines themselved were gradually superseded by other more conç>act and convenient symbols. Those, applied to the abacus with its primitive decimal system, led to the discovery of the value of position. Out of this again arose the Arcus Pythagorous or ’’written abacus*, with its accumulation of various series of numbers; and from t h i s , in quick succession, came th e new methods of decimal arithmetic; and lastly the invention of a sign to fill the *plade vide*, the *sunya*, or zero; . .

Dantzig believed that "the discovery of zero was an accident brought

about by an attempt to make an unambiguous permanent record of a

counting board operation.

5. The acquisition of new words in the English language.

Some of the words and expressions that have been absorbed into

the language as a result of wide use of the abacus are:

casting an account counter

calculate calculus

"borrow" "carry"

abacist to lay a wager (developed from: to lay the sum) cipher to read between the lines

166 Bayley, op. c it., X7, p. 71.

1^7Tobias Dantzig, Number, the Language of Science (New York: Macmillan Co., 1935), p. 31. 1 7 8

6. The techniques used in performing simple arithmetical operations.

In early algorisms, techniques were used which followed procedures

on an abacus. For example, Gerbert developed the following method

for addition:

M CXI

III 17 7 III 7 I I I II 17 7 7 71 I IX

Even after arithmeticians began using Hindu numerals to record num­ bers they used line-reckoning to perform the processes. In order to watch their progress they crossed the figures as used. The following example from Kobel is illustrative of this technique

69 Modern addition and subtraction patterns, along with the terms '’carry­ ing*' and "borrowing, " probably originated in processes used on an abacus. Early multiplication and division also showed the influence of the abacus. For example, the following method^^^ was used in order to multiply 4600 by 23:

G. Melvin, Education. A History (New York: John Day Co., 1946), pp. 137-8.

L. Jackson, Educational Significance of Sixteenth Century Arithmetic ("Contributions to Education," No., 8 Tnsw York: Teachers College, Columbia University, 19063 ), p. 46n, l^Ociiasles, ’’Regies de I ’Abacus," Comptes Rendus. X7I (Janvier- Juin, 1843), 221, 222, 234. 179

CM XM - M C XI

4 6 Multiplicand 1 8 Product of 600 by 3

1 2 Product of 4000 by 3

1 2 Product of 600 by 20

8 Productof 4000 by 20 - 1 ' P 8 Product total 2 3 M u ltip lie r To divide 900 by 8 the following technique^?^ was used:

CXI

2 D ifference 8 D ivisor

Dividend i 8 Product of the difference 2 by the denomination 9* 2 Product of the difference 2 by the denomination 1,

t * - Sum of the two digits 8 and 2 come from these multiplications. i Product of the difference 2 by the third denomination 1, 4 Product of the difference 2 by the 4th denomination 2, Remainder from the d iv is io n . 1 1 Denominations 9 2 ! Sum of the denominations. Quotient 2 1 1 t o t a l .

171: Ibid. . pp. 228, 229, 235. 18 0 In using an abacus, after the beads were shifted or the counters

displaced, there was no record to retrace or no possibility of re­ viewing the work. Therefore, a means had to be devised by which the results of the work could be checked. Thus the proofs of nines and

sevens were developed.

THE DISGOmMJANCE OF THE ABACUS

During the Middle Ages the western Europeans had been chiefly using the abacus for computation and the Roman numerals for the re­ cording of numbers. About the tenth century the coagileted system^"^^ of the Hindu-Arabic numerals began to make its appearance in various aspects of life. Some of the channels by which these new numerals are believed to have been introduced and disseminated to the European countries were:

1. The mathematical treatises of such writers as -

a. al - Khowarizmi, a Persian mathematician, who wrote

(c. 825) a small book ejq^laining the use of these numerals.

The 12th century Latin translation of this work, probably

^^%nith, o£. cit. . pp. 151-4» 173 '•^Having place value and the zero.

^74j>or a discussion of additional treatises see Suzan Rose Benedictÿ A Comparative Study of the Early Treatises Introducing into Europe the Hindu Art of Reckoning (^Concord. N. H. : Humford P re ss , 1914). 181 made by Adelard of Bath^^^ (c. 1120), was instrumental in

causing t h is system to be known.

b. Leonardo Fibonacci (of Pisa), who upon returning from a

voyage about the Mediterranean, wrote (c. 1202) his

Liber Abaci.

2. ,Commerce and trade during the early part of the 12th century.

Beginning with the 8th century the new numerals were establish­

ing themselves in the countries around the Mediterranean and

people were using them more and more in financial and commercial

transactions.During this period of time England traded with

th e se M editerranean c o u n tries and the men engaged in commerce

became acquainted with the numerals.

3. Scholars and schools (early in the 12th century).

As the new numerals became more comonly used in commercial trans­

actions, the schools began teaching their use. Also, as scholars

(such as Adelard of Bath and Fibonacci) traveled in countries

^^^Liber Algorismi de numéro Indorum (The Book of al-Khowarizmi, oh Hindu Number)7

^"^^Charles Singer in Yeldham, o£. c it. , p. 14.

^^^Howard Eves, Introduction to History of Mathematics (New York: Rinehart and Co., Inc., 195377 PP* 19-20. ^7%. E. Smith, Teadhing of Elementary Mathematics (New York: Macmillan Co., 1922), pp. 52-3.

l ’79ygidham, og. c i t . j p . 65. 182 along the Mediterranean, they came in contact with the numerals

and brought them back to their owh countries.

4* Calendars and almanacs composed in France, Germany, Spain, and

other southern parts of the continent.

These were sent to the monasteries throughout Europe and thus

helped to acquaint people with the numerals. Yeldham told of

an interesting feature of the preface to a calendar for the

year 1430 . . . "the writer reaching the hei^t of inconsis­

tency when he says that the year consists of 'ccc and sexty days

and 5 and sex odde howres', . . .

The spread and adoption of the Hindu-Arabic numerals was a slow process. From about the tenth century until well into the sixteenth century a struggle was waged between the abacists^^^ and the al- gorists.^^^ The chief^®^ abacists of this period were Gerbert (tenth century), Abbo of KLeuiy (c. 970), Heimannus Contractus (1013 - 1054),

Bernelinus (c. 11th century), Gerland (first half of 12th century), and Eadulphus of Laon (12th century). The outstanding^®^ algorismic text during this time was the so-called Algorisnms Vulgaris which is u su a lly a ttr ib u te d to John of Holywood (fl. c. 1230).

i-ÔÛYeidham, o^, c it. . p. 86, citing Harl. MS. 937. ^®^he advocates of the abacus as the best means of calculation. 182iphe advocates of the new numerals (the Hindu-Arabic) as a desirable substitute for the-abacus. ^®%teele, o£. c it. . p . x L ii. ^®^Yeldham, o£. c i t . . pp. 69-84. 183 The abacus was able to hold its own for raany centuries partly

because of the strong opposition to the Hindu-Arabic numerals. Many

people sincerely believed that operation with these new numerals was

actually inferior to the traditional methodO With the abacus. Others

objected to the numerals because of Origin. The Arabs and

Hindus were non-Christian; therefore, the Christians wanted nothing

to do w ith th e i r in v en tio n s. Laws were even made fo rb id d in g the use

of the numerals.

The acceptance of the new system of numerals was slow for other

reasons. Demilt said:

It was not needed in business since only results were recorded and the calculations could be quickly and accurately done by means of an abacus; there was no social need for the system for the age of admiration for statistics was not to begin for several c e n tu rie s.

Probably one of the chief reasons people refused to accept the

Hindu-Arabic numerals was that the Roman numerals reflected the structure of the medieval count ing-board. The combination of the

Roman numerals and the counting-board was a simple and effective way of perfoming the basic arithmetical computations needed by the ma­ jority of persons of that time.

In order to use the counting-board one needed to know only the following information:

1. The symbols for the Roman numerals.

^^^Glara DeMilt, "The Origins of Our Numeral Notation," School Science and Mathematics. XLYII (November, 1947)> 705-6, 184 2. The rules for the laying of a number (expressed in words or in

Soman numerals) on the counting-board, and conversely, the rules

for the reading of a number which had been laid on the counting-

board*

3* The principles 6f exchange, namely:

a, five counters on a line are exchanged for one counter in the

space immediately above the line.

b. Two counters in a space are exchanged for one counter on the

line immediately above the space.

4. The technique for counting to five.

As one w ill note, it was possible to record numbers and to per­ form the operations of addition and subtraction without having to learn any combinations.

On the other hand, if one decided to begin using the Hindu-

Arabic numerals he had the following difficulties:

1. The learning of a new set of numerals, including the "new­

fangled" zero.

2. The memorization of one hundred addition facts and one hundred

subtraction facts.

3. The new numerals were not designed to go with the counting-board,

30 a person would have had to transform a numeral from the Hindu-

Arabic to the Roman numerals before laying the number on the

board; and conversely, in reading a number laid on the board, one

would have to transform it from the Roman numerals to the Hindu-

Arabic numerals. 185 For exan$>ley assume one wished to record the number seventy-

eight upon a counting board. This number written in Roman numerals ia ucxvin.

C

Following the number as written, one immediately sees one L and lays

one counter in the space indicated by L; the two X's indicate the

placing of two counters on the line marked X; the one V indicates the

placing of one counter in the space marked V; and the three I's in­

dicate the placing of three counters on the line marked I. This is

quite simple and easy to follow. Whereas, seventy-eight expressed

in Hindu-Arabic numerals — 78 — gives no clue at a ll as to the

pattern to follow in laying the counters on the board.

Additional reasons for the slow acceptance of the Hindu-Arabic

numerals were: (1) prior to the invention of the printing press

(1450) knowledge was transmitted chiefly by word of mouth; (2) modem methods of conqputation, particularly those of multiplication and

division, were not developed until the 15th century;and

(3) cheap paper was not available until the nineteenth century — in fact, paper-making of any type was not introduced into Europe until,the 12th century.

18^ D. E. Smith and L. G. Karpinski, The Hindu-Arabic Numerals (Boston; Ginn and Co., 1911), pp. 136-7. ^^Loc. cit. 186 By overcoming the physical difficulties with the invention of cheap paper and of the printing press, the algorists began by 1500 to be winning the battle against the abacists. The discard of the abacus still was not sudden for it is the nature of human beings to make changes slowly. Italy was the first to cease using the abacus, 188 doing so even before 1450. The instrument was practically ex- 189 tinct in England by the middle of the 17th century, while

France,Gennany,^^^ and the Catholic Low Countries^^^ continued 192 its use even into the 18th century.

SUMMARY

In this chapter a study has been made of the abacus in its historical setting — the countries that have used abaci, the various foims which have been developed, the influence upon even present- day mathematics, and the great struggle man underwent before finally accepting the Hindu-Arabic numerals.

188 Barnard, o£. cit. . p. 105

^^^Ibid.. p. 208 ^90ibid.. p. 85. ^% bid.. p. 90. 192Tho following sources are excellent for information about the abacus: Moritz Cantor, MAthematische Beitrage zum Kulturlebeh der volker (Halle: H. W. Schmidt, 1863); Moritz Cantor, Vorlesungen Ober Geschichte der Mathematik (3 vols.; Leipzig: ^.G. Teubner, 1907-8); Alfred Nagl, "Die Rechentafel der Alt en," Sitzungsberichte der Kais. Akademie der Wissenschaften in Wien. 1914; Alfred Hagl, "Die Bechen- methoden auf dem Griechischen Abakus." Abhandlungen zur Geschichte der Mathematik. Leipzig, 1899. CHAPTER n A wmr HISrCRIGAI. SCRVET OP THE TEAGHIHG OF ARITHMETIC (6TH OaiTUSï B. C. TO I7TH GENTURT A. D.)

jc T ia r

T2m preceding ehapters have ahem hoar aan uaed tfai four phyaieal devieea — tally, knotted eerd, fiagera, and abaeua — to eoomunieate and record mambera and to perform the baaie mathematical eeayputatiena neceaaary for hia nay of life. In the earlieat tinea, infenation about the teehniquea of uaing theae devieea aaa transmitted in informal aettinga. Teung people learned by watehing and by liatening to their eldera. The first sehoola were believed to have been eatabliabed in cemnection with temples and aynagagues, chiefly for the purpoae of providing religioua inatruetion. Net until the time of the flouriah- ing of the ancient Greeks is there evidence of the beginnings of sehoola organised for puzposes other than the pezpetuation of reli- gieua beliefs.

TEAGHING THE USE OP THE FINGERS There is no extant record as to when sehoola were first eatab- liahed im Athmas, bat the educatiened. laws of Solon date back to the begimzdag of the sixth eontary B« C. The old Athezdan edueatiem prevailed until abeut 479 B. C. (the eloae of the Persian Wars), sad

IS? 188 mmny b#li#ve that during thla period of time (e. 600 B. C. - a. 479 B. C.) finger reekonimg m&e erne ef the mean# ef eonpatatien taught in the eehoola» Sehoola, aa fom al inatitutiena, mere eatabliahed in Berne about 300 B. C. In the Xudua (the moat elemeataiy aeheol), finger reckon- 2 ing aa a neana of calculation mas a part ef the curriculum.

^BUeood P. Cubberley. Hiatorr of Educatien (Bee Tork: Houghton M ifflin Co. i 1920), p. 27. W. Kane, ^n Baaar tomard a Hiatorr of Bdueatiwa (Chicago: Loyola Unlreraity Preaa> 193$)i pp« 42-3. Fredezdok Bby and C. P. Arrowood, The Hiatory and Phtln^pt^y of Bducationi ; Ancient liedieyal^ ; (Noe loi^cs Prentioe-Mall, Inc., 1942), p. 280. J. P, Mahaffy. Old Gre^ Educatien (Mem Torkt Harper and Brother#. 1Ô82), p. 56. — . K. J. Preemmn. Sehoola of Hellaa (Londonr Macmillan and Co.. Limited, 1932), p. 104. 8. S. Lwrie. Hiatoriaal Survey of Pre-Ghriatian Education (New Torkt Longaana, Green and Co., JL900), p. 258. B. C. Moore, The Story o f Diet ruction: the Beainninge (New Xezfci Macmillan Co> , 1 9 ^ pT S S * % . 6 . Geed. H iatorr o f Noatom Education (New York* Maemillam Co., 1947), pp. 47-8. Laurie, £g. e i|., p. 337. L. C. Karplnaki, Hiatorr of Arithmetic (New York* Band McNally and C o., 1925) j pp. 22-3. Luella Colo, Hiatorr of BAucatiom; ^eratea to Menteaaori (Nee York* Binehart and Co., 1950), pp. o2-3* B H> Wilda. Poimdaticna ojf Mbdem Bdttcation (New York* Farrar and Binehart, I n c ., 1939), P, 137. B. F. Butta. Cultural Hiatorr of Education (New York* McGraw- H ill Book C o., I n c ., 1947), pp. ll6yL25. Moore, on. eit^. no. 345-6. Cubberley, SSL* e^t.. pp. 64-5. W. Nk Bk Ball; Short Account of the Hiatorr of Mathematica (New York* Macmillao Co. , 1901), pk 117. Thomaa Woody, Mgc and Education B arlr S o cietiea (New York* Ihcmillan Co., 1949), p. 580. 189 Sin«« there are ee fee direct record# of finger reckoning being used by ea rly Greek# and Boaane, i t le neeeeeazy to r e ly upon in d irect reeords as reported in the literature of the periods. Sons authori­ ties believe that the latter serve to give a better picture of a people than many other sources. Kaynard eag>ressed this thought «hen she erotes That literature reflects life is a theexy often presented, and very videly aecepted. There are even those lAo claim for the imaginative record precedence over formal history as a true picture of a given period#) If one agrees with Maynard's thesis, the classical literature of the early Greek and Boman period (see pp. 68^99 and Appendices IV and 7) could well substantiate the writings of the authors aum- tiwaed in footnotes 1 and 2. One might safely assume that a practice ho widely used in busiaess transactions and in evexy day encounters would have been taught in the schools. For indicative of the eontimed teaching of finger reekomiag, again it is necessary to turn to the writings — this time, the textbooks — of the periods. The first w rittv instructions on finger reckoning available to present-day scholars were published in the seventh century A# D., and books continued including this phase of calculation until the I6th century (See pp.99*W*

TBACHmG THE USE OF THE TAUUT As far as the author of the present study could determine there was no indication of the use of the tally (the notched form) being

u Katharine Mayoiard, "Science in Early English literature," Isis. 1711 (1932), 95. 190 tamght in tha aohoola; but with aaj iaatrummt used by a# many gam- aratioas aa this ana aaa> tha young paapla aoald hava to laarn in soma nay tha basic principlaa of its aaa in ordar to be abla to carry on buainaaa tranaaetiona. Jankinson ingplled that thara might ha va baan a paried of training praraqoiaita to baing anplayad in tha Sm- glish Sachaqiaar during tha aaran-hnadrad-Toar pariod in which talliaa «era usadt • • .«a have bar a, if «a aan aatabliah tha axiatanea of a widoapraad ayatma, a piaca of tha scattarad and bbacura, but valuable avidanea for the aotivitioa* aapabilitioa and faahiona of a vary important paracn, the laieua littaratua. tha man of aduaation «he «aa not a clarie. The axiataaoa of a large body of aueh mam, aquippad aa we know they auat bava baan with what wa aheuld now ca ll a comnarical aduaation of quite ccnaidarabla range and finiah • •

TBACHING IHE USB CP m s QdlFU Bridansa would indioata that the Iheaa provided formal traimimg in the uao of the knotted cord (or quipu). Aa mentioned on page about 1200 A* 0. dâhoola «ara aatabliahad la idsLeb young man ro- ceivad the training nacaaaary to aaauma the rasponaibility of entering govanomant aarviea aa qnipocaaayoea (o ffic ia la in charge o f tha Inean Qaipua).

TEAÇHIHG THB USB OF m s ABACUS 3m Atbana L ittle ia known about tha teaching of axlthaatic in early

H ilary Janfcinaen, *iCadiaral Talliaa, Public and Private,** Arcbaoloaia.m iy (1925), 314. 1 9 1 Athenian schools, but according to available evidence several his- torians believed that for more difficult computation instruction was given in the use of the abacus. Oubberley,^ Freeman,^ and p Mahaffy stated that the form of abacus used in the schools was the type illustrated in Figure ^1,

In Rome

Butts^ and Oubberley^ believed that arithmetic, and particularly counting, was emphasized in early Roman schools. This was chiefly because citizens of the complex commercial and mercantile society of

Rome needed to keep business and household accounts. One of the

5 P. R. Cole, A History of Educational Thought (London; Oxford l&iiversity Press, 1931), p. 26, Lue11a Cole, A History of Education; Socrates to Montessori (Hew York: Rinehart"and Co., IP5577~p7~37e Eby and Arrowood, 0£, c it., p. 280. Sir PaullHarvey (ed. ), The_,Q^ord Companion to Classical lite r­ ature (Oxford: Clarendon Press, 19^1), p. Kane, og.. c it., pp. ii2-3, laurie, op, cit., p. 258, Sir John Leslie, Philosophy of Arithmetic (Bdi%urgh: William and Charles Tait, 1820),' p.' 95. Moore, 0£. cit., p. 138.

Cubberley, 0£, cit., p. 27. 7 'Freeman, 0£, c it., pp. lOh-5.

^Mahaffy, 0£, cit., p. 56. 9 B u tts, 0£, c it., pp. 116, 125. 10 Cubberley, 0£. cit., pp. 6U-5. 192 jMihods of counting nsac jtyj mccna of am abacas* Other writcrs^^ agrocd that the use of the abacas vac taught in the schools* Clarke told of the box that Bceam children brought to school aith thesi* The box obtained ariting-naterials, book-roUs, tablets, 12 and reokoning-stones (calcaliV . Comfirmatiom o f the la tte r part 13 of th is statement aae provided by Horace, iriio arete* * • •! oae it a ll to *ay father, aho, notaithatamdlng his short and narrow eireanstanees, disdained to pat me to MLavias's school. (tAore several groat men plac'd their Sràs, iriu» carry'd on their Axns their Coamters and Tables of the liemUily D&terest of several Sams, ot ahleh they aero oblig'd to.give in the Computation).^ In Other Earonean Countries In writing of education in Gaul during the fourth and fifth centuries i. D., HaaAoff stated that for advanced pupils there were

^^3lorian Cajozi, A History of liathenaties (New fork* MhcmiUan Co., 1897), p. 79. Cole, cit.. pp. 62-3. Philip Ouroe. History of Education (New Toik* Globe Book Co., 1921) , p . A l. Earl Fink, A Brief Hlmtory of Mathematics (Chicago* Open Court Publishing Co. , 1910), p. 2p, Good, jOg. j s ^ ., pp. 47^8. K am inski, o p . cit.. pp. 22-3. Moore, eg. si^ ., pp. 345-6. Wilds, m. s£t., p. 137. A. S. W ilkins, Homan Education fCambridge* U nlyersity Press, 1905), pp. 5>4. Woody, £g. £i^., p. 580. 12 , George Clarke, Tkg Education gg SÈâtifcSBLSà Borne (New York* Macmillah and Co., 189^ p. 55. ^%orace (65 - 8 B. C.) was a Homan poet and satirist. Horace Satires I. vl. 72. 1 9 3 special teachers (calculatores) who used the abacus or tabula»

In relation to the teaching of the abacus during the Middle Ages

Sedgwick and lÿler reported that " in 787 Charlemagne , . . order­ ed the establishment of schools in connection with every abbey of his realm , , , the mathematics taught in Charlemagne's schools would natu- 16 rally include the use of the abacus . , Ball agreed somewhat with

Sedgwick and %rler in a qualified statement he wrote about the abbey schools of this period. He said, " . . . mathematics, if taught at all in a school, was generally confined to the geometry of Boethius, the use of the abacus and multiplication table, and possibly the arithmetic of

Boethius; , . .

Cajori, however, did not believe that the use of the abacus would have been taught in Charlemagne's schools. He based his theory on his interpretation of the work of Alcuin (73^ - 80l^) who directed the prog­ ress of education in the great iYankish ïbçire. He wrote:

In the great sees [cathedrals] and monasteries he [Charlemagne] founded schools in which were taught the psalms, writing, singing, confutation (computus), and grammar, Efer computus was here meant, probably, , , , the art of confutation in general, Ebcactly what modes of reckoning were then employed we have no means of know­ ing, It is not likely that Alcuin was familiar with the apices or with the Roman method of reckoning on the abacus. He belongs to that long lis t of scholars who dragged the theory of numbers into theology.

^^Theodore Haarhoff, Schools of Gaul, a Study of Pagan and Chris- tian Education in the Last (%nkury of the Western SnpirelLondon: te^rdl œ i r s i ^ Rs-s^TTp^iTp. BF,------— l^W.T, Sedgwick and H,W, lÿler, A Short History of Science (Hew York: Macmillan Co,, 1939), p , 177* ^^B all, 0£, c it,, p, llj.1, iG C ajori, 0£, cit., p. 1 1 9 , 194 Tb# o f tb# ninth e#ntmry «laiplojed th# abaoiu, and thua helped to spread this font of arithmetie to the Western 19 nations* ^ By the tenth eentuxy Uie monasteries bad developed teo types of schools — the inner monastic schools for those intending to take the vows, and the outer monastls schools for laynsn. In the latter, sinple reckoning of the Roman type was tau|^t* This would have in­ cluded the use of the abacus*^ Oerbert (who la te r became Pope S y lv ester), when a young man, taught school at Rheims. During this time he used a form of abacus 2 l of his ema desiga. In surveying this period. Ball wrote s We may sum the matter up by saying that during the ninth and tenth centuries the mathematics taught was s till usually confined to that cosprised in the two works of Boethius together with the praetisal use of the abacus and the multi­ plication table, though during the latter part of ^ e tiow a wider range of reading was undoubtedly accessible,^ In 1283 in Florence there were six abacus schools in which per­ sons were trained in reckoning as preparation for a business eaxwer,^^ These schools might haro taught the use of the instrummt — the abacus — for it continued being used in Italy until the middle of

^^Fink. OP. cit.. p* 20. 20 jgabberlsy, £j^. c it.. pp. IgO-l, ^See Plate m .

^Ball, gi^,, p. 142. 23' ^^XjnnXjnn Thokndike, "Rlemsntary"Rlemsm and Secondary Education in th e ICLddle Ages,f gteeculun. %F (October, 1940), 400^. 195 th* I5tb omtarj. Or tlM aehools'oii^ have taught tb# uae of the HiBda-4rable anaberala* The aerehanta of Italy began oiing the netr notation aa ew ly aa the thirteenth eentoryj then, too, the word abacua changed ita meaning and beeame a ayn

Thia league became ao powerful that it kept arithmetic out of the common aehools for many yoara. In 1316 in Italy, "of 73 arte liated as auhject to a certain tax, the sixty-first ccopriaed ma at era of grammar and of the abacua, and thoae teaching boya to read and write." In the I6th century arithmetic was being taught in the sdieola by the method of the counters. The first regulation which introduced arithmetic aa a required subject was the Bavarian Schuelcrdnunak do 27 annf 1346, Aa taught at that time, arithmetic consisted chiefly of computation upon Unes with counters or a figare-c(mq»itation, with an oceasimaal use of finger-reckoning. OL Cajori, gi^., p. 1 2 9* ^ D . B. Smith, TeachiM of Elementary Hathamatiea (New Terkt Macm illan Co., 1922), pp. 62-3, 26 Thomdike, og. c i t . . X7, 400-81, citin g Giuasppa Manacorda, Stqrla della acuola in Italia* 11 medio eve (1914), I, 152. ^Plak, op. cit., p. 4 3 . 196

During th# r#ipi of Qa#«a Slisuboth (I55d - 1603)scmb # of th# town# pa#s#d laws stating that th# stnd#nts should b# taught to cast 28.29 accounts. This was to bo don# at least «me# a w##k, or on holidays and Saturdays^ or festival days. Th# 1668 edition of Beeorde's Grounde of Arte# contained a chapter on reckoning with counters.^ But, as stated cm page 186, the abacus bad practically disappeared in En^and by this date. By the 18th century the use of the instrument was discontinued in other western European countries. Thus the a lg o r is ts won the b a ttle . Europe ceased to use the abacus. The abacus which th# algorists banished when they brou^t into vopt# th# modem qrstem of reckoning was a remarkable in­ strument. Something concrete and real was lost in the victory of the algorists. When we attack our number problems by using automaticaliy the ways of thinking, saying, and writing digits that we have been taught, when w# do these things just because they are near and are thm st upon us, we are manipulating by symbolism and we often use it with little sense . . , It is all right to set up habits of manipulation. It is necessary in our world of today to compute accurately and rapidly, but our manipula tiens w ill be all the better and our coaputi^ a ll th# mbre swift and accurate if we come by these things through growth and a sense of their meaning.

i . M. Stowe, aueliih Grammar Schools in the Beian gg Queen beth ("Cmtribu^ions to Edlueatiom,^ No.22 [New To** Teachers ^llege/^Golumbia Onivelrsity, 1#8] ), pp. 106n, 107a, l^ n. ^S.G. Parker. Teactbook in Histonr of Itodem Blemeotarv Bchcation (Mew York: Ginn and C o., 1 9 1 :^ pp. 53-4». Florence A. Teldham, The Teaching of through Four Ydars (1535-1935XLondoni Georgs G. Harrap sndî'Co.i L td., 1936), p.33. R. Buckinc^iam, Eleim&tarv A rith n etie. I t s Meaning and Practicm (Mew York* Ginn and Co. . 19A7). P. 84. 197 SARLT FHILOSOFHEBS WHO ADVOCATED THE USB OF GWCRBTE METERIAIg IN TEACHING Th# marly mehool# war# •atabliah«d to educate the children o f the wealthier claeae#, and logiatlc^ was not alwaye included in the currioulnm for it was generally believed that only the tradeanan and aerrant elasaea needed the infoxnatien. But even with thia attitude an oecaaicmal voice epoke out advocating the nee of manipulative material# in the traiiing of youth. Probably the fir at person to mxpvna such an opinion was Plato (4277 - 347 B. C.). He wrotes It i# neoesaary then to say that the free-bozn oug# to learn of each of thoae subjects so much as the great mass of boys in Egypt learn together with their letters. For, in the fir at place, the m les relating to reckoning have been so artlesaly devised for children, that they learn it in sport, and with pleasure; (for there are) dietrlhutions of certain apples and ch a p lets, the same numbers being adapted to more and a t the sane time to fewer: . . .and, moreover, when playing, they mix phials of geld, and cofper, and eilver, and other tfediga of this Idnd, and some distribute them whole, adapting* as I said before, to their sports the use of necessary nuBdsers'; and thus they benefit those, who are learning to draw up and lead cut a m ie s, aM to arrange encampments, and to regulate a household, . . Another strong advocate of the use of objects in teaching was CcsMnius (a faasous w riter on education and a Moravian bishqp;

^^Ihe ancient Greeks divided the study of numbers into two ' branches* (1) arithmetic, idiich was the theoxy of numbers; and (2) logistic, which was the w t of calculating. These two branches were generally considered as separate subjects until the fifteenth can- tury Co* E Smith, History of Mathematics (New Toxic* Ginn and Co., 1925), H , p. 7.1 ^^Plato Imwg ai9. 198 . H# received hie in itial Inpulae tcward Senae-Aaàliaai^ from Vivea (a Spanlah acholar; 1492-1540). Eby and irrowDod statedt ' ...... 71ve* maa the firab to point oat from the atandpolnt of Pdyehologj> the function of aenae iapreaaiona, and to inaiat vqpon Inductive thinking aa oppoaed to the memoriterg, and authoritarian teadhing afaich waa unlveraally employed. Comeniua baaed hia philoaophy upon principlea advocated by Tivea, but aaa even more emphatic than Vivea in hia belief that a ll knowledge began by sensuoua perception. "In hia edueatiohal aorka may be found the firat promulgation of the principlea and plana of Object Teaching, and of a graduated mystem of inatruotion adapted 36 to the aanta of the age in which he lived." One of hia principlea aaa* Theae things, therefore, that are placed before the intsiligemce of the young, must be real things and net the ahadowa o| thhaga. I repeat, they muat be tfainaa* and by the term I mean determinate, real, and uaefhl things that cu make an impression cm the amaaea and on the im agination.^ In dlacuaaing the vemacular-aohool (the school for children between the ages of six and twelve), Comeniua said they should be taught "to count, with ciphers and with counters, aa far aa ia neceaaary for

3^ ' "The doctrine that an objective world with aenae qpalitiea exists indqcendent of cognition and that sensory experience y&elda direct and veritable knowledge ; applied to education by way of empha- aia on aenae training, the study of the physical mavirmoment, and the use of concrete objects aa illuatratlve materials in teaching." CC. V. Good, Dictionary of Education (New Torkt KeGvaw-Hlll Book. Co., Inc., 1945), p. 332.] ^^Frederick Eby and C. F. Arrcfwood, Develeoment of Modem Educa­ tion (New Terkt Prentiee-Nall, Inc., 1942), p. 56é ^N. A. Calkins, "The History of Object Teaching," Barnard's American Journal of Education. XII (1B62), 635. ^^John Amos Comeniua. The Great P idactia o f John Amos Comeniua (London* Adam and Charles llaCk, 1910), p.l84« 199 36 pM etieal pnrpoMa.* Thaae thr## am were not the enlx *veiees la the wHdemeee" during thle period of time, but they were tralLblaaere in thie basic philosophy. Their ideas were not heeded, had as the schools began using the Hlndu-drabie numerals, teaehing techniques receded farther and farther from the source of power which is available efaen nanl- palAtlTs nateriale are made the basis of arithmetic teaching. Smith agreed with this thought idiea he wrote* • . .in the enthusiasm of the first use of these symbols, the Chzdstian schools threw away th e ir abacus and th e ir numerical ccamters, and launched out into the use of Hindu figures, ând wmlë they saw that the old-style objectivo work was unnecessary for chleulatim, which is true, they did not see that it was essential as a basis for the coiaprshenaien of number and for the dereLopmnnt o f the elommtary ta b les o f a e r a tio n . Hence i t came to pass that a praiseworthy revolution in arithawtic broai^t with it a blamcwcrthy method of teaching. . ,39 asith described this "blamewerthy method of teachings as one in which the teacher required the students to define tarns and to leazn the rules in a mechanical manner.

SDlflttBX In this chapter a brief review was given of the teaching of the four physical devices — tally, knotted cord, fingers, and abacus — as a part of the curriculum in organiaed schools. Evidence in­ dicated that the fingers and the abacus were the only two of the devices taught Mdtensivoly in Europe.

266. % . X. Smith, Teaching of Blmmmtarar Mathenatics (Mew York* Macmillan Co., 1922), pp. 71-2. Ifoatloa «aa aise made of three of the early phUoaephere — Plate, Tivea, and Comeniua — «be adrocated that cenerete aateriala should be made the basis for leaRking, parbieularly «ith the younger children. With the introduction of the HindumArabic numerals, the schools ignored the reeMmchdatidh# of these hhnyaand of ethers she had similar beliefs. As a res«ü.t, teaching and learning became based on mechanical techniques «ith complete disregard for under­ standing. CHAPIBR 711

THE TSAGHIHG OF ARITHMETIC IH THE T M im ) STATES

(Early Colonial Days to 1935)

The system described by Smith (see page 199) was the one with

which the early colonists of America were familiar in their European

environment. Thus, it was only natural that when arithmetic was in­

cluded in the curriculum of the schools in the new country, it was

taught in a veiy m echanical way. The expression "when a rith m e tic was

included” is used, for many of the colonists brought with them old

world traditions and ideals, one of which was that "arithm tic was

not considered essential to a boy's education unless he was to enter

commercial life or certain trades,"^ Hot until I 699 was "cyphering"^

mentioned in the curriculum of the Latin grammar schools. The subject was optional in the schools until 1?89, when a law was passed in

Massachusetts and New Hançshire which made it one of the compulsory su b je c ts.

In tracing the development of the use of manipulative materials in the teaching of arithmetic in the United States, the years w ill be divided into the following periods:

% alter S, Monroe, Development of Arithmetic as _a School Subject ( "U,8 , Bureau of Education Bulletin," No. 10 [Washington, D. C, : Government Printing O ffice, 19173)» P» 5» 2The process of calculating, computing, or figuring arith­ m e tic a lly ,

201 202 1. The Ciphering Period — from early colonial days to 1821,^

2. The Period of the Influence of Warren Colburn — Active Period:

1821 t o 1857 .

3. The Period of the Influence of Vfarren Colburn — Static Period;

1857 to 1892.4

4. The Reflective Period - 1892 to 1911.^

5. The Life-Situations Period — 1911 to 1935.

6 . The Present Period — 1935 to 1956.

THE CIHIERIHG PERIOD — FROM EARLT CÔLOITIAL DAYS TO 1821

This period is given the name, the Ciphering Period, because of

the method prevalent in the teaching of arithmetic. Since practically

none of the students had textbooks the teacher had to dictate the prob­

lems, usually from his ovm ciphering book. The student brought from

home a book which consisted of a number of blank pages fastened togeth­

er. The teacher then set the pupil a "sum,” told him the rule for its

solution, and sent him back to his seat to work it out for himself.

When the student felt he had the problem worked correctly, he took it

to the teacher for approval. If approval was received, then he very

^This category was used by Plorian Cajori. The Teaching and History of Mathematics in the United States (” Bureau of Education C ircular o f Inform ation," Ho. 3 CWashington, D.C.: Government Printing O ffic e , I 890 I ) , p. 4 9.

4rh is category was used by Monroe, op. c i t . , p. 53»

^This category was used by Guy M. Wilson, Mildred B. Stone, and Charles 0. Dalrymple, Teaching the Hew Arithmetic (Hew York: McGraw- H ill Book Co., Inc., 1951), p. 25. 203 carefully and neatly copied it into his ciphering book. There were no

explanations or discussions; this same procedure was followed for each problem worked.

In the free schools of Hew England and New Amsterdam during the

second half of the 17 th century, children were taught not only to

"cipher" but quite frequently "to cast accounts."^ There were no textbooks adapted to the level of children; they were written chiefly for adults. The arithmetic books of this period were based on the deductive approach to learning, as indicated by the general order of treatment, namely, definition, rule with explanation, often a brief rule of proof, example with explanation, sometimes an.' exaüçle of proof, drill problems, and written problems.? The three most influential textbooks® were: Pike, Nicholas. A New and Complete System of Arithmetic. Newburyport, Massachusetts: printed and sold by John My call, 1788 ,

Daboll, Nathan. Daboll's Schoolmaster's Assistant. New London: Printed and sold by Samuel Green, 1800.

Adams, D aniel. The Scholars A rithm etic. Leom inster, M assachusetts: printed by Adams and Wilder, 1801.

^ L. C. KarpinskiV History of Arithmetic (New York: Rand ^cNally and Co., 192$), p. 82. ? H. L. Smith and Thomas Eaton, An Analysis of Arithmetic Text­ books (First Period - 1790 to 1820 ) T^Bulletih of the School of Ed­ ucation, Indiana University," Vol. XVIII ^Bloomington: Bureau of Cooperative Research and Field Service, Indiana University, 19423), pp. 50- 1.

® Lavada R atliff, "Historical Development of Methods in Arith­ metic in American Elementary Schools," (unpublished Doctor of Education dissertation. University of Texas, 1946), p. 31» 204 Sir John Leslie (1766 - 1832) of Scotland wrote a book in which a

large section was devoted to Palpable Arithmetic.^ In this section^®

he advocated the use of an abacus for developing number systems on the

binary scale, the ternary scale, . . . and the denary scale. Leslie's

work was an expression of his philosophy and belief as to what should

be done.

THE PERIOD OF THE INFLUENCE OF WARREN COLBURN ACTIVE HÎRIOD: 1821 TO 1857

A new period in the teaching of arithmetic began in 1821 with the publication of Warren Colburn's book First Lessons in Arithmetic 11 on the Plan of Pestalozzi. This was the first arithmetic in the

United States to introduce objective m aterials.H is book was unusual in that it suggested solving the first examples by using beans, nuts, and similar materials; there were no rules included; the pupils were introduced to a topic by means of practical examples; the examples were to be solved without the use of pencil, slate, or paper; the Hindu-Arabic symbols were not introduced until page 38 (except for the numbering of the problems); and methods of calculating with these symbols were not included.

^Arithmetic in which the numbers are exhibited by counters, or abbreviated representatives of the objects themselves.

^*^Sir John Leslie, The Philosophy of Arithmetic (Edinburgh: William and Charles Tait, 1820), pp. 15-102.

^ lln 1913 this book was s till being published by Houghton M ifflin .Company. 12jjonroe, op. c it . . p. 69. 205 In 1830 Colburn gave an address on the ’'Teaching of Arithmetic." In this address he stated:

At first, before he ^the student] is familiar with the addition and multiplication tables, some kind of counters seem to be necessary; but it is not important what they are . . . I believe the fingers do about as well as anything. If the scholar is allowed any helps of this kind, he should be left to manage them entirely by himself, and in his own way. Any helps by which the work is partly done for the scholar are certainly injurious. It is by his own efforts, that a child is to learn if he learns at all. You might, with as much propriety, expect that his muscles would be strengthened by seeing others exercise in the gymnasium as tb expect a child’s mind to be strengthened and improved when the teacher does the work for him. The teacher may assist him in understanding the question, but not in the opera­ tion —' not even in arranging his counters; for to do this, is to do for him the most important part of the so lu tio n . ^3

This philosophy, expressed by Colburn, was to have a revolutionary

effect upon the teaching of arithmetic and upon the texts of the United

States — a revolutionary effect which unfortunately lasted only a limited period of time.

About IÔ29 Uosiah Holbrook^^ (I788 - 1854) began to design and to have manufactured the first school apparatus that was inexpensive enough for schools to afford. He believed that apparatus was a necessity for the schoolroom. For arithmetic he manufactured a Numeral Frame (see

Plate XFI) with movable balls or counters. (This device was also called an abacus, counting frame, arithmometer, and arithmeticon.)

13a copy of this address is given in an article by Walter S. Monroe, "Warren Colburn on the Teaching of Arithmetic together with an Analysis of His Arithmetic Texts," The Elementary School Teacher. XII (June, 1912), 463- 80 .

14»»Josiah Holbrook," Barnard’s American Journal of Education, m i (March, i860), 229-56. 206

P late XVI, Numeral Frame

Brownell, o£. cit., p. 21. X o i

I. NÜKKRi^L PRAM:. Fio. a.

T he Numeral Frame was designed for Primary Schools, but has proved of nearly equal service in intermediate and grammar schools ; wherever young pupils require illustra­ tions to enable them fully to comprehend operations with abstract mathematical quantities, this frame furnishes tho readiest mode of giving the desired instruction. 20fl

This was to be used in teaching numeration, addition, subtraction,

multiplication, and division of whole.numbers, as well as represen­ tation and addition of fractions.

In a manual15 designed to accompany the apparatus, suggestions

were given as to the uses that might be made of the Numeral Frame.

In general, the teacher was to be the one who held the frame before

the class, performed the operations, and conducted the "recitation."

A warning was given that "children should not be permitted to handle

any apparatus except by express permission, and under the eye of the

teadher."^^ However, a few pages farther in the manual the suggestion

was made that "At leisure times let one of the class take the frame

and try to ’puzzle* the rest."^"^

After 1821 many primary books appeared which embodied the use of

objects. Some of the texts used pictures of objects, while others

represented examples graphically by means of marks or dots.^® In

15Franklin 0, Brownell, Teachers * Guide to Illustration. A Mamial to Accompany Holbrook* s School Apparatus (H artfo rd : Holbrook School Apparatus Co., li857) .

^^Ibid., p. 20. 17ibid.. p. 23. ^®Gottlieb von Busse, whose first works on arithmetic appeared in 1786 , was probably the first to use number pictures itzahlenbilder) systematically. He arranged them according to this plan:

fiv e [ j . 0 . Brown and Lotus D. Coffman, How to Teach Arithmetic (Chicago: Row, Peterson and-Co., 1914)* p* 1213” 209 19 addition to Colburn's book, two well-known textbooks of the period were:

Emerson, Frederick. The North American Arithmetic. Fart First. Containing Elementary Lessons. Boston: Lincoln and Edmands, 1829.

The title page of this edition of the book illustrated a

counting frame. The 18$6 edition^® had the same illustration

but no mention or use was made of the counting frame. The

latter edition used numerous illustrations (such as apples,

birds, stars, trees, pears, chairs, chestnuts, and dots) for

counting and for the four simple arithmetical operations.

Ray, Joseph. Ray's Arithmetic; Part First. Containing Simple Lessons for Little Learners on the Inductive ^thodof Instruction. Cincinnati; Winthrop B. Smith, 18AA»

The front of the book has a picture of a numeral frame

(the frame had 12 horizontal rows with 13 beads on each row).

In the Preface, Ray stated:

By means of the counters, which may be regarded as unit marks, used in connection v/ith each question, the pupils w ill not only be able to obtain the correct result, bub w ill, at the same time, see the reasons of the answers they give; . . •

In teach in g A rithm etic to sm all p u p ils , we recommend to the instructor to provide himself with a number of small cubical blocks, to be used as counters. Every school­ room should also be provided with an arithmometer and a black-board, though neither these or the blocks are absolutely necessary in using this book.

^^Monroepop. c it. , Chapter Till) discussed many of the series of textbooks that were used during this period. ^®A copy of this edition is in the library of Professor Nathan Lazar, Ohio State University. 21Q Tiïïhen the author first co/nmenced the preparation of this work, it was his intention to use pictures for illustrations. But on more mature reflection, they have been omitted, and simple counters used instead, for the following reasons: the use of pictures representing animate objects, has a tendency to divert the attention of the learner from the question under consideration; they occupy an unnecessary amount of space on the page; and lastly, the experience of the most successful teachers is opposed to their use in A rith m e tic .21

Ray varied from his expressed philosophy enough to lœe pictures

o f plums on page 5. But throughout the remainder of the book he was consistent in using stars, dots, balls, and counters to

represent pins, oranges, money, cherries, peaches and so on. In

this edition he used counters on 38 pages (of the 48 pages),

while in a later edition (I 857 ) counters appeared on 6 pages (of th e 96 pages).

An unusual and interesting book published in the United States in

1830 was by William Wilson, Vicar of Walthamstow (municipal borough,

N.E. Suburb of London). The book was A Manual of Instruction for

Infants* Schools ; with an engraved sketch of the area of an infants * school room and play ground. — of the Abacus ,22 pf a_ scheme of instruction, and the tables of number [adapted for infants* schools in the United States, by H. Wm. Edwards].

In a chapter on The Art of Arithmetic (pp. 6O-IO7 ), Wilson stated:

21yoseph Ray, Ray’s Arithmetic; Part First (Cincinnati; Winthrop B. Smith, 1844). P^ 4. ------^^The instrument that Vfilson called an abacus was used as a counting frame, according to the definitions in the present study. 23i)

Now, as nuxi*era have always a relation to things, it is requisite, as soon as possible, to fix the various attainments of the children, in the art of numbering, to sojne palpable object, which they may see or feel. Place the children, then, in the gallery, and let them count the balls, with your assistance on the abacus, . . . from one to one hundred, on the several rows,23

He advocateed that the abacus (see Plate XVII) be used for teaching addition, subtraction, multiplication, and division of whole numbers.

A unique feature of his instruction, particularly for this period of time, was that an aspect of arithmetic be one of having the children clap their hands in measured time while counting.

The most significant features of the arithmetic books of this period were the introduction of the following: books written specifi­ cally for younger children, mental or intellectual arithmetic, objec­ tive materials, and the inductive method.

Bronson Alcott^^ (the father of Louisa I/Iay Alcott) and Horace

Mann26 were two of the educators of this period who strongly advocated the use of objects in the teaching of arithmetic. The former commended

Holbrook's apparatus, but also indicated that "a thinking teacher cannot fail to observe that the world around him is full of

^%illiam Wilson, A Manual of Instruction for Infants' Schools (Nevj York: G-. and C. and H. Carvill, I 83 O), p. 61. (The American edition was published in IS 30, but the English edition was published in London in 1829») 2Ajjonroe, op. c it. . pp. 118-9.

25a . B. Alcott, "Primary Education,” American Journal of Educatim. I ll (January, February, 1828), 26-31, 86-94; Alcott, "Elementary Instruction", Amacioan Journal of Education, III (June, 1828), 369-74» 2%orace Mann, Lectures on Education (Boston: Ide and Dutton, 1845), pp. 28-9. 212

P la te XVII, The Abacus

W ilson, 0£. cit. , between pp. 68-9. Zlh

j

*

Jlllilvlil 21% apparatus.”2?

Not everyone believed in the philosophy advocated by Colburn and

his followers. An example of this was shown in a report made in

Boston in 1833 — a report on improvements in the primary schools:

In a few instances, a blackboard, a numerator (or abacus with balls to assist in teaching to count) and in one or two instances a few other objects for visible illustration are to be seen. But these, we found were positively forbidden by the General Committee.2& . . . We knovz that there are strong prejudices against "apparatus", and we regret that the manner in which it has been made, and presented, and abused, have given some ground; but we are surprised that a Committee on so important a subject, should utterly forget the value of objects for visible illustration, or should fail to discriminate between their use and their abuse.^9 Perhaps this attitude did not prevail very long, for liîann

(Secretary of the Massachusetts Board of Education), acting as

official representative of the Board, stated in a lecture given in

1837: At the last session of the Legislature, a law was enacted, authorizing school districts to raise money for the purchase of apparatus and Common School libraries, for the use of the children, to be expended in sums not exceeding thirty dollars, for the first year, and ten dollars for any succeeding year. Trifling as this may appear, yet I regard the law as hardly second in importance to any which has been passed since the year l647, when Common Schools were established.

B. Alcott, "Arithmetic." American Annals of Education. II (March 15, 1832), U7. 28rhe schools were under the direction of the General Committee on Primary Schools. 29««primary Schools of Boston," American Annals of Education. H I (December, 1833), 582-90. 3QMann. op. c it. , pp. 32-3* 215 In summarizing the principal achievements significant of this

period, R atcliff wrote;

Extensive use was made of inductive teaching; the process was from particular ideas to general principles . . . The child used objects to learn the combinations or to understand the processes; the objects used might be the fingers. The thing was to aliovj and encourage the child to think things out for h im s e lf.

THE PERIOD OF THE INFLUENCE OF WARREN COLBURN

STATIC PERIOD; 1857 TO 1892

The date 1857 denoted approximately the beginning of a static period in the teaching of arithmetic. No essential changes in content and method of teaching occurred, and no outstanding revisions or new textbooks appeared from about 1857 until near the close of the century.

The commendable techniques of Colburn — the use of objects and of the inductive method — which educators set out to imitate during the early part of the century, were soon overshadowed by the disci­ plinary function. About the middle of the century analytic methods began to dominate.

The textbooks of this period became deductive in form with emphasis upon the "rule.” Joseph Ray recommended objective materials (beans, grains of corn, pieces of crayon) for the younger pupils but cautioned against "frequent use of artificial aids," for it "tends to prevent the pupil from exercising his own intellectual powers, and thus, if carried too far, is productive of positive injury.

^^Ratliff, op. pit., pp. 105-6

3%tonroe, op. c it. , p. 125. 216

A teaching technique particularly emphasized was that of d rill.

A number of authors included d rill devices and d rill card exercises in their books.

Monroe and R atliff, who made surveys of the teaching of arithme­ tic, are in disagreement as to the extent of the use of objective materials during this period. The former stated:

The second P estalo zzian movement in the U nited S ta te s , usually known as the Oswego movement, emphasized almost ex­ clusively objective teaching. This movement, which dates from i 860 , appears to have had but little direct influence upon the teaching of arithmetic. There was only a slight increase in the use of objective materials in arithmetic a f t e r 1860.33

On the other hand, R atliff said:

Object teaching . . . was greatly affected by the work at Oswego, New York . . . Sheldon, in 1863, explained h is methods before the National Teacher's Association. After this, the Oswego methods became widely advertised and were very significant in influencing the methods used in the classroom . . . Objects had been rather generally used in primary arithmetic ever since the time of Colburn, . . . Formerly their use had been in helping the child to learn to count and to learn the simple combinations. They were now used in every stage of the child’s progress where they would help him to form concepts or understand operations.3^

During this period the American educational periodicals published several articles which advocated the use of tally objects (such as beans, counters, fingers, buttons, sticks and the numeral frame) in explaining numeration, and occasionally in,the teaching of the simple

33iÆoxmje, op. c i t . . p. 126.

34Ratlif f, cit. , pp. 121-3. 21%

arithmetical operations. ^ special form of numeral frame advocated

by one of the articles was the Arithmeticon^^ (see Plate XVI), It

consisted of an oblong frame within which were twelve horizontal rods ;

each rod contained twelve beads in a pattern of black and white, the

purpose of which is not clear.

In addition to advocating the use of tally objects, a few articles

were printed which suggested the use of an abacus. R ichards^? recom­

mended "exercises upon vaulting down numbers" in which a form of papeiv

35Thomas H ill, "The True Order of S tu d ie s," B arnard's American Journal of Education. VI (June, 1859)» 449-58. "Arithmetic [translated from Raumer's History of Pedogogyl." Barnard's American Journal of Education. .7111 (iVhrch, i860) , 170-84. James Currie, "Subjects and Methods of Early Education, Barnard * s American Journal of Education, IX (September, i860), 229-93» "/ W. E, Richards, "Manual of School Method for Teachers in the National Schools of England," .Barnard's American Journal of Education. X, (June, 1861), 501-30. "Conversations on Objects," Barnard's American Journal of Education, 21 (March, 1862), 21-52. Thomas Urry Young, "Subjects and Methods of Early Education," Barnard's American Journal of Education. XIII (March, I 863 ), 155-204, "School Architecture; Apparatus," Barnard's American J ournal of Education, XVI (September, 1866), 569-70» William F. Phelps, "Methods of Teaching for D istrict Schools, Lessons in Number," The National Teacher, I (October, I 870 ), 50-2» J. C. Greenough, "The Object of Arithmetical Instruction," The National Teacher, I (October, I 870 ), I 6- 8 . J. C» Greenough, "Elementary Instruction in Arithmetic," The National Teacher, I (November, 1870), 53-5» ^Subjects and Courses of Public Instruction in Cities," Barnard's American Jouimal of Education. XIX (I870 ), 469-576. N. A. Calkins, "How to Teach Addition," The National Teacher. I (March, 1871), 223- 5 » Edward A. Abbott, "Hints on Home Training and Teaching," Barnard's American Journal of Education, XXXII (1882), 5-122.

3^ichards, og. c it., 501- 30»

^^Richards, cit., pp» 501-530. 21$ and-pencil abacus was used, although he did not designate the technique

by this name. For example, the number 222 would be w itten:

h t u

2 2 2

The writer of the article "Comrersations on Objects" felt that the abacus might prove useful in teaching the first principles of arith­ metic. Abbott believed that elementary arithmetic "should be taught experimentally, first by means of the fingers, then with an abacus, chess-board, marbles, tin soldiers, counters, or other devices for representing numbers by concrete objects.He advocated the abacus for experimenting and making discoveries but not for the ordinary purposes of calculation.

John Stuart Mill (I806 - 1873), the English philosopher, believed in the use of objects in teaching arithmetic. He wrote:

The fundamental truths of that science [science of number] all rest on the evidence of sense; they are proved by showing to our eyes and our fingers that any given number of objects, ten balls for example, may by separation and re-arrangement exhibit to our senses all the different sets of numbers the sum of which is equal to ten. All the improved methods of teaching arithmetic to chil­ dren proceed on a knowledge of this fact. All v;ho wish to carry the child's mind along with them in learning arithmetic; all who wish to teach numbers, and not mere ciphers — now teach it through the evidence of the senses, in the manner we have d escrib ed . 40

^^‘'Conversations on Objects," Barnard's American Journal of Educatlon. XI (March, 1862), 21-52.

. 39Abbott, op. c it., pp. 5-122.

40j, s. M ill, A System of Logic (London: Longmans, Green, Reader, and Dyer, Ï5V5J, Ï , pp. 295-6. 219 / Coiupayre wrote a book which was tra n s la te d in to E nglish and pub­

lished in the United States. In this book he stated:

As a means of making a beginning in numeration, educators recom­ mend the use of small pieces of wood. As a matter of fact, all concrete objects are adapted to this purpose, and the choice is unimportant. The essential thing is, not to plunge the child all at once into the study of abstract numbers, but to resort at first to intuition, to intuitive computation; and for this purpose real objects should be employed, placed in the hands of the child, or points and lines drawn on the blackboard and presented to the pupil’s eye . 41

In addition to this he suggested that teachers might use numeral frames, but ôautioned against their overuse.

A more severe critic of the use of objects was Block, who wrote :

. . . our thinking cannot all be reduced to sensuous perceptions; the mind itself adds an element to sense, which is outside and beyond it, and whose growth is not helped by ways of instruction derived from caricatures of itself to be found in the writings of the sensuous school of thinkers. There are aspects of number which are not brought into clearness by being burlesqued with sticks or marks or scraps of paper, and which are best taught by enforcing the pupil to thoseaabstract mental operations which produce them, and which no illustration can more than vaguely adumbrate. We have a great deal to say about the necessity of making pupils think, and then have so little faith in the genuine thinking process that we shrink from it when it presents itself in its proper form, and fly to pictures and objectivities of all sorts to evade it. We must forever have things instead of ideas, objects in place of thoughts, exemplifications in lieu of law. It may be as well, perhaps, to induct the learner into the free exercise of his power of thought, to be satisfied with his know­ ledge without too much exercise of his imagination upon it, and let him discover that his thought is an avenue to truth as well as his eyes and e a r s , 42

/ 4lQabriel Compayre, Lectures on Pedogogy, trans. W. H, Payne (Boston: Heath and Co., 1889) , p. 383» 42Lewis J. Block, "Graphic Work in the Grammar School," Education, X (February, I890 ) , 357» 229

A ïrench jury which was chosen to judge exhibits at an Inter­

national Exposition in Vienna (IÔ 73 ), predominately held an unfavor­

able opinion of the numeral frame. Rambert, in speaking for the

group, stated that "This instrument corrupts instruction in arithmetic

. . . the child who counts only in this way loses his time, while the

one who has counted in his head has engaged in the most useful of e x e rc ise s,

R atliff summarized this period by saying that the objective of the teaching of arithmetic became chiefly that of mental discipline. Emphasis was placed upon d rill and formalized procedures.

THE HSELECTIVE PERIOD — 1892 TO I 9II

Three of the men who exerted a great influence upon the teaching of arithmetic during this period were Johann Friedrich Herbert (1776 -

1841 ), William James (I 842 - 1910), and John Dewey ( I 859 - 1952).

Herbert's principle of apperception ("New experiences are given meaning and interpreted by means of the ideas which one has obtained from his past experience and which are present in his consciousness at the time."^^) did much to counteract the emphasis upon the disci­ plinary function of Instruction in arithmetic; James advocated that the nature of learning was essentially an activity process; and Dewey

E. Buisson, Rapport sur L’Instruction Primaire a L*Exposl- tion Universelle de Vienne en YÔ73 (ifarlst iîmpriiiierie Nationale, 1675 ),P» 212, citing Rapport sur le groupe 3QCVI de L*Exposition de Vienne, p. 80. ^^Ratliff, op. cit., p. 150. 45Monroe, op. cit. , p. 128. 221 gave new emphasis to self-activity and to social participation upon the part of the child. As a result of the work of these men, text­ books began to use problems taken from practical situations and thus to stress the utilitarian value of arithmetic.

As a result of emphasizing the utilitarian value, the objective materials became more varied. An interesting and unusual textbook published in this period was The lirs t Steps in Number by Wentworth and Reed. The opening of the Introauction stated:

For a successful teaching of Number the teacher needs a great variety of objects. Blocks, splints, sticks, buttons, paper patterns, peas, beans, corn, spools, counters, shells, pebbles, horse-chestnuts, acorns, little tin plates, cups and saucers, tin money, are inexpensive and convenient to handle . . .

The teaching of Number as far as ten does not include the teaching of figures or other signs used in Arithmetic.^

This statement is of particular interest; on a basis of philosophy implied by the statement, no Hindu-Arabic symbol was used until page

197 — this constituted almost all of the first year's work for the child. QQestioning, disctiission, and work with objective materials the children were guided into discovering relationships among numbers.

Some of the writers of this period who advocated the use of manipulative materials were: 1. Badanes hi advocated the use of these materials, including the

abacus, in carefully planned and organized experiences.

G, A, Wentworth and E. M. Reed, First Steps in Number (Boston: Ginn and Go,, I 898 ), p . 1. hi Saul Badanes, "The (hrube Method o f Teaching A rithm etic" (unpub­ lished Doctoral dissertation. New York University, 189Ü)'* 222 2. Bain‘S® suggested using tangible objects — beginning with small

cones, such as balls, pebbles, coins, apples; then using larger

objects such as chairs, and pictures on a wall.

3 . DeGanao sa id :

Thinking vacuo is hard work; thinking in the concrete is a delight . . . Concreteness contributes perhaps more than any other single phase of instruction both to clearness and to vividness. It lays a foundation, therefore, for interest.

4 . Young presented to Americans a summary o f the view of fiv e o u t­ standing French educators. He stated:

The writers all concur in urging with emphasis the teaching of mathematics from concrete beginnings . . . They are all animated by the same spirit and urge: (a) the fundamental importance of the proper study of mathematics; (b) the con­ crete origin and the experimental relations of the subject; (c) the deadening effects of teaching on an abstract basis; and (d) the salutary results of beginning with the concrete . . . Ho recent French publication has come to my notice defending the contradictory assertions, so that it seems safe to say that the French expressions of theoretic thoughts on the teaching of mathematics trend today in the same general d ire c tio n as the American movement fo r th e in tro d u c­ tion of what have been called "laboratory methods" into teaching of mathematics.50

5 . Gajori wrote:

. . . the counting by groups of objects in early times led to the invention of the abacus, which is still a valuable school instrument. The earliest arithmetical knowledge of a child should, therefore, be made to grow out of his ex­ perience with different groups of objects; never should he

^^Alexander Bain, Education as a Science (Hew York: D. Appleton and CO., 1897), p. 288. ^9oh.arles DeGarmo. Interest and Education (Hew York; Macmillan C O . , 1904), pp. 114, 141. W. A. Young, "Some Recent Frencfi Views on Cone fete Methods of Teaching Mathematics." School Review. XIII-(March, I 905), 275-*% 223 be taught counting by being rexnoved from h is to y s, and (practically with his eyes closed) made to memorize the abstract statements 1/1=2, 2/1*3, e t c . 51

6. Gray52 believed that each pupil should have objective materials

at his desk with which to work while the teacher worked along

with the class using larger objects.

The American members of the In te rn a tio n a l Commission on the Teach­ ing of Mathematics asked Henry Suzzallo to make a critical study of

recent tendencies in methods of teaching mathematics in the first six

grades of the American elementary school. Some of the conclusions he

reached were:

1. The older reformers believed that it was not possible for the

pupil to have too much work with objective materials; the new

reformers believe that "objective presentation is an excellent

method at a given stage of immaturity in the special topic

involved, but it may be uneconomical, even an obstacle to effi­

ciency, if pushed b e y o n d . "53

2. A modern movement which has had a marked e ffe c t on o b jectiv e

teach in g i s th e movement toward " s e lf - a c tiv ity " on the p a rt of

the child. Formerly the work with manipulative materials was

51plorian Cajori; A History of Elementary Mathematics (New York: Macmillian Co., 1897), 17#8. 52john C. Gray, Number by Development (P h ilad elp h ia: 1. B, Lippincott Co., 1910). 5%exiry Suzzallo, "The Teaching of Primary Arithmetic," Teachers College Record. XII (March, 1911), 30. 224 done by the teacher while the child was merely a passive observer

. . . Under the in flu en ce o f t h is modern movement t-

Teaching now becomes "Indirect" rather than "direct"; the child learns through his own experience rather than through the statement of book or teacher, here the child's own thought and activity, not the teacher's, are conspicuously central in the teaching situation. The teacher stimulates the child into action; he suggests, guides, corrects, does everything in fact save obtrude his authority and opinion into the child's interpretation of his own experiences, 54

3. To a minor extent there is "provision for better transitions

from the objective presentation of applied problems to the

symbolic presentation of abstract e x a m p l e s . "55

In addition to discussing some of the modern movements in teach­

ing, Suzzallo listed what he considered to be several existing defects

in objective teaching. These were:

1. Indiscriminate use of objects,

2. A rtificiality of materials utilized.

3. Narrowness in the range of materials.

4. Inadequate variation of traditional materials.

5. Restricted use of diagrams and pictures.

6. Lack of unity in use of objects. Ratliff listed some of the principal characteristics of this period as follows:

5 4 i b i d . . X II, 37, 41.

5 5 i b i d . . X II, 47.

56ibid.. XII, 32-5. 225 1. More in te r e s t in th e c h ild as an individual, with special

attention to his physical, mental, and emotional needs.

_a. Counting on the fingers was not approved.

_b. Objecta were used to a limited extent.

e_. Student activity vi/as emphasized.

LIEE ACTIVITIES PERIOD — I 9I I TO I 935

Teachers continued the use of objective materials, but in a dif­

ferent framework. Objects and materials were used in relation to pro­

jects and to experiences in life situations. These projects and life

activities were to be ones in which the child's interest was whole­ hearted and purposeful.

R a tlif f summarized th e comments of se v e ra l w rite rs as to the

causes for failure in the teaching of arithmetic. They said failure

was due to a lack of a gradual transition from the concrete to ab­

stract number, lack of the development of an understanding of the

number system, lack of a development of generalizations with students,

too little attention to techniques of thinking, and too much formal

explanation,^® A foreign educator whose work had some influence in America was

Maria Montessori of Italy. She designed didactic material (education­ al apparatus) which played an important part in her program of in­ struction. Montessori believed that "the education of the senses :U

57Hatliff, 0£, p it.', pp. 197-9.

5^Ibid.. p. 295. 226 should be begun iaethodically in infancy, and should continue during

the entire period of instruction which is to prepare the individual for life in society#"59

For a rith m e tic she designed two forms of abaci — a sim ple form^® and a more advanced form (see Plate ). These abaci were to be

used by the child in discovering how to add, subtract, aM multiply whole numbers. (The order of the rods is the reverse of that in most

forms of abaci — the units' rod is at the top of the frame.)

Some of the educators of this period who favored the use of objective materials were the following persons:

1. Brovm and Coffman stated:

No experienced teacher . . . q.uestions the value of object teaching in the primary school. It is not only desirable but necessary that the pupil should be aided in his grasp of number by approaching it from the concrete. A genuine mastery of number in the early school years is gained by using it in a concrete manner . . .°^

Successful primary work today is almost intuitively associated with object teaching. No good primary teacher would think of attempting to teach numbers without a supply of objects . . . they are the schools' one best means of making concrete the abstract numerical concepts that up to the middle of the nine-

59Marla Montessori, The Montessori Method trans. Arthur I*iving- ston (New York: Frederick A. Stokes Go., 191?)* P» 221. 60i]ie simple form was sim ilar to the advanced form, bub differed in structure by having only the units', tens', hhndreds', and thousands' rods.

Brown and Coffman, 0£. cit.. p. 15.

62ibid.. p. 131. 227

Plate XVIII, iîontessori Advanced. Forxa of Abacus.

M argaret Drummond, The Psychology and Teaching of Hmnber (Yonkers- on-Hudson," Hew ^ork: World Book Co., 1925) » p. 75» Permission for use granted by the World Book Company# to

ffXJMOO

J

ABACUS, SHOWING 2,640,083 229 teenth. century were taught by purely memoriter methods . . . As children gain in . . . ability to organize and condense their experiences, the number of objects used in teaching decreases; but they never entirely disappear, for they are always of value in comprehending and interpreting new situations.o3

2. Punnett outlined a program of techniques by which the child, using

objective materials, discovers number relationships. She wrote: The teacher's aim . . . must be twofold: — (l) she must provide as great a variety as possible of things for the child to handle and use in various ways which involve numbering. It is by means of these experiences that number-relations vfill be learned. (2) She must encourage, as the child's growing power makes it possible, the application of these number-relations to events and experiences involving objects which are either no longer present in the schoolroom or have never been there, 3» McMurray advocated that the children themselves handle the abacus

and use it for learning to perform simple arithmetical computa­

tions. Se classified objective materials as seven basic types,

and then issued a word of caution:

In all those seven modes of concreting elementary number operations there is great danger of overdoing the matter by continuing too long in objective work, thus converting it into

^3lbid.. pp. 137 - 9. 64.Margaret Punnett, The Groundwork of Arithmetic. (New York: Longmans, Green, and Co., 1914), p. 39. ^%he seven types are: (l) physical objects, such as chairs, birds, houses, children, apples, fence posts, etc., (2) miscellaneous objects, such as toothpicks, buttons, beads, fingers, etc., (3) administrative devices such as the distribution of papers, books, and pencils to pupils, etc., (4) artificial illustrative materials, such as abacus, the counting frame, and the measured cubes, (5) standard units of denominate numbers, such as the foot, yard, gallon, etc., (6) number pictures expressed by dots or lines, and (?) bundles of splints or toothpicks for the teaching of the decimal scale. C 0. A. McliSirry, S p ecial Method in A rithm etic (New York: Macmillan Co., 1915), pp. 32-7.J 230 a routine. In all cases it is necessary to push on to the more rapid and abstract treatment of number, always keeping «the way open to a quick return,to the concrete, when there is lack of clearness in thinking.

Lindquist believed in the use of objective materials, but for a

purpose somewhat different from that of most educators of this

period. He stated his philosophy as follows :

There are really two ways of looking at this matter of object teaching; one is to base the idea of numbers upon concrete objects, so to speak, and a second to make use of objects, such as blocks, measuring, drawings, paper-folding, paper-cutting, and so on, for the purpose of motivating and clarifying the work in hand . . . The value of the second use of objects . . . can hardly be overestimat ed.

5. Overman considered i t e s s e n tia l th a t the f i r s t work w ith the

simple arithmetical operations be met in a concrete setting in

order that these facts may have meaning to the pupils. But he

issued a warning as follows;

It is not necessary to present all the addition, subtraction, Multiplication, fraction, and division facts concretely; indeed it would be a waste of time and would defeat the desired ends, as the pupils might form the habit of thinking objectively and be unable to handle the abstract number facts . . . It is just as bad to keep the pupils in the objective stage of development too long as it is to neglect it entirely."^

6. Klapper did not approve of the use of beads, beans, lentils, or

dots in rows and columns as objective materials; he considered

that they introduced "an artificiality into arithmetic that robs

^

various occupations (such as merchant or baker) or solve problems

based on classroom situations. He stated that, "We must ever be

mindful of the fact that unnatural objects are as unobjective as

mere verbal teaching," ^ And yet, after all this condemnation of

"artificial materials, " he suggested the teaching of each number

by having the student arrange sticks in imitation of geometric 70 figures drawn on the board by the teacher,

7, Thomson*^^ suggested, among other objective materials, using a

sinç)le abacus x-ilth fo u r v e rtic a l w ires, each w ire having te n

beads. The beads on the wire representing units were plain wooden

beads; the ones on the wire representing tens x^ere dull green;

those on the hundreds' wire, blue; and those on the thousands'

w ire, re d ,

8. Klein also advocated the use of the abacus, particularly for dis­

covering multiplication relationships. In addition to this he

s ta te d :

The child cannot possibly understand if numbers are explained axiomatically as abstract things devoid of content, with which one Can operate according to formal rules. On the contrary, he associates numbers with concrete images, Th^ are numbers of nuts, apples, and other good things, and in the beginning th ^

69 Paul Klapper, The TeacMng of Arithmetic (%w York: D, Appleton and Co,, 1921), p. %, 70 Klapper, o£, cit,, p, lij.^, 71 Jeannie Thomson, An Etoerim ent i n Number-Teaching (London: Longmans, Green and Co,,“T922}, pp, 6U^0, 232

can be and should be put before him only in such tangible f o r m . 72 9, Karpinski said:

The long-continued use of the abacus and of this visual form of representation of numbers testifies to the usefulness of tangible and visual aids in instruction. Teachers do -well to use such methods vjherever possible in instruction,73

10, Drummond used several types of objective materials including the

two forms of abaci designed by Montessori (see Plate XYIII),

She — Drummond — f e l t th a t:

In school , , . children of nine and ten would use it [the abacus] most profitably if they were allowed to meditate on its.signifi­ cance and to experiment freely with it. ^ts value would be gone, of course, if children used it to the word of c o m m a n d , 74

11, Hillegas believed that the first steps in teaching number should

be that of making use of classroom situations, such as keeping

attendance records, costs of luncheons, scores for games, and

similar activities, ^e advised teachers to "avoid the exclusive

use of any object such as sticks in the teaching of arithm etic, "75

12, Brownell wrote:

Under the persistent emphasis of psychologists since Pestalozzi, the school has come to realize in part that the basis for ab­ stract number is concrete number. It is much less common now

72 j 5g ii 3j. Klein, Elementary Ifethematics from an Advanced Standpoint, trans, E, B, Hedrick and G, A, Woble (New' York: Dover Publications, Qio datgl), p. 4 * 73Karpinski, op. cit. . p. 36,

7^Drummond, 0£ , c i t . , p, 76 , 75M110 B, Hillegas, Teaching ÜTumber Fundamentals (Philadelphia: J, B, Lippincott Co,, 192 5), p. 37. 233 than it was twenty years ago to introduce the pupil immediately and directly to abstract numbers in the form of additive com­ binations, the multiplication tables, and the like. There is usually a preliminary period in the first grade during which the pupil is given experience with numbers in concrete relations. Thus, for exaB5)le, he counts the number of pupils present or absent; he counts books, pencils, etc., in distributing them; he counts the materials that are needed for construction exercises; and so on.

To the extent of introducing practice in counting concrete numbers before beginning instruction in purely abstract numbers, the school has given some evidence of recognizing the need of a concrete b a sis f o r a b s tra c t number.'^® 13. Badanes was mentioned in the previous period in relation to a

comment that appeared in his dissertation. During this period

{1911- 1935) he and his wife developed textbooks for the younger

children. In a manual which accompanied one of the textbooks,

they expressed their basic philosophy in the following words :

The authors of this number primer point out that one of the causes of our pupils’ failure in number work is the fact that in our theory and practice of teaching we ignore the difference that exists between the thinking of a trained adult and that of a child: the one thinks in abstract symbols; the other, in the concrete and by means of imgggs. The authors base not only the initial steps of the learning process but the entire practice for the acquisition of knowledge and skill upon a great wealth of ingenious devices and concrete illustrations.77

14. Lennes^^ mentioned the use of blocks and sticks; but since he

does not go into much detail, one might assume he did not believe

7 ^ . A. Brownell, The Development of C hildren’s Number ^deas in the I*rimary Grades ('’Supplementary Educational Mdnographs," No. 35 [Chicago: University of Chicago Press, 1928]), p. 218.

77julie E. Badanes and Saul Badanes, T eacher's Book to Accompany a C h ild ’s Number Prim er (New York: Macmillan Co. , 1929) 1 p. v i. 7 ^ . J. Eennes, The Teaching of Arithmetic (New York: Macmillan Co., 1931), p. 127 . strongly in the use of manipulative materials,

15. Young defined his idea of concrete work as follows :

The problems are miade to relate to what comes within the range of his own [the child’sZl experience, to what he actually sees, or at least can easily understand, and, best of all, te his own activities. Only thus is the work really concrete.

The only mention he made of an abacus was in the back of the book

(page 386 ) in a section in which he suggested topics for Mathematical

Clubs. One of these topics was: The abacus and other counting m achines.

During this period an interesting magazine article e^peared in

v/hich the author made a psychological investigation of several math­

ematical prodigies. After analyzing the background of these children, he concluded that:

With extremely fevi exceptions, they were, in fact, the children of . . , men and women . . . with little or no education, . . .

It is no mere coincidence that Inaudi, Ifengiamele, Mondeaux, and Pierini were, in their earliest boyhood, sent into the fields to tend sheep day after day . . . Mondeaux . . . used to amuse himself by incessantly counting over heaps of pebbles and arranging them in different ways to repiesent sums in addition, multiplication and subtraction.

In the same way the influence of environomental conditions is plainly discernible in the case of other calculators, is the fact that they learned to calculate through play. Ampere like Mondeux, used pebbles for his self-education , . . Sidis . . . had been given several calendars by his father, . . . and for a long time these calendars were his principal means of amusement . , . Mantilla . . . played with calendars . . . the elder Bidder, . . • taught himself m ultiplication by means of self-devised toys.°^

W. A. Young, The Teaching of Mathematics (New York; Longmans, Green and Co.; 1931)» P* 210. Addington Bruce, "Lightning Calculators," McClure*s blagazine, XXXIX (September, 1912), 59A» 235 In his surrey of trends in teaching methods for the period prior

to 1911» Suzzallo stated that there had lately been some tendency to Si use hearing and touch in giving a concrete basis to teaching. The author of the present study found no indication of this tendency in

books and articles written prior to I 9II (other than tte book by

Wilson — see page 210), but did find indications of this practice in So two books written in the period 1911 to I 935. Overman suggested the

clapping of hands and the taking of steps to indicate numerical quan­

tities, Klapper outlined a multiple sense drill as follows :

(1) Visual Appeal; children point to pupils];: seats, desks, books, windowpanes, electric bulbs, etc,, to show how many two’s in e ig h t, how many fo u r’s in e ig h t, , . , e tc ,

(2) Auditory Appeal: Tap with pencil or rule, ring bell, clap hands and children recognize the combination , , ,

(3) Muscular Appeal: Children take steps, hop, sing a note, arrange fellow pupils, , , , etc,®3

SUMMARY

During this period the objective materials of an artificial type were discarded for concrete materials in life situations. These materials were to help in understanding, but then must be laid aside for work with the abstract numerals. Moreover, the aim of teaching during this period was conceived as one of broad social utilitarianism .

As a result of this point of view the problems used in classes in arithmetic were chiefly tsrue-to-life problems, if not actual problems.

^^Suzzallo, op. c it. , XII. p, 3A« '‘Overman, 0£_, c it,, p, 71.

83 K lapper, _0£, cit,. pp. 146-7. CHAPplH T ill

THE TjEACHH^G of AEEIHMETIC m THE TOUTED STATES

(1935 to 1956)

TEXTBOGEB OH AEITB3/IETI0

In order to deterioine the trends of this period of time a survey

was made of nine series of textbooks on arithmetic and of fifteen books

on the teaching of arithmetic. Tbe nine series of textbooks were:

Brueckner, Merton, and Grossnickle Series (John C, VJinston Co.), 1952- 55. Buswe11, Brownell, John, and Sauble S eries (Ginn and C o.), 1944- 5 5 . Carpenter, Clark, Swenson, Anderson, and Sauer Series (ifecmillan CO.), 1950- 53.

Clark, Junge, Clark, and Moser Series (World Book Co.), 1952-53.

Hartungj Van Engen, Mahoney, Riess, and Ehowles Series (Scott, Foresman and Co.), 1948-55.

Madden, Beatty, and Gager Series (Charles Scribner's Sons), 1955.

Morton, Gray, Spiingstun, and Schaaf Series (Silver Burdett Co.), 1947 - 5 2 .

XJpton, TJhlinger, and F u lle r S e ries (American Book Co. ), 194.9-52.

IVheat, Wheat, Kauffman, Douglas, and Koenker Series (Row, Peterson and Co.), 1951-53. In each of the above series, 1 a survey was made of the textbook.

hereafter, in referring to any of the series, the name of only the first author w ill be used.

236 237 or of the teacher's manual, or of both, for grades one including five.'^ These series were chosen for several reasons: th^r were the ones available, they were from major publishing companies, they were ones that had recent editions, and they were written by outstanding leaders in the field.

The following chart indicates the contents of the textbooks in relation to the devices and activities discussed in this study:

An X indicated the use of the device in some form. s Form of device or I0 I g d activity advocated: I & o I CO I 1 o IÜ I

1. Manipulative materials, X XX

(books, disks, pegs, milk

bottles, beads, shells,

buttons, counting-cubes,

countingsticks, etc,) g^. The Brueckner Series made extensive use of disks (buttons,

milk caps, checkers, — all disks alike in size, shape, and

color) in order to learn how to add, subtract, multiply,

and divide whole numbers.

Jb, The Carpenter Series had a unique feature in the book for

^The book for grade five of the Hartung Series was not available; Books I and H of the üüadden Series have not yet been published. 238

first grade. In the center of the book, between pages 52 and 53» are two heavy inset pages of stencils and counters. The stencils may be used in a new and attractiTe way to help the child to see a group as a unit. The push- outs, or figures and circles pushed out of the stencils, may be used as counters,^

r-l § 1 1 g Form of device or u "ë 1 activity advocated: 1 1 1 o ë â 1 , 1 î

2. Tally marks. XXX XX XX X

_a. Tally marks were used

for the following pur­

poses : to represent

numbers, to teach di­

vision, and to check

division. These three

techniques were not.

however, used by each

of the series.

3. F ingers X X X a. In relation to the use of the fingers, the authors of the

Carpenter Series stated:

hale Carpenter and Mae Knight Clark, Teacher's Guide for How Many? .Arlbhntetic 1 (New York: Macmillan Co., 1953), P* 1* 239 Should any teacher be worried about the "dangers" of having children count their fingers, let her rest assured that it is no more "dangerous" than the use of any other available concrete aid to learning. The fact that we have a number system which uses a base of ten (a decimal system) goes back to our possession of ten fingers (digits). Why not allow children to use such a convenient and readily available aid to learning about our wonderful number sys­ tem? Of course we w ill not want children to go on perma­ nently solving their number problems by finger-counting methods; when such counting is no longer needed as an aid, it can be and is easily discontinued, often with no adult suggestion.4

In th e Wheat S eries th e follow ing statem ent was made:

• . . the fingers serve a useful purpose in comparing small groups, and certainly they are most convenient. Although some teachers object to the use of the fingers in finding answers, the method has great value and is worth serious consideration.5

A 10 1 g Eorm of device or U t g 1 activity advocated: 1 1 1 o â I â 1 0

4-. Kinesthetic responses. X XX X

(rhythmic beats, bouncing a

ball, skipping, tapping,

clapping)

^Dale Carpenter and Esther J. Swenson, Teacher's Guide for Arith­ metic 2 The World of Numbers (New York: ilacmillan Co., 1953), p. 20,

^Margaret Leckie Wheat and H arry Grove Tiïheat, Manual f o r Row- Peterson Arithmetic. Primer and Book One (White Plains, New York: Row, Peterson and Co., 1952), p. 79» 240

s § § Form of device or ;g activity advocated: pp a I M I I

5, Notched stick and knotted

s tr in g .^ 6. Numeral frame (or counting

fram e). X

(10 rows of 10 beads each.)

_a. The authors of the

Buswell S eries recommend

using this device for 1 serial counting; ordinal

counting; addition,

subtraction, multipli­

cation, and division of

one-and two-place

numbers,

7» Counters (beads or clothes­

pins) on a wire or line. X

(9 counters, 10 counters,

20 counters, 100 counters,

6-Tn the Buswell series a list was given of Children's Activities involving number. This list was adapted from one given in an article by Ifiarian J. Wesley, "Social Arithmetic in the Early Grades," Child­ hood E ducation. XI (May, 1935)» 367-70. 241

0) B § Id> g Form of device or I t :§ I activity advocated; I S Ü I*

variable number of counters)

The Buswell Series use

the counters for the four

fundamental operations;

the Carpenter Series, the

Brueckner Series, and the

Clark Series for repre­

sentation; and the Wheat

Series, for representation,

addition, and subtraction.

8. Use of a form of abacus x

(paper-and-penc i l , Carr’s

open-end abacus, Spitzer

abacus, place-value

pockets or chart)

a.. The Brueckner Series begin the study of place value in grade

three. For studying place value they suggest the use of an

abacus in grade four. The device has four vertical rods with

ten beads on each rod.

Other devices used to teach place value are card holders

(a single card in the ones* section represents one unit; a 2U2 bundle of ten cards in the tens* section represents one ten),

a framework of a vertical-form abacus with Hindu-Arabic

numerals (see page 267)» and Carr’s open-end form of abacus (see page 267 ).

K The authors of the Buswell Series advocated that an abacus

(a form having three vertical rods, each rod having ten

beads) be used for more capable children only in order to

teach them addition. Instructions are also given for making

the Spitzer form of abacus.?

Additional devices suggested for teaching place value are

pocket charts (a bundle of 100 cards in the hundreds* section

represents one hundred, a bundle of 10 cards in the teas*

section represents one ten, and a single card in the ones(

section represents one unit); a place chart in which a symbol

( ^ ) in the tens' section represents me bundle of 10, and a

single mark ( 1 ) In the ones' section represents one unit;

and a framework of an abacus (vertical form) with Hindu- Arabic numerals,

G_, The Carpenter Series use a device in grade one which is called

an abacus, but, according to the present study, is a horizon-

tal-rod counting frame. The device is suggested for use when

the child reaches the point of being ready to study about

7Guy T. Buswell, William A. Brownell, and Irene Bauble, Teaching Arithmetic We ^eed (New York: Ginn and Co., 1955)» pp. 331-2, 243

"five"; the principle of place-value has not been taught.

^n grade two the abacus is listed as classroom equipment

to be used for developing the decimal idea of number. Addi­

tional devices suggested for teaching the principle of place

value in grades three and five are a framework of an abacus

(vertical form) with Hindu-Arabic numerals, place value

pockets or charts (using cards on which the Hindu-Arabic

numerals are printed), and coins (pennies and dimes).

In the Clark Series the recommendation was made that an open-

end abacus be used for symbolizing three-place numbers. Other

devices used for teaching the principle of place value are:

(1) coins (pennies and dimes);

(2) two tin cans, one labeled units and the other labeled

tens. One Counting stick in the can labeled tens

represents 1 ten, and one counting stick in the can

labeled units represents 1 unit.

The Hartung Series use a framework of an abacus (vertical form) to teach the principle of place value. This form of abacus is used with tally marks (tally marks of same color in all sections; also red tally marks in tens* section and black tally marks in ones* section), small sticks, and Hindu-

Arabic numerals.

A second device used to teach the principle of place value is a box divided into two compartments. One large.ball is used in the left-hand section to symbolize one ten, and one small ball is used in the right-hand section to symbolize s a one u n it .

A third device for teaching the principle of place value

consists of a rod threaded through a small card. This card

serves to separate the tens' section from the units' section,

A large cardboard disk is used on the left side of the card

to symbolize one ten, and one small cardboard disk is used on

the right side of the card to syiabolize one unit.

A fourth device is that of using coins (pennies, nickels,

and dimes) in combination with a framework of an abacus

(vertical form) in which tally marks and Hindu-Arabic

numerals are placed, jf. The Madden series used the following devices to indicate

place value; coins (pennies and dimes); a pocket chart with

one bundle of ten tickets in the tens' section to represent

1 ten, and a single ticket in the ones' section to represent

1 unit; a framework of an abacus used with horizontal marks

— one mark in the hundreds' section to represent 1 hundred,

one mark in the tens' section to represent 1 ten, and one

mark in the ones' section to represent 1 unit; and a frame­

work of an abacus used with colored counters — one red

counter in the hundreds' section represented 1 hundred, 1

yellow counter in the tens' section represented 1 ten, and

one blue counter in the ones' section represented 1 unit.

This Series also pictured the vertical line form of abacus

with pebbles used in the spaces for counters, and a vertical 245 rod form of abacus (10 beads on each, rod) which they called

a counting frame but used as an abacus (according to the

definition of abacus in the present study).

The authors of the Morton Series suggested a horizontal-rod

abacus (4 wires with each wire having 10 beads) for use in

teaching place value. Other devices suggested for this

same puipose were: coins (dollars, dimes, pennies); a

framevjork of an abacus with Hindu-Arabic numerals; one

bundle of ten sticks to represent 1 ten, and a single stick

to represent 1 unit; and a pocket chart with which one

bundle of 100 cards in the hundreds * section represents 1

hundred, a bundle of 10 cards in the tens' section represents

1 ten, and a single card in the ones' section represents

1 u n it,

_h. The Upton Series, in teaching the meaning of three-figure

numbers, used a bundle of 100 sticks in the hundreds* section

to represent 1 hundred, a bundle of 10 sticks in the tens'

section to represent 1 ten, and a single stick in the units'

section to represent 1 unit. Additional devices used to

teach the idea of place value were: coins (dimes and pen­

n ie s ), and p e n c ils xvrapped in bundles o f 10 p e n c ils each,

For historical interest a picture and a brief discussion of

the Chinese suan pan was included.

The tVheat Series use a framework of an abacus (vertical form)

quite extensively. It is used viith Hindu-Arabic numerals,

horizontal lines, diagrams of pebbles, diagrams of buttons 246

or vjooder. discs, and dots symbolizing the number of hundreds,

tens, and units. This Series advocated the use of this

device for the representation and for the addition of numbers.

Other devices used to indicate place value aie pocket

charts {a bundle of ten cards in the tens* section represents

one ten, and a single card in the ones' section represents

one ujjit), and a horizontal-f orm abacus vd.th wooden discs

(one color of disc is used on the tens' line and a different

color of disc on the units' line),

A CRITIQUE OF THE DEVICES USED IN TEB8E SERIES

TO INDICATE RLACE VALUE

In discussing the principle of place value, most of the series use forms of devices which represent the tens and the hundreds in a manner which seems questionable to this author. These devices were:

1, Using 10 cards (or 10 sticks or 10 pencils) in the tens' section

to represent one ten, and one card (or 1 stick or 1 pencil) in

the ones' section to represent one unit,

2, Using a red tally mark (or a yellow counter) in the tens' section

to represent one ten, and a black tally mark (or a blue counter)

in the ones' section to represent one unit,

3, Using a large ball (or a large disk) in the tens* section to

represent one ten, and a small ball (or a small disk) in the

ones' section to represent one unit,

4, Using one dime to represent one ten and one penny to represent

one u n it . 247

In the Vi/heat Series the following statement is made: "The abacas

uses the same kind of counters in different positions to show different

values. We use the same figures in different positions to show dif- 8 ferent values." This basic principle of place value, as interpreted

by the writer of the present study, has been violated in each of the

instances listed above.

The Hartung Series avoid most of the errors listed above by using

the following techniques:

1. In working with groups of objects the students are directed to

arrange the objects in groups of ten. Then they are to "repre­

sent the objects by tally-marks in a one-to-ten correspondence

for the piles of 10 and a one-to-one correspondence for any

rem ainin g on es. "9 In most of the illustrations for the series,

for each group of 10 objects, one black tally-mark is shown in

the tens' section, and for each single object, one-black tally-

mark is shown in the units' section.

2. In working with coins, the coins are arranged in the follovfing

combinations to represent a pile worth ten cents: one dime,

2 nickels, 1 nickel-and-5-pennies, and 10 pennies. Then a frame­

work of an abacus is used for recording purposes, For each pile

%arry Grove Wheat, Geraldine Kauffman, and'Harl R. Douglas, Row-Peterson Arithmetic; Book Four (White Plains, %w York: Row, P e te r s o n and C o ., 1952), p. 244*

^Maurice Hartung, Henry Van Fngen, and Catherine Mahoney, Numbers in Action, Teachers' Edition (Chicago; Scott, Foresman and Op., 1955T, p. 196. 2^8 w orth lOçl one tally mark is placed in the tens' section, and for each pile worth lç( one tally mark is placed in the units' section.

According to the principle of place value as discussed and ex­

plained on pages lOU including 107, the form of counter used in each

place (the units' place, the tens' place, the hundreds' place, and so

on) must be exactly alike in every respect (color, size, shape, material). For example, if a single card is used in the units' column to represent one unit, a single card must be used in the tens' column to represent one ten, and a single card must be used in the hundreds' column to represent one hundred. Otherwise, the basic principle of place value is violated.

An exception to the above restrictions might be the use of color in many of the modern forms of abaci. The counters may be of different colors if designed in this way for aesthetic purposes only, and if no reference is made to a color as a determining factor of the value of a counter.

Another point that should be mentioned is that, according to the inteipretation in the present study, a counting frame (or numeral frame) is not an abacus, but is a form of structured tally. The structure of both the counting frame and the abacus may be identical.

For examplei most counting frames consist of horizontal rods with a definite number of beads on each rod. The Persian abacus (see page il6) was of this form. The determining facto# is the way in which the instrument is used; if the principle of place value is applied, then the instrument is an abacus, according to the definition adopted 249 in the present study,

STAaEMENTS OF EHILOSOEHT IIT THE NINE SiEIES

The basic philosophy expressed in each of the series of text­ books and manuals is as follows:

1. Brueckner Series10 ■ *

The authors of Heady f or Numbers believe that the Arithmetic classroom should be a learning laboratory. In this laboratory the children should engage in a wide variety of concrete, meaningful experiences in which numbers play an essential role • , ,

Under teacher guidance the children should vjork with manipulative materials and visual aids in such a way that numbers w ill be made meaningful to them,^^

• , . studies of how children learn arithmetic show that the most effective learning takes place if the following steps are taken in the order given: [The first three steps are given,] _a. Learning by doing (manipulating actual materials). _b. Learning by seeing (visualization of the complete operation), c_, Earning by discovery , , ,12

The following proverb shows how the child best learns: ’What I hear I may forget What I see I may remember, But what I do I will k n o w , *13

lOff^he Winston Arithmetics is the only series of textbooks ever published that fully utilizes both manipulative and visual materials to clarify and extend meanings and to aid the learner in discovering facts, meanings, and generalizations,*’ (L, I, Brueckner, Eldo L, M erton, and F, E. G rossnickle, Teachers Guide f o r D iscovering Numbers (Philadelphia: John C, Winston Co., 1953)» p. 6, ^^Elda L, Merton and L, J, Brueckner, Teachers Edition. Ready for Numbers. Grade One (Chicago: John C, Winston Co,, 1955)/ P-

^Ibid,, p, 27.

^3loc, c it. 250 2. Buswell Series —

The purpose basic to this series is to provide the kinds of experience that develops truly meaningful and functional concepts. Typically these experiences are organized in the following sequence: 1. Manipulative experiences with real or natural objects (the children themselves, chairs, pencils), 2. Manipulative experiences with representative objects (smaller objecta, such as pegs or buttons or marks on paper, that are intrinsically less distracting and at the same time more readily manageable), 3. Experiences in identifying the idea in pictures of real or natural objects, and/or in reproducing the idea in th e same medium. 4. Experiences in identifying the idea in semi-concrete pictures (regular patterns of dots or crosses), and/or in reproducing the idea in these same ways. 5. Experiences in identifying the idea in abstract form (in examples with numbers) or in verbally described situations (word problems), and/or in reproducing the idea in these same ways.14* 3. Carpenter Series —

"Meaningful concrete activities should always form the

major portion of the young child's number experiences,"15 4. Clark Series —

learning arithmetic results from experiences in counting groups of things, in separating and combining groups, in comparing groups, learning proceeds from experiences with things (the concrete) to the representation of things (the pictorial), then to the use of number symbols. Through these stages pupils learn the nature of arithmetic . . . Meaning is given to arithmetical facts and processes through the use of concrete materials.1"

14Bu3vrell, Brownell, and Sauble, op. cit. , pp. 9-101

15carpenter and Clark, op. cit., p. 3*

l^John R. Clark, Charlotte W. Junge, and Caroline Hatton Clark, Teacher's Guide. Humber Book 1 (Yonkers-on-Hudson, Hew York: World Book G o., 1952), inside front cover. 251 5» Hartung Series —

The program of Which this "picture number book" is a part provides a firm foundation on which to build the work with numbers in the elementary school. Instead of depending upon learning throu^ reading or d rill with abstract number symbols, the emphasis throughout is upon visual experiences which are to be supplemented by activities using actual objects.^7

Research suggests that children think in terms of actions. They think in terms of what they have seen happen or have caused to happen to objects, such as blocks, markers, toys, animals, or people. This conforms to the well-knovm fact that children learh through the senses of touch, sight, hearing, and smell, . . . the elements of meaning are based largely upon experiences — direct contact with objects through the senses of touch, sight, h earin g , and sm ell. Number meanings are no exception , . . 6. Madden S eries —

Many studies support the value of using manipulative and graphic materials in developing meanings for number and number processesi^

The learning sequence in The Scribner Arithmetic series begins usually with a life experience that is normal for chil­ dren. The experience is analyzed with the aid of concrete materials . . .

The classroom should be a learning laboratory containing many materials for ineasuring, counting, visualizing processes, and developing mathematical ideas . . .20

l?Maurice L. Hartung, Henry Van Engen, and Catharine Mahoney, Numbers in Action (Chicagoî Scott, Foresman and Co., 1955), P» 145«

^% bid.. pp. US-9. ^^Richard Madden, leslie 8. Beatty, and William A. Gager, The Scribner Arithmetic Book 3 (New York; Charles Scribner's Sons, p.-;:' 322. ' ■ ^ - ^^Richafd Madden, Leslie Beatÿy, and William A. Gager. The Scribner Arithmetic Book 4 (New York; Charles Scribner’s Sons, 1955), p. 321. 252 7. Morton Series —

It is understood that the arithmetic program in Grade One should be rich in concrete experiences for the pupil. At this level, manipulative activities with real objects, rhythmic activities, and dramatization play an important part in laying foundations of understanding.^^

Children need to have many and varied experiences with con­ crete materials before practice on abstract number combinations is begun. Research studies have revealed that children cannot bridge the gap between the concrete and the abstract if the transition from the one to the other is abrupt . . .

Concrete materials can make a significant contribution to understanding, particularly in the early grades. If used effectively there — that is, with the realization that they are a means to an end, not an end in themselves — Jiiany diffi­ culties can be avoided in later grades.22

8 . Upton Series —

Introductory work in primary number is based upon the every­ day activities of children . The first materials of instruction consist of real objects such as apples, coins, beads, and pencils, which the children count, compare, or combine as the situation requires. Through these objective materials basic number mean­ ings and concepts begin to be formed, Real objects, however, cannot be used indefinitely. Materials suitable for this work are not available in large variety. Objects like books are too bulky for easy handling while others like shells may be distract­ ing due to some element of interest other than number. To get a wide variety of interesting materials that can be easily used, it soon becomes necessary to replace real objects with semiconcrete materials such as pictures of animals, apples, and flowers, along with number pictures such as dots . . . These semiconcrete

^^Robert Lee Morton and ^ r l e Gray, Teacher ' s Guide to Accompany Maki% Sure of Arithmetic; Grade One (New York: Silver Burdett Co., 1948 ) » p. 1» 22^ 0bert Lee Morton, and Others. Iilaking Sure of Arithmetic. Teacher's Guide for Grade Rive (New York; Silver Burdett Co., 1952), : p. \ 1 2 ' 253 materials are of the greatest service in the teaching of primary number.

9» Wheat S eries - -

At every poiht the pupil does the work. He arranges. He notices. He considers. He makes up his mind. As he works, so he learns. As he works, he makes what he learns his own. As he w o r ^ , he knows what he le a r n s , and he is sure of what he knows. ^4

Several of the devices suggested in the Brueckner Series are ones

that were designed by one of the authors (Grossnickle) in collabo­ ration with two other persons. A booklet published by the company tint distributes the instruments gives a description of the instruments, suggestions for their use, and the advantages of using them.

In discussing the instruments, the following statement is made:

By using the devices these men developed, the child can acquire every major concept and discover every relationship in arithmetic taught in the elementary grades. These devices are not a substitute for d rill, but they do make d rill enlightened and therefore more effective. They give the child a vehicle of expression and a variety of organized experiences which will come to th e a id of h is memory, should memory f a l t e r . 25

A pupil learns the meaning of a number by manipulating objects so that he can have a visual picture of it. About 8 ? peroeeht of one’s sensations are visual.^®

B. Upton and Margaret Uhlinger, I Gan Add. Arithmetic Workshop Book Two- (Hew York: American Book Co., 1950)i inside front cover.

24-Wheat and Wheat, op . c i t . . p . 9»

25”Kumber as the Child Sees I t.” A series of Devices for Teaching Arithmetic Meaningfully. Designed by Foster E, Grossnickle, William, Metzner, and Francis A. TEade. Distributed by the John C. Winston Co., inside front cover.

26ibid. , [pages unnumbered]. 254 One of the devices discussed in this booklet is a Modernized

Abacus. This form has four vertical rods, each rod having ten beads.

The to p bead on each w ire i s th e same co lo r as th e beads on th e next

succeeding wire (to the left). In discussing the instrument the

following statement is made: "A vertical abacus is superior to one with horizontal beads, because the position of each row corresponds to the place value of our written numbers,

This form of abacus (the Modernized Abacus) has the following two lim itations:

1. With ten beads on a rod, one cannot directly add numbers whose

sum i s g re a te r th an 10,

2, The author of the present study questions the technique of having

,the top bead on each rod the same color as the beads which

represent the next higher order. This would tend to direct the

attention of the student away from the fact that the entire

group of 10 beads on any rod is equivalent to 1 bead on the rod

on the left,

BOOKS m THE TEACHING OF ARITHMETIC

The following books were used as representative, for this period

^*^Ibid. , ^pages unnumbered (1935 to 1956), of books on the teaching of arithmetic Brueckner and Grossnickle, How to Make Arithmetic kîeanlngful (19A7). Buckingham, Elementary Arithmetic. Its Ileanihg and Practioe. (19A7) . Drummond, learning Arithmetic by the Montessorl Method (19A7). (This book is an English publication but Is included because it pertains to the Montessori method.) R osenqulst, Young C hildren Leam to Use A rith m etic. (l9y!^9). Stem, Children Discoyer Arithmetic. (19A9). Stokes, Teaching the Meanings of Arithmetic. (1951). Vflieat, How to Teach Arithmetic. (1951)• ' Wilson et al. Teaching the ))îew Arithmetic. (1951). Harding, Functional Arithmetic: Photographic Interpretations. (1952), Hickerson, Guiding Children's Arithmetic Experlenoes. (1952). Morton, Teaching Children A ritW etlc. (1953). Clark and Eads, Guiding Arithmetic Learning. ( 1954). H ollister and Gunderson, Teaching Arithmetic in Grades I and II. (1954). ------Spitzer, The Teaching of Arithmetic. (1954). Harding. Explorations in Arithmetic, (1955).

Each of the above, books was checked to determine If the authors

advocated the use of tally objects and of forms of abaci. The fol­

lowing information was obtained:

1. All but one of the books advocated the use of manipulative

^^Although the earliest books used in the survey were published In 1947 » some of the authors had earlier books. These books were the follow ing: a. Robert Lee Morton, Teaching Arithmetic in the Elementary School (3 v o ls .; Hew York: S ilv e r B urdett Company, 1937 - 39)• _b. Harry Grove Wheat, The Psychology and Teaching of Arithmetic (Boston: D, C. Heath and Company, 1937)• _c. Guy M. Wilson, Mildred B. Stone, and Charles 0. Dalrymple, Teaching the Hew Arithmetic (New York; McGraw-Hill Book Company, Inc., 1939rr John R. Clark, Arthur S. Otis, and Caroline Hatten. Primary Arithmetic through Experience. (Yonkers-on-Hudson, New York: World Book Company, 1939). _e. Herbert F. Spitzer, The Teaching of Arithmetic (New York: Houghton M ifflin Company, 194-8) . The basic philosophy In the earlier books is, in the main, the same as that in the later books. 256 materials in the form of tally objects such as cards, splints,

discs, bead frames ( 9-b ead , 10-bead), buttons, tops, checkers,

metal washers, and similar materials.

2. The following books advocated the use of a form of abacus (as

defined In this study);

_a. Brueckner and Grossnickle —

In this book the statement was made that, "A donating block

and an abacus are two Inexpensive devices which should be

used In all primary grade c l a s s e s , "^9 T^e forms o f abaci suggested are a two-rod form (10 beads on each rod) for

beginning number work, and a five-rod form (9 beads on each

rod) for later work. The authors recommended using the

abacus for the representation, comparison, addition, sub­

traction, multiplication, and division of numbers.

A pocket chart was used to teach the principle of place

value. With this device a single card In the ones’ pocket

represented one unit; a bundle of 10 cards in the tens’

pocket represented one ten; and a bundle of 100 cards In the

hundreds’ pocket represented one hundred.

_b. Buckingham —

On pageL'196 a quotation was given from this book which

expressed the author’s philosophy concerning the value of

29Leo J. Brueckner and Foster E. Grossnickle, How to Make Arithmetic Meaningful (Philadelphia; John C. Winston Go., 1947)» p. 176. “ — — 257 the abacus. With a stateüient of this type one would then

have expedted him to discuss techniques by which the abacus

could be used in teaching, but he did not,

Drummond —

The author advocated the use of the two forms of abaci

designed by Montessorl (see Plate XVIII) for addition and

subtraction. Another device by which the four simple arith­

metical operations (addition, subtraction, multiplication,

and division) were taught, consisted of single beads, bars,

squares, and cubes. Por example, the number 2631 would be

represented by a single bead, 3 b a rs, 6 squares, and 2 cubes,

arranged in order from right to left, respectively.

S tern —

This author was extremely ctltical of the vertical-rod form of abacus. Instead of this she used two other devices for teaching place value. One form consisted of a block-holder having a slit across the center. Cards having the Hindu-

Arabic numerals printed on them were used with the holder.

A card with a 3 printed on it and placed on the right-hand side of the holder represented 3 units. A card with a 3 printed on it and placed in the slit along with a card having a 0 printed on it, and so placed that the card with a

3 was to the left of the card v/ith the 0, rep resen ted 3 te n s . Another device consisted of a single cube to represent one 258

u n it, a 10-blook (a single long block marked into ten equal

blocks) to represent one ten, and a liundred-s-quare (a square

block made up of ten 10-blocks) to represjenb one hundred, and so on,

e« Stokes ——

In describing the form of abacus prefenred, this author

sta te d :

We do not favor use of the ordinary abacus as much as do some teachers. Different wires for different place values do not seem as real to us as, say, larger beads for tens placed alongside smaller beads on the same w ire .30

We think this latter form is much better than the abacus where tens are on one wire and ones on another j as the digits in a number stand side by side, so should the representations of these values be placed in adjacent positions,^ He also advocated that "the child be given large buttons,

pebbles, or the like, to represent the tens' units and

materials of the same kind but smaller to represent the ones' ,"32

A third form of device to represent numbers was made up of

sticks — the sticks used in the tens' place were to be larger

than those used in the ones' place.

30(j, Hgiyjton Stokes, Teaching the ^%anings of Arithmetic (Hew York: Appleton - Century-Crofts, Inc., 1951)» p.337»

31ibid, . p, 376. 32x,oc. c i t . 259 f # Vftisât ——

Vftieat used the paper-and pencil form of abacus quite

extensively (using Hindu-Arabic numerals and horizontal

lines as counters). He advocated its use for the repre­

sentation, and for the addition of numbers. Wilson etaal —

The only reference to the abacus in this book was a sugges­

tion that it could be used as a topic for an appreciation

unit for the few who miglit choose to do one.

h. Harding (Functional Arithmetic) —

In this book several forms of abaci were illustrated. These

were an open-end abacus, a type used in schools in Cuba (8

wires bent at r i^ t angles, each wire having 12 beads), the .

Abacounter (see pages 269-72), the zoo-abacus (five vertical OO wires having 3 plastic animals on each wire), a traditional and a modern form of Japanese abaci, a Ohinese abacus, a model

of a Roman abacus, a Russian abacus, a model of a Persian

abacus, and four types of commercially available abaci.

i_, Hickerson —

The author made the following statement relative to the use

of the abacus in teaching arithmetic:

The ancient abacus and the modern calculating machine should be available for those children who have already acquired an understanding of the place-value aspect of

33The zoo-abacus was designed by Professor Nathan Lazar of Ohio State University. 260 the number system. These machines should not be used for the purpose of ’teaching* the number system . Rather, they should be used for the purpose of illus­ trating other ways of computing besides the method of committing to memory the different number combinations and knowing techniques of computation.34 Morton —

This author illustrated an abacus which had one feature dif­

ferent from other forms of abaci used in schools at the

present time. The abacus consisted of 10 vertical rods,

each rod having 9 beads. The unusual feature was that a

decimal point and two commas were represented on the abacus.

The decimal point was so placed that numbers as small as

thousandths of a unit could be symbolized,

k. H ollister and Gunderson —

These authors suggested using the abacus, place-value dia­

grams (referred to in this study as paper-and-pencil abacus),

and a place-value chart for the teaching of place value.

In the latter device a single card in the ones' place

represented one unit; a bundle of 10 cards in the tens' place

represented one ten; and a bundle of 100 cards in the

hundreds' place represented one hundred.

S p itz e r —

The author designed a form of abacus that is known as the

Spitzer abacus, ^t is made of 3 vertical wires having 10

34j, A. Hickerson, Guiding Children’s Arithmetic Experiences. (New York; Prentice-Hali, ^ncT, 1952), p. $6 . 261 beads on each wire.^^ In discussing the abacus as a teaching

device,. Spitzer stated;

For teaching the positional value of numbers, the abacus has no superiors among devices, ^t is also useful in the study of addition and subtraction and in minor ways in the study of multiplication and division. There are several forms of abacuses, each of which has some special advantage.3°

Some of these other forms of abaci that he suggested were: a

paper-and-pencil form of abacus (using Hindu-Arabic numerals

in the spaces, or horizontal marks on the lines), the Chinese

abacus, the Japanese abacus, a Roman pebble type of abacus,

a Number Form of abacus (vertical rods on which individual

counters slide, each counter having a Hindu-Arabic numeral —

from one including nine — painted upon it), and a Board

Form of abacus (vertical grooves in which counters slide).

m, Harding (Explorations in Arithmetic) —

In this book the author suggested using two types of abaci,

namely, a computing abacus having IS or 19 beads per wire,

and a 9-bead abacus.

Twelve of the fifteen books selected for this survey advo­ cated the use of some form of abacus in the teaching of arithmetic.

The authors did not, however, agree on the various aspects of the use

3Herbert F. Spitzer, The Teaching of Arithmetic (New York: Houghton M ifflin Co., 1954), P» 49* 3^Ibid. . p. 333. ' 262 of the abacus, such as the grade levels in which the abacus should be

used, the forms of abaci to use, the mathematical principles that

should be taught by the use of the abacus, and the value of the use of the abacus,

A CRITIQUE OF THE BOOKB IN RELATION TO TECHNIQUES

OF TSACaiING WITH FORMS OF ABACI

Brueckner and Grossnickle had a section on the historical aspects

of the abacus (pp. 36-39)» j^n this section Incorrect Ideas were given

of two of the forms of abaci. The first Illustration was described In the following words: "The abacus pictured at the right Is of Roman origin. Today, the Chinese and Japanese use a counting device based on t h i s pattern.The Illustration given was that of a Chinese abacus (see Plate XI) with the rods In a horizontal position Instead of a vertical position. This error was not as serious as the one described below.

The next two Illustrations on pages 37 and 38 were entirely misleading. They were supposed to represent the foim of abacus used during th e Middle Ages by money changers. The follow ing statem ent was made: "When a customer brought money to the place of exchange, he laid the coins on the counting board to be dounted."^® The form of abacus used by money changers was basically the type Illustrated in Figure 12

37Brueekner and Grossnickle, op. c lt. , p. 37»

3^Loc . c l t . 263, -- used with or without lines. This was a loose-counter form of

abacus, not a fixed rod with a fixed number of counters as illustrated

in the book by Brueckner and Grossnickle. A counter placed on a line

(or a counter placed opposite a Iyer, if lines were not used) of the

abacus, represented 1 pence, or 1 shilling, or 1 pound, or 10 pounds,

or 100 pounds, or 1000 pounds, according to the line on which the

counter was placed. The lines were arranged in the ordered form (from bottom to top) named above. A counter placed in a space between the lines. or between the lyers (not placed on lines or rods as indicated in Brueckner and Grossnickle), represented five times the value of a counter on the line immediately below the space.

The first example given to be represented on the counting board w as 3657» This was given as an abstract number with no indication of the denomination of money this represented. As can be seen from

Figure 12, one must know the unit with which one is dealing (pence, shilling, or pound).

Assume that the authors of this text wished to indicate on a counting board an arrangement of counters that would symbolize 3657 sheep. The form of counting board would then be of the type shown on p ages 127-8* The number 3657 would be indicated as shown in the following figure: , . M 0 ^ 0 ______(3000) D 0 ( 500} 0 J 100) 1 # ( 50)

X ______V 9 ( 5 ) I A A ( 2 ) 264

The couaters representing 500, 50, and 5 are in spaces, not on lines.

The following chart lists the books which indicated place value

in a manner which the author of the present study considers question­

able. The device used is also included in the list.

Book , Device

Brueckner and Grossnickle Pocket Chart

Drummond Single bead, bar, square, and cube.

Stern Single cube, 10-block, and 100-square.

Stokes A small bead and a larger-sized bead. A small stick and a larger stick.

Hollister and Gunderson Place-value chart.

In each of the above cases the basic principle of place value was violated. This principle requires that whatever type of counter is used to represent one unit in the units' column (or section), exactly the same type and the same number of counters must be used to represent one ten in the tens' column, and exactly the same type and the same number of counters must be used to represent one hundred in the hundreds’ column.

A CRITICISM OF THE ABACUS AS A TEACHING DE7ICE

Stern entitled a chapter (Chapter XII) in her book; The Prison

Bars of the Counting Frame [she used the words counting frame and abacus synonymously]. In this chapter she stated:

The ancients invented this device [the abacus] in their first groping with numbers . . .

With the introduction of our numerals [the Hindu-Arabic numerals"] and of positional notation, the counting frame or 265 abacus disappeared and the human Mnd was freed to investigate numbers as far as his intelligence could carry him.39 In another part of her book she described the types of materials that

could be used to construct the number 73Ô. The following materials

represent the number: 7 hundred-squares, 3 10-rows, and Ô sm all

cubes. She advocated this type of representation because the pieces are of different structure, thus revealing the magnitude of each quantity. She continued the discussion by stating:

Here is the advantage of our materials over the abacus, in which the number 738 is represented by 7 beads on the third wire, 3 beads on the second, and S beads on the first; what we see is nothing but 18 beads of the same size at different places. With the abacus, therefore, it is impossible to develop a full concept of the clear-cut structure of ournnalaber system. The materials that Stern suggests for representing the number 738

are effective for use while one is working in the "tally stage" of teaching the meaning of number, but they do not serve the purpose of teaching the principle of place value. %n fact, these materials vio­ late the principle. For the 7 hundred-squares (each square a section,

10 by 10, made up of 100 small squares), as she interprets their value, would represent the number 700 regardless of their position. And the

3 tens-row (each row consisting of 10 squares) would represent the number 30 regardless of their position.

These materials could be used to precede the teaching of the

^^Oatherine Stern, Children Discover Arithfiietic (New York: Harper and Brothers, 1949), pp. Il6, ll8.

AOIbid., pp. 266, 268. 266 principle of place value. But the abacus, particularly the vertical

form, serves as effective preparation for the introduction of the

Hindu-Arabic numerals. As Stem states (in the last paragraph quoted

on page 265), ", , . the number 738 is represented by 7 beads on the third wire, 3 beads on the second, and S beads on the first; . .

and it is this very form of representation which provides a transi-

S,to(,K, pi le. fi »n tional step to the Hindu-Arabic numerals.

COHTEMPORAHT ïOmiS OF ABACI

Some of the textbooks and the books on methods suggested the use of forms of abaci that are modern counterparts of historical abaci.

Two of these — the open-end abacus and the paper-and-pencil abacus — are illustrated and discussed on the following pages. Three other forms aie also included — the Abacounterthe P f e il D e v i c e , 43 and the Arithmetic 13ayy44

4^Loc . c i t . 4^A discussion of the Abacounter appeared in an educational monograph (see page 270 , footnote 52).

43a discussion of the Pfeil Device appeared in a Master of Education Report (see page 273» footnote 53).

44a discussion of the Arithmetic Tray has not as yet appearèd in p r in t. 267 1. The open-end abacas*45*46

The open-end abacus consists of a platform base in which

dowel rods are inserted in a row. With this device it is neces­

sary to use counters such as ring-checkers, or carroms, so as to

be able to slip the counters over the rods.

The abacus shows twenty-three

This abacus has the same advantages as any abacus of the form

which uses an unlimited number of counters, with the additional

advantage of being a device which is easy and inexpensive to make.

2. A paper-and-pencil abacus. 47

This form of abacus, as the name implies, consists of making

a replica of an abacus by drawing lines upon paper. This could

45Louise A Mayer, "The Scarbacus or Scarsdale Abacus," Arithmetic. Teacher. I I (December, 1955)» 159» 46^11ce Rose Carr* "Using a 'Calculator! to Develop Basic Under- âtandings and ^eanings in Arithmetic," Mathematics Teacher. XLIII (May, 1950), 195.

47see Harry Grove Wheat, How to Teàch Arithmetic (Evanston, Illinois: Row, Peterson and Go., 1^51’)» PP. 92-3» 147-9. 268

be done in three basic ways:

TEM6 vjMrrt> TEMS* UWIT6,

: : : —— Z. 3 o e . o e .

The abaci show twenty-three.'

The paper-and-pencil abacus has the advantage of providing

a sijnple and inexpensive way for each student to have a form

of abacus upon which he can work the problems himself. Another

advantage of this form of abacus is that it makes a very

effective testing device.

The Arithmetic Tray. 48

The Arithmetic Tray is made up of two sections; (l) on

the left — a section on which loose counters, such as metal

washers, or checkers, may be used as tallying devices; and

(2) on the right — a section on which vertical strips form

columns for an abacus.

^%he Arithmetic Tray was designed by Professor Nathan lazar of Ohio State University, 269

UWOe.C,AMI qooo

O e.

O o o o o o o o o 026/5 Sl|Z€.D

Twenty-three is shown the "long way" in the tallying section on the left, and is shown the "short way" in the abacus section on the right.

This device has the advantages of (1) allowing the student to

arrange his own groups in recording numerical quantities and in

performing arithmetical operations the "long wajï" — the groups

are not already structured for him; (2) allowing the student to

in d ic a te num erical q u a n titie s in two ways — th e "long way" in

the tallying section, and the "short way" in the abacus section}

and (3) allowing the student to perform arithmetical operations

on the abacus section. The Abacounter.5®»51

49spme persons prefer that the organised form be in terms of rows of ten counters each.

50The abacounter was designed by Professor Nathan Lazar of Ohio State University. 5^The following thesis indicates many possibilities for using the Abacounter: Maiy Helen Games, "The Use of the Abacounter for Clari­ fying A rithm tical Meanings in the Junior High School," (unpublished Masterâs thesis. The Ohio State University, 1952). 270 The abacounter is a device designed to make the facts, concepts, and operations of arithmetic . . . understand­ able. The abacounter is made up of two sections : (l) on the left — a counting frame, or counting rack, consisting of horizontal rods and beads; (2) on the right — a simplified abacus, consisting of vertical rods and beads . 52

mît1fîîP“ .... 1 1tr— till 1 iiiii

111! 1 11111 nn 1 Hill --- Httt--HtH-

The counting frame and the abacus each show twenty-three.

The counting frame on the left consists of twelve horizontal

rods with each rod containing ten beads. Thus it is possible to

use this section of the abacounter as a tallying device.

The simplified abacus on the right consists of seven vertical

rods with each rod containing twenty beads. The advantage of

having twenty beads on each rod is that this simplifies opera­

tions if one wishes to add numbers whose sum is greater than 10.

For example, if one wishes to add 49 and 29, it is possible to

^^Nathan Lazar, "A Device for Teaching Concepts and Operations Relating to Integers and Fractions,” Arithmetic 1949. ed. G. T, Buswell and M. L. Haitung («Supplementary Educational Monographs,” No. 70 [" Chicago : University of Chicago Press, 1949J), ?» 87 . Indioate both nuiiibers at the saxne time before performing the

operation of addition. Then after pushing all of the counters

t e n s UKirrs TENS UNITS

^ INCUCftfTEO := NOM&ER. : £ POStT'ON

Srocic piLfe Po s i t i o n srrep \ £>Tep 2.

together one finds he has 18 counters on the units’ rod. Ten

of these counters are pushed down into the stockpile and re­ placed by pushing up {from the stockpile] a counter on the tens'

rod. The requit is seven counters on the tens’ rod and eight counters on the units’ rod.

Another advantage of having 20 beads on each rod is that this arrangement makes it possible to work examples in vjhich the students need to ’’carry 2”. For example, with the problem

15 / 26 / 39» it is possible to symbolize the three numbers simultaneously, in this way the student w ill be less inclined to generalize that 1 is the only number that can be "carried.”

The advantage of having a counting frame and an abacus combined in one device is that the student may indicate in two ways (the ”long way” on the counting frame and the ’’short way" 272

on the abacus) any number including 120. (Numbers la rg e r

than 120 may be symbolized on the abacus section only.) This

technique enables the student to comprehend more easily the meaning of a given number, and in addition to develop an

appreciation and an understanding of the principle of place v alu e.

In the diagram on page 270, the units' rod is the first rod on the right, •'■t is possible to place the units' rod in any position and thus use the rods to the right of the units' rod to represent decimal numbers. For example, if the units' rod were the third rod from the right, then it would be possible to represent tenths and hundredths.

Since the Abacounter is a combination of a counting frame and an abacus it is important to have the beads on the counting frame section the same color as the beads on the units' rod of the abacus section, if the Abacounter were an abacus only, then it would probably be best, other than for aesthetic purposes, to have the beads on all of the rods the same color.

Then too, the Abacounter is usually used in combination with a paper-and-pencil form of abacus, which has no color.

Any misinterpretation that the student might make from the use of color in the former, is offset by the lack of color in the latter. 273 5. The P f e il Device. This device is a miniature set of steps. These steps are

divided longitudinally into three equal sections. At the rear

of the top tread are grooves into which plywood strips —

marked units, tens, hundreds — are set. The counters for the

device are one-inch cubes.

The abacus shows twenty-three

The principal advantage of this form of abacus is that it is

possible to indicate simultaneously as many numbers as there are

steps. The sinç)le arithmetical operations may be performed on

this device as on any other form of abacus.

53Designed by Arthur Pfeil, Instructor in IJniversity School, Kent State University. Por an explanation of the use of this device see Arthur P feil, "A Description and Comparative Analysis of Some Devices Which Have Been Used to Give leaning to Integers and the Operations with Integers,” (unpublished Report, Master of Education, The Ohio State University, 1954)» In addition to his own device, Pfeil illus­ trates and discusses the following; Carr Device (open-end abacus), Dunfee Device, Urossnickle Device (Place Value Pockets), Spitzer Device (Tens Block)Stern Devices (Counting Board, Humber Cases, U nit Blocks, Pattern Board), Abacounter, Modernized Abacus, Modern confuting Abacus (Ideal School Supply Co.). 274 THE PKENCIPAL GHâRACraSRISTIO OF TEE PGESENT PERIOD (1935 TO 1956).

If one were to give this period a name that would typify the

aspect of arithmetic which was stressed, it would have to be a name

that referred to the Meaning Theory. There are probably as many

interpretations of the meaning theory as there are persons writing

about the theory. A few of the outstanding articles written on the

subject were by the following educators: 1. Buckingham —

This author explained the terms significance, meaning, and

insight. In speaking of meaning he said:

. . . the meani^ of number, as I understand it, is mathematical. J-n pursuit of it we conceive of a closely knit, quantitative system.

Under the heading of meaning I include, of course, the rationale of our number system. The teacher who emphasizes the social aspects of arithmetic may say that she is giving meaning to numbers. I prefer to say that she is giving them significance. [By the significance of number I mean its value, its importance, its necessity in the modern social orderj. In my view the only way to give numbers meaning is to treat them mathematically. . . . significance and meaning are nothing without an experiencing n atu re. 54

2. Brownell —

Brownell thought of meaning as having two facets. The aspect thab

Buckingham referred to as the "significance” of number, Brownell

called the "meaning for" arithmetic ; the aspect to which Bucking­

ham applied the term "meaning," Brownell applied the expression

"meaning of" arithmetic. The latter author grouped the meanings

^^B. R. Buckingham, "Significance, Cleaning, Insight — These Three", Mathematics Teacher. 2XXI (January, 1938), 26-7» 275 of arithmetic tmder four categories: a. The basic concepts,

K The understanding of the fundamental operations,

a. The more, important principles, relationships, and

generalizations of arithmetic,

jd. The understanding of our decimal number system and its use in

relationalizing our computational procedures and our algo­ rism s, ^5

3» Johnson —

This writer divided the meanings of arithmetic into three

broad categories:

Structural Meaning: meaning of arithmetic as a coherent

system .

(1). The meaning of the number symbols themselves.

(2). The meaning of place value and its relation to the

number system.

(3)» Number re la tio n s and th e i r meanings.

(a ). Concepts, generalizations and principles in arithmetic.

Jb, Functional Meaning: Meaning for arithmetic. [.This answers

the question; What is arithmetic for?] jB, Operational Waning; Has to do with the understanding of

processes [includes rationalization and also the way the

^% illiam A. Brownell, "The Place of Meaning in the Teaching of Arithmetic," Elementary School Journal. XLTII (January, 1947)» 257-8. 276 process i s done]]»^^ 4 . Van Ingen —

This author wrote:

In any meaningful situation there are always three elements. (1) There is an event, an object, or an action. In general terms, there is a referent. (2) There is a symbol for the referent. (3 ) There is an individual to inte^ret the symbol as somehow referring to the referent. i>V Van Engen believed in the operational theory of meaning and expressed this idea as follows;

It is important to note that the symbol only produces an intention to act and that the act need not, in itself, take place. However, if the individual is challenged to demonstrate the meaning of the symbol, then the action takes place . . . he gives a definition by example — an example of actions taken which are appropriate for that symbol, . . . 5S

A w riter who was apprehensive about the interpretations of the

meaning theory was Weaver. He stated:

There is no doubt that the rise of the "meaning theory" has accelerated the use of representative materials in instruct­ ional activity. These materials are intended to make arithmetic more meaningful, ^t is unfortunate that many persons have misinterpreted the implications of this fadt. Consequently they have harbored one or more of several misconceptions: (l) that the mere use of any concrete or semiconcrete materials guarantees meaningful instruction.

^^1. Té Johnson, "What Do We Mean by W aning in A rithm etic?," Mathematics Teacher. XLI (December, 194Ô), 362.-7.

57h . Van Engen, "An Analysis of %aning in Arithmetic, I," Elementary School Journal XLIX (February, 1949), 323»

^ ^ I b i d .. XIXS, 324. 277 (2 ) that instruction cannot be meaningful without the use of these representative materials, and ( 3 ) that these materials minimize, or even replace, the use of abstract number symbolism,59

In regard to the first misconception, he said that the mere use

of the materials was not sufficient, but that the materials must be

especially appropriate to the meaning to which they are applied, and must be used with skill and discrimination.

In regard to the second misconception, he stated that it was

unnecessary to use these materials for each of the numerous extensions

of the skills; such a practice would deny children the use of higher

levels of thinking through which meaning may be derived.

His response to the third misconception was to state that

conventional number symbols and computational algorisms need to be

supplemented, not replaced, by lower levels of representation.

V/eaver has taken an extreme point of view. In the teaching profession there are usually persons who misinterpret the principles of a philosophy of,teaching. But one should not describe their practices as if they were general.

The writings surveyed for this chapter ^ve no indication that the educators in the field advocated the "misconceptions” that Weaver discussed in bis article. Teachers are noted for their willingness to "follow the textbook" and to accept the writings of leaders in the field. Knov/ledge of this policy makes it difficult to believe

59j. Fred Weaver, "Some Areas of Misunderstanding abcxit Meaning in Arithmetic". Elementary School Journal. LI (September, 1950)» 38. 278 that the situation described by Weaver is a general one.

The educators in the field do not recommend that concrete

materials be used "for each of the numèrous extensions of the skills

but that the beginnings of number work be with these types of materi­

als and that whenever necessary the student could return to them for clarification.

Nor do the educators imply that the concrete materials should

replace the Hindu-Arabic numerals. It is generally accepted that one of the goals of the teaching of arithmetic is to enable the student to understand the number system and to work effectively with the numerals; the above goal can be achieved by beginning with concrete m a te ria ls .

Weaver stated that the practice of trying to teach arithmetic meaningfully accelerated the use of representative materials. This acceleration might be reflected in the number of articles written about these materials during the Present Period. In the following chart information is given as it was found in the Education Index:

Number o f a r t i c l e s Number o f listed under the articles heading: Arithmetic listed under (Subheading: Teach- the heading: Period of Time _ ing aids and devices ). . Abacus

January, 1935 to June, 1935^^ 8 0 July, 1935 to June, 1938 18 0

^9a list of these articles is given in Appendix# VI.

^^It should be noted that this is for six months only. 279 Number of a r tic le s Number of listed under the a r tic le s heading: Arithmetic listed under (Subheading: Teach- the heading: Period of Time (Coat * d ) _lng a id s and deTrioesV Abacus. " Ju ly , 193Ô to June, I941 12 0

Ju ly , 1941 to June, 1944 12 1

Ju ly , 1944 to June, 1947 11 1

Ju ly , 1947 to May, 1950 17 1

June, 1950 to May, 1953 30 3 62 June, 1953 to March 2, 1956 47 6 If the number of articles on a topic is indicative of teaching

trends and practices, then one could assume that the use of "aids

and devices" has greatly increased during the last twenty-one years.

The table also shows, more particularly, an increased interest in the use of the abacus.

SIMMHT OF THE PERIOD

The authors, in general, are in agreement that arithmetic should be taught in such a way as to be meaningful to the student. At the present time, it is believed that arithmetic can be meaningful to the child if a classroom situation is provided in which the child himself uses manipulative materials, under careful and skilled guidance from the teacher, in order to make his own discoveries of the relationships which are basic to the number system.

^2it should be noted that this is for 33 months, while the majority of the ones listed above are for 36 months. 280

The textbooks for the early grades tisiially consist of semicon­ crete materials only - - materials which are principally pictures.

Since pictnres are static, it is impossible to demonstrate effectively problems involving motion.

As mentioned in the above paragraph, the textbooks usually contain semiconcrete materials only; the accompanying manuals contain the suggestions for the use of manipulative materials. With this practice there is the conger of teachers ' neglecting to use the manuals, and as a result confining the beginning of number work to experiences with semiconcrete materials. This handicap could be overcome to a certain extent by including in the textbook as an integral part of the work, problems which involve manipulative materials. GHAFTIB IX

SmWARI AND SUGEmanONS

SBHMIBT firüitiie» indioatas that in tha Naollthie Aga aaa reaehad that atag# in h is davalopnant in which ha bacana a hardanan« As ha accum­ u lât ad poaaaaaions ha naadad a technique for recording and fo r com­ municating nuabara* Per this parpoaa ha began using the materials around him that ware plentiful and convaniant — material a such as pabblaa, aticka, a h alla, and barriaa. Chapter I I (The TUlly) shows how mankind in a l l parts of the world laaznad to taka aatariala of this type and to use them officiantly for the representation of numbers, and to a certain extant, for cosputation# In first th$ea matariala ha matched oge tallji-objact (such as a pebble or a stick) with maa of the objects ha was counting. Later, whan ha naadad to count larger quantities ha learned that ha could match jQS laraa tal3j«-obiaet (or one tally-objact of a certain color) with JBS eroup o f ob je c ts ha was counting, and match one small t a lly - blijact (dr one telly-ebjact of a different color) with oga of the objects ha was counting. As man acquired new sk ills and laaznad to make rope and cord, he began to use t h is type o f m aterial fo r a more permanent form o f record keeping^ Chapter III (The Knotted Card) gives am account of the ma#qr différent peopics who used this material for recording numbers, and the variety of ways in shich they u#ad it. In fbet, one 2&L 282 grw ip -M. tlM Xiieas o f P«ra>» d«T«I^p«d a teehnleal form o f bookkoopiag (prterilÿ roeordlng of mutbors | bj use of the knotted eerd» or the aeiim &a they called it . With Uie qaipa they progressed into the nett level of men* s developswnt in the represen­ tation of amerioal quantities ^ they used the priaeipls of place value. %oy arranged the knots representing units in one row, the knots representing tens In another row parallel to the units' row, and the knots representing hundreds in a third res parallel to ths first too rose. At the tine that man nas looking for materials to use to assist him mith his record keeping, he discovered that his osn fingers served as one of the most convenient means of coamnnieating and representing numbers, and even of computing vith numbers. In Chapter 17 (The Ungers) the author diseussea this uidely-practiced system of finger notation and fin g er rec&wning — a system ubich had far-reaching effects upon man's eultuxnl life . Che finds evidence of the earlier use of finger notation and finger reckoning in classical literature, in the nuBUiral uords of many languages, in the base of the majority of number systenm, and in early numerical notation. The three types of devices just discussed — the tally, knotted cord, and fingers — as used by swst peoples fbr recording and ocm- patlng devices, were net based on the idea of place value. When man developed intellectually to the stage at which he grasped this technique, he then devised an instrument which is called the abacus. Chapter V (The Abacus) gives a historical review of the abacus and 283 . describes "ttie various fonos that have been developed in different countries. One finds evidence of the influence of this instrument even until the present time. This influence can be detected in lite r­ ature, in some of the words in the English language, and in some of the techniques used in performing simple arithmetical operations.

One form of the abacus is also credited with having been instrumental in the spread of the Hindu-Arabic numerals and in the invention of the value of position and of zero'.

As would be expected, p h y sical devices as commonly used by adults as the four discussed in the present study the tally, knotted cord, fingers and abacus - - became the basis for the teaching of arithmetic to children in the schools of the various periods. The use o f the fin g e rs and th e abacus was tau g h t much more ex ten siv ely than the use of the other two devices. Chapter VI gives a brief his­ torical survey of the use of these devices in the teaching of arith­ metic from the time of the founding of schools in ancient Greece

(c, 6G0 B, C, ) until the l?th century A, D, Also included in this chapter is a succinct discussion of the beliefs of three of the early philosophers who advocated the use of concrete materials in teaching.

In the next two chapters (Qiapters VII and VIII) the writer traces the development of the use of concrete materials in the teaching of arithmetic in the United States, The former includes the period from early colonial days until 1935, During this time the techniques of teaching of arithmetic fluctuated in a manner m «dghb b# deaerlbed ma foUonrst (1) tlié a i i o f Mefaanieal ^Oehntqoem; (2) m lialtod aso of objoetivo m&torlAl#; (3) mphmal# om dimciplino (w±bh its acooppanyiog aoejwtnieal toehaiquos) ; (4) belief in self-eetirity on the part of the ehild; and (5) phasis upon life situations as a basl# of activities in arithnetic. In Chapter VIII the trends in the use of objective materials for the years 19^g to 1956 are indicated. In order to detersdne these trends a survey eas made of nine series of textbboks cm arith­ metic and of fifteen books on the teaching of arithnetic publl^ed in the United States during these years. The principal characteristics of the period (1935-195&) as nanl- fested In these books «érea 1 . Bsphasis upon teaching fo r neaning* 2. Increased use of objective naterlals, eith the child himself using the materials. 5. The laboratory abroach in idileh the child makes his osn discover­ ies of asahber relationships» under the skilled guidance of the teacher.

POSSIBUB SUQSBSriGNS W&Bt THE TEACHING OF AHITHNKriC Since the present study eas primarily historical in nature the suggestions have no eaqperimsntal basis. But often the story of the past helps to point W&e nay to the future. This r ^ r t seems to indicate that the human race has performed arithmetic computations for a longer period of time mithout numerals than mlth numerals. This seomiog paradox may have Implications for the 285 pr#amt"d»y t#aehimg of aritheàti#. Pariaapa tho efaHdraa are being introduead to tba Hindtt>^rftble nuamrals too aarly. Prinitlva nan had to bagin hia keeping of nuaerieal raeorda bj uaing tha aioplaat mthoda and aatariala availabla «•> a ta llj egretaa. Tha child* too, parhapa naada to begin hia noabar work with a tally ayataa. Tha author of tha praaent atudy doaa not baliava in tha "Sooial fiaoapitulation^ thaory, but thia la a aituatlon in whlnh it edL|^t ba padagoginally adviaabla to follow i t . Tha aajority of teactbooka of tha praaent time obearaa thia basic prino^la of wli* a tally ayatea, but are inclined to rush alapst iaaadiataly into a preaantation of the Hindu-drabie nuaarala. Tha tally ejatea doaa not aarra as an adequata prépara- I tion for thaaa nuaarala. Parhapa thare should ba an iataraadlata step — that of using a form of abacus. Sine a the Hindu^rabie nuaaral aystea, baaed upon tha principle of place value, was an out­ growth of the abacus, the abacus cam thus sarva as a direct transi­ tion from th* tally aystea to tha nuaarala and place value.

SUGCSSTIOIS m k 3BQPMCB HT THE TKAGHIl» OF ARIfHUBTlC On a basis of the Idaaa axpraaaad la the pracading paragraphs, the following suceasaiva types of eaqpariencas are auggaated for tha baginniüag muabar o f woxt o f children: 1 . Xaqparianeaa with unatructurad foaaa o f ta lly ^ b ja c ta . a. Objecta in Ufa aituationa, such as claaarooa aatariala (chairs, peacila, t^ka, shaata of paper). 2S6 b« R^plicaa of real objects» such as statuettes of animals, toy automobiles, and miniature chairs. £• Geunters, such as washers, buttons, bottle tops, and cardboard c ir c le s . 8emi-ecnorete materials^ such as pictures of birds, animals, and d ots, e ân unstructured bead rod. (This is a rod containing approximately 30 to 50 beads. The number of bead» may eary as required.) 2. Experiences with a structured counting frame where the counters are grouped In tens. 3. Experiences with a form of abacus with which loose counters are used, such as the abacus-section of the arithmetic tray or an open<^end abacus. 4* Eq>eriences with a form of abacas having a fixed number of counters, such as the Abacounter. 5. Experiences with a paper*>aad-pencil form of abacus, a. Horisontal lines used as counters. Hindu-Arabic numerals used as counters. With a background of thb tppe indicated above, the child should then bel prepared to use the Hindu-Arable numerals without using an abacus form. Even thouggh the above suggestions are listed in a nudiered order, the author did not intend to imply that with the introduction of new types of experiences one discarded former types of experiences. The 287 #pir&l method of toechlag would be the preferred method to uee. For eomuiple, work with the abaeue section o f the arithm etie tray should be preceded by and used simultaneously eith ea^erienees with the tallying seetion. Similarly, work with the abacus section of the Abacounter should be preceded by and used simultaneously with exper­ iences with the counting-frame section*

SUGŒSTIW8 FOR THE USE ŒTHE ABACUS IN THE TEACHING OF ARITHMBriC A portion of the present study was based upon a surrey of nine series of textbooks on arithmetic and of fifteen books on the teaching of arithmetic* This surrey showed that nob a ll educators in the area of mathematics hare realised the potentialities of the abacus* This instrument may be used for the following purposest 1* to teach ths prlociple of place value* 2* to qnobelise numbers (in teg ers and decimal fractions}* 3* to represent relationships among numbers, including thefire fundamental laws of arithmetic (the commutative laws of addition and multiplication, the associative laws of addition and multi- piication, and the distributive law)* 4* to perform the four simple arithmebical operations (addition, subtraction, multiplication, and d iri^ n ) with integers and with decimal fractions* 5* to represent numbers and to perform simplear ith h stic a lopera­ tions with number systems other than the decimal number system* 288 Th# àb&cua^ i f used fo r thooo piurpoa»B, mould then oefvo e f f e e t it è lj for the usual mork in the prlaary grades, for uork ulth the laws of arithnetio and with other nand>er bases la junior high school, and for romedial wozk in junior high Ochool. Many teachers are inclined to think of the terns "arithmetic" and "numerals" as synonymous, to think th at i t i s im possible to have the former without the latter. The present study shows that the ex­ perience of man does not support this tenet. Perhaps, then, the implications derived from the present study may be that the teacher need not feel impelled to rush the child into beginning his work with the Hindu-Arabic numerals, but may proceed throuÿi a carefhlly planned sequwce which consists of experiences with the tally, with the abacus, and than with the numerals.

BBCQHIEmMTlQIIS FOR OTHER STUDIES ... ’J, 1* Since the author of the present study reads French and SpaniA only, she was unable to consult sources in other foreign languages in which there is a wealth of informât ion pertaining to the use of Idiysical devices that are discussed in this study. A study of a nature similar to this one could be enriched ioaeasurably by the use o f the unexamined sources. 2. , Another study m i^t be a comparative erqperimental study using two groups of first-grade children. One group would follow ths se­ quence presented in a modem textbook on arithmetic, the usual sequence being one in which the Hindu-Arabic numerals are introduced 289 the first ten pages* The other grpap noald follow the sequence suggested in the present study in idiich the intreduction of the Hindu- Arabic mmerals is delayed, 3* Another possible study would be to investigate the way in which the Abacus is used in the schools of China and Japan, particularly seeking to determine if it ie primarily a recording and ccsg>uting device to be used in lieu of paper and pencil. /

APFENDIZ I

290 Apparoat I ADKQIlSnATm BSTORlf (An irm m apeeoh A«XlT«r«d by Charles Dickens on juns 27, lÔ55)ï Ages ago a savage node of keeping accbaats on notched sticks vas introdneed into thé Coart of gnchequer, and the aecounts were Irapt, mach as Bobinsen Crusoe kept his calendar on the desert iiCABd. m thé course of considerable revolutions of time,. .. a multitude of aceoanta#ts, book»keepen; and actuaries, wnre bite and died. S till official routine inclined to.',thèse notched sticks, as if th v V^e pillars of the constitution^ and still the Enchequef teoounts continued to be kept on certain ipUnta of elm wood called "tSllies." In the reign of George III an teqSlry was made by some revoluttonaxy iqplrit, whether pens, ink,, and paper, slates and pencÉlI^ being IS eidatenee, this obst%ate adherence to an obsolete custom oug^ to be continued, iwad idiether a change oug^t not to be effected» All the red taps in the country grew redder at the bare mention of this bold and original conception, and it took till ld26 to get these sticks abolished. In 1S34 it was found that there was a considerable accumulation of them; and the question then arose, what was to be done with such wbxn^ut, wexnreaten, rooten old bits of wood? I dare say there was a vast amount of minuting, manoranduming, and dsepatch-bojdag, on t h is mighty subject. The sticks were housed at Westminster, and it would naturally occur to any intelligent perswa that nothing could be easier than to allow them to be carried away fo r firew o o d by th e m ism b le people who lire in that nei^iborfapod. However, th ^ never had been useful, and official reutino required that they never should be, and so the order went forth that they were to be privately and confidentally burnt. It came to pass that they were burnt in a stove in the House of Lords; the House of Lords set fire to the House of Commons; tee two houses were reduced to ashes; architects were called in to build others; we are now in the second million of the cost thereof; the national pig is not nearly over the stile yet; and the little old woman, Britannia, hasn't got heme to-night.

4)iekene, "Administrative Reform," Sneeches (London* lUchael Joseph, Ltd., [no datej ), pp. 16S-7d.

291 AFPSnXEZ I I

292 # p p m m n m BSraOSMCSS TO FIMDOt S D G O I^ IN CLASSICAL LITEBATORE

Pl&ntasï MjUL«e Glori**i# er thé Braggart Ikrrlor. 201-206. (Bomb 6; B. C .|* Jttft look a t him . . • Baota h is la f t hand on bia lo f t thigh» and raekona on tha flmgara of his rif^ht hand*

Orid* Pontic Boiatlaa. ii. iii (Boman poat» 43 B. G. - 17 A. n»%. Bat noc'-n-daya ovary mo lovoa hia own intoroat» and ha roekona» on hia anzioas fingers» lAat may tarn oat asafal to himaolf.

Ovid. Fasti, ii. 123. (Pootieal troatiao on tho Roman Calondar) Mor had tha aneionbs aa many ealanda as wo havo now: thoir year was short by two montha. • . Honeo through ignoraneo and lack of acianea they reckoned laatros» each of sAieh was too short by tan months. A year was eoantod lAon tho moon had rotamod to tho fa ll for tho tenth time; that nnnbor was than In groat honour» whothar boeaaaa th at i s the nnnbor of the fingers by which wo are wont to count, w bocaaao a woman bfings forth in twice five mmtha» or bocauao tho namorala ineroaso up to ton» and from that wo start a fresh round.

Seneca. Lottara to Lnciliua. 38. 10. (Roman Stataaman and Philosopher» c. 4 B.C. - 6^ A. D.) Ho toachoa mo to rockm and pats my fingers at tho aarvico of avarioa ^stoad of teaching no that each calculations are wholly irralovant . . .

pialogaaa. Da Ira» 11. 33. . . .what if it be an invalid nonoy-londar . . . who can no Icngsr use hia hands to count with» . . .

293 294 Pliny, üàtttgal Hirtory. %X%I7. 33. (Boain Seholarÿ 23-79 â« D.)« • . • and also by tha twe-facad Jaaaa, dadioatad by King Kunaf nhiah la abrahippad aa Indiéàtihg var and paaca, tha ilngara of tha Statua baiog so arrangad as to indleata tha 355 day# of tha yoar,^^ , , . Të&tnllianl. Apoloaatlcna. adx. (la tin ehureh fat bar, «rota this in 197 4. D .). Wa should have to a a ttla ooraalyaa do«n to many document a «1th ca lcu la tin g movamaats of tha fiagars> . . .

D lo(n ). Bosmn History» laod.* 32. (Boiail^iijtoriA^iMd administrator» «rota this betuaan 200 and 222 4. D.;. Thau upon his [Hhrcus3 rttum to Borne ha mada an addrass to tha paopla; and «hila ha «as saying» among other things, that ha had baem absent many yaara, thay cried out, "eight," and indlnatad this also «ith thair hands, in order that thpy mig^ reeeiya that number of gold pieces for a banqiant.

Quintilian. Inatltutas ^ Oratory, xi. 3* 117» (HoBMin xhatorician of tha 1st century 4.D. ). Soma ramsrks or fhults in the managamant of tha hands must ba%ddad* at le a s t on sUch fa u lts as ara incidant to SSparianoSd spaaksrs; for as to the gestures of • • . forming the nusbar five hundf^ by bonding the thumb, • . •

Lucian. Timw. 13. (Greek prose «rite, fl. 2nd eentury). That «as «by you presented yourself to us palled and fu ll of «orries, sdth your fingers dafonsad from tha habit of counting on them, • • •

4pulaius. 4nolosia. 09. (Boman philosopher and satirist, 2nd century). If you had isaid thirty idian you meant tan, it might be supposed thàt you |ad made a mistake in the gesture of cal^lation, and that you had pressed your fingers together [« lol instead of making them form a c iw la [ « 3 0 l. But tha numhar forty i s one 295

#hieh is tha «aalMb to asqtrass for it ia indioat ad by tha opan hand; and ahan yon iaeraasad i t by h a lf, tha JsLstaka could not ba blamad OB tha gastura of your fimgara.

liaerobias. Cbn—ptary on tha Praam of Scioio. ii. 11. 17* (Latin wrltar and philosophar, fl. e. 400 A. D.) Ona has only to count on hia fingars to find how many ÿaars romain in a tvantiath part Cof a sorld-yaarl aftar subtracting tha 573 yaars fromPoomlus' death to Seipio*s campaign.

Hacrobitts., Las Satunialas. i, ii, and ix. L*axaaq>la da PlAten m'antorisa done a na paa ealeular sur las doigts las annaas da mad ddnviraS; . . .

Sidonitts. Lattars. ix. Latter IX. (Latin wrltar, f1. 455 - 475 A.Ù.)

. . • Ghrysippus counting sith clanehad fingars, . . .

Bachtal C"Mngar-Gounting among tha Bomans in tha Fourth Cantury, " Clasaicsl fhilologr. 17 (Januazy, 1909), 25-311 discussad soraral aanons of Augustina (bidSop of Hippo, 354 * 430 A .P .) in oriiich ha used rafarancas to fizgar notation in such a way that ona must infar that his congragatien in tha North African Churdh «as familiar with tha system. The following sermons ware listed t Sermons clx x r. 1. ccxlTiii. 3* c c x ltx . 3* c e l. 3 . e c l i . 5 -7 . cclii. Ô-11. celx x 7 . c l . 1 . ccxlTiii. 5. Traotatus in lohannis Bsangalium exxii. 7. aMirratienas ip Paalmos riix. 9. âPFiaiDiz 111

296 4P F E ^IIIZ I ïàBLY m i r a s ABOUT THE ABACQS Ihrtlanu# Capella,^ a Latin wzitar mbo Aouriahod about 475 4

1 Also known as Felix Capella and may hare lired in the 4th century. 2 Isidori Hlspalensis aplscooi ^ig^h^ librl riwinti ex anti- Quitatp eruti. Basel, 157?/ - 3 Charles Singer in Florence A. Teldham, Storr o f Beckon^g in the Middle Ames. (Lcndon: George G. Karrap and Co. L td., 192o), p .l3 .

^ . teldham, gg. cit.. p. 36. 5 South central France 6 The Latin Text and French trm nslatioa (w ith comments) of a portion of Gexhert* s writings are giren by Chasles* "Analgm et Ex­ plication du Traité de Gerbert. " Comptes Rendus. %FI (F erler, 1843)> 281-99.

297 298 Gerbert alee déelgiaed en abacas (see Plate HX) ehleh eas diylded into thirty eoloans of irtileh three «ere reserved for free* tioaai ooagpatations. With this iastrunsnt he used eoonters. On each eoanter a symbol (see Plate XX) «as InioribeÀ* 4 single counter in­ scribed «ith tbs symbol for three (instead of a group of three counter) signified 3, or 30, or 300 depending upon «hether it «as placed in the units*, the tens', or the hundreds' column, reapectlrely. Sobert Becorde (c. IjilO - 1558) «rote one of the best acceants of reckoning «ith unmarked counters in Pert II of his Greunde of 8 trtes. first published in 15A2« His books «ere the most influential &i(^i.sh mathematical publications of the I6th century. Ae fMm of abacus he used in his eaplanations had horizontal lines, omitted superfluous elements, and had a star beside the thousands' line to guide the eye. Figure 54 illustrates the type of abacus he used. On Figure 55 the counters are anranged to represent 1543. Wo more than four counters «ere placed on a line. When the number of counters on a line reached five Uwy «ere replaced by one counter in the space above the line. Ho more than one counter «as placed in a space. When the nuaber of counters in a space reached t«o they «ere replaced by one counter on the next line above the space.

1 -These sy ^ ls «ere known as aoAes. notas, figures, signs, characters, o-laments. or einhers. BoWthius (475^24) is supposedly ereditml with theirintroduction [D. S. Smith and L. G. Kaïpinsid, HindU'mêrabie Wumeimls (Boston* Ginn and Co.. 1911)» p* 119.J A... " Sections from this work are g iv # in Yeldham, gp. pp. 45-6; F. P. Barnard^ th e Cisting-Gounter and the Counting-BoardTTOxfoxdt CUrenden Pz«ss, 1914), pp. 256?61; and Robert Steele» Barjiert Arithmetics in Rnalish (London: Oxford Bniversity Press, 1922)pp.52-65< 299

P late XlZt Diagram o f an ibaeaa Uaed in &i£^nd in the . Yaar 1111.

Floranee A. X^ldham^ stmey ija thi Mittila Amaa (Londdn: Qaorg# G. m^rra#^a^ Co. litd ., 192677 iap* 3 M . Peraiaaion fa r naa gppantad by Gaor)ga 0 . Harrap and Cb^ mit

Diagram of an Abacus usbd in England in the Y ear iiii The upper part is used for integers and the lower part for fractions 301

Plate %%. Sjnbols üsed on Gerbert*s Countere

PlQrenee A . Teldhaa. The Story /of Reekmalna In the Middle Aaea Harrap. and Co. * Ltd, p. 37. FezttiaaiAa for uad granted by George G. Harraÿ ind Co. Ltd. The signs of the numbers head the columns in the draw­ ing of the abacus.

I 5 ft ^ y F A Ô 2 0 I 2 3 4 5 6 7 8 9 I®

Vw o 303

I o o c» o

\ O Ç ?

F ig . 54 ïôria of Abacus Used by Robert Recorde.

Fig. 55 The Board Shows 1,543* îrooi Robert Records, The Srouad of Artes (London: Reynold Wolff, 1542). Figs. 54-55. Form of Abacus Used by Robert Recorde. 304 a âddlticMUJ. important mritèra aho laeludad eaplBaatloos of th# usa of iho abaott» «oret 10 fod olay, John, Printer of 4a Introduction of Alaoriano. ^ leam to rËBkoa w tb Pea or mrth the coaator». London. 1574.

■ 1 1 ,1 2 ...... Betaehj Gragorlne, Margarita PhileaoDkLea. Freibu%%;, 15@3* ■ ^ J Caeanua, Johànnea, M Éorithnae L ineaila P roieetiliu m . Vienna* 1514*

14 . Silioeae* Joannea Maz^^e* ürithaetlea. Paria. 1526. Kb*bel* J a c o b .Ain nea geordnet Raehenbieehlin anf den llniea mit Recheapfepingea. Aagsbarg, 151L.

^Other men aho «rote about the abaeua are Mated la Susan Boae Beaediety 4 Ccapara4iaé Study of the Early Treatiaea into Mareae the Hindu-Art o f Reckoning cCooeord. N.H. t The Runferd Praaa* 1% 4), pp. 10. IS: D. E. Smith. Raw A rltbnetiea CBoatoni Gian and Co.* 190S), pp. 36, 68, 70i 74, 123, 125, 138, 140, 151, 152, 155, 156, 165, 167, 170, 180, 181, 182, 183, 188, 195, 250, 252, 268, 269, 2?1, 284, 295, 300, 3Q2i 303, 353, 361, 4l2, 435, 482; L. Jackaoa, Bdttcatioaal of Sixteenth Century Arithmetic ("Contriba- iiena to Sducatloa,” Mo. sllNew fork* Teachers College, Columbia (IniToraity, 1906 ] ), p. 181; Eyaa Thorndike, Science and ThoaAt in the Plfteeath Century (Mew Torkt Coluiabia Uaiyeraily Preaa, 1 ^ ) , pp. 151-60; and Teldham. gg. c i t . . pp. 92-6.

^^Baraard, o p . c i t . . pp. 2 6 6 - 7 1 . pp. 271-5.

1 2 ' Smith, OP. c it.. pp. 82-3.

B a r n a r d , op. cit.. pp. 275-80. 14 Ibid. , pp. 280-92. I t pp. 292-301. 16 , Smith, o p . cit.. pp. 100, 103, 106. 365 17,16 d« Jioya, Jttaa P e n s , Twttado de Math—a tlc a s . A lcala. 1573* 19,20 ;' .- v ' Trénchant, Jan, L'Aritbmatiqne de Jan tranehant . . .Arac L*Arb dai C alcttlT anx Getone. LTona. is 7 6 . 91 ^ Legendre, Francola, L*ArttttaetlQue en sa Perfectien. Parle, 1733*

17 Barnard, cit.. pp. 301*3. IB Sbith, jgg. c i t . . pp. 308, 310, 3L1. 1 9 Barnard, s g . c i t . . pp. 304*9* 20 Smith, pp. ci^,. p. 322. ^^mard, gg. cit.. pp* 313-7. 4FFEMDZX 17

306 ippama vr BEFE8B1CES TD THE iBACUS IN (SEEK UTEBAOUBE

Eristophanes. Clottda. 18-20. (This play was given in 423B.C.) (4487 - 3807 B.C., Athmian draaatlst) Sirepaiadaat Boyl light a laaip. And fetch my ledger* now I*11 reckon up Who are my créditera, and W*at I owe them.

Diogenea Laertiua. Llvea o f Eminent Phileaophera; Solon. 1. 59. (4127 - 323 B .C ., Greek cynic phlloaopher) Solon (arehon 594 B.C.) He uaed to aay that thoae who had Influence with tyrant a were like the pehblea employed in calculaticma; for, aa each of tlmpebblea repreaented new a large and now a email number, so the tyremta would treat each one of thoae about them at one time aa great and famoua, at another aa of no account.

Theophrastus, Characters. x It , xxUl, and aodr. (c. 371 - 387 B.C., Greek phlloaopher and naturalist)

x It . Stupidity. Stupidity, to define It, la a slowness of mind In word and deed; and the stupid man he, that after he has cast ap an account, w ill ask one that aits by vriiat it comes to; . . . ism . Pretentiousness. . • • and i&em strangers are sitting neat him he w ill ask one of them to east the account, and reckox^ig it in sums of tm , twenty- five, and fift^]0 aaalgn plausible na#Os to each sum given, and make it as much as thiee thousand pomid.

307 3 0f xxir. Arrogance. He need to say that «hen this nan cones to a reckoning nith you he commands hie page to do the counting and adding and set the sum down to your account.

Plutarch. L ires: Cato the Younger. Ixx* (46? - 120? A. D ., Greek essa y ist and biographer} « • • in his death struggle fe ll from the couth and made a loud noise by overtumlng a geometrical abacus that stood near. APPEKDS V

309 Apmmn V SEFEB^CES TO TKE ABACUS IN fiOUIV LITERATURE

Ifereus Talarins ttixtialls. Book II. 48. (e. 40 - e. 102^ RcMum •plgraaaatiat) A tavam er, and a batcher and a bath, and a barber, and a draugbtH>oaxd and placaa, . . .

Daelana Junlaa Juvenalis (Juvenal} (607 - 1407, Reman poet and satirist) Satire ix. 40, Well, set out the counters, call in the lads vith the reckoning board, . • • ga^ire xi. 1^2. So destitute am I of ivory that n ^ h ir my dice nor counters are made of it ; . .

310 APPENDIX VI

311 ipPENbn 7 1 ARTICLBS usm m m EDDCATlm INBEÎ ROM JiWiRT# 193$ TO IttRCH £,1956, om m tm HEADING* iBAcas H, p. Opitger. *ibaeu 0 in the Teaching ofAlrlthaetio > " Elementary jg&aai, (Pebreary, 1942)y^% P. 1C. Rich. "Nimber Combinat lone Self-taught with Home-made Abaeue,** Journal of muQatina. dOTIII (March. 19A51. 88-9. Nathan Laaar* ^ e v io e [Ababounter] fo r Teaching Coneepte and Operations Relating to Integers and PYaotions.f Arithmetic 19A9. ed. G. T, Bttsvell and ic. L. Hartang. ( "Supplenoataxy Bdueatlonal IGanographs, " No. 70 é) Chicago t University of Chicago Press, 1949. Vera Sanford. "Counters* Computing i f You Can Count to F ire," Nbthe­ matics Teacher. XLIH (November, 1950), 360-70. T. Ian. "Chinese Abacus," Itethematios Teacher. ILITI (December, 1950), 402-4 , P. S. Hamthone, "Making an Abacus, School Science and Mathematics. U (Ihroh, 1951), 227-a. B. A. Suelts. irCounting Devices and Their Uses," Arithmetic Teacher. I (Pebmary, 1954), 25-30. R. C. â a ith . "Abacus; Working Brawing," In d u strial Arts and Vocational BdttCatlon. ILIII (October. 1954). 277. 0. Jenkins. "Larry and the Abacus; Story," Arithmetic Teacher. I (October, 1954), 21t4 . A. P. Schobt. "Believe It or Not, They Now Love Arithmetic," Parents IfaagLeg, m (May, 1955), 44-5. R. W. Plewelling. "Abacus as an Arithmetic Teaching Device," Arith­ metic Teacher, i l (November. 1955). 1G7?-11. L. A. Mayor. "Searbacus or Scarsdale Abacus," Arithmetic Teacher. H (December, 1955), 159. The folloering article appeared too late to be included in the Education Index of March, 1956* II. Vere DeVahit. "The Abacus and Multiplication," Arithmetic Teacher III (March, 1956), 45;

3 i? BIBLIOGSAFHÏ

313 BIBLIOGRAPHY

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"Addition on the Abacus-Japanese Style," Time. XLVIH (November 25, 1946), 35.

Alcott, A. B. "Arithmetic", American Annals of Education. II (iferch 15, 1832), 147.

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"Almanac," Encyclopedia B ritan n ica (1953 é d .). I , 668-70.

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Yeldham, Florence A. Story o f Reckoning in Middle Ages. Londcm: George G. Harrap and C o., L td ., 192%. . Teaching of Arithmetic through Four Hundred Years 11535-1935). London: George G. Harrap and Co. L td ., 1936. Yoahlno, Y. The Japanese Abacus Explained. Tokyo. Japan: Xyo Bun Kwan, 1 937 . Young, J.W.A. "Some Recent French Views on Concrete Methods of Teaching Mathematics," School Review. UIX (March, 1905), 275-9. Young, J.W.A. The Teaching o f Mathematics. New York: Longmans, Green and Co., 1931. AUTOBIOGRAPHY

I, Julia Elizabeth Adkins, was born in Wayne, West Virginia, on

June 26, 1910, I received ny secondary school education in the public schools of Ceredo, West Virginia, % undergraduate training was obtained at Marshall College, Huntington, West Virginia, from which I received the degree Bachelor of Arts in 1930,

I taught for seventeen years in public high schools in West

Virginia, was a high school principal for two years, and taught for two years in the laboratory school at Marshall College, During this time I received the degree Master of Arts from Ohio State University

(l9i).3) and also studied at the University of Kentucky, and at Teachers

College, Columbia University,

In 19^2 I began working on my doctorate at Ohio State University,

For the last three years I have also been supervisor of student teachers in secondary mathematics.

336