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 the Knotted 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 century A. 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 of Arithmetic . 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 ...........................................................................