Abstract
Despite the voluminous prior research on the history of the abacus before the 20th century in East Asia, little has been studied on the educational ideas contained in the descriptions of manipulation of this counting device to perform elementary operations. The aim of this chapter is to explore the educational ideas related to the execution of the most elementary operation—addition—on abacus presented in the documents from the middle 15th century to early 20th century in China and Japan. The analysis focuses on different aspects related to the manipulation of this counting device in order to unveil their educational ideas. The results of this study show that the education of abacus addition experienced a historical change around the Meiji Restoration (1868), from an era of the use of verses to an era of the elaboration of “techniques of fingers,” dealing with positions of fingers on instruments, and their movement. This study may be of importance in providing an understanding of the education of abacus addition in China and Japan before the early 20th century, as well as in proposing a method of analysis for exploring the educational ideas of other elementary operations performed on the abacus or other counting devices in the history of mathematical education.
A large part of this chapter is based on my doctoral dissertation defended in the Paris Diderot University (University Paris 7) in 2013. I would like to thank the editors, especially A. Volkov, and two anonymous reviewers for their constructive comments and suggestions.
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- 1.
Here and below I use the term “fingering” to refer to the method of using a particular finger to move a given bead or group of beads to perform an operation on an abacus.
- 2.
The illustrations of the abacus depicted in two other abacus books published in the same period—the Computational Methods with the Beads in a Tray (Panzhu suanfa 盤珠算法, 1573) by Xu Xinlu 徐心魯 (dates of life unknown) and the Compass for the Computational Methods ( Suanfa zhinan 算法指南, 1604) by Huang Longyin 黃龍吟 (dates of life unknown)—have only one bead in the upper compartment and five in the lower, in each column. Since they were used to illustrate the results of operations, it is very doubtful that the illustrations correspond to the structure of the concrete abacus used by the authors. Besides, in the procedures of division, it is more reasonable to have two beads than one in the upper part from the point of view of necessity and convenience (Chen 2013, pp. 19–47). In brief, even if this 1 + 5 type instrument existed during that period, it seems that it was less used than the 2 + 5 types found in China and in Japan before the 20th century.
- 3.
Lishou is a legendary person who mastered arithmetic and was the historiographer of the mythical Chinese sovereign, the Yellow Emperor (Huangdi 黃帝), dated around 2600 B.C.
- 4.
Translated from Xu (1573, p. 1143). The original text reads (the chapter author’s punctuation): 如本行下五子俱已在位。今又要上一。則下無一可上。故於上面下一。是五。復於下面去四。故上得一。.
- 5.
According to the extant Chinese abacus works published before the 20th century, the second upper, or the highest, bead was never used in the execution of an addition using the decimal system.
- 6.
Translated from Xu (1573, p. 1143). The original text reads: 如本位子滿在位。又要加一。卻無一可加。故幾退去九子。卻于上位還一子。當下位十子。卻正一也。.
- 7.
It was mentioned earlier in the chapter that the second upper, or the highest, bead was never used in the execution of an addition using the decimal system. The result of the addition, 13, therefore cannot be set directly in the working column.
- 8.
In addition to these two phrases, Xu Xinlu explained in the same way the conditions for the use of the two phrases, “2 lower 5 remove 3” and “2 withdraw 8 advance 10,” which will not be presented here (Xu 1573, p. 1143).
- 9.
The example is extracted from the four rightmost places of the operation 246,913,578 + 123,456,789, called Method of the third addition (Disan shangfa 第三上法) in the Computational Methods with the Beads in a Tray (Xu 1573, p. 1144).
- 10.
This specific mode of movement of beads might not have been created initially for didactical purpose, however, it is found in the very first page of Xu’s book in an exercise of addition, where the illustrations of an abacus representing the results of the operations together with the versified rules of addition were provided to teach its readers how to move beads to perform certain operations on an abacus. This context suggests that the mode of movement of beads found in the book can be related to the educational ideas of the author.
- 11.
Those who are not familiar with the arithmetic operations on a Chinese abacus may wonder why the 2nd upper bead and the 5th lower bead were still kept on the instrument if these two beads were not supposed to be used in addition. Actually, in the operations of multiplication and division performed according to the ancient methods used on the Chinese abacus, it happens that the number appearing in a column in the process of the operation may become larger than 10 (for example, 11 to 17 in the operation of division by a one-digit number (Chen 2013, Sect. 1.2.2.). If an abacus of type 1 + 4 was used, the operator from time to time had to keep in mind, temporarily, the numbers larger than 10 produced in the process of the operations. This would be very inconvenient and may have caused mistakes.
- 12.
Unlike Computational Methods with the Beads in a Tray, which was entirely dedicated to computation on an abacus, these two works presented computational methods performed on an abacus as well as counting rods, the principal mathematical instrument used in China before the second half of the 16th century. To be precise, it is considered by contemporary historians that the computational methods of the four elementary operations presented in these two works were performed on the abacus while the other more advanced mathematical methods, such as extraction of square roots, were performed using counting rods (for a more detailed discussion, see Hua (1987, pp. 66–73)).
- 13.
These two verses in the two books were presented together with the other two verses related to the operations of subtraction—“Verses on breaking five” (Powu jue 破五訣) and “Verses on breaking ten (Poshi jue 破十訣).” Since one of the four phrases of “Verses on breaking five,” that is, “without 1 remove 5 return 4 below” (無一去五下還四), can be applied only to some operations performed on the abacus, these four verses, as, therefore, the entire collection of rhymed instructions, could not have been created at first for counting rods and then applied to the abacus, but must have been intentionally created for the abacus, even though the two verses related to the operations of addition can be applied correctly to both the abacus and counting rods (Chen 2013, p. 54).
- 14.
Translated from Wu (1450, p. 16); Wang (1524, p. 358). The original text reads: 起五訣。一起四作五。二起三作五。三起二作五。四起一作五。成十訣。一起九成十。二起八成十。三起七成十。四起六成十。五起五成十。六起四成十。七起三成十。八起二成十。九起一成十。
The Chinese character qi 起 used in the title of the “Verses on erecting five” qiwu jue 起五訣, and in every phrase of these two verses, has various meanings in Chinese, I therefore translate it to different words in English according to its different contexts. For the name of the verse qiwu jue, I translate qi, whose meaning is rather close to “use” or “build” in this context, as “to erect;” the qi used in the phrases, whose meaning is rather close to “take off,” I translate it as “to remove,” such as “1 remove 4 make 5.”
- 15.
Translated from Li (1705, p. 134). The original text reads 六起四下還一成一十. This phrase is one of the three versified phrases for the cases of adding 6 to the column, the other two phrases are 六上六 and 六起四成一十.
- 16.
The four texts were H.N. Robinson’s (1806–1867) The Progressive Primary Arithmetic (1862), H.N. Robinson’s First Lessons in Mental and Written Arithmetic (1870), Ch. Davies’ (1798–1876) Primary Arithmetic (1862) , and Davies’ Intellectual Arithmetic (1858).
- 17.
Endō Toshisada was the author of A History of Mathematics in Great Japan ( Dai Nippon Sūgakushi 大日本数学史, 1896, reprinted in 1918, 1960, and 1981), the first important book on the history of traditional Japanese mathematics, wasan 和算. He was a practitioner of wasan and began to learn Western mathematics after the Meiji Restoration.
- 18.
It refers to the name of the table of addition shown in Fig. 8, which was never found in the older Japanese abacus books.
- 19.
Translated from Endō (1878, pp. 19a–b). “… 最モ緊要ナルハ相加ノ数正ニ十ニ満ツルモノトス…學者既ニ加算九九ヲ諳記セハ宜ク之ヲ盤上ニ施スコトヲ習ハスヘシ….”
- 20.
In the book, the author did not inform the user which finger to use.
- 21.
This does not mean that the “finger movements” had not been taught by instructors beforehand.
- 22.
However, it cannot be concluded that the verses were not taught orally any more.
References
Bréard, Andrea. 2014. On the transmission of mathematical knowledge in versified form in China. In: Scientific Sources and Teaching Contexts Throughout History: Problems and Perspectives, Boston Studies in the Philosophy and History of Science 301, ed. A. Bernard and C. Proust, 155–185. New York etc.: Springer.
Chen Yifu. 2013. L’étude des Différents Modes de Déplacement des Boules du Boulier et de l’invention de la Méthode de Multiplication Kongpan Qianchengfa et son Lien avec le Calcul Mental. PhD dissertation. Paris: Université Paris Diderot-Paris 7.
Endō Toshisada 遠藤利貞. 1878. Sankajutsu jugyōsho 算顆術授業書 [Textbook of the Art of Counting beads]. Tokyo: Tokyo shihangakkō 東京師範学校. National Diet Library Digitized Contents Database, http://dl.ndl.go.jp/info:ndljp/pid/829098.
Horiuchi, Annick. 1994. Les mathématiques japonaises à l’époque d’Edo. Paris: Librairie Philosophique J. Vrin. [For an English version of this work see Horiuchi 2000].
Horiuchi, Annick. 2000. Japanese Mathematics in the Edo Period (1600–1868). A Study of the works of Seki Takakazu (?–1708) and Takebe Katahiro (1664–1739). Basel: Springer.
Hua Yinchun 華印椿 (ed.). 1987. Zhongguo zhusuan shigao 中國珠算史稿 [Sketch of a history of computation with beads in China]. Beijing: Zhongguo caizheng jingji chubanshe 中國財政經濟出版社.
Jami, Catherine. 1998. Abacus (Eastern). In Instrument of science: an historical encyclopedia, ed. R. Bud, and D. Warner, 3–5. New York/London: Garland.
Kawamoto Koji 川本亨二. 1968. Gakuseiki no shuzan kyōiku ni kansuru ichikōsatsu 学制期の珠算教育に関する一考察 [Survey on the Abacus Education in the Period from 1872 to 1879]. Kyōikugaku zasshi 教育学雑誌 [Journal of Educational Research] 2: 23–35.
Ke Shangqian 柯尚遷. 1578. Shuxue Tonggui 數學通軌 [Avenue to Mathematics]. Reprinted in Guo Shuchun 郭書春 ed. 1993. Zhongguo kexue zhishu dianji tonghui 中國科學技術典籍通彙. Shuxue juan 數學卷 vol. 2, 1167–1212. Zhengzhou: Henan jiaoyu chubanshe 河南教育出版社.
Li Gong 李塨. 1705. Xiaoxue Jiye 小學稽業 [Course of Minor Learning]. Beijing: Zhonghua shuju 中華書局, 1985. Reprint.
Suzuki Hisao 鈴木久男 and Toya Seiichi 戶谷清一 (ed.). 1960. Soroban no rekishi そろばんの歴史 [History of the Soroban]. Tokyo: Morikita Publishing Co., Ltd. 森北出版株式会社.
Tamaoki Tetsuji 玉置哲二 and Ishibashi Umekichi 石橋梅吉. 1913. Shuzan seigi 珠算精義 [Essential of Computation with the Beads]. Tokyo: Buneikan 文永館. National Diet Library Digitized Contents Database. Available at http://dl.ndl.go.jp/info:ndljp/pid/952188.
Uegaki Wataru 上垣渉. 1998. Gakuseiki ni okeru sanjutsu kyōkasho no taiyō 学制期における算術教科書の態様 [On a Phase of Textbooks for Arithmetic in the Period from 1872 to 1879]. Nihon sūgaku kyōiku gakkaishi Sūgakukyōiku 日本数学教育学会誌 数学教育 [Journal of Japan Society of Mathematical Education, Mathematical Education] 80(6): 89–96.
Uegaki Wataru上垣渉. 2000. Meiji chūki ni okeru shuzan no fukkōundō ni kansuru ichikōshō 明治中期における珠算の復興運動に関する一考証 [A study on the Reconstructional Movement of the Calculation with Abacus (Soroban) in the Meiji Middle Period]. Miedaigaku kyōikugakubu kenkyūkiyō Kyōiku kagaku 三重大学教育学部研究紀要 教育科学 [Bulletin of the Faculty of Education, Mie University, Educational Science] 51: 1–20.
Uegaki Wataru 上垣渉. 2001. Shogaku sanjutsusho no tanehon ni kansuru saikôshô 小學算術書の種本に関する再考証 [Re-study on the Source Book of Shogaku sanjutsu-sho: Elementary text of arithmetic]. Nihon sūgaku kyōiku gakkaishi 日本数学教育学会誌 [Journal of Japan Society of Mathematical Education] 76: 3–16.
Volkov, Alexeï. 2001. L’abaco (Abacus). In Storia Della Scienza (Encyclopedia on History of science, in Italian), vol. 2 (Cina, India, Americhe), gen. ed. S. Petruccioli, volume eds. K. Chemla, F. Bray, D. Fu et al., 481–485. Rome: Istituto della Enciclopedia Italiana.
Wang Wensu 王文素. 1524. Xinji tongzheng gujin suanxue baojian 新集通証古今算學寶鑒 [Precious Mirror of Mathematicsm newly collected and provedly]. Reprinted in Guo Shuchun 郭書春 (ed.). 1993. Zhongguo kexue zhishu dianji tonghui 中國科學技術典籍通彙. Shuxue juan 數學卷 vol. 2, 337–971. Zhengzhou: Henan jiaoyu chubanshe 河南教育出版社.
Wu Jing 吳敬. 1450. Jiuzhang Suanfa Bilei Daquan 九章算法比類大全 [Great Compendium of the Comparison and Classification in Nine Chapters of Mathematical Methods]. Reprinted in Guo Shuchun 郭書春 (ed.). 1993. Zhongguo kexue zhishu dianji tonghui 中國科學技術典籍通彙. Shuxue juan 數學卷 vol. 2, 1–333. Zhengzhou: Henan jiaoyu chubanshe 河南教育出版社.
Xu Xinlu 徐心魯 (ed.). 1573. Panzhu Suanfa 盤珠算法 [Computational Methods with the Beads in a Tray]. Reprinted in Guo Shuchun 郭書春 (ed.). 1993. Zhongguo kexue zhishu dianji tonghui 中國科學技術典籍通彙. Shuxue juan 數學卷 vol. 2, 1143–1158. Zhengzhou: Henan jiaoyu chubanshe 河南教育出版社.
Zenkoku shuzan kyōiku renmei 全国珠算教育連盟 [League for Soroban Education of Japan]. 1971. Nihon shuzanshi 日本珠算史 [History of computation with beads in Japan]. Tokyo: Akatsushi shuppan Co., Ltd. 暁出版株式会社.
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Chen, Y. (2018). The Education of Abacus Addition in China and Japan Prior to the Early 20th Century. In: Volkov, A., Freiman, V. (eds) Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators. Mathematics Education in the Digital Era, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-73396-8_10
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