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Didactical Dimensions of Mathematical Problems: Weighted Distribution in a Vietnamese Mathematical Treatise

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Scientific Sources and Teaching Contexts Throughout History: Problems and Perspectives

Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 301))

Abstract

The paper focuses on a series of problems on weighted distribution found in the Vietnamese mathematical treatise Ý Trai toán pháp nhất đắc lục 意齋算法一得錄 (A Record of What Ý Trai [=Nguyễn Hữu Thận] Got Right in Computational Methods) compiled in 1829 by Nguyễn Hữu Thận 阮有慎 (1757?–1831?). An analysis of the problems suggests that they may have been designed by the author of the treatise in order to introduce the concept of weighted distribution and to justify the general algorithm for solving the problems of this type.

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Notes

  1. 1.

    The “pre-modern” Asian mathematical traditions mentioned here include the mathematical practices and related texts that either antedated the encounter with Western mathematics that happened in the seventeenth century in China or remained uninfluenced by it, as in the case of traditional Japanese mathematics wasan 和算of the seventeenth to nineteenth centuries.

  2. 2.

    In this paper, the pinyin transliteration system is used for Chinese terms (including titles of treatises and personal names), the Quốc Ngữ transliteration is used for Vietnamese terms, and the Hepburn romanization system is used for Japanese terms. The so-called “traditional Chinese characters” (and not their simplified versions used in the People’s Republic of China and, in certain cases, in Japan) are used everywhere in this paper except the Bibliography.

  3. 3.

    Chemla and Guo 2004: 4.

  4. 4.

    In traditional Japan, professional mathematicians published collections of problems idai遺題(lit.: “problems left [for solution of the reader]”) without providing their solutions to challenge fellow mathematicians, see Horiuchi 1994: 40, 65, 114–116, 158–167.

  5. 5.

    The present collection of papers contains contributions providing argument for, as well as against, this assumption in various historical contexts; see, for example, the papers of Christine Proust and Anne-Marie Chartier.

  6. 6.

    What is known about the social positions of those who studied and taught mathematics, for instance, in traditional China from the second half of the first millennium to the early second millennium AD amounts to a fragmented picture of a rather complex network of individuals and institutions, including the state-run School of Computations (see below).

  7. 7.

    Series of Babylonian algebraic problems all having the same answer were discussed in Vaiman 1961: 174–175; Høyrup 1990: 330 also mentions such problems and claims, without providing references though, that “it was recognized already in the early 1930s that Babylonian ‘algebra’ problems were constructed from known solutions.” I am thankful to Dr. Evgenii I. Slavutin (Moscow) who kindly drew my attention to this fact in a series of inspiring discussions in the 1980s.

  8. 8.

    This is the case of the series of problems on weighted distribution from Chapter 3 of the Jiu zhang suan shu: an analysis of a source dated of the late first millennium AD shows that the order of the problems in all the extant editions differs from that found in an earlier (and now lost) edition of the text (Volkov 2011).

  9. 9.

    The earliest printed editions of the Chinese mathematical treatises used for educational purposes are dated of the late eleventh century; one can therefore only speak about their established versions starting from that time. As far as the extant Vietnamese mathematical treatises are concerned, the majority of them are handwritten, and it is unknown whether their printed versions ever existed.

  10. 10.

    Sun 2000: 138; Lee 2000: 515, n. 230.

  11. 11.

    See des Rotours 1932: 139–142, 154–155; Siu 1995: 226; Siu and Volkov 1999; Volkov 2013.

  12. 12.

    It is possible that earlier editions of at least nine of these textbooks had been used for instruction in the School of the Northern Zhou dynasty (Volkov 2014). Descriptions of the mathematics curriculum of the Tang State University and the identification of the textbooks with the extant mathematical treatises as well as information about their authors can be found in Siu and Volkov 1999; Volkov 2012, 2014.

  13. 13.

    The extant treatises of this period not used in the School are the recently unearthed mathematical texts of the late first millennium BC as well as the mathematical manuscripts found in the Dunhuang caves (Gansu Province, China). However, it is quite possible that these texts were also used for educational purposes.

  14. 14.

    Des Rotours 1932: 141–142, for a discussion of the procedure see Siu and Volkov 1999; Volkov 2012.

  15. 15.

    On the memorization test see des Rotours 1932: 30–31, 141, n. 2; Siu and Volkov 1999: 91, n. 41; Lee 2000: 142.

  16. 16.

    Siu and Volkov 1999; a piece of evidence supporting this hypothesis is found in a Vietnamese mathematical treatise; see Volkov 2012.

  17. 17.

    This conjecture was made in Siu and Volkov 1999 and discussed in great detail in Siu 1999, 2004.

  18. 18.

    CDW 1990: 991.

  19. 19.

    YH 1993: 1048–1050; for English and French translations see Lam 1977: 11–14 and Friedsam 2003: 58–63, respectively; see also Zhou 1990: 397.

  20. 20.

    Sun 1996: 517; Lee 2000: 521; Sivin 2009: 147.

  21. 21.

    Sun 1996: 517; Yang 2003, vol. 2: 819–820.

  22. 22.

    Li 1955: 279–280.

  23. 23.

    Volkov 2002: 373.

  24. 24.

    A translation and an analysis of this model examination paper can be found in Volkov 2012.

  25. 25.

    Volkov 2002, 2005, 2006.

  26. 26.

    Volkov 2008, 2009.

  27. 27.

    CDW 1990.

  28. 28.

    YT 1829 . The title of the treatise contains an allusion to the Chinese proverb (chengyu) “qian lü yi de” 千慮一得 (“Out of one thousand ideas [of a silly man] one is right”).

  29. 29.

    Trần and Gros 1993, 2: 51; Trinh 2007: 509. I am grateful to Nguyễn Thị Dương who kindly sent me a translation of the biographical notice on Nguyễn Hữu Thận from Trinh 2007.

  30. 30.

    See also the notice no. 4505 in Trần and Gros 1993, 3: 708–709.

  31. 31.

    Some Vietnamese authors recently suggested 1757–1831 for his lifetime.

  32. 32.

    Trinh 2007: 509–510; see also Trần and Gros 1993, notices 1707, 3091, 4505 (2: 50–51; 3: 22, 708–709); Liu et al. 2002, notices 0123 (1: 22), 1954 (1: 384), and 2772 (1: 526).

  33. 33.

    This interpretation of the term 侍郎 used in the biography is based on its Chinese equivalent; see Hucker 1985: 426–427, notice 5278.

  34. 34.

    See Hucker 1985: 306–307, notice 3630; 410–411, notice 5042.

  35. 35.

    Trần and Gros 1993, 3: 22, notice 3091.

  36. 36.

    Trần and Gros 1993, 2: 286, notice 2165.

  37. 37.

    Trần and Gros 1993, 1: 776–777; Liu et al. 2002: 21.

  38. 38.

    YT 1829, preface: 1b.

  39. 39.

    See, for example, Brousseau 1997: 176. By “mathematically identical problems” I mean the problems whose generic solutions can be written with the same mathematical formula.

  40. 40.

    This interpretation of the division S ÷ n is quite naturally dubbed “distribution division” in the literature on didactics (Gravemejier 1997: 335), since several objects are (equally) distributed among several sharers.

  41. 41.

    Problems with such conditions are found in various traditions; Chinese mathematical treatises, for example, contain problems in which fractional numbers of individuals are supposed to be obtained, see Volkov 2011.

  42. 42.

    YT 1829, 3: 2a. The original manuscript YT 1829 contains commentaries written in smaller characters, and since the first page does not contain any mention of commentator(s), I assume that the commentaries were added by the author himself; some of the commentaries quoted below are consistent with this hypothesis. When quoting the original treatise I reproduce the commentaries in a smaller font size, in translation they are placed in angular brackets <like this>. The text quoted below contains several corrections made on the basis of comparison of YT 1829 with another manuscript copy of the treatise preserved in the library of Han-Nom Institute, Hanoi (call number VHv.1184). Added characters in the text and added words in the translation are placed in square brackets. The translation is mine.

  43. 43.

    The Vietnamese monetary units used here are quan貫,bách陌, and văn文; they had the following relative values: 1 quan = 10 bách = 600 văn.

  44. 44.

    The mathematical identity justifying this answer is (10,500–24) ÷ 36 = 291.

  45. 45.

    YT 1829, 3: 2a–b.

  46. 46.

    “Return to the source” is a standard term meaning a verification of the numerical solution.

  47. 47.

    This instruction may have prescribed performing the operations in reverse order. See a verification of this type provided in Problem 3 below.

  48. 48.

    The Vietnamese author is not the first to suggest sharing an amount among a fractional number of sharers, it had been done many times well before him. One of the earliest examples can be found in Problem 18 of Chapter 1 of the Jiu zhang suan shu: 61/3 coins are to be shared among 31/3 people (Chemla and Guo 2004: 167).

  49. 49.

    The visual representations shown in Fig. 10.2 as well as in the other figures below are not reconstructions of the representations of the Vietnamese learners (for which no evidence exists); they are provided for the modern reader to depict the underlying models of distribution imposed by the choice of parameters. The actual representations employed by learners in the traditions discussed, theoretically, may have differed considerably from the suggested here.

  50. 50.

    On the use of the counting rods in Vietnam see Mikami 1934; Volkov 2009, 2013.

  51. 51.

    YT 1829, 3: 2b–3b.

  52. 52.

    The author is using the expression 幾分之幾, literally meaning “a number [m] of parts obtained by division of a unit by an integer number [n]”, an equivalent of the modern expression “m/n”.

  53. 53.

    The author uses here the term 全人 he probably coined himself; it can be understood as “entire/complete person” as well as “the person [having a share expressed with] integer [number]”. The ambiguity of the term may have been intentional and was meant to reflect the possibility of the two interpretations discussed below.

  54. 54.

    This multiplication by seven may have been perceived by the reader of the treatise as analogous to the simultaneous multiplication of the “dividend” and “divisor” by ten in the previous problem.

  55. 55.

    YT 1829, 3: 3b–4a.

  56. 56.

    YT 1829, 3: 4a–5a.

  57. 57.

    YT 1829, 3: 5b–6a.

  58. 58.

    The Vietnamese treatises available to me do not contain sequences of problems similar to those found in A Record.

References

Primary Sources

  • CDW. 1990. Cheng Dawei 程大位. Suan fa tong zong jiaoshi算法統宗校釋[Unified origin of counting methods, with emendations and explanations], ed. Mei Rongzhao 梅荣照and Li Zhaohua 李兆华. Hefei: Anhui jiaoyu.

    Google Scholar 

  • YH. 1993. Yang Hui 楊輝,Cheng chu tong bian ben mo乘除通變本末[Alpha and omega of the continuity and transformation of the counting methods]. In Guo Shuchun 郭書春ed. Shu xue juan數學卷 [Mathematical section] of Ren Jiyu任繼愈ed. Zhongguo kexue jishu dianji tonghui中國科學技術典籍通彙[Comprehensive collection of primary materials on Science and Technology in China]. Henan Education Publishers 1: 1047–1072.

    Google Scholar 

  • YT. 1829. Nguyễn Hữu Thận 阮有慎Ý Trai toán pháp nhất đắc lục意齋算法一得錄[A eecord of what Ý Trai [=Nguyễn Hữu Thận] got right in computational methods]. Manuscript A.1336 preserved in Han-Nom Institute, Hanoi, Vietnam.

    Google Scholar 

Secondary Sources

  • Brousseau, Guy. 1997. Theory of didactical situations in mathematics. Dordrecht/Boston: Kluwer Academic Publishers.

    Google Scholar 

  • Chemla, Karine and Guo Shuchun. 2004. 九章算術. Les Neuf chapitres. Le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.

    Google Scholar 

  • Des Rotours, Robert. 1932. Le traité des examens, traduit de la Nouvelle histoire des T’ang (chap. 44–45). Paris: Librairie Ernest Leroux.

    Google Scholar 

  • Friedsam, Manfred. 2003. L’enseignement des mathématiques sous les Song et Yuan. In Éducation et instruction en Chine, Les formations spécialisées, vol. 2, ed. C. Despeux and C. Nguyen Tri, 49–68. Paris/Louvain: Editions Peeters.

    Google Scholar 

  • Gravemeijer, Koeno. 1997. Mediating between concrete and abstract. In Learning and teaching mathematics: An international perspective, ed. T. Nunes and P. Bryant, 315–345. Hove: Psychology Press.

    Google Scholar 

  • Horiuchi, Annick. 1994. Les mathématiques japonaises à l’époque d’Edo. Paris: Vrin.

    Google Scholar 

  • Høyrup, Jens. 1990. Algebra and naive geometry. An investigation of some basic aspects of Old Babylonian mathematical thought. Altorientalische Forschungen 17(2): 27–69, 17(2): 262–354.

    Google Scholar 

  • Hucker, Charles O. 1985. A dictionary of official titles in Imperial China. Stanford: Stanford University Press.

    Google Scholar 

  • Lam Lay Yong. 1977. A critical study of the Yang Hui suan fa. Singapore: NUS Press.

    Google Scholar 

  • Lee, Thomas H.C. 2000. Education in traditional China: A history. Leiden/Boston: Brill.

    Google Scholar 

  • Li Yan 李儼. 1955. Tang Song Yuan Ming shuxue jiaoyu zhidu 唐宋元明數學教育制度 [The system of mathematics education [in China] of the Tang, Song, Yuan, and Ming [dynasties]]. Zhong suan shi luncong中算史論叢 [Collected works on the history of Chinese mathematics]. Beijing: Keji Publishers. 4: 238–280.

    Google Scholar 

  • Liu Chun-Yin 劉春銀, Wang Xiaodun 王小盾, Trần Nghĩa 陳義 (eds.). 2002. Yuenan Han Nan wenxian mulu tiyao越南漢喃文獻目錄提要 [Annotated catalogue of the Vietnamese materials in Han and Nom]. Taibei: Institute of Chinese Literature and Philosophy, Academia Sinica.

    Google Scholar 

  • Mikami Yoshio 三上義夫. 1934. Annan-no ichi sansho-ni tsuite 安南の一算書に就て [On one mathematical book from Annam [= Vietnam]]. Gakko sugaku學校數學 14: 3–11.

    Google Scholar 

  • Siu Man-Keung. 1995. Mathematics education in ancient China: What lesson do we learn form it? Historia Scientiarum 4(3): 223–232.

    Google Scholar 

  • Siu Man-Keung. 1999. How did candidates pass the state examinations in mathematics in the Tang dynasty (618–917)? – Myth of the ‘Confucian-Heritage-Culture’ classroom. In Actes du Colloque “Histoire et épistémologie dans l’éducation mathématique”, 321–334. Louvain: Université catholique de Louvain.

    Google Scholar 

  • Siu Man-Keung. 2004. Official curriculum in mathematics in ancient China: How did candidates study for the examination? In How Chinese learn mathematics: Perspective from insiders, ed. Fan Lianghuo, Wong Ngai-Ying, Cai Jinfa, and Li Shiqi, 157–185. Singapore/River Edge: World Scientific.

    Google Scholar 

  • Siu Man-Keung, and Alexei Volkov. 1999. Official curriculum in traditional Chinese mathematics: How did candidates pass the examinations? Historia Scientiarum 9(1): 85–99.

    Google Scholar 

  • Sivin, Nathan. 2009. Granting the seasons. New York: Springer Science + Business Media LLC.

    Google Scholar 

  • Sun Hongan. 1996. Zhongguo gudai kexue jiaoyu shilüe [Brief history of science education in ancient China]. Shenyang: Liaoning jiaoyu Publishers.

    Google Scholar 

  • Sun Peiqing 孫培青 (ed.). 2000. Zhongguo jiaoyu shi中國教育史 [History of education in China]. Shanghai: Huadong Shifan Daxue Publishers.

    Google Scholar 

  • Trần Nghĩa, and François Gros. 1993. Catalogue des Livres en Han-Nôm. Hanoi: Nhà xuất bản khoa học xã hội.

    Google Scholar 

  • Trinh Khac Mạnh. 2007. Nhà xuất bản Văn hóa thông tin (Tái bản có chỉnh lý và bổ sung). Hanoi: Viện khoa học xã hội Việt Nam, Viện Nghiên cứu Hán Nôm.

    Google Scholar 

  • Vaiman, Aizik Abramovich (Вайман, Айзик Абрамович). 1961. Шумеро-вавилонская математика III—I тысячелетия до н. э. [Sumerian and Babylonian mathematics from the third to the first millennium BC, in Russian]. Moscow: Oriental Literature Publishers.

    Google Scholar 

  • Volkov, Alexei. 2002. On the origins of the Toan phap dai thanh [Great compendium of mathematical methods]. In From China to Paris: 2000 years transmission of mathematical ideas, ed. Y. Dold-Samplonius, J.W. Dauben, M. Folkerts, and B. van Dallen, 369–410. Stuttgart: Franz Steiner Verlag.

    Google Scholar 

  • Volkov, Alexei. 2005. Traditional Vietnamese mathematics: The case of Lương Thế Vinh (1441–1496?) and his treatise Toan phap dai thanh (Great compendium of mathematical methods). Traditions of knowledge in Southeast Asia, vol. 3, ed. U Kyi Win, 156–177. Yangon: Myanmar Historical Commission.

    Google Scholar 

  • Volkov, Alexei. 2006. State mathematics education in traditional China and Vietnam: Formation of the “mathematical hagiography” of Luong The Vinh (1441–1496?). In Nho Giao o Viet Nam/Confucianism in Vietnam, ed. Trinh Khac Manh and Phan Van Cac, 272–309. Hanoi: Social Sciences Publishing House.

    Google Scholar 

  • Volkov, Alexei. 2008. Mathematics in Vietnam. In Encyclopedia of the history of science, technology, and medicine in non-western cultures, 2nd ed, ed. H. Selin, 1425–1432. Dordrecht/Berlin: Springer.

    Chapter  Google Scholar 

  • Volkov, Alexei. 2009. Mathematics and mathematics education in traditional Vietnam. In The Oxford handbook of the history of mathematics, ed. E. Robson and J. Stedall, 153–176. Oxford/New York: Oxford University Press.

    Google Scholar 

  • Volkov, Alexei. 2011. Weighted distribution in Vietnamese mathematics. Paper delivered at the 13th international conference on the history of science in East Asia, Hefei, P.R. of China, 25–29 July 2011.

    Google Scholar 

  • Volkov, Alexei. 2012. Argumentation for state examinations: demonstration in traditional Chinese and Vietnamese mathematics. In History and historiography of mathematical proof in ancient traditions, ed. K. Chemla, 509--551. Cambridge: Cambridge University Press.

    Google Scholar 

  • Volkov, Alexei. 2013. An early Japanese work on Chinese mathematics in Vietnam: Mikami Yoshio’s study of the Vietnamese mathematical treatise Chi minh toan phap指明算法. In Seki, Founder of modern mathematics in Japan: A commemoration on his tercentenary, eds. E. Knobloch, H. Komatsu, and D. Liu, 149–172. Tokyo etc.: Springer.

    Google Scholar 

  • Volkov, Alexei. 2014. History of mathematics education in Oriental Antiquity and Middle Ages. In Handbook on the history of mathematics education, ed. A. Karp and G. Schubring. New York etc.: Springer (forthcoming).

    Google Scholar 

  • Yang Xuewei 楊學為 (ed.). 2003. Zhongguo kaoshi shi wenxian jicheng中國考試史文獻集成 [Collected materials on the history of examinations in China]. Beijing: Gaodeng jiaoyu Publishers.

    Google Scholar 

  • Zhou Dongming 周東明. 1990. ‘Xi suan gang mu’ yu Yang Hui de shuxue jiaoyu sixiang 《习算纲目》 与杨辉的数学教育思想[The ‘syllabus for studying mathematics’ and Yang Hui’s philosophy of mathematics education]. 华中师范大学学报 (自然科学版) [Journal of Central Normal University, Natural Sciences Series] 24(3): 396–399.

    Google Scholar 

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Acknowledgment

I would like to express my gratitude to the Institute for Advanced Study (Princeton, NJ, USA) where my research on Vietnamese mathematical treatises was conducted in 2007; I also gratefully acknowledge the support obtained in the framework of the project “Multiculturalism in Monsoon Asia” (季風亞洲與多元文化) of the National Tsing-Hua University (Hsinchu, Taiwan) throughout the work on the paper. I would like to thank Christine Proust and Alain Bernard for their encouragement and numerous suggestions, as well as the participants of the Workshop “Textes et instruments scientifiques anciens élaborés dans un contexte d’enseignement: situations, usages, fonctions” (Paris, France, December 15–16, 2008) for their comments on the first draft of the paper; my thanks also go to Richard Kennedy who patiently polished the English of the final version.

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    Alexei Volkov

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    Alain Bernard

  2. Laboratoire SPHERE UMR 7219, Université Paris 7 - CNRS, Paris, France

    Christine Proust

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Volkov, A. (2014). Didactical Dimensions of Mathematical Problems: Weighted Distribution in a Vietnamese Mathematical Treatise. In: Bernard, A., Proust, C. (eds) Scientific Sources and Teaching Contexts Throughout History: Problems and Perspectives. Boston Studies in the Philosophy and History of Science, vol 301. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5122-4_10

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