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Powell, a mathematician, theoretical physicist, logician, writer, and political commentator in the United States for all levels of mathematics, reading comprehension, SAT/ACT tutoring, science, English grammar and composition, and history to students in grades 4 through 12 including In maths, the UK is ranked 27th, slipping down a place from three years ago, the lowest since it began participating in the Pisa tests in 2000 In reading , the UK is ranked 22nd , up from 23rd I'm bilingual in Vietnamese and English in fluent writing, speaking, and listening. Richland Collegiate High School of Math Science Engineerin Associate's degree Pre-Nursing Studies. 2020 - Most international schools start recruiting in the fall, with the bulk of openings in winter and early spring. You have plenty of time to sort through job offers and pick your best location, or change your mind. For best results, we recommend to allow yourself a year to plan and execute your move. Teaching abroad is a big market. App Vay Tiền Nhanh. Mathematical Communities Published 06 August 2020 The Mathematical Intelligencer volume 42, pages 50–58 2020Cite this article 532 Accesses 1 Altmetric Metrics details Access options Buy single article Instant access to the full article PDF. 39,95 € Price includes VAT Ukraine NotesSee Văn Đạo chủ biên. Giáo sư Lê Văn Thiêm, NXB ĐHQG Hà Nội edited by N. V. Dao, Professor Lê Văn Thiêm; VNU Hanoi Publisher, Grothendieck. Mathematical life in the Democratic Republic of Vietnam. Report presented on December 20, 1967, at the invitation of the Mathematics Department of the Faculty of Science of the University of Paris; translated from the French by N. Koblitz. Available online at T. Hoa and T. V. Nhung. Vietnam Institute for Advanced Study in Mathematics. Asia Pacific Mathematics Newsletter 23 2012, 19– H. Khoai. On contemporary mathematics in Vietnam. Seki, Founder of Modern Mathematics in Japan A Commemoration on His Tercentenary, edited by E. Knobloch et al. Springer Proceedings in Mathematics & Statistics 39, pp. 375–383. Springer, Koblitz. Recollections of mathematics in a country under siege. Math. Intelligencer 123 1990, 16– Koblitz. Random Curves Journeys of a Mathematician. Springer, Thiem. Beitrag zum Typenproblem der Riemannschen Flächen. Comment. Math. Helv. 20 1947, 270– Legrandjacques. A colonial university for South-East Asia? the Indochinese university in Hanoi 1906–1945. Kyoto Review of Southeast Asia issue 22, Young Academics Voice, October 2017. Available online at Schwartz. A Mathematician Grappling with His Century. Translated from the 1997 French original by Leila Schneps. Birkhäuser, Volkov. Mathematics and mathematics education in traditional Vietnam. In Oxford Handbook of the History of Mathematics, edited by E. Robson and J. Stedall, pp. 153–176. Oxford University Press, Volkov. Argumentation for state examinations demonstration in traditional Chinese and Vietnamese mathematics. In The History of Mathematical Proof in Ancient Traditions, edited by K. Chemla, pp. 509–551. Cambridge University Press, referencesAcknowledgmentsThis article is based on a talk presented at the 12th International Conference on Mathematics and Mathematics Education in Developing Countries ICMMEDC, Vientiane, Laos, November 1–3, 2019. I was encouraged by Graeme Fairweather and some other participants to publish it. I would like to thank him for carefully reading and helping me to edit the article. Neal Koblitz also gave me useful comments. I have met Neal many times and discussed with him aspects of the development of mathematics in Duy Phượng sent me several references, and Hoàng Xuân Sính sent me a copy of the photo with Grothendieck. I also would like to thank the referee for careful reading and suggestions for further informationAuthors and AffiliationsHanoi Mathematical Institute, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307, Hanoi, VietnamLê Tuấn HoaAuthorsLê Tuấn HoaYou can also search for this author in PubMed Google ScholarCorresponding authorCorrespondence to Lê Tuấn informationPublisher's NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community” is the broadest “schools” of mathematics, circles of correspondence, mathematical societies, student organizations, extra-curricular educational activities math camps, math museums, math clubs, and more. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and should be uploaded to or sent directly to Marjorie Senechal, and permissionsAbout this articleCite this articleHoa, The Development of Mathematical Research in Vietnam at a Glance. Math Intelligencer 42, 50–58 2020. citationPublished 06 August 2020Issue Date December 2020DOI Discover the world's research25+ million members160+ million publication billion citationsJoin for free Helaine Selin Editor Encyclopaedia of the History of Science, Technology, and Medicine inNon-Western Cultures Third Edition With 2463 Figures and 138 Tables  Editor Helaine Selin Hampshire College Amherst, MA, USA ISBN 978-94-007-7746-0 ISBN 978-94-007-7747-7 eBook ISBN 978-94-007-7748-4 print and electronic bundle DOI Library of Congress Control Number 2015957805  Springer Science+Business Media Dordrecht, 1997, 2008, 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by SpringerNature The registered company is Springer Science+Business Media Dordrecht 2818 Mathematics in Vietnammodels pp. 191–225. Hillsdale, NJ Lawrence Erlbaum Associates. Joseph, G. G. 2000. The crest of the peacock Non- European roots of mathematics. London Penguin Books. Kaleva, W. 1995. The cultural context of mathematics Education development in Papua New Guinea. Papua New Guinea Journal of Education, 31, 143–149. Koch, G. 1983. The material culture of Tuvalu. Suva, Fiji Institute of Paciﬁc Studies, University of the South Paciﬁc. Laycock, D. C. Observations on number systems and semantics. Papuan languages and the New Guinea linguistic scene. Ed. S. A. Wurm. New Guinea area languages and language study. 1975. Vol. 1. Paciﬁc linguistics, series C – No. 38, 219–33. Lean, G. A. 1991. Counting systems of Papua New Guinea. Lae, Papua New Guinea Department of Mathematics and Statistics, Papua New Guinea University of Technology. MacGregor, G. 1937. Ethnology of Tokelau Islands. Honolulu, HI Bishop Museum. Me´traux, A. 1936. Numerals from Easter Island. Man, 36253–254, 190–191. Philp, H., & Kelly, M. R. 1977. Cognitive development in Papua New Guinea – some comparative data. The Australian Journal of Education, 21, 256–267. Pospisil, L., & De Solla Price, D. 1976. Reckoning and racism. The Journal of the Polynesian Society, 85, 382–383. Riesenberg, S. H. 1972. The organisation of navigational knowledge on Puluwat. Journal of the Polynesian Society, 81, 19–56. Ross, A. S. C. 1936. Preliminary notice of some late eighteenth century numerals from Easter Island. Man, 94–95. Rumsey, A. 2001. Tracks, traces, and links to land in Aboriginal Australia, New Guinea, and beyond. In A. Rumsey & J. Weiner Eds., Emplaced myth Space, narrative, and knowledge in Aboriginal Australia and Papua New Guinea pp. 19–42. Hono- lulu, HI University of Hawai’i Press. Seidenberg, A. 1962. The ritual origin of counting. Archive for History of Exact Sciences, 2, 1–40. Taylor, C. B. 1957. Hawaiian almanac. Honolulu, HI Tongg Publishing Company. Tyerman, D., & Bennett, G. 1832. Journal of voyagesand travels by the Rev. Daniel Tyerman and George Bennett, Deputed from the London Missionary Society, to visit their various stations in the South Sea Islands, China, India, & c. between the years 1821 and 1829. Boston Crocker and Brewster. Winkler, C. 1901. On sea charts formerly used in the Marshall Islands, with notices on the navigation of these islanders in general. Smithsonian Institute Report for 1899, 54, 487–508. Wolfers, E. P. 1971. The original counting systems of Papua and New Guinea. The Arithmetic Teacher,18, 77–83. Mathematics in VietnamAlexei Volkov Center for General Education and Institute of History, National Tsing-Hua University, Hsinchu, Taiwan IntroductionPrior to the late twentieth century, only a few efforts were made to locate and study the extant Vietnamese mathematical treatises. The ﬁrst attempt was made by the Japanese historian of mathematics Mikami Yoshio 三上義夫 1875–1950 who in his paper Mikami, 1934 analyzed the contents of the Vietnamese treatise Chỉ minh toán pháp 指明算法 Guide Towards Understanding of Computational Methods and provided several quotations from it; the current whereabouts of the book studied by Mikami are unknown. For the ancient and medieval Chinese personal names, terms, and titles of treatises, I use the pinyin transliteration system adopted in Main- land China and in European sinology. For Viet- namese terms, I use their original form in Chinese characters adopted in Vietnam since the ﬁrst mil- lennium AD together with their Vietnamese transliteration in Quốc ngữ 國語 system and, in some cases, with their Chinese reading in pinyin. For personal names, titles of books, and technical terms in Chinese, I provide traditional Chinese characters and replace them with their simpliﬁed versions currently used in the of China only when referring to publications in which the sim- pliﬁed versions were originally used. His quota- tions suggest that the treatise Mikami had at his disposal was either identical with, or textually close to, the extant treatise Chỉ minh lập thành toán pháp 指明立成算法 Guide Towards Understanding of the Ready-made Computa- tional Methods see below Volkov, 2013a. In 1938, the Chinese mathematician and historian of science Zhang Yong 章用 1911–1939 visited Hanoi and explored the collection of old Viet- namese books preserved in the French School of Mathematics in Vietnam 2819the Far East E´cole franc¸aise d’Extreˆme-orient. Unfortunately, soon after this visit he passed away and his ﬁndings remained unpublished. Zhang Yong’s study of the discovered materials was summarized by Li Yan 李儼 Li, 1954. General works by colonial French scholars, such as Huard and Durand 1954, pp. 120, 144, 1963, pp. 538, 540 and by modern Vietnamese authors Tạ, 1979, contained only scarce and often unreliable information on traditional Vietnamese mathematics Volkov, 2002, p. 375. The ﬁrst systematic attempt to describe the extant Viet- namese astronomical and mathematical texts was made in 1991 by Han Qi 韓琦 Han, 1991; in his work he used the partial copies of Vietnam- ese treatises made by Zhang Yong in Vietnam, while the original texts preserved in Vietnamese and French libraries remained unexplored. As for the publications on the history of mathematics in Western languages, only a short paragraph was devoted to the topic by Martzloff who drew upon secondary sources Martzloff, 1997, p. 110. Soon after that, I started publishing papers based on my study of the original mathe- matical treatises preserved in Vietnam and in France Volkov, 2002, 2004, 2005, 2006, 2009, 2012, 2013a, 2014a, 2014b. Vietnamese Mathematics An OutlineIn China, a state-run School of Computations Suan xue 算學 or 筭學 was open during the Northern Zhou 北周 dynasty 557–581, yet there are reasons to suggest that a prototype of the School already existed during the Northern Wei 北魏 dynasty 386–534 Lee, 2000, p. 515; Volkov, 2014a, p. 58. In some sources, the character suan 筭 is used instead of the character suan 算 in the titles of treatises or names of educational institutions. The original meanings of these two characters were different according to the etymological dictionary Shuowen jiezi 說文解字 by Xu Shen 許慎 AD 55?–149?, the character suan 筭 meant “counting rods,” while suan 算 meant the operations performed with them. The School was re-established during the Sui 隋 581–618 and Tang 唐 618–907 dynasties and operated, with interruptions, until the early ninth century in two metropolitan cit- ies, Chang’an 長安 now Xi’an 西安 and Luo- yang 洛陽 Li, 1933 [1977], pp. 260–264; Volkov, 2014a, p. 63.. Vietnam which at this time occupied a relatively small area in the North of the present-day Vietnam was domi- nated by various Chinese dynasties from 111 BC to AD 544 and became a province of the Chinese Empire from 602 to the fall of the Chinese Tang dynasty in 907; it remained under control of ephemeral Southern Chinese dynas- ties even longer, until 938. This means that in the ﬁrst millennium AD gifted Vietnamese students, at least formally, had the same right to be admit- ted to the metropolitan educational institutions as the candidates from other Chinese provinces, and the history of mathematics education in Vietnam prior to 938, technically, is identical with that of mathematics education in the rest of China. When Vietnam gained independence from China in AD 939, the Vietnamese state took shape based upon the blueprint of its Chinese counterpart, and a State University 國子監 M Viet. Quốc Tử Giám, Chinese Guozi jian, liter- ally “Directorate [of Education of] Sons of State”, an equivalent of the Chinese University of the Tang dynasty, was established in 1076; for the location of this institution in Hanoi, see, for example, Volkov, 2013b, p. 119, Fig. 1. There are records about the metropolitan examinations in “computations” 算 Viet. toán, Chinese suanthat took place in 1077, 1261, 1363, 1403 or 1404, 1477, 1507, and 1762, yet the contents of these examinations, with one exception, are unknown Volkov, 2013b, p. 121, 2014a, p. 68. An attempt to reconstruct the examination proce- dure is presented below. In China, the second century of the rule of the Song 宋 dynasty 960–1279 was marked by sev- eral attempts to restore mathematics education of the ﬁrst millennium. The mathematical textbooks were reedited and block-printed in 1084; a new governmental School of Computations was established in the same year. It was closed and opened again several times, and ceased to func- tion in 1120 Volkov, 2014a, p. 66. After the 2820 Mathematics in VietnamMathematics in Vietnam, Fig. 1 The block-print Cửu chương lậpthành t´ınh pháp 九章立成併法. A copy pre- served in the Institute of Han-Noˆm Studies, Hanoi call number transfer of the capital to Lin’an 安 modern Hangzhou in 1127, no attempts to restore the School are known, except the reprinting, in 1200–1213, of the set of mathematical textbooks of 1084 Friedsam, 2003; Lee, 1985, pp. 96–102; Li, 1933 [1977], pp. 273–280; Volkov, 2014a, pp. 66–68; Yang, 2003, vol. 2, p. 120. The fact that the ﬁrst mathematics examination in the independent Vietnam took place in 1077, that is, before the re-establishment of state mathematics education in China, may suggest that in the elev- enth century the Vietnamese mathematics educa- tion and examination system functioned according to the Chinese regulations of the ﬁrst millennium AD. However, the mathematical textbooks supposedly used in Vietnam at that time for mathematics instruction were lost, most likely, prior to the early ﬁfteenth century. One of the possible reasons for this loss may have been the destruction of books by ﬁre in the metropol- itan area during the military conﬂict with Champa in 1371. Champa 占城, Vietnamese Chieˆm Thành, Chinese Zhan Cheng was a state located in the Central and Southern part of present-day Vietnam; it was defeated by Viet- nam in a series of wars and ceased to exist as an independent political entity in 1832. Another reason for the loss of books may have been their massive removal from Vietnamese librar- ies and transfer to China during the Chinese occupation from 1407 to 1427. After this period, two successive orders of the newly established Leˆ 黎 dynasty prescribed to look for ancient books all over the country and to deliver them to the court, yet there is no available information concerning any mathematical books found dur- ing these campaigns. Moreover, a large number of books were destroyed or lost during the battles that took place in the capital in 1516 and 1592 Boudet, 1942, p. 233. The state of mathematics and mathematics education in Vietnam of the eleventh to the early ﬁfteenth centuries thus can- not be reconstructed on the basis of the extant sources, even though some hypotheses can be advanced, mainly on the basis of comparison with contemporaneous systems of mathematics education in China, Korea, and Japan; for an outline of these systems, see, for example, Volkov, 2014a. The last mathematics examination mentioned in historical documents took place in 1762, yet a relatively large number of the extant mathemati- cal treatises dated of the nineteenth and even of the early twentieth century suggest that the tradi- tional Vietnamese mathematics education sur- vived the French invasion which started in 1859 and existed in some form until the early twentieth century. In particular, the reign of the Emperor Minh Mệnh Minh Mạng明命 temple name Thánh Tổ 聖祖, personal name NguyễnPhu´c Kiểu 福晈 or NguyễnPhu´c ảm 福膽, 1791–1841, r. 1820–1841 was the time when traditional Chinese scholarship of the Ming 明 dynasty 1368–1644 was highly appreciated. And even though some elements of the European mathematics and astronomy may have been known to Vietnamese court astronomers in the seventeenth century directly from Western mis- sionaries or via Chinese translations of Western books, the Vietnamese treatises of the nineteenth and even early twentieth century still looked Mathematics in Vietnam 2821aresurprisingly similar to Chinese mathematical texts compiled prior to the advent of the Western mathematics to China in the early seventeenth century. On the interaction between the European missionaries and Vietnamese astronomers, astrologers, and geomancers indiscriminately referred to as “mathematicians”, see Borri, 1631, pp. 178–190, 1931, pp. 372–381; Baldinotti, 1629, p. 196; De Rhodes, 1854, pp. 111–113, 185; Volkov, 2008. The catalogs Cổ Học ViệnThụ Tich Thủ Sáchis a variant title of the well-known classic Jiuzhang suanshu 九章算術 Computational Procedures of Nine Categories also known as “Mathematical Procedures in Nine Chapters”, the Suanfa tongzong 算法統宗 Interrelated Ori- gins of Counting Methods 1592 by Cheng Dawei 程大位 1533–1606 listed as Suanfa tongzun 算法統尊 printed in the year gui-si 癸巳 of the era Wanli 萬曆, 1593, the Gougu yinmeng 勾股引蒙 Introduction for Beginners to [Methods] of Right-Angle Triangles by Chen 古學院書籍守冊 Inventory of ˙ Boo ks [Pre- Xu 陳訏 1650–1722, the collected works of served] in the Hall of Ancient Learning, NộiCác Thụ Mục 內閣書目 Catalog of Books [Pre- served] in the Inner Pavilion, NộiCác Thủ Sách內閣守冊 Inventory [of Books Preserved] in the Inner Pavilion, and Tụ Khueˆ Thụ Viện Tổng Mục 聚奎書院總目 General Catalog of the College of Advanced Scholarship of the Imperial library currently preserved in the Institute of Han-Noˆm Studies Hanoi contain the titles of a number of Chinese books on mathematics. Among them there are the treatises of the ﬁrst millennium AD Zhoubi suanjing 周髀算經Computational Trea- tise on the Gnomon of the Zhou [dynasty], Sunzi suanjing 孫子算經 Computational Treatise of Master Sun, Wucao suanjing 五曹算經 Computational treatise of Five Departments, Haidao suanjing 海島算經 Computational trea- tise [Beginning with a Problem about a] Sea Island, Xiahou Yang suanjing 夏侯陽算經 Computational Treatise of Xiahou Yang, and Qigu suanjing 古算經 Computational Treatise on the Continuation [of Traditions] of Ancient [Authors]; in the catalog Cổ Học ViệnThụ Tich Thủ Sách 古學院書籍守冊 all these treatises ˙ Mei Wending 梅文鼎 1633–1721, namely, 梅氏曆算全書 Complete Books on Calendar and Computations of Mr Mei 1723 and 梅氏叢書輯要 Compiled Essentials of the Collected Books of Mr Mei 1759, the Yang Hui suanfa zhaji 楊輝算法札記 Reading Notes on the Com- putational Methods of Yang Hui by Song Jingchang 宋景昌 dates unknown; active in the mid-nineteenth century printed in 1842, and the Zhiming suanfa 指明算法 Guide Towards Understanding of Computational Methods. The author and date of publication of the latter treatise are unknown, but it is quite possible that this book M was not identical with the aforementioned Viet- namese treatise Chỉ minh toán pháp 指明算法 studied by Mikami Yoshio, because there existed several Chinese mathematical treatises bearing the same title; for instance, the catalog of the old Chinese mathematical books Wu & Li, 2000 lists six treatises with the same title Zhiming suanfa 指明算法 published during the Ming and Qing dynasties p. 366. One more comment is in order here the catalog Cổ Học Viện Thụ Tich Thủ Sách lists the treatise Xiangjie dated in the 41st year of the Chinese Qianlong jiuzhang˙suanfa as “a block-print [made] in the 乾隆 reign period 1776, and this date may suggest that these copies were based on the edi- tions of the respective treatises preserved in the Chinese Imperial collection Si ku quan shu 四庫全書. Another group of Chinese mathematical treatises listed in these catalogs consists of later mathematical works such as the Xiangjie jiuzhang suanfa 詳解九章算法 Detailed Expla- nations of the Computational Methods of Nine Categories 1261 by Yang Hui 楊輝 1238?–1298? the Jiuzhang suanfa 九章算法 year wu-chen of the era Shaoxing of the Song dynasty” 刻宋紹興戊辰年; this year corresponds to AD 1148, which is inconsistent with the con- ventional dates of life of Yang Hui. The Vietnam- ese Imperial library also preserved a number of Chinese works on calendar and mathematical astronomy, yet it remains unknown how much the Chinese books from this impressive collec- tion were available to the Vietnamese scholars interested in mathematics and mathematical astronomy. 2822 Mathematics in VietnamThe Extant Vietnamese MathematicalTreatises This section contains an annotated and alpha- betically ordered list of the extant Vietnamese mathematical treatises currently preserved in libraries and in private collections. The book conventionally considered the oldest, the Toán pháp đạithành no. 16 below, is credited to the authorship of one Lương Theˆ´ Vinh 梁世榮 1441–1496?, a high-rank ofﬁcial of the Leˆ 黎 dynasty 1428–1789 see below. Another book no. 3 in the list was compiled in the early eighteenth century. The other dated books were published either in the nineteenth century or even in the early twentieth century. The remaining books are not mentioned in the bibliographic chapter of the ạiViệt thoˆng sử大越通史 Complete History of the Grand Viet published in 1749 by the famous Vietnamese literatus Leˆ Quý oˆn 黎惇 1726–1784?, and therefore one can be tempted to date all the remaining books as compiled not earlier than the late eighteenth century Gaspardone, 1934, p. 149; Tran, 1938, pp. 97–98. On the other hand, the mathematical contents of all these books are similar to that of Chinese mathematical treatises antedating the introduc- tion of European mathematical methods to China by the Jesuits in the early seventeenth century. 1. Bu´t toán chỉ nam 筆算指南 Compass of Handwritten [lit. “Brush”] Computations This text was authored by one NguyễnCẩn 瑾 also written as 謹 elsewhere; dates unknown and revised by Kiu OánhMậu 喬瑩懋 1854–1912, the two extant block-printed copies are dated 1909. The book is formally devoted to the explanation of the Western method of written calcula- tions the practitioners of Vietnamese math- ematics traditionally used counting instruments such as counting rods and abaci to perform arithmetical operations, but actu- ally it covers a rather wide range of topics characteristic of traditional Chinese and Vietnamese mathematics. The book contains ﬁve chapters quyển 卷. Chapter 1 is entirely devoted to detailed explanations of four arithmetical operations performed in written form and exempliﬁed with relevant prob- lems. Chapter 2 contains problems of various types, from simple division to solution of simultaneous linear equations. Chapter 3 deals with the “measuring ﬁelds,” that is, calculation of the areas of plane rectilinear and curvilinear ﬁgures, and chapter 4 is devoted to square and cube root extraction. The concluding chapter 5 deals with right- angle triangles and is mainly devoted to methods of distance surveying with one or several gnomons using the methods found in ancient and medieval Chinese mathematical treatises. The Institute of Han-Nom Studies hosts a manuscript titled Toán pháp 算法 Computational methods authored by NguyễnCẩn 謹 and revised by Kiều Oánh Mậu 喬瑩懋; the copy is dated 1909 Tran and Gros 1993, v. 3, p. 352, no. 3791. It is possible that this is a manuscript copy of the Bu´t toán chỉ nam 筆算指南. 2. Chỉminh lậpthành toán pháp 指明立成算法 Guide for Understanding of the Ready-made Computational Methods This treatise completed by Phan Huy Khuoˆng 潘輝框 in 1820 opens with a picture of an abacus which is an exact reproduction of the picture found in the aforementioned Chinese mathematical treatise Suanfa tongzong by Cheng Dawei. The picture is followed by tables representing powers of 10, monetary units, units of length, weight, and volume. The following pages contain 32 diagrams of various “ﬁelds’ shapes,” that is, various rectilinear and curvilinear plane ﬁgures; similar shapes can be found in the Suanfa tongzong, even though the list found in the Vietnamese treatise differs in spots. The treatise contains a “mock examination problem” with a solution and explanations; it was translated and discussed in the work of Volkov 2012. 3. Cửu chụơng lậpthành t´ınh pháp 九章立成併法 Ready-made Methods of Addition of Nine Categories Mathematics in Vietnam 2823As the conventional dates of publication of the inspected editions suggests namely, 1713 and 1721, this is the earliest extant Vietnamese mathematical text here the word “earliest” refers to the date of physical production, printing, of the text and not to the date of its actual compilation. The treatise is relatively short and written partly in Noˆm and partly in classical Chinese. There exist block-printed editions of the treatise Fig. 1 and one manuscript version of it was altered by a later editor, arguably, in the early nineteenth century. The catalogue of Tran and Gros 1993, v. 1, pp. 375–376 suggests that the treatise was compiled by one Phạm Hữu Chung 范有鍾, literary pseu- donym 字 Phu´c 福; however, the name of the author is speciﬁed in one of the editions as Phạm Hữu Chổng ? 范有偅 and the pseudonym as Phu´c Cẩn 福謹, while another source provides the pseudonym of the author of the treatise as phonetically identical yet written with a different character Cẩn 福菫. The treatise is not subdivided into chapters and consists of short sections devoted to dis- cussions of various topics such as multiplica- tion table, calculation of areas of plane ﬁgures, proportional distribution, and operations with common fractions, often in versiﬁed forms. The treatise also contains a number of mathe- matical problems presented in the traditional format “problem – answer – solution.” 4. Cửu chương lậpthành toán pháp 九章立成算法 Ready-made Computational Methods of Nine Categories The author and actual time of compilation of this treatise are unknown; there exists a manuscript copy of it made in 1899 and pre- served in the National Library of Vietnam Hanoi. The text introduces the names of the “large numbers,” a multiplication table, and computation of areas of rectilinear and curvilinear ﬁgures. 5. Cửu chương toán pháp 九章算法 Computational Methods of Nine Catego- ries; another title Cửu chương toán pháp lậpthành 立成 Ready-made Computational Methods of Nine Categories. The catalog of the collection of the Insti- tute of Han-Nom Studies Hanoi lists two manuscript copies of this treatise produced in 1882. It opens with the section titled “Nam toán”南算, “Southern Vietnamese computations” containing explanations of arithmetical operations, a list of names of the large powers of 10, and the multiplication table 9 x 9. Next comes a long “historical” section in Noˆm containing explanations of basic mathematical operations and a legend- ary history of Vietnamese mathematics orig- inating from the traditional Chinese mathematical curriculum. The following sec- tions contain rhymed descriptions of various mathematical methods, such as extraction of square roots, reduction of common fractions, calculations needed for conversion of units of measure and monetary units, calculations of areas of plane ﬁgures, simple and weighted distribution. In this part, short rhymed descriptions are accompanied by rel- evant detailed computational procedures. The next section contains a large number of mathematical problems of various kinds, M including several problems of indeterminate analysis, such as, for example, the classical problem concerning the unknown number of rabbits and roosters having together 36 heads and 100 feet. 6. ạithành toán học chỉ minh 大成算學指明 Guidance and Explanations for the Great Compendium of Mathematical Learning The treatise is authored by one Phạm Gia Kỷ 范嘉紀, a governmental functionary dates unknown, and edited by Phạm Gia Chuyeˆn 范嘉瑼 b. 1791, a doctoral tieˆ´n s˜ı 進士 degree holder since 1832, a scholar from the state University Quốc Tử Giám. The treatise preserved in the Han-Noˆm Insti- tute is a meticulously handwritten text with- out a cover, yet the title and the names of the compilers are mentioned on the ﬁrst page of the copy. The dates of compilation and of publication are not speciﬁed. The treatise contains 20 subsections; some of them ﬁt into traditional Chinese classiﬁcation of “Nine categories,” for example, section 2824 Mathematics in Vietnam16 is devoted to the method of double false position. However, certain sections appear rather original; for example, section 1 con- tains a classiﬁcation of rectilinear solids containing 15 different types, to the best of my knowledge, not found in the extant Chi- nese treatises. 7. ại thành toán pháp 大成算法 Computational Methods of Great Compendium The cover page of this manuscript treatise preserved in the Institute of Han-Noˆm Hanoi is missing, and the title Cửu chụơngtoán pháp 九章算法 that appears on its sec- ond page refers to the multiplication table and thus hardly can be the title of the treatise. Another title, ạithành toán pháp 大成算法, appears on p. 5a of the manuscript, yet it remains unclear whether this is the original title of the treatise. The text is writ- ten in classical Chinese yet contains a num- ber of paragraphs in Noˆm explaining certain procedures. It is not subdivided into chapters and contains descriptions of arithmetical operations, measure units, and a list of prob- lems devoted to the calculation of areas of plane ﬁgures, simple and pondered distribu- tion, calculation of land taxation, currency conversion; square and cube root extraction procedures are not found among the topics discussed. The latter fact suggests that this treatise is different from the Cửu chươngtoán pháp 九章算法 of 1882, despite the fact that these two books are listed together in Tran and Gros 1993. 8. & 9. Lập thành toán pháp 立成算法Ready- made Computational Methods An anonymous manuscript of unknown date in classical Chinese bearing this title is preserved in the Institute of Han-Noˆm Stud- ies Hanoi. It contains a standard introduc- tory part devoted to arithmetical operations and a multiplication table followed by a long section devoted to the computation of the areas of plane ﬁgures accompanied by rele- vant diagrams. The remaining part is devoted to mathematical problems of various types. The library of the Institute of History Hanoi Mathematics in Vietnam, Fig. 2 Page 75 of the Thống toˆng toán pháp 統宗算法Picture courtesy of the National Library, Hanoi hosts another manuscript cataloged under the same title Lập thành toán pháp; however, a cursory analysis shows that these two texts are not identical. The actual title of the latter treatise and the date of compilation remain unknown, since the ﬁrst pages of the manu- script are badly damaged while the last pages are missing. 10. Thống toˆng toán pháp 統宗算法 Counting Methods of Interrelated Origins This is a manuscript of unknown date authored by one Tạ Hữu Thụờng 謝有常 dates unknown and preserved in the National Library Hanoi Fig. 2. Its title makes an obvious allusion to the aforementioned Chinese treatise Suanfatongzong 算法統宗 Interrelated Origins of Counting Methods 1592 by Cheng Dawei 程大位 1533–1606. Indeed, cer- tain parts of the Chinese treatise are quoted verbatim, as, for example, the versiﬁed Mathematics in Vietnam 2825rules of calculation of areas of plane ﬁgures, the problem of two walkers, and so on. Nevertheless, the compiler of the Viet- namese treatise considerably modiﬁed sec- tions of the Chinese prototype and added a large number of problems not found in the original, adapted the Chinese original to the Vietnamese measure units, and provided his explanations in Noˆm. 11. Toán điềntrừ cửu pháp 算田除九法 Nine Methods of Division for Computation [of Areas of] Fields A manuscript copy of this anonymous treatise of unknown date is preserved in the Institute of Han-Noˆm Studies Hanoi. The treatise, as its title suggests, is devoted to calculation of areas of plane ﬁgures. The text is written in classical Chinese, yet con- tains commentaries in Noˆm. 12. Toán học cách tr´ı 算學格致 Exploration [of Things] and Extension [of Knowledge] in the Science of Computations The ﬁrst page of the manuscript preserved in the Institute of Han-Noˆm Studies Hanoi is missing and the treatise was listed under the bogus title Toán pháp 算法 Computational methods in the catalogue of Tran and Gros 1993. However, the actual title of this treatise is reproduced at the beginning of each chapter; it reads Toán học cách tr´ı Hoàng Phong Dụ gia thụ ch´ınh bổn 算學格致黃豐裕家書正本, that is, “The rectiﬁed copy of the manuscript of the Exploration [of Things] and Extension [of Knowledge] in the Science of Computa- tions [preserved] in the family of Mr Hoàng Phong Dụ.” The name of the actual compiler s thus is are unknown. According to the table of contents, the book originally contained four chapters and an appendix discussing matters related to construction works. However, the extant copy contains only chapters 1–2 and the ﬁrst four pages of chapter 3; the rest is irretrievably lost. The opening chapter is devoted to numerical nota- tion, execution of arithmetical operations with a counting instrument an abacus is mentioned but not pictured, and metrological units. It also contains a number of problems illus- trating these topics, such as distribution of a given sum of money among a given number of persons. Chapter 2 is devoted exclusively to the calculation of the areas of rectilinear and curvilinear plane ﬁgures accompanied with detailed explanations. Chapter 3 is devoted to extraction of square and cube roots. The contents of chapters 4 and 5 cannot be restored on the sole basis of their titles provided in the table of contents. 13. Toán học để uẩn 算學底蘊 Secrets of the Science of Computation Since the ﬁrst page of the treatise is miss- ing, the name of the author is unknown. The title Toán học để uẩn under which the treatise is found in catalogs is apparently determined solely on the basis of the line “Table of Con- tents of the Secrets of the Science of Com- putation” 算學底蘊目錄 found on the ﬁrst page of the only extant manuscript copy of the treatise preserved in the Institute of Han-Noˆ m Studies. The date of the compila- tion is uncertain either, but, since the name of the reign era of the emperor Gia Long 嘉隆 M r. 1802–1820 is mentioned in the manu- script, one can suggest that the treatise was compiled no earlier than 1802. The book contains 6 chapters devoted to the following topics arithmetical operations; computation of areas of plane ﬁgures; extraction of square and cube roots; a brief presentation of the methods according to the traditional Chinese classiﬁcation 九數 “Nine [types] of numerical [computations]”; construction works; and a mathematical theory of music. A cursory analysis of the book reveals that it contains a large number of quotations from the aforementioned Chinese treatise Suanfa tongzong 算法統宗 Interrelated Ori- gins of Counting Methods 1592 by Cheng Dawei 程大位. 14. Toán học ta^m pháp 算學心法 MentalMethods of the Science of Computation The cover page of the manuscript copy of this treatise hosted in the Institute of Han-Noˆ m studies is missing and its title is restored on the basis of the title of the preface 2826 Mathematics in Vietnamsigned by Hoàng Phong Dụ 黃豐裕 in 1850. The table of contents lists ﬁve chapters oper- ations with numbers, calculation of areas of plane ﬁgures, square and cube root extrac- tion, computations related to construction works, and computations related to land tax- ation. However, the chapters are not clearly separated and their titles are inserted in the body of the text, which suggests that the original layout was altered by the copyists. The treatise also contains items not corresponding to the ﬁve announced topics, such as methods of remote surveying. 15. Toán pháp 算法 Computational Methods The actual title of the book remains unknown; the bogus title “Computational Methods” was apparently given to a manu- script with missing ﬁrst pages by the copyists or by librarians on the basis of its contents. The author and the date of the com- pilation are not known either. A manuscript copy and a microﬁlm of the book are pre- served in the Han-Noˆm Institute call number The book is not subdivided into chapters; it contains a sequence of 250-odd problems devoted to topics including calcu- lation of surfaces, applications of the right- angle triangles, extraction of square and cube roots, among others. A cursory analysis shows that the book is but a copy-and-paste style compilation made on the basis of the Chinese mathematical treatise Suanfatongzong 算法統宗 Interrelated Origins of Counting Methods 1592 by Cheng Dawei 程大位. For instance, problems devoted to the calculation of areas of plane rectilinear and curvilinear ﬁgures, the related computa- tional procedures as well as the related geo- metrical diagrams of the Vietnamese treatise are found in chapter juan 3 of the Suanfa tongzong, problems devoted to root extrac- tion and solution of polynomial equations were copied from chapter 6 of the Chinese treatise, and so on. 16. Toán pháp đạithành 算法大成 Great Com- pendium of Mathematical Methods The date of compilation and the name s of the compilers are unknown. Mathematics in Vietnam, Fig. 3 The cover page of the Toán pháp đạithành 算法大成preserved in the collection of the Institute of Han-Noˆ m Studies, Hanoi A treatise with the same title was credited to the authorship of Lương Theˆ´ Vinh 梁世榮 1441–1496?, a functionary of the Leˆ 黎 dynasty 1428–1789, in his biographies see below; however, the authorship and the actual date of compilation of the treatise preserved in the Library of the Institute of Han-Noˆ m Studies Hanoi are unknown, and some parts of the treatise can make one doubt that the book was compiled in the ﬁfteenth century. There exist two manuscript copies of the treatise, one produced prior to 1934 Fig. 3 and its copy made in 1944. The treatise is not subdivided into chap- ters and contains 138 problems, if one counts the problems per se as well as several pro- cedures that do not correspond to any partic- ular problems; the presence of those most probably resulted from loss of parts of the text. Some geometric problems are not Mathematics in Vietnam 2827explicitly stated, but introduced with a dia- gram of a ﬁgure with given dimensions. The problems found in the treatise are related to such popular topics of traditional Chinese mathematics as the partitioning proportional distribution, proportions, “rule of three,” rule of double false position, square root extraction, calculation of the areas of recti- linear and curvilinear plane ﬁgures, calcula- tion of volumes of solids, conversion of monetary and metrological units of various types, indeterminate analysis, and “numeri- cal divination.” The treatise also contains a large and relatively independent section devoted to land taxation for more details, see below; see also Volkov, 2002. 17. Toán pháp đềcu o ng 算法提綱 Presentation of the Key Points in Computa- tional Methods The title Toán pháp đề cụơng 算法提綱 of this treatise hosted in the National Library Hanoi is not the actual title of the book. The ﬁrst pages of the manuscript is are miss- ing and Toán pháp đề cụơng is but the sub- title of its ﬁrst remaining section. The ﬁrst part of the treatise contains a very detailed explanation of the operations performed with the abacus and provides numerous diagrams representing conﬁgurations of beads on the abacus the instrument featured is the stan- dard Chinese 2+5 beads abacus with 11 bars. The remaining part of the manuscript is a compilation featuring nine categories of Chi- nese traditional mathematics most likely made, as a cursory analysis suggests, on the basis of the Chinese treatise Suanfa tongzong算法統宗 by Cheng Dawei 程大位. 18. Toán pháp ký diệu 算法奇妙 Mysteries of Computational Methods The title of this anonymous treatise was established by Tran and Gros 1993 most likely on the basis of the ﬁrst page of the single manuscript copy preserved at the Insti- tute of Han-Noˆm Studies in Hanoi. However, both the table of contents and the opening part of the treatise following it specify its title as Tập thành chụ gia huyễ n ngh、itoán pháp 集成家幻儀算法 Complete collection of the counting methods [using] magic devices of all schools [of mathematics]. The date of compilation of the treatise is uncertain; how- ever, since it contains an appendix featuring the taxation norms adopted during the reign of the Emperor Minh Mang 明命 r. 1820–1841, one can conclude that the received manuscript copy was made no ear- lier than 1820 it remains unclear whether there existed a printed edition of it. The treatise is not subdivided into chapters, even though the title page possibly added later suggests that the treatise originally contained “three chapters united in one.” The received manuscript contains a short introductory sec- tion providing information about the numer- ical notation, units of measures, and other auxiliary topics; the main section contains a number of problems, tables, and explanations related to the traditional topics such as cal- culation of areas of plane ﬁgures, operations with common fractions, and extraction of square and cube roots. 19. Toán pháp quyển 算法卷 Volume on Com- putational Methods M This mathematical work signed by one ỗ ức Tộ 杜德祚 was completed in 1909. The book opens with a multiplication table and a set of prescriptions for operating a counting instrument, and contains problems related to proportional distribution, calculation of vol- umes of solid ﬁgures, calculation of areas of plane ﬁgures, extraction of square roots, cal- culation of the harvest collected from a ﬁeld with a given surface, etc. 20. Tổng tụ chư gia toán pháp đạitoàn 總聚家算法大全 Great Compendium of the Com- putational Methods of All Schools The manuscript copy preserved in the Han-Noˆm Institute is incomplete; it contains only chapters 3 48 problems and 4 32 prob- lems, and the ﬁrst two pages of an Appendix containing one problem. The problems deal with weighted distribution, calculation of volumes, and other topics relevant mainly to construction works, mass labor, and other issues related to administrative duties. It is possible that the treatise contains problems 2828 Mathematics in Vietnamproposed at state mathematics examinations chapter 3 opens with a mention of such an examination. 21. Tru`ng d´ınh Toán học chỉ nam ta^n bieˆn 重訂算法指南新編 New Edition of the Re- established [text of the] Compass for Methods of ComputationsA manuscript copy of this anonymous treatise is preserved in the library of the Institute of History Hanoi. It contains prob- lems dealing with calculation of areas of plane ﬁgures, extraction of square and cube roots, applications of right-angle triangles to remote surveying including the method involving two gnomons and its justiﬁcation with diagrams, use of gnomons for land surveying, as well as other methods. The text mentions the system of measuring units established during the reign of the Emperor Gia Long 1802–1820 and is written in clas- sical Chinese, thus appearing to be compiled in the ﬁrst half of the nineteenth century; however, it contains a large number of Western-style calculations written with Ara- bic numerals inserted into the text, thus giv- ing the impression that its authors or at least its later editors or copyists were well familiar with the Western methods, and the extant copy was produced much later. 22. Ý Trai toán pháp nhất đắc lục意齋算法一得錄 A Record of What Ý Trai Understood Correctly in Methods of Computation This treatise was completed in 1829 by NguyễnHữu Thận 有慎 [=NguyễnÝ Trai] b. 1736?-? who occupied high positions in the government and stayed in China between 1809 and 1813 where he obtained books related to astronomy and communicated with local astronomers. For more informa- tion on the author and the contents of the treatise, see Volkov, 2014b, pp. 254–255. The compilation consists of eight chapters devoted to the basic arithmetical operations performed with a counting instrument, metrology, computation of areas of plane ﬁgures, weighted distribution, “extraction of square roots” in modern terms, solution of quadratic equations, properties of the right-angle triangles especially those related to their use for remote surveying, “extrac- tion of cube roots” numerical solution of cubic equations, and other topics of the traditional mathematical curriculum. The book contains the author’s explanations concerning the numerical operations to be performed, the etymology of the mathemati- cal terms, the rationale of the procedures, and thus may have been used as the mathematical manual for professional training of govern- mental astronomers and mathematicians. My paper Volkov, 2014b contains annotated translations of excerpts related to the method of weighted or proportional distribution. The Structure of a Mathematical Treatise The Example of the Toa´n pha´p đại thành There are two manuscript copies of the treatise Toán pháp đạithành, both found in the Han-Nom Institute, Hanoi; their call numbers are and When manuscript was pro- duced is unknown but it certainly happened prior to 1934, while the date when the copy was made 1944 is written on its front page; a comparison of the copies suggests that the MS is a copy of Neither manu- script has a preface or a postface, or any other data which would suggest the date when the trea- tise was compiled or would specify the identity of the authors. The name of the presumed author “Doctor Lương Theˆ´ Vinh” is written only on the ﬁrst page of each manuscript next to the title; however, it is possible that this page was added later Gaspardone, 1934, p. 149, n. 1. The treatise is compiled in the traditional “Chinese” way, as a collection of problems with numerical answers given along with the proce- dures algorithms for their solution. There are also several procedures that do not correspond to any particular problems; most probably this is the result of loss of parts of the text containing the corresponding problems and answers. The total number of problems in the treatise amounts to 138. Some geometric problems are not explicitly Mathematics in Vietnam 2829stated, but introduced with a diagram of a ﬁgure with given dimensions. In one case neither numerical data nor a problem accompanied an algorithm, yet the algorithm may well be a frag- ment from the famous Chinese “Sunzi remain- ders problem.” The text of the treatise can be subdivided into eight parts Part 1 problems 1–35 contains problems devoted to partitioning, and, in particular, to division. Part 2 problems 36–42 contains prob- lems devoted to the calculation of the areas of plane ﬁgures a square, a rectangle, a ﬁgure approximated by the area of a trapezium, a cir- cle, and a segment of a circle. Part 3 problems 43–69 contains problems devoted to propor- tions, the rule of three, and the rule of double false position, as well as to rather simple cases of multiplication and division. This part also includes a method for calculating the height of an object when the height of another object and the length of the shadows of both objects are given. Part 4 problems 70–85 contains prob- lems devoted to root extraction and to an auxil- iary algorithm used for the conversion of monetary units of one type into another. Part 5 problems 86–93 is a sequel to Part 3. The reader is asked to solve problems on the calcu- lation of interest and on multiplication and divi- sion. However, there is a problem devoted to the calculation of the volume of a solid ﬁgure and a fragment on divination. Part 6 problems 94–131 is related to various subjects, such as calculations of the areas of various ﬁgures. Here one ﬁnds such shapes as rectangles, circular seg- ments, a “horn of the bull,” circles, “drums,” ellipses, rings, an “eye-lid” or “eye-brow,” the intersection of two circles, an isosceles triangle, a trapezium, a rectilinear ﬁgure com- posed of several adjacent trapezia, a quadrilat- eral with four given sides, and the ﬁgure formed by two adjacent squares. The remaining prob- lems in this group are devoted to the extraction of square roots, calculation of the volumes of rectilinear solids, and to the conversion of met- rological units. Part 7 does not contain mathe- matical problems; this is a large independent text devoted to land taxation. Part 8 problems 132–138 embraces various problems devoted to “numerical divination,” the calculation of the height of a tree when the length of its shadow is given, a rhymed solution to a problem of indeterminate analysis, and the calculation of the area of a quadrilateral. Several “earmarked” mathematical problems and methods found in the Toán pháp đại thành studied in order to suggest the possible origins of the contents of the treatise were discussed in the work of Volkov 2002. The analyzed topics included 1 the counting instruments to be used; 2 the 9 x 9 multiplication table featured in the treatise; 3 the lists of the so-called large numbers; 4 rhymed algorithms for computation of areas; 5 problems of remote surveying; 6 problems related to numerical divination; and 7 problems on indeterminate analysis. The results of the comparison of the abovementioned mathematical methods, problems, and instru- ments with their Chinese counterparts can be summarized as follows for more details, see Volkov, 2002 1. The lack of explicit references to the abacus M in the treatise suggests that either the Com- pendium was compiled before Vietnamese mathematicians were acquainted with any Chinese books devoted to abacus calcula- tions, or it was compiled later solely on the basis of Chinese and Vietnamese mathemati- cal books written prior to 1573 when the ﬁrst extant Chinese mathematical treatise devoted especially to the use of the abacus, Xu Xinlu’s 徐心魯 Counting Procedures for Pearls on a Plate Panzhu suanfa 盤珠算法, was published. 2. The text of the treatise does not contain any information conﬁrming that it was indeed authored by Lương Theˆ´ Vinh, the ﬁfteenth- century literatus and governmental ofﬁcer. 3. It is not impossible, however, that the book may well have been a compilation made exclusively on the basis of mathematical trea- tises compiled in China prior to the late ﬁf- teenth century and later available in Vietnam. However, the compilation of the treatise involved a substantial “localization,” that is 2830 Mathematics in VietnamfLthe adaptation of the problems and methods to local measure units, currency, tax system, as well as to the names of speciﬁc local objects mentioned in the problems plants, drugs, kinds of food, animals. 4. The seeming similarity between certain methods in the Compendium and in the Chi- nese treatise Suanfa tongzong 算法統宗 1592 by Cheng Dawei 程大位 does not nec- essarily mean that the Vietnamese text was compiled on the basis of the book of Cheng. The similarity can be explained by the fact that Cheng himself based his manual on numerous mathematical texts compiled in the thirteenth to sixteenth centuries available to him, ﬁrstly and most importantly, the treatises of Yang Hui 楊輝 ﬂ. ca. 1275 and Wu Jing 吳敬 ﬂ. ca. 1450. It is not impossible that the compilers of the Compendium also had access to these treatises, or to other older Chinese treatises containing similar materials and later lost. The preliminary exploration of the contents of the Vietnamese treatise Toán pháp đại thành presented in Volkov 2002 thus did not permit a clear picture of its origins. An exploration of the materials related to the life and activity of its presumed author, Lương Theˆ´ Vinh, thus appeared necessary to establish the origins of the book. The obtained results of this exploration Volkov, 2005, 2006 are summarized in the following section. Lương Theˆ´ Vinh The Case of a “Mathematical Agiography”籙 Records of Great Examinations Through Generations. All these texts present rather short descriptions of Lương’s ofﬁcial career and spec- ify his birthplace and his ofﬁcial duties within the Hàn l^am 翰林 Academy. The second and third treatises mention a mathematical book he wrote, yet provide different titles for the book. The sec- ond and the third biographies mention Lương’s diplomatic activities, the details of which, how- ever, remain unspeciﬁed. The second group of the extant Lương’s biog- raphies focuses primarily on the supernatural cir- cumstances of his life and death. A short description of these biographies is found in Volkov 2005, and the translation of one of them is published in Volkov 2006. The prelim- inary analysis of the legends suggests that the early legends of Lương were created in two nonintersecting social circles that can be dubbed “Palace” and “Village”, yet both groups of leg- ends portrayed him as possessing supernatural powers or divine origin. One can conjecture that the reason he became associated with mathemat- ics may have been related to his ofﬁcial duties during his lifetime such as, for example, his par- ticipation in diplomatic activities and in military operations against the Cham, probably related to his work in cartography, brieﬂy mentioned in some of his biographies. His legendary capacities of “counting” and “measuring” in a broader sense in this case would have merged with his established status of supernatural being and thus may have made him the patron saint of profes- sional mathematicians by the early eighteenth century Volkov, 2005. The temple devoted to Lương Theˆ´ Vinh 梁狀元祠 located in his native village Cao HươngNam inh Province, Vụ Bản district hosts the The biographies of Lương can be provisionally statue o ˙ ương, his portrait, details of his ofﬁcial subdivided into two categories, the “historical biographies” and the “legendary accounts.” The “historical biographies” of Lương Theˆ´ Vinh are found in the collections of biographies a˘ng khoa lục登科錄 Records of Successful Exam- inees by Nguyễn Hoa˜n 俒 1712–1791 et al., in the manuscript entitled 登科錄本 Manuscript copy of the Records of Successful Examinees, and in the treatise Lich đạiđạikhoa lục歷代大科 ˙costume boots and hat, and a number of Impe- rial edicts related to the establishment and func- tioning of the temple Volkov, 2005. The ofﬁcial portrait of Lương preserved in the temple Fig. 4 depicts him as a state functionary without making any visual reference to the miraculous circum- stances of his birth and life; only one inscription mentions a mathematical work he presumably authored. Mathematics in Vietnam 2831Mathematics in Vietnam, Fig. 4 The portrait of LươngTheˆ´ Vinh preserved in his temple, Cao Hương village Conclusions The available materials do not suggest any reli- able picture of traditional Vietnamese mathe- matics prior to the ﬁfteenth century, and no information is available concerning the transfor- mation of the system of mathematics education that most likely resulted from the Chinese occu- pation of Vietnam in the early ﬁfteenth century. However, numerous sources suggest that math- ematics and mathematical astronomy occupied in Vietnamese society a position similar to that of its Chinese counterpart in the ﬁrst millennium AD and remained a discipline supported by the state until the early twentieth century. The extant Vietnamese mathematical trea- tises were produced, most likely, during the period from the early eighteenth to early twenti- eth centuries on the basis of older Vietnamese mathematical treatises which presumably were based upon their Chinese counterparts of the Ming dynasty 1368–1644. A preliminary investigation of the extant treatises suggests that the Vietnamese mathematicians were not aware of the elements of the contemporaneous Western mathematics introduced by Western missionaries in China in the seventeenth and eighteenth century, or were not willing to men- tion them in their texts. Instead, the Vietnamese mathematical treatises appear rather similar to the corpus of the “practical” or “popular” Chi- nese mathematical treatises of the late Yuan 1279–1368 and Ming dynasties dramatically different from the high-level mathematical texts devoted to the higher degree polynomial algebra of the late Song 960–1279 and early Yuan dynasties. In Ming dynasty China, tradi- tional mathematics was transformed into an applied discipline used for practical ends by low-level state ofﬁcials, merchants, and artisans, while remaining the subject of the research conducted by isolated scholars without the ideo- logical and material support of the state. It was no longer one of the subjects of the state exam- inations, as it had been during the Tang 618–907 and Song dynasties. In Vietnam, on the contrary, the discipline remained embedded in the framework of the old Chinese-style state education, employed traditional didactic prac- tices, and was linked to the bureaucratic hierar- chy via the system of state mathematics M examinations Volkov, 2014b. ReferencesBaldinotti, J. 1629. Histoire de ce qui s’est passe´ e`s royaumes d’Ethiopie.. . [par le P. Emanuel Almeida] et de la Chine.. . [par le P. Emanuel Diaz] avec une briefve narration du voyage qui s’est fait au royaume du Tunquim nouvellement descouvert, tire´esdes lettres adresse´es au R. P. ge´ne´ral de la compagnie de Je´sus. Traduites de l’italien. Paris S. Cramoisy. Borri, C. 1931. Relation de la Cochinchine. Bulletin des amis du vieux Hue´, 183–4, 285–402. Borri, C. [Christoforo] 1631. Relation de la nouvelle Missiondes Peres de laCompagnie de Iesus av Royavme de l’italien dv Pere ChristoﬂeBorri Milanois,qui fut vn des premiers quientrerent en ce Royaume. Par lePereAntoinede la Croix, de la mesme Compagnie. A Rennes, chez Iean Hardy. Boudet, P. 1942. Les archives des empereurs d’Annam et l’histoire annamite. Bulletin des amis du vieux Hue´,293, 229–259. Cadie`re, L., & Pelliot, P. 1904. Premie`re e´tude sur les sources annamites de l’histoire d’Annam. Bulletin de l’E´colefranc¸aise d’Extreˆme-Orient, 4, 617–671. 2832 Mathematics in VietnamDe Rhodes, A. 1854. Voyages et missions du pe` re Alexandre de Rhodes de la Compagnie de Je´sus en la Chine et autres royaumes de l’orient. Paris Julien, Lanier et Cie. Friedsam, M. 2003. L’enseignement des mathe´matiques sous les Song et Yuan. In C. Despeux & C. Nguyen Tri Eds., E´ducation et instructionen ChineVol. 2, pp. 49–68. Paris/Louvain, Belgium Editions Peeters. Gaspardone, E. 1934. Bibliographie Annamite. Bulletin de l’Ecole franc¸aise d’Extreˆme orient., 341, 1–173. Han, Qi 韩琦. 1991. Zhong Yue lishi shang tianwenxue yu shuxue de jiaoliu 中越历史上天文学与数学的交流 The interaction between Chinese and Vietnamese astronomy and mathematics in the past. Zhongguo keji shiliao 中国科技史料 122, 3–8. Huard, P., & Durand, M. 1954. Connaissance du Viet- Nam. Paris/Hanoi, Vietnam Impre´merie Nationale/ Ecole Franc¸aise d’Extreˆme-Orient. Huard, P., & Durand, M. 1963. La science au Viet Nam. Bulletin de la Socie´te´ des e´tudes indochinoises, Nou- velle Serie 383–4, 531–558. Hucker, C. O. 1985. A dictionary of ofﬁcial titles in imperial China. Stanford, CA Stanford University Press. Lee, T. 1985. Government education and examinations in Sung China. Hong Kong, China/New York The Chinese University Press/St. Martin’s Press. Lee, T. 2000. Education in traditional China A history. Leiden, The Netherlands Brill. Li, Yan 李儼. 1933 [1977]. “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]. [Originally published in Kexue zazhi 科學雜誌 Science Maga- zine, vol. 17 1933, no. 10, pp. 1545–1565; reproduced in Li Yan 李儼, Zhong suan shi luncong 中算史論叢 Collected works on the history of Chi- nese mathematics. Beijing Zhongguo Kexueyuan, 1954–1955, vol. 4, pp. 238–280, and in Li Ziyan 李子嚴 =Li Yan 李儼, Zhong suan shi luncong 中算史論叢, Taibei Taiwan shangwu, 1977, vol. 4, pt. 1, pp. 253–285. In this paper the reference to page numbers are given according to the Taiwanese reprint of 1977.] Li, Yan 李儼. 1954. Zhang Yong jun xiuzhi Zhongguo suanxue shi yishi 章用君修治中國算學史事 The heritage of Mr. Zhang Yong’s work on the resto- ration of the history of Chinese mathematics. Li, Yan. Zhong suan shi luncong 中國算學史論叢 Collected papers on the history of Chinese mathematics Vol. 1, pp. 135–146. Taibei, Taiwan Zhengzhong shuju. Martzloff, 1997. A history of Chinese Springer. Mikami, Yoshio 三上義夫. 1934. Annan-no ichi sansho-ni tsuite 安南の一算書に就て On one math- ematical book from Annam [=Vietnam]. Gakko sugaku 學校數學, 14, 3–11. Siu, & Volkov, A. 1999. Ofﬁcial curriculum in traditional Chinese mathematics How did candidates pass the examinations? Historia Scientiarum, 91, 87–99. Tạ, Ngọc Liễn. 1979. Vài ne´t về toán học ở nuớc ta thời xua Some Features of Vietnamese Mathematics in Pre-modern Times. T、imhiểu khoa học ky~ thuật trong lichsử Việt Nam The Study of Science and Techno˙logy in Vietnamese History pp. 289–314. Hanoi, Vietnam Nhà Xuất Bản Khoa Học Xa˜ Hội Social Sciences Publishing House. Trần, N., & Gros, F. Eds.. 1993. Catalogue des Livres en Han Noˆm [bilingual edition]. Hanoi, Vietnam Nhà Xuất Bản Khoa Học Xa˜ Hội Social Sciences Publish- ing House. Tran, V. G. 1938. Les chapitres bibliographiques de Le- qui-Don et de Phan-huy-Chu. Bulletin de la Socie´te´ desEtudes Indochinoises Saigon, Nouvelle Se´rie, 131, 7–217. Volkov, A. 2002. On the origins of the Toan phap dai thanh Great Compendium of Mathematical Methods. In Y. Dold-Samplonius, J. W. Dauben, M. Folkerts, & B. van Dallen Eds., From China to Paris 2000 years transmission of mathematical ideas pp. 369–410. Stuttgart, Germany Franz Steiner Verlag. Volkov, A. 2004. History of ideas or history of textbooks Mathematics and mathematics education in traditional China and Vietnam. In Horng et al. Ed., Pro- ceedings of Asia-Paciﬁc HPM 2004 Conference His- tory, culture,and mathematics education in the new technologyera, May 24–28, 2004. Taichung, Taiwan Department of Mathematics Education, National Taichung Teachers College, pp. 57–80. Volkov, A. 2005. Traditional Vietnamese mathematics The case of Lương Theˆ´ Vinh 1441-1496? and his treatise Toan phap dai thanh Great Compendium of Mathematical Methods. In U Kyi Win Ed., Tradi- tions of knowledge in Southeast Asia Part 3, pp. 156–177. Yangon, Burma Myanmar Historical Commission. Volkov, A. 2006. State mathematics education in tradi- tional China and Vietnam Formation of “mathemati- cal hagiography” of Lương Theˆ´ Vinh 梁世榮 1441–1496?. In Trinh K. M. & Phan V. C. Eds., Confucianism in Vietnam pp. 272–309. Hanoi, Viet- nam Nhà Xuất Bản Khoa Học Xa˜ Hội Social Sci- ences Publishing House. Volkov, A. 2008. Traditional Vietnamese astronomy in accounts of Jesuit Missionaries. In L. Saraiva & C. Jami Eds., History of mathematical sciences, Portu- gal and East Asia III The Jesuits, the Padroado and East Asian Science 1552–1773 pp. 161–185. Sin- gapore, Singapore World Scientiﬁc. Volkov, A. 2009. Mathematics and mathematics educa- tion in traditional Vietnam. In E. Robson & J. Stedall Eds., Oxford handbook of the history of mathematics pp. 153–176. Oxford Oxford University Press. Volkov, A. 2012. Argumentation for state examinations Demonstration in traditional Chinese and Vietnamese Mathematics in Vietnam 2833Mathematics of the Hebrew People 2833 mathematics. In K. Chemla Ed., The history of math- ematical proof in ancient traditions pp. 509–551. Cambridge Cambridge University Press. Volkov, A. 2013a. 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 E. Knobloch, H. Komatsu & D. Liu Eds., Seki, founder of modern mathematics in Japan A commemoration on his tercentenary. Springer pro- ceedings in mathematics & statistics Vol. 39, pp. 149–172. Tokyo, Japan Springer. Volkov, A. 2013b. Astrology and hemerology in tradi- tional Vietnam. In Robert & P. Marsone Eds., Stars and fate Astrology and divination in East Asia Extreme-Orient Extreme-Occident 35 pp. 113–140. Paris PUV. Volkov, A. 2014a. History of mathematics education in Oriental Antiquity and Middle Ages. In A. Karp & G. Schubring Eds., Handbook on the history of math- ematics education pp. 55–70, 79–82. New York Springer. Volkov, A. 2014b. Didactical dimensions of mathemat- ical problems Weighted distribution’ in a Vietnam- ese mathematical treatise. In A. Bernard & C. Proust Eds., Scientiﬁc sources and teaching contexts throughout history Problems and perspectives Boston studies in the philosophy and history of sci- ence, Vol. 301, pp. 247–272. Dordrecht, The Nether- lands Springer. Wu, Wenjun 吴文俊 series ed. & Li, Di 李迪 volme ed.. 2000. Zhongguo shuxue shi daxi 中国数学史大系 Encyclopaedia of the History of Chinese Mathematics. [Supplementary volume 2] Zhongguo suanxue shumu huibian 中国算学书汇编 Comprehensive catalog of Chinese mathematical treatises. Beijing Shifan Daxue Publishers. Yang, Xuewei 杨学为 Ed.. 2003. Zhongguo kaoshi shi wenxian jicheng 中国考试史文集成 Collected mate- rials on the history of examinations in China. Beijing Gaodeng jiaoyu Publishers. Mathematics of the Hebrew People Tony Le´vy A “Hebrew mathematical text” is any text or work whose language is Hebrew usually written in Hebrew characters, and whose content is mathematical in a narrow sense, that is, does not include astronomy apart from relevant mathematical sections, astrology, or calendar calculations. Apart from a few passages that are to be found in biblical and postbiblical rabbinical literature and which are relevant to the history of mathe- matics number words and fraction words, prac- tical rules of geometry, the oldest mathematical tract in Hebrew is the Mishnat ha-Middot, by an unknown author. This tract gives practical rules for the measurement of areas and volumes, and then deals with the measurements middot of the Tabernacle erected by the Jews in the desert. It has been recently shown that its composition was probably inﬂuenced by the geometrical part of al-Khwārizmī’sAlgebra. This tract remained unknown to most medieval Jewish scholars and its Hebrew mathematical terminology was of no consequence. Two famous scholars are central in the elev- enth and twelfth centuries in Spain Abraham bar Hiyya ca. 1065–1145 and ▶Abraham ibn Ezra 1092–1167. This period saw the actual birth of Hebrew mathematics. Abraham bar Hiyya, also called Savasorda latinized from the Arabic ṣāḥib al-shurta, ﬂourished in Barcelona in Christian Spain, but was probably educated in the Arabic kingdom M of Saragossa. Bar Hiyya wrote books in Hebrew on mathematics, astronomy, astrology, and phi- losophy. He clearly indicated that his Hebrew compositions were written for Jews living in southern France in Hebrew, EreṣSarfat who were unacquainted with Arabic scientiﬁc culture and unable to read Arabic texts. Bar ḥiyya can thus rightly be considered the founder of Hebrew scientiﬁc culture and language, and speciﬁcally the father of Hebrew mathematics. We know of two mathematical compositions by Bar ḥiyya the extant parts of a scientiﬁc encyclopedia and a geometrical compilation. The ﬁrst of these books, Yesodey ha-Tevuna u- Migdal ha-Emuna The Foundations of Science and the Tower of Faith, is presumably an adap- tation from some unknown Arabic composition; the geometrical and arithmetical parts are extant. The study of its content sheds some light on eleventh century mathematical literature in West- ern Islamic lands and its diffusion. The ḥibbur ha-Meshiḥa we ha-Tishboret The Composition on Geometrical Measures enjoyed ResearchGate has not been able to resolve any citations for this publication. Alexei VolkovThe 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. Alexei VolkovIn 1934 Yoshio Mikami 1875-1950 published a paper devoted to the Vietnamese mathematical treatise Guide [towards] Understanding of Calculational Methods Chi Minh Toan Phap. His analysis of several topics discussed in the treatise representation of numbers with counting rods, format of multiplication table, generic problems of various categories, etc. allowed him to advance hypotheses concerning the origin and the time of compilation of the treatise. The book studied by Mikami nowadays is not available. In the present paper the author examines Mikami's work and provides a description of the Vietnamese mathematical treatise Chi Minh Lap Thanh Toan Phap by Phan Huy Khuong preface 1820 textually close to that investigated by MartzloffFor the English language edition, this completely unique book of Martzloff has been fully revised and updated. It includes many new recent insights and illustrations, a new appendix on Chinese primary sources and a guide to the to the bibliography. From the reviews "This book ranks with the most erudite Asian publications, and is the most informative and most broadly informed on its topic in any language." N. Sivin, China Quarterly "⋯ crammed with insights, cautionary tales and a great deal of information about current research ⋯ will surely become a standard reference for students, teachers, and researchers alike", J. N. Crossley, Annals of Science "⋯ a truly scholarly and balanced exposition ⋯ a book that the reviewer believes belongs in the library of every university or college, as well as in that of every individual seriously interested in the history of Chinese mathematics", B. L. McAllister, ZBfM "Martzloff History demonstrates clearly that while the Chinese were adept in applying their mathematics to a host of practical problems, including astronomy and engineering as well as commercial transactions, they also paid attention to algorithmic techniques, methods of calculation, geometric constructions, and even certain purely logical problems. But above all, what sets this book apart from the usual histories of mathemathics in any language, Chinese or Western, of any period or country is its emphasis first on context, then on content, in describing the long history of Chinese mathematics ⋯. It is primarily the question of context that Martzloff approaches directly. Perhaps the greatest contribution his book makes is the chance it offers to consider issues of cultural context as significant, determining factors in the history of mathematics. Thus, Martzloff tries to get inside the Chinese mind, to explain how and why mathematics developed as it did in China, and often in ways strikingly different from its Western counterparts. Although he does not always account for these differences, he succeeds admirably in describing them, which results in a refreshingly rich sense of its evolution as well." Joseph W. Dauben, Historica Mathematica 20, 1993. © Springer-Verlag Berlin Heidelberg 1997, 2006. All rights reserved. Alexei VolkovThe paper is focused on astrology that is, observations and mantic interpretations of positions of celestial bodies, in particular those resulting in solar eclipses and hemerology predictions made on the basis of calendar in traditional Vietnam. The author provides a short description of the history of institutions dealing with astronomical observations and with their astrological interpretations, and discusses the extant Vietnamese materials relevant to the topic. Attention! This job posting is 803 days old and might be already filled. Other Information Location Hà Nội Date Posted 2021-03-31 Category Academic English Communication / General English IELTS / TOEIC / TEFL Teaching Math / Science Teaching Job Type Full-time Are you willing to accept and support qualified teachers currently outside of Vietnam? No Nationality of candidate American, Australian, British, Canadian, Irish, New Zealand, South African Experience 0 - 1 year Candidate Requirements Bachelor's Degree, TEFL certification, Master Degree in TESOL or equivalent Where is the employer located Hà Nội Salary 2,300 net USD/month Description Job requirements Native English speaker. Bachelor’s degree or higher in the Science, Math, English/ESL, educational field is plus. Have a qualification in English Language Teaching or PGCE or TEFL/TESOL/CELTA or equivalent. Job description Teacher will be arranged to teach in a private schools with the following position Primary and secondary students. Subject of teaching Mathematics Working location Ecopark Urban area. Working hours 20 teaching hours per week from Monday through Friday, daytime, morning and afternoon. BENEFITS Salary 2,300 net USD/month Work permit Will be covered, Lunch free Lesson plan provided and teaching assistant in the class. Text Book Provided Get All New Job Notification Create email alert and get first jobs that arrived. I originally moved to Vietnam to pursue a dream of living in Asia and teaching English, but over the past few years, my role as a teacher has morphed from teaching English as a second language to teaching Science and Maths. I actually studied Biology at university, so when the opportunity came along to utilise my degree and teach science and maths at a Vietnamese public school, you can bet I jumped at it! This is an opportunity that many people are not aware of, so I thought I’d lay out all the details here for anyone who also has a background in science or maths that they want to make use of. Where can you teach science and maths? I taught science and maths through a company called EMG Education, which supply teachers for public schools in both Ho Chi Minh City and Hanoi. As a secondary teacher at EMG, you specialise in either the science and maths program or the English program. Primary teachers teach all three subjects; science, maths and English. To teach within the secondary science and maths program you must meet the same requirements that every other teaching English position in Vietnam asks for, with the addition of a background in science or maths. A science/math background is not required for the EMG primary program. By teaching the science and math program at EMG, you are essentially teaching kids how to speak English through specialist content. This is called content integrated language learning CLIL and it looks like it might be the future of language learning. I loved my time teaching science and maths in Vietnam. I felt more challenged and engaged in the lesson content than I did teaching English, which I think shone through in my teaching. EMG follows the UK science and maths curriculum, with content similar to GCSEs. I was always so impressed by the knowledge and ability of the students – they were keeping up with the UK curriculum despite English being their second language. EMG provides full lesson plans, materials and textbooks, so there is actually little prep required outside of familiarising yourself with the materials. I taught grade 6, the first year of secondary school. For some of my students, it was their first time learning about science, let alone studying it in English. My 100+ students showed curiosity and excitement in every class; It was beautiful to see their wonder at learning about the world through science for the first time. We studied a wide variety of topics across all three sciences, including plant biology, the elements, geology, forces and space. Most science lessons included an interactive experiment which was so fun to do together. Maths was more challenging for me to teach as I’m pretty terrible at maths and have completely forgotten everything I studied at school. But challenges are good and by the end of the year, I had grown to love my maths classes. As with science, we studied a range of topics, some that my students already understood in Vietnamese and others that were brand new to them. Decimal numbers, fractions, prime numbers, 3D shapes and graphing are some of the topics I remember teaching. Obviously, I taught a lower secondary grade, so the topics were fairly simple but they do get more and more complex the higher the grade you teach. Normally EMG will assess your background knowledge and place you accordingly, so if you have a masters in mathematics, you’re probably going to be asked to be a higher grade specialist math teacher. Teaching for EMG Education As already mentioned, I loved my time teaching science and maths. I left this position for two main reasons; personality clashes and lifestyle, not because of the teaching or work environment. Pros of EMG Education ✅ Because you only have 4 different classes, you end up spending a lot of time with the kids each week which allowed us to build up a strong bond with your students. ✅ The content is fun and engaging to teach, and students are engaged and interested. ✅ The offices where teachers prep classes are super social which is perfect if you are new to the city. ✅ The academic coordinators who escort teachers to classes have almost native level English a higher ability than I found at my language centre’s equivalent position. Plus they were all amazing people and I have lots of fond memories working together. Cons of EMG Education ❌ I received little-to-no developmental support or teaching guidance during my time at EMG. This was probably just my case, as I came to EMG with a lot of teaching experience and was quickly put into their top pay bracket so I guess they thought I was doing a good job already. It does make me wonder if teaching at public schools is better suited to more experienced teachers who don’t need as much support. ❌ You are required to stay in the office until 5pm, even if your classes finish at 330pm. For me, this was a lot of dead time when I could have been working on the blog I ducked out most days, coming back to clock out at 5pm. ❌ It was challenging to take vacation time during the semester. It’s not impossible but it is discouraged, with an emphasis put on taking time off in the summer months when classes don’t need to be covered. ❌ When I worked at EMG, the science and maths department was very male-dominated, and at that, I worked with some very immature individuals. The Lowdown on teaching at EMG Most secondary math and science teachers have 8 lessons a week; 4 science and 4 maths. Plus two free blocks to prepare, so a total of 10 blocks. Working hours are scheduled between 8am – 5pm; most of my classes started at 730am in the morning and then again around 2pm. Most class sizes range from 30-35 students, always with an assistant in the room. The largest class size I heard of at EMG was 37 students. ✍ For more information on teaching English in Vietnam, check out my Complete Guide to Teaching in Vietnam. Launched the largest online school teaching English through Mathematics and Science in Vietnam Online course summary With a large number of students, iSMART Online School is the largest online school in Vietnam that provides intensive English training through Mathematics and Science. In the 2021 – 2022 school year, the school plans to recruit 16,000 more elementary school students across the country for courses. Students can learn and improve English – Mathematics – Science with a team of experienced Vietnamese and foreign teachers. In the next 5 years, iSMART Online School aims to provide an international standard quality educational environment on a digital platform to 1 million primary and secondary school students. iSMART Online School provides training programs with options Independent learning with an account, Online interactive class with a teacher live class. The program content is built closely with the Kemdikbud program to foster, strengthen knowledge and improve students’ linguistic thinking skills. The curriculum at iSMART Online School meets the accreditation standards of the Education Research Institute of Ho Chi Minh City University of Education. HCM. The digital lecture system won the 2020 Sao Khue Award for superior technology products in the field of Education and Training. In addition to mastering communication skills, iSMART Online School students have the ability to think logically in English and knowledge of mathematics and/or natural sciences, helping them to easily transition to bilingual programs and later international economics. Specifically, iSMART Online School students can continue their education to America’s online Global Ivy School to receive an American baccalaureate diploma. Students enjoy the lesson Students have a clear learning path according to each academic year, each academic year has 2 terms. Classes are small, with homeroom teachers who regularly monitor student progress. Parents will receive teacher evaluations to understand their child’s learning progress. Throughout the year, iSMART Online School students will be able to participate in many exciting extracurricular activities such as competitions English Champion, Scientist Squad, Australian Math Olympiad – AMC, Math and Science Festival… iSMART Online School is run by the Board of Directors and a team of experts from iSMART Education and the EQuest Group including Principal, Mr. Jacques Souliere – more than 25 years of experience in education in many countries, Ms. Annabelle Vultee – former CEO of EF Education First, Mr. Bach Ngoc Chien – Deputy General Director of iSMART Education… Mr Jacques Souliere – Principal of iSMART Online said “Students get the foundation and basic English skills when they study at the Vietnamese Ministry of Education and Training’s English program. And this foundation will help them learn English better, through learning English. Mathematics and Science subjects. Only special English is taught can help students learn sufficient skills to study and then work in an international environment.” iSMART Online School is currently enrolling students in grades 1 to 5 for the 2021-2022 school year, fall semester. Tuition fees from 750,000 VND/ online account/ year, 650,000 VND/ month for online interactive classes with foreign and Vietnamese teachers. iSMART Online School is the online version of iSMART Education – a unit that teaches English through Mathematics and Science based on digital lectures in more than 400 schools across the country with over 100,000 students enrolled in the 2021-2022 school year. You are reading the article Launched the largest online school teaching English through Mathematics and Science in Vietnam at – Source Mathematics Education plays a key role in the ongoing education reform in Vietnam, which commenced with the renewal of the curriculum and textbooks from primary to secondary and high schools. In the world, Didactical Situations in Mathematics and The Realistic Mathematics Education have been widely and effectively applied in the Netherlands, America, France, Indonesia, etc. This article presents some cultural characteristics of Mathematics Education in Vietnam and the results of initial research on these two theories, providing some models of teaching situations and examples for pilot implementation, initial survey the ability to apply Didactical Situations in Mathematics in Vietnam, providing some suggestions for the renovation of Mathematics curriculum and textbooks in Vietnam at may be subject to copyright. Discover the world's research25+ million members160+ million publication billion citationsJoin for free Journal of Physics Conference SeriesPAPER • OPEN ACCESSRealistic mathematics education RME and didactical situations inmathematics DSM in the context of education reform in VietnamTo cite this article Nguyen Tien Trung et al 2019 J. Phys. Conf. Ser. 1340 012032View the article online for updates and content was downloaded from IP address on 25/10/2019 at 0144 Content from this work may be used under the terms of the Creative Commons Attribution licence. Any further distributionof this work must maintain attribution to the authors and the title of the work, journal citation and under licence by IOP Publishing LtdInternational Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi mathematics education RME and didactical situations in mathematics DSM in the context of education reform in Vietnam Nguyen Tien Trung1, Trinh Phuong Thao2, Tran Trung3 1 Editor, Vietnam Journal of Education, 4 Trinh Hoai Duc, Hanoi, Vietnam 2 Teacher, Thai Nguyen University of Education, Vietnam 3 Assoc. Prof., Senior Teacher, Vietnam Academy for Ethnic Minorities, DreamTown-COMA6 urban area, road 70, Nam Tu Liem district, Hanoi, Vietnam E-mail 1nttrung 2trantrung Abstract. Mathematics Education plays a key role in the ongoing education reform in Vietnam, which commenced with the renewal of the curriculum and textbooks from primary to secondary and high schools. In the world, Didactical Situations in Mathematics and The Realistic Mathematics Education have been widely and effectively applied in the Netherlands, America, France, Indonesia, etc. This article presents some cultural characteristics of Mathematics Education in Vietnam and the results of initial research on these two theories, providing some models of teaching situations and examples for pilot implementation, initial survey the ability to apply Didactical Situations in Mathematics in Vietnam, providing some suggestions for the renovation of Mathematics curriculum and textbooks in Vietnam at present. 1. Introduction Mathematics is a science that arises, develops from real life and serves as a practical tools to solve reality problems. Therefore, it is necessary that mathematics teaching be linked with the reality in society as well as in learner‟s life. Hence, how mathematical teaching is linked to practice, what extent is appropriate in the context of mathematical education reform in Viet Nam, what adjustments are required in mathematics teaching are the questions that need investigating and resolving. At present, the Ministry of Education and Training of Vietnam is organizing a major educational reform, commencing with the renovation of the curriculum and textbooks from primary to secondary and high school. It is projected that in principle, there will be only one general education program in the whole country and possibly a range of textbooks, approved under the regulations of the Ministry of Education and Training. The general education curriculum after 2018 will be designated in the aim of developing learners‟ competencies. As explained by experts and the Ministry of Education and Training, the current program was designed in the direction of content development. In the context of general education curriculum reform, Mathematics curriculum also needs to be renovated towards learners‟ competency development. The researchers believe that this innovation should be implemented comprehensively in terms of objectives, programs, textbooks and teacher training. This study focuses on the Theory of Didactical Situations or the Didactical Situations in Mathematics DSM and the Realistic Mathematics Education RME in order to propose some implications for the current reform of the mathematical education program in Vietnam. International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi Mathematics teaching in Vietnam in the context of education reform Mathematics Curriculum and Mathematics Textbook after 2018 MoET Maths Cirriculum Outcomes [10] Mathematics Education aims to develop students core qualities; general competences, and mathematical competence with the core components of including competences of mathematical reasoning, mathematical modeling, Maths problem solving, mathematical communication, competence to use mathematical tools and media. Mathematics education also develops key knowledge and skills and creates opportunities for students to experience and apply Mathematics to real life. Mathematical education connects mathematical ideas, Mathematics and other disciplines, and Mathematics and real life. Mathematics education is conducted in many subjects such as Mathematics, Physics, Chemistry, Biology, Technology, Informatics, Experimental Activities, etc., in which Mathematics is the core subject. In this study, we focus on proposing some implications for the development of the Mathematics curriculum in general, in the context of education reform in Vietnam. However, it is advisable to discuss the current teaching context in Vietnamese schools. Systems of Vietnam Education Vietnam education system is shown in the following diagram Figure 1. Vietnam education system The role of Curriculum and Textbook Teachers and students mainly use textbooks as the core materials in the classroom in teaching Mathematics Mathematics textbook are divided into several chapters containing lessons and a mathematics lesson often has some formal mathematical definitions, theorems, regulations or formulae [8]. There is one series of mathematical textbooks in school despite the fact that students live in different regions [8]. However, there are some sets of books that are used exclusively in some elementary schools, or in some provinces, such as books for ethnic minority students translated into ethnic minority languages, Maths book for students in some international schools based on the international curriculum, in parallel with the current textbook. Recently in the Math textbooks, there have been more real life related exericses, which is just somehow related to real life problems rather than actual real life ones. It is common that problems in Math textbooks are generated by modelling real life problems, in stead of examining the actual real life ones. High School Education Grade 10 to 12, Vocational Education Grade 6 to 9 Basic Education Grade 1 to 5 Vocational Education International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi Classroom Organization Teachers often spend much less time on teaching concepts and theories, mainly focusing on teaching rules, methods, and problem solving skills. This is primarily because current assessment tools mainly focus on testing basic knowledge and skills of the lessons in textbooks through solving exercises. At present, in order to approach the new cirriculum, in the test of some schools, some Departments of Education and Training have included Math questions that require students‟ application to other subjects, to reality or real life related situations in end-of-term, end-of-year or entrance exams at all levels. According to Le Tuan Anh [8]. students usually have to take part in some extra-lessons to keep up with the curricula or pass examinations. Mathematics lessons in the textbooks as well as examinations in schools offer few examples or applications relating to real life or real world [3]. Vietnamese students often struggle to apply mathematical knowledge in reality [8]. In spite of some changes in the content of mathematic textbook, a formula or a theorem is often presented as follow + Step 1 Content of a formula or a theorem; + Step 2 A proof of the formula or theorem; + Step 3 Application of the formula or theorem in some pure mathematics examples. Similarly, a concept is often performed as follows + Step 1 Definition; + Step 2 Some examples of the concept; + Step 3 Characteristics [8]. Although there have been various changes in the practice of mathematics teaching these days, some following situations are still common 1 The teacher mainly imparts content and knowledge, pupils learning by the examples; 2 The majority of teachers generally prefer explanations, lectures and samples with frequent incidental questioning; 3 The teachers are not monitored, they do not help pupils to create problems and „occupy‟ new knowledge [3]. 4 Mathematics teachers usually use „training for exams‟ method to help their students carefully practice forms of problems which usually appear in examinations; 5 One „real goal‟ of teaching mathematics in school is to help students score higher on examinations [8]. The second situation greatly affects Mathematics teaching and learning. A session usually lasts 45 minutes, while a lesson from one to three or four hours. It is up to teachers to appropriately adjust the teaching content up to 30% different to the original subject teaching schedule and lesson plans. However, the motivation for innovating teaching content and techniques is not high and popular so teachers tend to closely comply with the prescribed teaching content distribution according to the regulations of the Ministry of Education and Training. Mathematics teachers usually use pieces of chalk, one blackboard, ... and students have the specific desk in the classroom for semester or school year [8]. Teachers sometimes use overhead projectors, computers, beamers, videos and other tools for their teaching in case of the good teacher contest. In the primary classroom, contrary to the description of the secondary classroom that “students do not have the chance to play games, which can help them to learn mathematics” [8], students often play at least one game in each Mathematics lesson. At present, more and more primary and secondary students want to participate in the Maths Competition in English language organized by national and international organizations, for example, International Kangaroo Math Contest Violympic online International Mathematics Assessments for schools etc. 3. Research questions and method The research questions are  Is it possible to combine and apply the DSM and RME theories in the process of reforming Mathematics education in Vietnam?  Can Vietnamese students learn Mathematics based on models and examples of applying these two theories in Mathematics teaching? To answer the two above questions, we studied about DSM and RME, developed some models of Maths teaching situations by combining the research results on the two theories, proposed some International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi some teaching situations relevant to the Vietnamese cultural and practical context. The research was then conducted using the proposed teaching situations in order to give some initial recommendations for the application of the two theories in the process of reforming Mathematics education in Vietnam. These situations were employed to teach some groups of students 140 students at 12th grade in some high schools in some provinces including Bac Giang, Thai Nguyen, Hai Duong, Hanoi. The findings help us to make an initial evaluation of the effects or obstacles in promoting mathematical education in practice in the context of Vietnam. To access the student's compentence and readiness to learn Mathematics in practice and by practice, in each situation, we provided a work sheet with the given tasks for students, which were then collected and evaluated in terms of ability to recognize and identify the situational tasks and Maths problems; Maths modeling compentence, the ability to solve the modelled problems and compentences to apply knowledge and make the conclusion in the real life contexts. The statistics of the number of students who could solve the tasks and the degree of completion of the tasks allow us to find out the outstanding tendencies and difficulties in teaching Maths in connection with real life among Vietnamese high school students. The analysis of some typical answers to the assigned tasks partly helps to determine the level of students‟ task achievement and the difficulties in certain contexts. Subsequently, it helps to identify the cause of those difficulties. Hopefully, we then may propose suitable recommendations for teaching Mathematics in the current context of Vietnam. 4. A brief overview about Didactical Situations in Mathematics and Realistic Mathematics Education Some basic concept in Didactical Situations in Mathematics The theory of Didactical Situations is stated by the psychologist, educationalist Guy Brousseau. Subsequently, a variety studies created a school called Didactical Situations in Mathematics. In this school, knowledge is constructed in various forms. According to Guy Brousseau [6], stated knowledge is usually “hidden the “true” functioning of science, which is impossible to communicate and describe faithfully from the outside, and replaces it with an imaginary genesis”, and, “to make teaching easier, it isolates certain notions and properties, taking them away from the network of activities which provide their origin, meaning, motivation and use. It transposes them into a classroom context. Epistemologists call this didactical transposition.”, and to have effective teaching, the production and teaching of mathematical knowledge requires an effort to transform this knowledge into institutionalized knowledge, a depersonalization and a decontextualization that tend to blot out the historical situations which had presided over their appearance. There are two main important concepts adidactical situation and didactical situation. In the process of teaching, the teacher need to provoke the expected adaptation in her students by a judicious choice of problems that she puts before them. These problems, chosen in such a way that students can accept them, must make the students act, speak, think, and evolve by their own motivation. Between the moment the student accepts the problem as if it were her own and the moment when she produces her answer, the teacher refrains from interfering and suggesting the knowledge that she wants to see appear. Not only can she do it, but she must do it because she will have truly acquired this knowledge only when she is able to put it to use by herself in situations which she will come across outside any teaching context and in the absence of any intentional direction. Such a situation is called an adidactical situation. Each item of knowledge can be characterized by a or some adidactical situations which preserves meaning. In the adidactical situation, teacher‟s specific intentions are hidden and students can function without teacher intervention. We can say that in the learning process, students face to face the adidactical situation with the support of teacher is the context of didactical situation. Discussing the learning process, Brousseau said that Knowing mathematics is not simply learning definitions and theorems in order to recognize when to use and apply them [6]. And the work of International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi is imagining and presenting to the students situations within which they can live and the knowledge will appear as the optimal and discoverable solution to the problems posed. And these are three types of situations called situation of action; situation of formulation or situation of communication; situation of validation. These situations is presented clearly in the famous example the game “the race to twenty” which created by Guy Brousseau. The Situation of action lays the essential foundation for the explicit models and formulations which follow. The Situation of action provide feedback to the student on which to base, and against which to test, his models. In general, formulation occurs in Situations where the student has a certain amount of information, but either needs more information than she can come up with on her own or does not have the means of taking action on her own, and in order to proceed must communicate with other members of the class. If the groups become argumentative, the next Situation may develop while the group planning sessions are going on. In any case, it will do so in the following one. This is the Situation of validation. Realistic Mathematics Education The Realistic Mathematics Education RME developed by the Freudenthal Institute is also known as “real-world mathematics education” [15]. RME aims at enabling students to apply mathematics. In RME, this connection to reality is not only recognizable at the end of the learning process in the area of applying skills, but also reality is conceived of as a source for learning mathematics. Just as mathematics arose from the mathematization of reality, so learning mathematics has to originate in mathematizing reality. Van den Heuvel-Panhuizen, M. [17]. Even in the early years of RME, it was emphasized that if children learn mathematics in an isolated fashion, divorced from their experiences, it will quickly be forgotten and the children will not be able to apply it [4]. “In RME, mathematics is viewed as a human activity which connects mathematics to the reality. Reality refers to mathematics that is relevant to everyday situations and problem situation that are real in student‟s mind” [9]. And according to Lu Pien Cheng, the real-life context problem refer to problems embedded in real life situations that have no ready-made algorithm [9]. According to Freudenthal, mathematics was not the body of mathematics knowledge, but the activity of solving problem and looing for problems, and, more generally, the activity of organizing matter form reality or mathematical matter – which called “mathematizing” [4]. And he clarified what mathematics is about “There is no mathematics without mathematizing” [4]. So the teacher need to find out the context, create the context which support student to construct mathematics knowledge. There are some suggestions for teachers find and create contexts for mathematics teaching context in history of mathematic; context in real life primary students‟ life games, shopping, saving and using money, film,...; social issues traffic, weather forecast, lottery, ...; integrated education mathematics in Physical, Chemistry, Informatics Technology, etc.. It is possible to point out some crucial principles in the studies of Mathematics teaching in the light of RME  + Activity Principle The learner is considered as an active subject in the teaching process whose their activity is the key factor to the outcome of this process. Therefore, the best way to learn Maths is by solving Maths problems;  + Reality Principle. Learners must be able to apply Mathematics knowledge to solve practical problems and mathematics education should start from meaningful practical situations with learners to give them opportunities to cooperate those meanings into the mathematical structures in their minds.  + Level principle emphasizes cognitive development through various levels of mathematical learning from non-mathematical contexts involving knowledge, through symbols, diagrams, to pure mathematics content of knowledge. Models are very important as a bridge between informal experiences, the mathematical context involved, and pure mathematical knowledge. The models here can be understood as mathematical modeling. International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi + Intertwinement principle Learners are placed in a variety of situations in which they may perform various types of tasks intertwined reasoning, calculating, statistics, algorithms conducting, etc., using a lot of Mathematics knowledge and tools from different disciplines, even other sciences.  + Interactivity Principle encourages interpersonal and group activities to create opportunities for individuals to share their skills, strategies, discoveries, ideas, etc. with other learners. In return, they can benefit from others for cognitive and personal development.  + Guidance Principle is described as a process of guided reinvention in mathematics instruction. Specifically, teachers need to design scenarios or situation or context that are potentially rich in activity, of which the implementation can create meaningful cognitive leaps for learners. 5. Some recommendations for teaching mathematics in the direction of RME and DSM Renovating the maths teaching program in direction of RME and DSM by designing appropriate teaching situations From the conceptual and theoretical analyses of the above-mentioned mathematical schools, we believe that in their classrooms, teachers can innovate the teaching programs by designing teaching situations as follow. The teaching situations employed here are not necessarily derived from the DSM perspective. Every teaching situation, designated by combining the approach of the two theories, fulfils the following requirements Each situation is a dual context consisting of two interlocking contexts a knowledge context designed by the teacher the context in which the students handle a performance task, and the knowledge discovering requirement may not be explicit; and the second context is a classroom setting with teacher - student interaction with cultural relevance and adaptability. Each situation must include all the basic types of situations as described in the DSM situation of action, situation of communication and situation of validation. Each situation is designed to be either started or terminated in practice, in one of the four following type of situations. Figure 2. Some types of Maths teaching situations Situation for the application of concepts in the real context Situation modeled by personal mental activity Situation as envolving the social circumstances Situation for constructing new concepts or theorems in internal mathemtics International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi Examples of some teaching situations and assessments of students' compentences through teaching situations Some example of teaching situations. Situation 1 Model 2. Your school is organizing a sport games, and you are assigned to work as a coach of triathlon. The athletes are required to swim and run from point A to point B is as shown in the figure below. The swimming speed is m/s, the running speed is m/s and the distance between the two banks AH = 200 m, HB = 1000 m. Advise the athlete the best strategy for the situation. Figure 3. The athletes are required to swim and run from point Perform the two following tasks  Task 1. Describe possible ways to perform the athlete's task.  Task 2. What is the best way to perform the task? Why? Situation 2 Model 2. The water wheel is a device of Vietnamese farmers in the mountainous North Westhern region. Its shape is formed by a reel, which is a large wheel with the five meter diameter. The rotating axis is made of solid bamboo trunk with hundreds of "spokes", forming a stable frame for the water reel. The reel rim is about 80 cm wide, with the nests in place to halt the water to creat a thrust of the treadmill, and the inclined tentacles tied up with an angel of about 30 degrees. Connected to the water wheel is often the gutters leading back to local houses or fields, often made from the half-spilt bamboo trunk. Each water has 40 bamboo tentacles, each tube is 60 cm in length, 20 cm in diameter, of which the shell thickness is about 1 cm. Figure 4. water wheel Let's do the following two tasks  Task 1 Explain how the water wheel in the above figure can collect water?  Taks 2 How many rounds does the water wheel rotate to collect enough water for the pool with 3 m3 in capacity? International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi 3 Model 4. Here is the rules and prizes of Vietlot a new form of digital lottery in Vietnam. Rules 1. Playing one line of numbers Players select 6 out of 45 numbers from 01 to 45 to form a line of numbers to play. 2. Playing multiple lines of numbers  Players choose 5 numbers from 01 to 45 bao 5, the sixth number will be selected randomly by the software system from the remaining 40 numbers, which will form 40 lines of numbers. Compare your lines with the winning one in the selected draws to determine the prize.  Players select 7 numbers bao 7 to 15 numbers bao 15 or 18 numbers bao18 from 01 to 45. Then, the software system helps players create all 6-digit combinations from the selected numbers to form the lines of numbers to play. 3. Playing in multiple draws Players select the draw and are entitled to play in up to 6 consecutive draws. In each draw, there are 4 prize tiers and there is 1 draw in each prize drawing session to select a winning line of numbers composed of 6 number from 01 to 45. Prizes  Playing one line of numbers Players select 06 numbers to form a line of numbers. Each draw ticket costs 10,000 VND.  Playing multiple lines of numbers Draw ticket cost 10,000 VND per play, multiplied with the number of selected lines. Table 1. Rules and prizes of Vietlot a new form of digital lottery in Vietnam Projected prize proportion Minumun 12 billions VND and added value Note  is the matching number with the winning line, in any order. In the case that a winner's tickets win multiple prizes, the winner will only be awarded with the highest prize. In case there are multiple winners of special prize, the special prize is shared in proportion with the winners' ticket values. As for the First to the Third Prize, the prize value is calculated by the number of draws each draw ticket costs 10,000 VND multiplied with the prize value in each draw. Question 1. Calculate the probability of winning special prize and other prizes when playing Vietlot. Question 2. Analyze the advantages of the two current lottery types the traditional National and Provincial Lottery and Digital Lottery Vietlot. Advise lottery players the more favourable type of lottery and the optimal playing. Question 3. Study some international types of lottery that you know and compare with the lottery types in Vietnam in terms of prizes, winning ratio and other aspects. Question 4. Why do people like playing lottery? Are playing lottery and lotto betting Lô-Đề is an important and even indispensable human‟s need? Why did the lottery company launch the digital lottery Vietlot in the South of Vietnam? Question 5. Select and do one of two following tasks International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi If you are a lottery advocate, write a paragraph that analyzes its advantages reminds players about the pitfalls objectively and scientifically; 2. If you are an opponent of lottery playing, write a paragraph that analyzes the disadvantages and advantages objectively and scientifically. Question 6. If you were an investor, which lottery company would you choose invest between these two type of abovementioned lottery? Findings and discussion Situation 1 Task 1. All the students were able to describe some of the strategies to carry out the given task. However, all the proposes solutions included swimming from A to H then running to B; swimming from A to C and then running to B, in which the point C was identified mainly on the drawing without any special features; or swimming straight from A to B. Task 2. All the test takers failed to fulfil this task. Most of the assignments submitted were either incompleted or incorrect using their backdground knowledge of planar geometry Figures 5., 6., 7., 8. below. In other words, many of the student's assignments focused on composing the images to work out the shortest distance. As a result, students neglected the spped when calculating the swimming or running time. Thus, students can not perform the modelling task, which means they failed to transform the situational task into a Math problem. Figure 5. Figure 6. Figure 7. Figure 8. Moreover, two proposed solutions to the task were revealed included either figuring out the shortest path by drawing the diagram first and then calculating the required time in that scenario Figure 9., or pointing out some possible scenarios in the diagram, then calculating the time under each circumstance, then selecting the option with the shortest time Figure 10.. Figure 9. Figure 10. It is shown that all 12th graders in this survey were not be able to model the situations, and of course failed to solve the problems in the given context. In addition to the above mentioned reasons, their teachers also said that students were not familiar with such Maths tasks due to the lack of proper practice with that task type. Moreoever, students are not motivated enough to solve those practical Maths problems as these problems are not included in important examinations. International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi 2 Task 1. Many students could not explain how water could be collected. A clear and well-structured explanation as in figure 11. below is quite rare. Figure 11. Task 2. Many students can not relate or refer to calculating the capacity of water in the inclined tube rather than the filled tube. Thus, many students couldn‟t explain how to collect the water because they made mistakes by calculating the full capacity of the tube rather than considering all the factors in the real situation Figure 12, and Figure 13. Figure 12. Figure 13. The given task is considered an interesting, yet challenging one. Unfortunately, in reality, it turned out to be such a huge challenge with all student participants in the research that all of them left the problem completely unsolved or failed to transform the situational task into a Maths problem. If they couldn‟t even understand the questions, it‟s certain that they couldn‟t answer it. Situation 3 Question 1. Most students could not answer this question by writing „I don‟t understand.‟ or answer incorrectly. The reason identified is that although the instruction is very specific, the students still could not understand the rules of the game. Also, despite being given plenty of time, they still did not try asking any lottery agents or players to understand the rules of the game. Thus, they failed to model the situation in the task using their own understanding of the lottery rules, which affects the subsequent questions. In all the students‟ answer sheets, only one worksheet was found with the correct answer to question 1 as shown below Figure 14. However, as the answer was too brief, we conducted an interview to ask the student about the way he did it. It was found out that the student understood the question, the rules of the lottery game as well as the basic concepts such as accordance, combinations or permutations and managed to apply them. International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi 14. Question 2. Because many students failed to answer Question 1, very few students succeeded in answering this question correctly. Some students still answered question 2 without answering Question 1 Figure 15.. Interestingly, they all gave the same advice which was not to play, because the winning ratio of the lottery in Vietnam was too small. Another interesting finding is that when asked about the playing strategy, they all replied "do not play" Figure 15., and if you play, you should choose Vietlot Figure 16.. On the other hand, we also see the unclear rules of the game in the answer of a student to the first question when he wrote that Vietlot's special prize is 12 times worthier than the special prize of the tradition Northern Lottery Figure 16.. This is not true. The students could not project the cases of multiple winners and did not know how to calculate the unstated prize of Vietlot. Figure 15. Figure 16. Question 3. Collecting some answers like "I do not know", "Most lottery types are similar because the winning chances are very small", ... Figure 17., Fig. 18., we followed up by asking the students in person what lottery types they refered to with "most”. It was revealed that the students were aware of neither the rules of both lottery games mentioned in the task nor those of other types. When being asked, "Did you use the Internet to search for the rules?", all students answered similarly "no" or “I did not think of that way". This points out the lack of practical knowledge and skills and the habit of using the internet as an important searching engine. Although Internet-connected PC and smart phones are widely available with many students, they still haven‟t got this habit and competence. Figure 17. Figure 18. Question 4. With this question, although students could not provide complete answers, the submitted ones were quite satisfactory. They are expected to study and discuss the belief in “good luck” as well the the subtle distinction in the culture and lifestyle between Northerners and Southerners in Viet Nam regarding playing lottery. Students were able to partially explain the reasons for playing the lottery such as "want to spend less, get more" Figure 19., Figure 20. and the difference in lottery players‟ behaviour in the North and the South. For example, they mentioned that Southerners do not have the habit of saving as the Northern people Figure 20.. All the answers we collected stated that playing the lottery is not an essential need. International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi Figure 19. Figure 20. Question 5. There were very few answers to this question. All the submitted answers warned people against playing lottery given the explanation that "the chance to win is too small" Figure 21., Figure 22. and the money spent would be much more than the amount earned Figure 21.. Interestingly, there was a student who compared the lottery with lotto betting - a form of illegal gambling based on the daily results of the official lottery draw- and stated that the chances to win in lotto betting might be much stronger! Figure 21. Figure 22. Question 6. Most of the students choose Vietlot as the investment plan but they could not explain their choice. When we interviewed some students, they still did not point out specific or convincing justifications for their option Figure 23.. Their answers even implied the misunderstanding between investing in lottery companies and investing in playing lottery. Figure 24.. Figure 23. Figure 24. 6. Conclusions and recommendations In all the situations mentioned, the students were always given plenty of time, not only in class but also independent self-study time. The knowledge we asked is just basic knowledge rather than complex knowledge, which requires complex mathematics transformational techniques or belongs to the field of probability and requires derivatives to identify the maximum and minimum values of the functions. However, the findings of the initial research point out various concerns about the students‟ competence to apply mathematics into practice including 1 Students have not yet adapted to the proposed Maths task types as well as the organising methods of practical learning activities. 2 Students are not proactive in exploit relevant information in Maths problems, in life when encoutering new task type and knowledge they are familiar to in school. 3 Students' reading comprehension is generally limited; 4 The given contexts have not been fully and profoundly exploited and examined by the students. International Annual Meeting on STEM Education I AM STEM 2018IOP Conf. Series Journal of Physics Conf. Series 1340 2019 012032IOP Publishingdoi Student's modelling competence is not adequate. Therefore, we would propose some recommendations for developing the mathematics cirriculum as follow 1 It‟s crucial that the Maths cirriculum take into account and introduce the requirements and regulations for developing learners‟ modeling compentence in a more practical way. In the context of Vietnam, the Mathematics program needs to be designed in a more open and practical way. One solution would be learning from the effective international Maths cirricula in other countries such as the Netherlands, USA, Singapore or Indonesia; 2 In the Vietnamese context, the implementation of school curriculum and classroom programs needs to be more flexible to further promote the teachers‟ creativity in designing classroom or school programs more effectively and progressively; 3 It is necessary to consider and systematically innovate assessment methods in major mathematics examinations throughout the school year or provincial and national examinations in the aim of assessing learners‟ modeling competence and reading comprehension ability because assessment form and content play an important role in exam preparation and teaching and learning methods. 4 Maths teachers need to update their teaching content in the direction of enhancing the application and connection of Maths knowledge and real life in order to to help students use Mathematics and realise the significance of Maths; 5 It‟s necessary to develop of students‟ reading comprehension ability in teaching Mathematics. 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For this reason, educators need to find out the context, create appropriate contexts support students to construct mathematical knowledge. some other suggestions for educators to find and create contexts for learning mathematics, namely the history of mathematics in context; context in real life life of elementary school students games, shopping, saving and using money, movies,...; social issues traffic, weather forecast, lottery, ...; integrated education mathematics Physics, Chemistry, Information Technology, etc. Trung, Thao, & Trung, 2019. ...... Gusnarsi, Utami, and Wahyuni 2017, Putri, Isrok'atun, and Kurnia 2017, Hasan, Pomalato, and Uno 2020, Ariati and Juandi, 2022 the role of the educator must be able to carry out the RME approach. According to Trung et al. 2019 several important principles of teaching Mathematics based on RME as follow. The first, activity Principle. ...A. M. Mega PurnamatatiHerlina UsmanElih YunianingsihThe purpose of the research was to provide knowledge of the effect of the Realistic Mathematics Education RME approach and motivation on students' mathematical reasoning abilities. This quasi-experimental study employed a pretest-posttest control group design, focusing on fifth-grade students at SDN Lulut 05, Klapanunggal District. Class VA was selected as the experimental group, while class VB served as the control group using purposive sampling. Data analysis involved the N-gain formula and t-test. The results shown 1 There is a difference in the increase in students' mathematical reasoning ability after being given the RME approach indicated by the N-gain value of in the medium category and for the experimental class in the low category, then the control class in the t-test independent so that the Sig value is obtained. actions, processes, objects, and schemas APOS theory is a constructivist learning theory created by Dubinsky based on Piaget's epistemology and used to teach math worldwide. Especially the application of APOS theory to the curriculum of a mathematics class helps students better understand the concepts being taught, which in turn contributes to the formation and development of mathematical competencies. With the aid of the APOS theory and the activity, classroom discussion, and exercise ACE learning cycle, this study sought to ascertain the effect of teaching derivatives in Vietnamese high schools. In this quasi-experimental study at a high school in Vietnam, there were 78 grade 11 students 40 in the experimental and 38 in the control classes. As opposed to the control class, which received traditional instruction, the experimental class's students were taught using the ACE learning cycle based on the APOS theory. The data was collected based on the pre-test, the post-test results and a survey of students' opinions. Also, the data that was gathered, both qualitatively and quantitatively, was examined using IBM SPSS Statistics Version 26 predictive analytics software. The results showed that students in the experimental class who participated in learning activities based on the APOS theory improved their academic performance and attitudes. Additionally, it promoted the students' abilities to find solutions to problems about receiving increasing attention from mathematics education scholars, there has not yet been any overall understanding of the current state of realistic mathematics education RME. To address this gap, this study aims to provide a review of 288 studies on realistic mathematics education from the Scopus database between 1972 and 2019. Using descriptive and bibliometric analyses, this study addresses four research issues as follows i the total volume, growth trajectory, and geographic distribution; ii the most influencing authors and research groups; iii the most influencing sources journals, books, conferences; and iv the most important topics. Several implications for not only mathematics education scholars but also other stakeholders, including policymakers, school managers, mathematics teachers, may not be considered in this study.

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