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An Artefactual Approach to Ancient Arithmetic

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An Artefactual Approach to Ancient Arithmetic An Artefactual Approach to Ancient Arithmetic IRENE PERCIVAL Can attefacts from a civilisation long dead help arithmetic come alive? This sounds contradictory, yet my recent wotk with young stndents seems to suggest that this can indeed be the case. More than a decade ago, the French Institutes for Reseanh on the Teaching of Mathematics (the IREMs) pub­ lished a set of papers (Fauvel, 1990) promoting the use of historical documents to 'humanise' mathematics classes- but these mticles focused on students in secondary education. On the other hand, several excellent texts ( e g. Reimer and Reimer, 1992) have recently been published to facilitate the introduction of historical mathematics to the elementary classroom, but they make little or no reference to actual arte­ facts My work with elementaty school students combines the documentary approach of the IREMs with two other atte­ factnal approaches: stndents' own construction of objects and documents imitating those studied and using ancient calcu­ Figure I Narmer~ Macehead lating devices, albeit in modern reconstructions I have explored these approaches in several series of latter latet in the session, we looked at several problems eruichment classes during the past tluee yeats At first, my taken from Chace's transliteration of the Rhind Papyrus motivation was to complement the work on ancient civili­ (1927/1979) and the students were able to identify other sations covered in social studies classes by twelve- and hieroglyphics whose repetition marked them as possible thirteen-year-old students in their final yeat of elementary number symbols school in Bdtish Columbia, Canada These classes covet IJII many aspects of life in the ancient world, but mathematics '" i L:JJ, is rarely mentioned I considered this to be a regrettable w 'P omission, so I read several social studies texts in mdet to "'00 'Gl 0 OJ , , :::nn :; .9plfl 2 ..999.9!.! ... be able to relate the mathematics, patticulatly the arithmetic, w u... ~m of ancient civilisations to the stndents' other knowledge A 0 detailed account of such an integration was the basis of my "' nnnn•• 5I ,Jp::1 • .... .,.oj J MSc. thesis (Percival, 1999) Since then, I have also taught w w n p this material to students in other grades, in both large and ,,99H .~. ~ 1r 4 111199!0 Hft small groups, and have found a high level of interest in the t 'db number systems used by other civilisations D.)Ill '. 5 "So much stuff, it could be ., numbers" t 5~~ Egypt and Mesopotamia often seem to head the list when Jlli.999JI11 r !J ltl Y99Jlll U ancient civilisations are mentioned and my courses are no :!6 exception. We began5000 years ago at the start of the Egypt­ ian dynasties with the macehead commemorating Pharoah Figure 2 Problem 79 from the Rhind Mathematical Papyrus Narmer 's victories in battle (Budge, 1926/1977, p. 5) I explained the nature and purpose of this object to my For some of the civilisations stndied, such artefacts enabled stndents and challenged them to 'fmd the numbers' Almost the students to determine the value of individual symbols innnediately, I was told that "there's so much stnff there [in Although the problem from the Rhind Papyrus shown in the bottom right corner], it could be to do with numbers" I Figure 2 (p. 137) could perhaps have led to conjectures for was smptised how quickly the students reached this con­ these values, a fairly high level of mathematical sophistica­ clusion: the numbers they use every day belong to a ciphered tion would be needed to justify such guesses. My stndents were able to see powers of seven in the left-hand column system and yet they had no difficulty assuming the tally when they knew the meaning of the symbols, but the length nature of an oldet system. Does this reflect earlier school of time taken to do so suggests that this pattern was not work in tallying or is it really an intuitive idea? The tally sufficiently obvious to confirm speculations on the value of nature of so many early number systems would suggest the 16 For the Learning of Mathematics 21, 3 (November, 2001) FlM Publishing Association, Kingston, Ontario, Canada the symbols However, the idea of students acting as 'math­ The students were able to detetmine the value of the indi­ ematical archaeologists' was fundamental to my course, so vidual ciphered symbols and to consider how these symbols I designed a puzzle in which six Egyptian nwnbets had to be had developed from the emlier hieroglyphics One observant paired up with theit modem equivalents: a vety wugh boy noted that eight hunched was represented by tom dots approximation to the Rosetta Stone. 'This led to some valu­ with a line underneath, with similar notations for six able conjecturing and, in one case, a heated discussion broke hunched and two hunch·ed, and shouted excitedly "they're all out as to the value of a patticular symbol. times'd by two" .. Othet students investigated how the seven One student suggested that it meant 100,000, but two oth­ single lines of the hiewglyph fm seven could have devel­ ers, who rarely had the courage to contradict the first in oped into something I ather like the modem nwnetal 2 The standard mathematical tasks, each pwduced mguments ensuing discussion highlighted the hwnan actions involved designed to show that it represented I 00 This Egyptian in the development of mathematics This humanist view­ work, which was new to all of them, seemed to place the stu­ point is rmdy considered by either students or teachers, but dents on an equal footing and the voices of the second and is a rnajor focus of my course. third students held such a ling of confidence that the first The students enjoyed deciphering the ancient docwnents student finally backed down and were inttigued by the imagery of the hieroglyphics The contrast between the modem place-value system and They produced a variety of explanations as to what the the tally-like Egyptian system made this matching puzzle symbols represented, based on theii knowledge of Egyptian a revealing exercise and the resulting pairs of notations civilisation Many of their suggestions were close to those provided useful data when I later asked the students to focus suggested by the 'expetts', but the 100,000 symbol remained on the similarities and differences between the two systems 'a panot' to them, despite being told Menninget 's conunent These compruisons led to a discussion of ciphered numbers that: as opposed to the repeated matks of the tally system and also the Egyptian symbol fm 100,000 is a tadpole, such as helped solidify the students' understanding of place value wiggle by in countless nwnbers in the mud of the Nile and the role of zero in out present numeration system when the water retreats within its normal banks after As mentioned above, the problem shown in Figure 2 pro­ the yemly flood.[. ] Its name, hfn, also means 'innu­ vided an interesting exercise in pattern recognition, merable' (1969, p 122) patticulmly as the fourth line (left-hand side) contains a mis­ take by the ancient copyist Discussion of this etror, in which Elsewhere (Irons and Bumett, 1995), we read that: only three spiials were chawn instead of tom, not only high­ the 100,000 symbol tepresented the nwnbet of tadpoles lighted an advantage of a ciphered numbet system, but also in a mudhole (p. 13) focused on the difficulty in intetpreting ancient docwnents, thus combining the mathematical and 'archaeological' Ihis led to the question of 'which mudhole?' I he students aspects of the course decided it would be that of 'a camel's hoofptint', and this Decoding is a task which appears to interest most proposal was followed by some valuable work on estima­ students. At its most elementary level, it merely requires tion, which did indeed suggest that 100,000 tadpoles would implementing a given one-to-one correspondence between fill a volwne of approximately that size. Thls diiection was two sets of symbols, but even fhls can be a valuable mathe­ not one which I had platmed, so I have to admit to feelings of matical exercise for young students .. However, in my ancient relief when the estimation gave reasonable values! Relating arithmetic course, the decoding often requited more sophis­ the mathematical symbols to the flooding of the Nile, and ticated fhlnking, as is illusttated by the next step of the the camel's hoofpiint, again reinforced the 'hwnan-made' Egyptian work. I showed the students the original hieratic nature of mathematics: unlike our own number symbols version (p 137) of the Rhlnd Papytus problem (see Figme 3) whose miginal significance is mostly lost in antiquity, the they had previously decoded from hiewglyphics (see Egyptian symbols cleady show signs of human intetven­ Figm·e 2), and asked them to match the 'Egyptian hand-wtit­ tion in their origins ing' with the translitetation It could be mgued that reading and wtiting hieroglyphics is nothing new, that it is an integral part of any social studies course about Egypt However, I feel that a study of mathematical hiewglyphics is patticularly beneficial, as students can then read relatively large sections of Egyptian text, a difficult task fm prose passages which usually contain a much wider vmiety of symbols When I first taught fhls course integrating atithmetic into the Egyptian part of the social studies curriculwn, I gave my students chy reeds from which to make their own sheet of papyms. Concmrent with fhls ptactical wmk, they leamt Egyptian methods of addition, subtraction, multiplication, division and fractions and were able to compose hiero­ glyphic calculations to copy onto theii personal 'Egyptian attefact' This home-made papytus proved to be a powetful Figure 3 Original hieratic script of Problem 79 motivatm.
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