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Handbook on the History of Mathematics Education pp 27–53Cite as

Mathematics Education in Antiquity

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Abstract

This chapter is derived from highly heterogeneous sources both in their nature and in their geographic and chronological distribution. These sources represent different environments and refer to different cultural and institutional codes. Whereas ancient sources do not describe a coherent picture of teaching mathematics in Antiquity, some details from the better documented educational contexts of Mesopotamia, Egypt, and the Greco-Roman World provide impressionistic insight into these traditions. This approach shows that modern knowledge of these contexts is limited and that even the kinds of questions framing the topic depend strictly on the nature of the surviving sources.

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Notes

  1. 1.

    This methodological approach is developed in Bernard and Proust (2014); see in particular the introduction.

  2. 2.

    The presence of school tablets in a house is not always a proof that this house served as a school; in particular, school tablets may have been brought from other places to be reused as construction material. Thus, the archaeological context must be analyzed carefully for each context. See, for example, the case of the “schools” in Ur analyzed by Charpin (1986, pp. 432–434) and Friberg (2000), the case of the houses of “Aire II” of Tell Haddad analyzed by Cavigneaux (1999, pp. 251–252), the case of “House F” in Nippur analyzed by Robson (2001, pp. 39–40), and the case of the house of the “gala-mah” in Sippar-Amnânum analyzed by Tanret (2002, p. 5). About bins for recycling tablets, see Tanret (2002, pp. 145–153).

  3. 3.

    The studies of the curriculum in the edubba are mainly based on Nippur sources; from the abundant literature on the subject, see Cavigneaux (1983), Civil (1985), Tinney (1999), Vanstiphout (1996), Veldhuis (1997), Robson (2001), George (2005), Proust (2007), and Delnero (2010).

  4. 4.

    About the role of memorization in learning process and transmission, see Veldhuis (1997, pp. 131–132, 148–149) and Delnero (2012).

  5. 5.

    One example of text not clearly linked with teaching is the famous tablet Plimpton 322 (see Britton et al. 2011); other examples are found among the so-called series texts, which are lists of problem statements written on numbered suites of tablets (see Proust 2012).

  6. 6.

    This typology comes from the classification of tablets used in OB Nippur for learning Sumerian literary (Tinney 1999, p. 160).

  7. 7.

    Literal translation based on Neugebauer and Sachs (1945, p. 70). The passages written in Sumerian in the cuneiform text are represented by plain font and passages written in Akkadian by italic font. The measurement units used are 1 ninda ≈ 6 m, 1 gin ≈ 1.7 dm3, and 1 še ≈ 0.04 g.

  8. 8.

    These tablets are six catalogue texts conserved at Yale University and two related procedure texts (including YBC 4663). See Proust (2012).

  9. 9.

    Rochberg (2004, Chap. 6), Robson (2008, Chap. 8), Clancier (2009, pp. 81–103, 205–211), Steele (2011), Ossendrijver (2012, Chap. 1), and Beaulieu (2006).

  10. 10.

    For a reliable guide to the bibliography and contents of the specific texts, see Clagett (1999).

  11. 11.

    For an accessible discussion of “The House of Life,” see Strouhal (1992, pp. 235–243).

  12. 12.

    For ancient scholarship and the history of texts and their transmission, see Reynolds, Leighton and Wilson (1968).

  13. 13.

    For the question of our sources of knowledge on ancient mathematics, see Fowler (1999, pp. 199–221), particularly the list in pp. 268–275.

  14. 14.

    For Greek literature, see Fowler, op. cit. For Latin technical literature (corpus agrimensorum, on which more below), see Dilke (1971, 128ff) as well as Chouquer and Favory (2001) (esp. Chap. 1).

  15. 15.

    Including copying, annotating, and writing memoranda, summaries, and abridgements (epitomai)

  16. 16.

    For more detail, see, for example, Aelius Theon’s Progymnasmata, which is basically a handbook for teachers of rhetoric.

  17. 17.

    On the uncertainty of the date and context of Euclid – uncertainty that already dates from Late Antiquity – see Vitrac (1990, pp. 13–18).

  18. 18.

    Vitrac (1990, pp. 114–148). Other treatises by Euclid besides the Elements might be more legitimately suspected to contain some kind of exercises in demonstration or in the technique of analysis (Pseudaria and Dedomena, respectively); see Vitrac (1990, pp. 21–23).

  19. 19.

    Not only were they teachers, but, in the case of late Platonist commentators like Proclus, they considered themselves as the successors (diadochoi) of a Platonic tradition that included such famous mathematicians as Euclid or Nicomachus. Thus, according to his biographer Marinus, Proclus believed he was the reincarnation of the latter.

  20. 20.

    Pappus probably did not author the collection as such, but only the constituent individual treatises which were put together long after Pappus’s time.

  21. 21.

    For more detailed discussions of this event and Pappus’s account of it, see Knorr (1989, pp. 63–76), Lloyd 1996, Cuomo (2000, 127ff), and Bernard (2003).

  22. 22.

    He answers not only by demonstrating his own capacity to analyze the shortcomings of the construction but also by suggesting that the students could have proceeded otherwise if they had possessed more knowledge of the underlying problems.

  23. 23.

    Entitled “Proclus or On Happiness” = Vita Procli. This “biography” is better termed a hagiography. For the nature of Marinus’s discourse, see Vita Procli XLI-C (Saffrey and Segonds).

  24. 24.

    Such an approach to mathematics is already distinctly represented by Theon of Smyrna in the second century A.D. (Delattre 2010). For the noninstitutionalized framework of Late Antique education, see Derda et al. (2007, pp. 177–185) (E. Szabat) See also Watts (2006).

  25. 25.

    This particularity is best explained by the fact that this period is characterized by, among other things, the violent confrontation of various cultural and didactic models, especially between Christian and pagan models, which led each party to highlight and effectively represent these values.

  26. 26.

    H.I. Marrou, in his short discussion on the teaching of elementary calculation, already warned against the too easy identification of papyri with mathematical content as corresponding to school exercises (Marrou 1964 6, note 10, pp. 398–399). Modern discussions confirm this.

  27. 27.

    This standard and traditional view is found, among others, in the influential syntheses of Marrou (1965), Bonner (1977), and Clarke (1971).

  28. 28.

    For a more detailed account of the contents of each level, see Cribiore (2001, Chaps. 6, 7 and 8, pp. 160–244); in those chapters, she focuses on only the basic contents of each level. See also the lucid and updated synthesis provided in Szabat (2007), with many references to the debates on these issues.

  29. 29.

    The term “grammaticus” should not be understood as equivalent to our modern “grammarian,” which now designates a distinct discipline. Although the latter was first constructed in antiquity, the competence of the “grammaticus” as a teacher extended much beyond mere “grammatical” analysis of literary and poetic texts: this teaching included a thorough initiation in the reading and analysis of a characteristic corpus of poets and classical writers. See Szebat (2007, pp. 185–187) for a synthetic summary and Kaster (1988) and Cribiore (2001, pp. 185–219) for more detailed explanations.

  30. 30.

    Some “idealized” accounts allude to the existence of such “separate” professionals, but these accounts are uncertain and ambiguous. There is some, albeit scanty, evidence of such mathematical teaching at the secondary level. See Kaster (1983, p. 335) and Cribiore (2001, pp. 40–42).

  31. 31.

    For a discussion of the thorny and interesting issue of the “technical” terminology of ancient education, see again Kaster (1983, pp. 329–331) and Szabat (2007), with references to other studies on the same subject.

  32. 32.

    On the analysis of Marrou’s precautions on this issue, see the insightful discussion of Kaster (1983, p. 324).

  33. 33.

    Booth paid attention to the situation in first-century AD Rome. His theses (Booth 1979) are conveniently summarized in Kaster (1983), who expands on his argumentation.

  34. 34.

    For an efficient summary, see Szabat (2007, pp. 178–181), who draws on previous studies, esp. Kaster (1983).

  35. 35.

    Kaster (1983, p. 346). The same point is made in Cribiore (2001, Chap. 1) (pp. 15–44) for the sole case of Hellenistic and Roman Egypt and Szabat, op.cit. Even the imperial state did not heavily intervene in educational institutions. At best, laws would oblige cities to finance municipal chairs, without intervention in and regulation of their study. The majority of teachers, though, worked privately and directly depended on fees from students and their parents.

  36. 36.

    Kaster (1983, pp. 337–338) summarizes the “positive” reasons to believe that Booth’s model is better adapted in general to antiquity, although it should not be viewed as an alternative model applicable to all ancient situations.

  37. 37.

    On the uniformity and strong identity of the grammatikoi’s teaching, the classic study is Kaster (1988).

  38. 38.

    This idea of “concentric” studies, in which the same elements and methods are retrieved at each level but with a different depth and difficulty, is central to the argument of Cribiore (2001).

  39. 39.

    This point is made in Szabat (2007, pp. 180–181) and Cribiore (2001), Chap. 1 (on school accommodations) and Chap. 2 (on teachers).

  40. 40.

    Fowler (1999)gives the sole extensive discussion on the papyrological evidence concerning mathematics.

  41. 41.

    Cribiore (2001, pp. 180–183). This short discussion is devoted to the question of the acquisition of numeracy at the elementary level.

  42. 42.

    For an account of ancient Greek astrology, the standard reference remains Bouché-Leclercq (1899). See also the more recent Barton (1994), especially pp. 134–142 as far as astrological training is concerned.

  43. 43.

    For an updated extensive study on this question, see Hadot (2005), especially the fourth “étude complémentaire,” pp. 431–455, concerning mathematics. Note, however, that Hadot has a tendency to reduce any ancient mathematical teaching to being basically dependent in all cases on a philosophical ideal, an idea which is somewhat open to criticism.

  44. 44.

    See Isocrates’s ideas in Antid. pp. 258–269. For Quintilianus, see Inst. Orat. I.10, especially pp. 34–49.

  45. 45.

    Even by those, like Firmicus, who obviously had little command of the mathematical contexts of their art.

  46. 46.

    These commentaries can indeed be seen, at least in part, as conscious imitations of the Almagest; see Bernard( 2014) on this point.

  47. 47.

    As far as the corpus agrimensorum is concerned, the work of Toneatto (1994-5) has drastically improved our understanding of its history; see Chouquer and Favory (2001, Chap. 1).

  48. 48.

    Is Hero’s Dioptra, for example, an exercise in land surveying, as the kinds of problems treated therein strongly suggest, or the skillful description of an instrument, as the preface and many technological details indicate? Is Vitruvius’s treatise merely a work on monuments and house-building, or also on machine-building (book 10), the science of sundials and astronomy in general (book 9), and many related subjects, as the contents suggest?

  49. 49.

    Like Hyginus “gromaticus,” the second Hyginus or Siculus Flaccus. For translations of these authors, see Campbell (2000) or the various annotated editions published in French by J.Y. Guillaumin, in particular Guillaumin (2005, 2010).

  50. 50.

    It has even become commonplace in scholarship on these kinds of sources that they represent didactic efforts and are scholastic “manuals,” a qualification difficult to dismiss because of the vague and multifarious meanings of this category. The idea is discussed and nuanced in Chouquet and Favory (2001, p. 38).

  51. 51.

    Acerbi and Vitrac forthcoming: introduction, A4. A preliminary version of this detailed analysis is available online on hal-SHS http://hal.archives-ouvertes.fr/hal-00473981/fr/ (consulted 5.1.12).

  52. 52.

    For a list of such documents, see Fowler (1999, pp. 269–276). The list is focused on tables rather than problems.

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Author information

Authors and Affiliations

  1. Centre Alexandre Koyré, UMR EHESS, Paris-Est University (UPE), Paris, France

    Alain Bernard

  2. Laboratoire SPHERE, CNRS & Université Paris Diderot, Paris, France

    Christine Proust

  3. University of Idaho, Moscow, USA

    Micah Ross

Authors
  1. Alain Bernard
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  2. Christine Proust
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  3. Micah Ross
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Correspondence to Alain Bernard .

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Editors and Affiliations

  1. Teachers College, Columbia University, New York, New York, USA

    Alexander Karp

  2. Universität Bielefeld Institut für Didaktik der Mathematik, Bielefeld, Germany

    Gert Schubring

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Bernard, A., Proust, C., Ross, M. (2014). Mathematics Education in Antiquity. In: Karp, A., Schubring, G. (eds) Handbook on the History of Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9155-2_3

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