Babylonian Numeration: A Mini-Primary Source Project for Pre-service Teachers and Other Students

When teaching any new topic, instructors face a dizzying number of choices about how to do so. Among these is: how much scaffolding should be provided?  At the risk of oversimplifying, answers to this question lie along a spectrum. At one end, some traditional teaching methodologies give students all of the theorems, methods, or rules relating to the topic at hand. At the other extreme, some “Discovery Learning” methodologies provide students with almost no information, instead offering an extended series of leading questions that encourage them to build their own knowledge. The purpose of this article is not to revisit the complex question of the benefits or drawbacks of these methods, but to provide an example at one end of these extremes that makes use of the history of mathematics: the mini-Primary Source Project Babylonian Numeration.

Exploring the Babylonian numeration system in general presents a marvelous opportunity, for students who are not familiar with it, to challenge basic assumptions about numerical representation. While this system shares certain features with the more familiar Hindu-Arabic base-10 numeration system (e.g., the use of position to convey the value of each symbol), it differs significantly in other respects (e.g., base 60, use of only two distinct symbols) that cause the two systems to look quite dissimilar.

Many books that teach numeration using Akkadian cuneiform take a more traditional approach in laying out the rules for base-60 numeration; some even use Hindu-Arabic numerals to represent the values within each “place value” (the Hindu-Arabic numeral 72, for example, might be written as “60 12” in the Babylonian system). The Primary Source Project Babylonian Numeration, at the other extreme, provides students no clue as to how the system works, no familiar numerals to hold on to, and not even a clean depiction of the numerals themselves. Thrown in at the deep end, as it were, students must find their own way to land.

The project itself is quite straightforward: students are given a representation of a tablet and asked to work out for themselves both the structure of the Babylonian system and the mathematics depicted on the tablet. Instructors are left to decide the actual visual format of the tablet that they wish to present to students, which could take several forms. Consider the following two representations of a Babylonian tablet:

(Readers are invited, at this point, to work out for themselves the mathematics depicted.)

The image on the left is, of course, a photograph of the front side of the tablet on which the PSP is based: Tablet HS 224 in the Hilprecht Collection at the University of Jena (photographer C. Proust, Cuneiform Digital Library Initiative). The image on the right is an attempt to faithfully represent the tablet as it has been preserved: Image 26 on Plate 16 in [Hilprecht 1906]. The two images contain similar information; the one on the right, however, is easier to read.

We could, of course, go a step further and clean up the symbols for the student, as in the following:

There may be good reasons to do this, but in practice the author has found that letting students grapple with the missing information only increases their level of productive struggle, and increases their pride when they work out the mathematics involved. Indeed, the students, who usually succeed in determining what’s going on—possibly with a few hints—are generally thrilled with their accomplishment.

The complete project Babylonian Numeration (pdf) is ready for student use, and the LaTeX source code is available from the author by request. Instructors are welcome to change the image and to make different choices regarding the presentation of the tablet if they wish. If they do, of course, the author would love to hear the results of their experiment. A set of instructor notes that explain the purpose of the project and guide the instructor through its goals is appended at the end of the student project.

This project is the seventeenth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources appearing in Convergence, for use in courses ranging from first-year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection also offers dozens of other mini-PSPs and a similar number of more extensive full-length PSPs.

Acknowledgments

The author is grateful to Dr. Christine Proust (Centre National de la Recherche Scientifique and Paris Diderot University), for providing information about and permission to use her photograph of the tablet in this article and the mini-PSP itself. The development of the student project Babylonian Numeration has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

Hilprecht, Hermann Vollrat. 1906. The Babylonian Expedition of the University of Pennsylvania. Series A: Cuneiform Texts, Volume 20. Philadelphia: University of Pennsylvania Press.