In Spring 2012, when I was teaching our upper-division Topics in the History of Mathematics course, I wanted students to interact somehow with primary sources. This point was driven home when a guest lecturer from the history department explained what primary sources were and their importance to the discipline of history. Also, I wanted students to do more reading and writing about mathematics (in addition to solving historical math problems), along the lines of what Klyve, Stemkoski, and Tou described in their comprehensive overview for Convergence of several different ways to use primary sources (and in particular the Euler Archive) in teaching and learning, "Teaching and Research with Original Sources from the Euler Archive."
The research paper assignment that I crafted was admittedly somewhat terse and vague in its wording: “To engage with primary documents (or translations thereof) to answer a math history question.” Since the class is usually composed of only ten to fifteen math majors and minors, my plan was to meet with each student individually and facilitate a choice of primary source material that appealed to her or his interests. Then together we would tailor a relevant question (mathematical or historical or both) for the student to answer. That way, I could point each student toward a problem that would be challenging, but still achievable.
In addition, each student had to choose a source that was different from those selected by other students in the class, which I hoped would lead to more personal investment in their topics, and would avoid the possibility of students working on the same topic and copying each other’s work. Students could choose sources that we had studied or mentioned in class, they could peruse source books or journals in the library, or they could search online. The assignment was worth 20% of the course grade, with an expected length of about six pages. Though it was announced on the syllabus on the first day of class, topic selections were not finalized until after spring break, which meant that students had about one month to complete the assignment.
While most students did a good job reporting on their primary sources, there was less success with the second part of the assignment: engaging with the source material. Most of the students who were otherwise performing well in the course simply explained the mathematics of their primary document and did not attempt to extend the mathematics or draw connections to modern formulations of the same topic. Primary sources selected by these students included some of the more famous documents:
- Leibniz’ proof of the Fundamental Theorem of Calculus;
- Germain’s correspondence with Gauss;
- Euler’s paper on the Königsberg bridges;
- Nash’s dissertation on game theory; and
- various authors’ work on Fermat’s Last Theorem, starting with Fermat’s own infamous comment in the margin of his copy of Bachet’s translation of Diophantus.
These students wrote interesting reports, but really did not formulate and answer a mathematical or historical question of their own, and so I call this the “report” group. To be clear, the reports were often well written and novel, but strictly speaking, they did not engage with the material in the way I had tried to outline in the assignment.
Interestingly, the students who were more likely to extend beyond the primary source material – those comprising what I call the "engaged" group – were also more likely to struggle with the other aspects of the course. Like the students in the "report" group, these students selected some famous works.
- One student chose the Babylonian tablet Plimpton 322, and tried to link it to ancient architecture and fractions, but the paper suffered from a lack of consideration of scholarship on the tablet, such as Eleanor Robson’s “Words and Pictures: New Light on Plimpton 322.”
- Another student wrote about Lagrange’s explanation of Lagrange multipliers in his Mécanique Analytique and created some examples applying Lagrange’s techniques.
- The author of a paper on Gauss’ Disquisitiones Arithmeticae demonstrated how modular arithmetic could be applied to cryptography.
- And one paper on magic squares, both the early Chinese Lo Shu square and the work of Seki Takakazu (Seki Kowa) in 17th-century Japan, included a derivation of the formula for the sum of any row or column in a normal magic square of order n.
This article will examine in more detail two representative student papers not yet mentioned, one from each group, which deal with lesser-known mathematical documents. The “report” paper describes Turing’s unpublished work on phyllotaxis, while the “engaged” paper discusses an original number theory article by Euler. Comparing these two examples of student work will effectively serve as an assessment of the assignment overall, and will point out several ways to improve this assignment in the future.