One of the advantages to this kind of an open-ended assignment is that it can be tailored to individual students at their level of mathematical ability. The two students whose work is featured in this article are probably in the average to good range, and so I wanted them to work with mathematics that was more within reach than, say, Leibniz’ proof of the Fundamental Theorem of Calculus. Turing’s mathematics was accessible and Euler used only algebraic techniques in this paper, and so both students made good choices in this regard. Each student explained the mathematical content of her selected source material well.
Some might argue that the mathematical content in these two selections was not “challenging” enough for an upper-division math course, and at first I would have agreed with them. But while reading mathematical papers filled with algebraic equations has become second nature for us professional mathematicians, it is far from being so for undergraduates, even for math majors. Thus, even if the mathematics in these papers was minimal or “easy,” it was still a challenge for each student to try to understand and fill in the missing steps, and I feel that both of them also did well in this aspect.
To contrast the students’ performances, however, the first student found a document that interested her, but she did not extend or apply the mathematics beyond its initial scope. Rather, her paper read like a report on phyllotaxis and Turing, albeit a fascinating one. On the other hand, the second student had difficulty finding a document that interested her, but once she had a topic, she did stretch herself by applying the formulas in that paper to create her own unique example, thereby engaging with the content in a way that I wanted students to engage. To me, this assignment should embody the best of both worlds, by encouraging students to seek out primary sources of personal interest to them and then to converse with the ideas presented therein to put their own intellectual stamp on someone else’s work.
Assignments like this can help to make mathematics and its history more meaningful to students as well as to faculty untrained in the history of mathematics. Indeed, some of the shortcomings in the students’ writing can be attributed to my own inexperience with giving such assignments. There is much to be gleaned from this experience.
- Task Clarification. To make sure each student poses and answers a mathematical history question, I need to be more explicit about what I mean by “engaging” with a primary source, ideally by sharing some model projects. To be clearer, I want students to understand and explain the mathematics in their primary source, and then extend or apply that mathematics to something outside their primary source. Next time, I will point students to what I feel are some particularly good examples of engagement, such as: “Paradigms and Mathematics: A Creative Perspective” by Matthew Shives, a student of Betty Mayfield at Hood College, and also the winner of the HOM SIGMAA 2013 Student Paper Contest; or “The Moore Method: Its Impact on Four Female PhD Students” by Jackie Selevan, a student of Sloan Despeaux at Western Carolina University. Several other good examples can be found in Despeaux’s article, "SMURCHOM: Providing Opportunities for Undergraduate Research in the History of Mathematics,” for Convergence. There are also two outstanding examples of joint faculty-student projects directed by Adam Parker of Wittenberg University, “Peano on Wronskians: A Translation” and “An Analysis of the First Proofs of the Heine-Borel Theorem,” which can be found in Convergence here and here, respectively.
- Document Selection. To get the students to choose the primary sources themselves, I plan to start earlier. And to help the students select appropriate source material, I will require them to fill out a brief checklist of basic properties of the document, such as: author, publisher (if published), date, location, provenance, etc. This will enable the student (and me) to catch those sources that may not meet the strictest definition of “primary.” On a related note, I plan to focus the entire course more narrowly on primary sources, and so that should provide the students with a wider variety of “acceptable” examples during the semester which they can then examine in more detail for their research papers.
- Early Draft. In addition to having students peruse the model examples listed above, I also will require a draft to be turned in several weeks before the final version, so I can make sure that they are trying to extend or apply the mathematics beyond its original scope. I can also check the quality of their secondary sources, as students do not always have the information literacy required to distinguish peer-reviewed sources from those that are not peer-reviewed.
- Rubric Development. As my goals for the assignment become clearer, this would also be a good time for me to develop a rubric that outlines what I am looking for in a way that is clear to the students and that will make the grading of the research papers objective, consistent, and understandable. It could also help toward ongoing assessment efforts in the larger context of the math major.
Admittedly, many of these changes are fundamental features of a good writing assignment, and probably come second nature to those more experienced than I. But perhaps, like my students, I needed to make a few mistakes of my own in order to learn my lesson. By adopting these improvements in the future, I hope to push my students not only to select and report on a primary source, but also to engage more deeply with it and to enjoy a more positive learning experience overall.