In the 1990s, Reinhard Laubenbacher and David Pengelley began advocating for the inclusion of primary historical documents in teaching undergraduate mathematics. They ran workshops, published articles, and wrote two books on implementing this pedagogical idea. They claimed, in the Preface to the first book, “Nothing captures the excitement of discovery as authentically as a description by the discoverers themselves.”

Of course, one cannot merely hand an undergraduate Cauchy’s Cours d’Analyse and expect much good to come from it. A student would need to be taught how to read it and asked questions that forced them to wrestle with the contents. The instructor would need to have some expertise in what Cauchy is doing and why he is doing it, and know how to read two-hundred-year-old mathematics and be able to anticipate the difficulties and misinterpretations students would generate. It is easy to believe in the Pengelley–Laubenbacher thesis and to be tempted to implement it. It is equally easy to do a terrible job through lack of preparation and expertise.

Enter TRIUMPHS, the elaborately near-acronymic NSF-funded project on TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources. The seven principal investigators have assembled an extensive collection of Primary Sources Projects (PSPs) that can be plugged directly into an existing course to give students the experience of, as Abel said, “study[ing] the masters.” Each PSP has been carefully designed, classroom-tested, and peer-reviewed. Each features excerpts illuminating a core mathematical idea from the standard undergraduate curriculum, which are imbedded in explanatory commentary and interwoven with student tasks. The idea is to teach the mathematics in a historically rich way, not to teach the history of mathematics.

The volume by Barnett, Ruch, and Scoville features extensive instruction in using PSPs in an undergraduate classroom and each module includes Notes to Instructors that provide historical context and pedagogical advice. The modules cover standard topics in real analysis, topology, and complex analysis and they range in length from experiences covering one or two class days to three or four weeks of class time.

David Pengelley, after launching the movement of using primary sources in undergraduate instruction, got deeply interested in reading such sources himself and immersed himself in the unpublished writings of Sophie Germain. At the same time, he began experimenting with inquiry-based learning in his own teaching, combining these two interests in an innovative undergraduate number theory course that included readings from Germain. Asking his students to follow in Germain’s auto-didactical footsteps seemed like a natural way for them to learn the material. There are two, parallel, investigations occurring simultaneously in the course. The student reader is trying to discover and understand the elementary number theory Germain is using while also trying to understand the path she is taking towards Fermat’s Last Theorem. Pengelley is a deft teacher, a gifted expositor, and highly skilled at IBL instruction. The student is led, gently, to investigate illuminating examples through more or less elementary computations, and then asked to consider what Germain must have thought when performing these same computations. By the end, the students are operating at quite a sophisticated level both mathematically and historically. There is no other book like this; students in a course taught from it will be taken on an extraordinary intellectual journey.