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Teaching undergraduate real analysis can be a delicate balancing act between intuition and rigor. On the one hand, the whole point of the course is to erect the formal machinery underlying the calculus in a rigorous way. On the other hand, if you operate solely in a formal way, it is notoriously difficult for the average undergraduate math major to appreciate what you are doing and why you are doing it. You need to include intuition and motivation and explain the need for the formality. And the details of the rigor can seem to students to have been handed down from Olympus, where did the definition of continuity come from? Who ever thought that up, and how? It can be tempting to motivate the material by presenting it in its historical context. The problem is that most of us don’t know the details of the historical development and they can be difficult for nonexperts to master.

Enter the elaborately anagrammatical NSF-funded TRIUMPHS project: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources. The project, organized by a collective of mathematicians and historians of mathematics, developed a collection of Primary Source Projects (PSP) designed to be inserted into a course that develops a topic with historical motivation and context. Every PSP contains an essay providing context, an excerpt or excerpts from one or several historical documents, and a skillfully scaffolded series of student exercises to help the student unpack the excerpt and relate it to the modern understanding of the topic. Importantly, the point is not to teach the history, the point is to teach the relevant mathematical idea in a rich historical context that provides motivation. Every PSP has been rigorously edited and classroom-tested in multiple institutions. Each also includes several pages of teaching notes and recommendations for instructors along with helpful suggestions for further reading.

This volume contains two dozen PSPs for use in real analysis, topology, and complex analysis classes. For example, in one PSP we can read D’Alembert’s early attempt to define the limit concept. This is entirely verbal, illustrated by the example of polygons circumscribing and inscribing a circle, and it explicitly requires approaching the limit from a single side. There are eight exercises presented after this excerpt focused on making D’Alembert’s description precise by our standards and contrasting it to the modern definition of a sequential limit. In another PSP, we read Abel’s letter to Holmboe in which he describes divergent series as “devilish” and challenges Holmboe to explain what goes wrong with the Fourier sine series for at In yet another, students are asked to unpack the Darboux–Hoüel correspondence in which Darboux goads Hoüel to reconcile the latter’s explanation of a derivative with the behavior of . at zero. There are, as noted, two dozen of these, but you would be cheating your students if you never showed them Euler’s derivation of presented here. He starts by asking us to consider the expression for “infinitely small” After proving that when . then , he asks us to consider what choice of , will give It is full of beautiful Eulerian algebraic prestidigitation and the student tasks are to explain and unpack all the wizardry. .

Use of these modules enables a nonhistorian to present, in a historically accurate way, the development of core ideas in analysis, topology, and complex analysis. The student tasks required here are deep, rich, and challenging. This is inquiry-based learning through a historical lens with an extraordinary level of care given to effective pedagogy.