By Amy Ackerberg-Hastings and Janet Heine Barnett

“Quickly, children, gather around the radio! The greatest mathematician of our time is speaking!”

OK, it is almost certainly not true that David Hilbert’s 1930 radio address attracted a Super Bowl-sized audience—it was only broadcast locally in Königsberg, Germany, and it lasted a mere four minutes—but his 373 words have inspired and challenged mathematicians around the world ever since they were delivered.

When he gave this speech, Hilbert (1862–1943) had just retired from the University of Göttingen at age 68. His hometown of Königsberg prepared a celebration that included an invitation to lecture to Gesellschaft der Deutschen Naturforscher und Ärzte (the Society of German Natural Scientists and Physicians), because of course we honor academics by asking them to do more work. Hilbert’s achievements were towering: an axiomatization of geometry, a synthesis of algebraic number theory, the Basis Theorem in invariant theory, research into problems in integral equations and mathematical physics, and the statement of 23 open problems in mathematics that helped to shape the direction of mathematical research for more than a century.

As his accomplishments suggest, Hilbert believed that philosophy, mathematics, and intellectual culture were intertwined. Thus, the portion of his lecture that he delivered over the radio a few days later argued that mathematics is the bridge that holds theory and practice together. He reeled off a dizzying list of references in the course of four minutes, from Galileo to the Russian novelist Leo Tolstoy. He argued forcefully that every problem in mathematics is solvable. And, he ended with the statement that still resonates with many mathematicians:

Wir müssen wissen,

Wir werden wissen.

(We must know, We will know.)

The audio of Hilbert’s radio speech and transcripts in German as well as English translation (prepared by Jim Smith) can be shared with students in a variety of classes to get them talking about the meaning and nature of mathematics and its role in society. Is every problem in mathematics in fact solvable? What sorts of problems can’t mathematics solve? What does it even mean for a problem to be solvable in the first place?

The words of mathematicians who impacted history can support students’ mathematical learning in more direct ways as well. Consider, for example, the student project “Braess’ Paradox in City Planning: An Application of Multivariable Optimization” by Kenneth M Monks, based on a classic 1968 paper in which Dietrich Braess (b. 1938) first studied the counterintuitive fact that the removal of an edge in a congested traffic network can result in improved flow. That’s right—closing down a busy street can actually get you to school early! As the project guides them through Braess’ paper, students witness how a problem from outside of mathematics (one they may even care about) can be turned into a problem that can be studied (and perhaps solved) using mathematical techniques. They then analyze Braess’ examples using standard optimization techniques from a multivariable calculus course. A first-hand view of the mathematization process, additional practice with the standard techniques they are studying, and exposure to a second optimization framework are all packaged together in a classroom-ready, open-access project.

Ken’s project is one of a growing collection of “mini-primary source projects” that are based on the original writings of mathematicians from various world cultures and historical time periods and that are designed to teach standard mathematical topics from across the current undergraduate curriculum. A second collection of similar primary historical source projects designed specifically for discrete mathematics and computer science is also available. By inviting students to enter the thinking processes of the creative mathematicians who first posed (and sometimes solved!) the problems that led to today’s mathematics, these projects teach mathematical content from a historical perspective that provides context, motivation, and direction for student learning in ways that standard textbook treatments cannot.

Communication is especially central in the learning experience offered by primary source-based projects. Obviously, students and instructors must read and interpret the primary-source material itself. Additionally, they are led into reflecting on changes in the meanings of specific words in ways that help to solidify an understanding of today’s terminology. In a similar way, they translate procedures originally presented in verbal form into today’s symbolic notation, and they encounter other types of writing that serve as bridges from where students are now to where they need to be in order to succeed in their current and subsequent coursework. All of these learning outcomes often arise most organically via in-class, small-group discussions.

Further, like other resources for teaching mathematics using its history, those we have described above place human faces on the subjects that we ask our students to learn. In particular, historically-based stories and activities reveal the fact that mathematics is shaped by real human beings who lived in particular times and places. This awareness can open up formal or informal conversations about who is able to participate in mathematics, towards what end, and with what consequences, and it can foster discussions about why and what mathematics matters to various groups within our society.

For readers looking to bring the benefits of using history in their own teaching—or those just wanting to know more about what those benefits might be—additional stories and classroom activities can be found in Convergence, the MAA’s refereed, online journal for using the history of mathematics to teach mathematics. An index of classroom-ready resources makes it easy to find materials and ideas for using history to inspire—and challenge!—students in any course. And who knows? Maybe the next David Hilbert will be among them.

Amy Ackerburg-Hastings is a historian of technology and science who specializes in the history of mathematics education, particularly in the United States and Scotland. She can often be found attending her teenager’s soccer matches, playing the organ, or walking the aptly-named Ping and Pong.

Janet Heine Barnett is a retired mathematics professor who remains active in the NSF-funded project Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS). In her spare time she enjoys bird photography, dancing, and travel.

Together, Amy and Janet edit Convergence. Both can be reached at convergence@maa.org.