Primary Historical Sources Enrich Undergraduate Mathematics Instruction
By: Erin R. Moss, Co-Editor of DUE Point, Millersville University of Pennsylvania
The purpose of TRIUMPHS (TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources) is to develop, test, and evaluate projects for students based on primary historical sources related to core topics in today’s mathematics curriculum. These projects take a “guided reading approach” to primary source material by intertwining a series of student tasks with excerpts from a primary source, to help students actively engage with and explore the mathematics in those excerpts. In this way, the project teaches mathematical content, from a historical perspective that motivates and provides direction to student learning. Principal Investigator Dr. Nicholas Scoville describes more below.
How has using primary historical sources transformed your instruction?
When I was an undergraduate taking a first course in point-set topology, I could “do the math” but had lost the forest for the trees. I didn’t really know how point-set notions fit into the larger picture of mathematics. Utilizing primary sources in the classroom has helped me address this barrier for my students. After engaging with primary sources, math is no longer seen as abstract, isolated results based on definitions handed down to us from the Platonic realm. Students see the organic development of concepts, and, just as importantly, that human struggle developed these concepts.
What makes a primary source text appropriate for classroom use, given that some are so difficult to read?
In one sense the writing is more difficult because it tends to be archaic. Some early primary sources even use only words to describe equations! But in another sense, primary sources can be easier to read because mathematicians of antiquity are not afraid to expound upon, muse about, or explain their ideas in colloquial language. Mathematical writing in modern textbooks is concise and technical, yet often non-intuitive for students. But in primary sources, ideas are generally explained in an intuitive manner, and it is then up to the student to rigorize the idea. Students are encouraged to ponder, “Does my definition convey what the author is trying to say? Does it hold up in the ‘obvious’ cases? What about special cases?”
How do you design exercises around a text?
Many times, the exercises write themselves. Sometimes this happens because historical authors gave definitions or proofs in keeping with the standards and conventions that were in place during their own time, which often differ from how we practice today. This can provide a great opportunity for students to fill in the details. Other times, the mathematics in the primary source needs to be unpacked. For example, the project “Primes, Divisibility, and Factoring” by Dominic Klyve is based on a primary source in which Euler wrote, “It is known that the quantity an + 1 always has divisors whenever n is an odd number or is divisible by an odd number aside from unity.” Based on this simple statement, Klyve has seven exercises that force the student to dig deeply into what Euler is saying. Two such exercises are “What is unity? Find all n up to 16 for which n is an odd number or is divisible by an odd number. How else could you describe this class of numbers?” and “Formulate a conjecture about how to find a non-trivial divisor of an + 1 for any a when n is an odd number other than unity or is divisible by an odd number other than unity.”
What are students’ reactions to studying the primary source material?
The reactions tend to be mixed. Some students are initially quite skeptical of primary source projects. They don’t like the archaic language, having to do lots of reading, or even the deviation from the “way math is supposed to be taught.” Other students, especially those to whom lecture-style teaching does not appeal, love primary source projects. It places the math they are learning in a larger context and provides them a glimpse of the humanistic side to mathematics, a side that is almost non-existent in modern mathematical pedagogy.
What other opportunities are there for faculty to learn more about TRIUMPHS?
Recordings of prior webinars and links to publications can be found on our website, as can a browsable list of 70+ available projects (https://blogs.ursinus.edu/triumphs/). We are always looking for people to test our projects in the classroom, and we provide support and guidance throughout the implementation process.
Learn more about NSF DUE 1524065
Full Project Name: Collaborative Research: Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS)
Abstract: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1524065&HistoricalAwards=false
Project Contact: Dr. Nicholas Scoville, PI; nscoville@ursinus.edu
*Responses in this blog were edited for length and clarity.
Erin Moss is a co-editor of DUE Point and a Professor of Mathematics Education at Millersville University, where she works with undergraduates from all majors as well as graduate students in the M.Ed. in Mathematics program.