A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources

Regular readers of Convergence are unlikely to need to be convinced of the value of studying primary sources in order to understand the development of mathematics. Essentially all good scholarship in mathematics is necessarily based on primary sources. But, as many have come to recognize, primary source materials are also valuable in learning the subject.  Thanks in part to the work of David Pengelley [11], Jerry Lodder [10], Janet Barnett [3], and others [1,2,5,6,12], ideas about the incorporation of primary sources in the teaching of university mathematics have become increasingly well-known over the last few decades. Dozens of mathematics faculty have given talks recently at national meetings about this pedagogical innovation, and several articles in Convergence have described details of such an approach [7,8,9,13,14], and even provided collections of classroom-ready student projects based on primary sources [4]. 

An outside observer, seeing this significant increase in the attention paid to the role of primary sources in teaching, may think that the movement to promote this method of teaching has been successful, and that these techniques are now widely used.

Some of us, however, share the view that the techniques are not nearly mainstream enough. That faculty with an interest in primary sources can use them in their teaching is well and good, but we are convinced that there are so many benefits derived from their use that we would like to see them available to all instructors of university mathematics.

         
Students work on a Primary Source Project under the supervision of Janet Heine Barnett at a
TRIUMPHS Site Tester Workshop, held at the University of Colorado, Denver, during September 2016.

Five years ago, at a workshop for the NSF-funded project Learning Discrete Mathematics and Computer Science via Primary Historical Sources in Las Cruces, New Mexico, a group of us started planning a large project: a national, funded, extended effort to build on the successful work done over the previous decades to promote this work of teaching with primary sources and help to make it more common. We would do this by continuing to prepare ready-to-use curriculum materials in the form of classroom projects based on primary source materials through which students could learn a wide variety of standard topics in university mathematics classes. If more such projects were available, we reasoned, they might significantly reduce the perceived difficulty some instructors may have in using original sources for the first time, especially if this broader collection of projects included a variety of shorter projects. Furthermore, if we combined the work of disseminating these projects with a careful research plan, we might learn something important about the benefits or barriers of using such an approach.

In 2015, our group, which had by this time expanded to a team of seven, secured a Collaborative Research grant from the National Science Foundation of about $1.5 million over five years to run this grand experiment on a large scale. We have been working actively since then first to write and edit these “Primary Source Projects”, or PSPs, then to test them in classrooms – either our own or in those of colleagues at other institutions.

The grant effort, TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (or TRIUMPHS), is well underway, and we encourage readers to find out more about its current status at the TRIUMPHS website. In addition to administrative notices governing the work of the grant, we maintain there an updated set of PSPs, available for perusal and download.

We’d especially like to call attention to a new set of "mini-PSPs" which takes our work in an exciting new direction. These shorter Primary Source Projects are designed to be completed within one or two class periods, and to give instructors already familiar with primary sources more flexibility in their use. We plan to publish a series of these mini-PSPs in Convergence over the next few years - beginning with the mini-PSP The Derivatives of the Sine and Cosine Functions - and we invite the mathematical community to read and use them.

If readers do decide to use projects in their classes, of course, we would love to know about it – contact the project’s author or any member of the TRIUMPHS team in advance, and we’ll be happy to talk with you. And have fun exploring them!

Subset of TRIUMPHS PIs and Advisory Board Members: David Pengelley, Diana White,
Danny Otero, Dominic Klyve, Janet Heine Barnett, Kathy Clark, Nick Scoville.

Acknowledgments

This material is based upon work supported in part by the National Science Foundation under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

References

1. J. Barnett, G. Bezhanishvili, J. Lodder, and D. Pengelley, Teaching Discrete Mathematics Entirely From Primary Historical Sources, PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies, 26 (2016) no. 7, 657-675, DOI 10.1080/10511970.2015.1128502.
2. J. Barnett, J. Lodder, and D. Pengelley, Teaching and Learning Mathematics From Primary Historical Sources, PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies, 26 (2016) no. 1, 1 - 18, DOI 10.1080/10511970.2015.1054010.
3. J. Barnett, Learning mathematics via primary historical sources: Straight from the source’s mouth, PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies: Special Issue on the Use of History of Mathematics to Enhance Undergraduate Mathematics Instruction, 24 (2014), no. 8, 722–736, DOI:10.1080/10511970.2014.899532.
4. J. Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D. Pengelley, I. Pivkina, D. Ranjan, and M. Zack, Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science, Loci: Convergence (July 2013), DOI 10.4169/loci003984.
5. J. Barnett, H. Leung, J. Lodder, D. Pengelley, and D. Ranjan, Designing Student Projects for Teaching and Learning Discrete Mathematics and Computer Science via Primary Historical Sources, in Recent Developments on Introducing a Historical Dimension in Mathematics Education (V. Katz and C. Tzanakis, eds.), Mathematical Association of America, Washington DC, 2011,  187–198.
6. J. Barnett, J. Lodder, and D. Pengelley, The pedagogy of primary historical sources in mathematics: Classroom practice meets theoretical frameworks, Science & Education: Special Issue on the Philosophy and History of Mathematics in Mathematics Education 23 (2014), no. 1, 7–27.
7. K. M. Clark, 'In these numbers we use no fractions': A Classroom Module on Stevin's Decimal Fractions - Overview and Introduction, Loci: Convergence (January 2011), DOI:10.4169/loci003333.
8. A. Dematte, Introducing the History of Mathematics: An Italian Experience Using Original Documents, Loci: Convergence (February 2010), DOI:10.4169/loci002856
9. D. Klyve, L. Stemkowski, and E. Tou, Teaching and Research with Original Sources from the Euler Archive, Loci: Convergence (April 2011), DOI 10.4169/loci003672.
10. J. Lodder, Networks and spanning trees: the juxtaposition of Prüfer and Borůvka. PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies: Special Issue on the Use of History of Mathematics to Enhance Undergraduate Mathematics Instruction, 26 (2016) 24 (2014), no. 8, 737 -751, DOI: 10.1080/10511970.2014.896835.
11. D. Pengelley, Teaching with primary historical sources: Should it go mainstream? Can it?, in Recent Developments on Introducing a Historical Dimension in Mathematics Education (V. Katz and C. Tzanakis, eds.), Mathematical Association of America, Washington DC, 2011, 1–8.
12. D. Ruch, Creating a Project on Difference Equations with Primary Sources: Challenges and Opportunities, PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies: Special Issue on the Use of History of Mathematics to Enhance Undergraduate Mathematics Instruction, 24 (2014), no. 8, 764–773, DOI: 10.1080/10511970.2014.896836.
13. N. Scoville, Georg Cantor at the Dawn of Point-Set Topology, Loci: Convergence (March 2012), DOI: 10.4169/loci003861.
14. L. Stemkoski, Investigating Euler's Polyhedral Formula Using Original Sources - Original Sources: Who, What, Where, How, Convergence (April 2010), DOI:10.4169/loci003297.