by Janet Barnett, Colorado State University – Pueblo; Dominic Klyve, Central Washington University; Jerry Lodder, New Mexico State University; Daniel Otero, Xavier University; Nicolas Scoville, Ursinus College; and Diana White, Contributing Editor, University of Colorado Denver
Mathematics faculty and educational researchers are increasingly recognizing the value of the history of mathematics as a support to student learning. The expanding body of literature in this area includes recent special issues of Science & Education and Problems, Resources and Issues in Undergraduate Mathematics Education (PRIMUS), both of which include direct calls for the use of primary historical sources in teaching mathematics. Sessions on the use of primary historical sources in mathematics teaching at venues such as the Joint Mathematics Meetings regularly draw large audiences, and the History of Mathematics Special Interest Group of the Mathematical Association of America (HOMSIGMAA) is one of the largest of the Association’s twelve special interest groups. In this blog post, which is adapted from a recent grant proposal, we explore the rationale for implementing original sources into the teaching and learning of undergraduate mathematics, and then describe in detail one method by which faculty may do so, namely through the use of Primary Source Projects (PSPs).
Teaching from primary sources has long been common practice in the humanities and social sciences [11, 17]. Reading texts in which individuals first communicated their thinking offers an effective means of becoming mathematically educated in the broad sense of understanding both traditional and modern methods of the discipline [9, 18]. The use of original sources in the classroom promotes an enriched understanding of the subject and its genesis for instructors as well as students.
In contrast to many textbook expositions, which often present mathematical ideas in a distilled form far removed from the questions that motivated their development, original sources place these ideas in the context of the problem the author wished to solve and the setting in which the work occurred. Problems and the motivations for solving them are more apparent and natural in primary sources, and the works of these thinkers are more compelling than traditional textbook expositions. Exposing the original motivations behind the development of “esoteric” mathematical concepts may be especially critical for placing the subject “within the larger mathematical world,” thereby making it more accessible to students [22]. Further, primary texts seldom contain specialized vocabulary (which comes with later formalism), thereby promoting access to the ideas by students with varied backgrounds.
Precisely because they give students the opportunity to interpret results as they were originally presented and then reformulate them in modern terms, original source readings encourage robust understanding of mathematics. Engagement with the original problems from which concepts arose has been observed to invite questions of a different nature than students generally pose in more traditional approaches as it provides students with a basis for making their own reflections and developing their own judgment, thereby helping them to see how to develop ideas and reason with them on their own [12, 13]. Drawing on Sfard’s work on “mathematical discourse” and “commognition” [23], Kjeldsen and Blomhøj have further suggested that the reading and interpretation of original sources may even be essential for raising students’ awareness of what constitutes a proper definition, or deciding whether a solution is correct and complete [16], a key feature of success in the study of advanced mathematics. Jankvist further proposes that original source materials may help with the “transition problem” between educational levels [14].
Finally, the reading of original texts provides a means of responding to calls within the Science, Technology, Engineering, and Mathematics (STEM) education community to engage undergraduate students in authentic research at the frontiers of the field. While the importance of such engagement for socializing and retaining students within the disciplinary community has been noted by Project Kaleidoscope Director J. Narum [20] and others, the present research frontier in mathematics is generally too far removed from the undergraduate experience to make this possible. Engaging students with texts that represent the state of mathematical knowledge at an earlier stage of development can, however, provide undergraduates an opportunity to experience research at a frontier stage as it was practiced by some of the greatest mathematical minds throughout time.
Despite the benefits of primary source materials detailed above, and granting the wide availability of such materials via published collections and web resources [6, 8], there are significant challenges to incorporating primary sources directly into the classroom. Using secondary historical sources, such as [15], may suffice to reap some of the benefits of the original works; however, use of such sources carries its own difficulties, including the risk of placing too much emphasis on learning the history of mathematics per se, as opposed to using that history to support the learning of mathematics.
One approach to addressing these issues is through PSPs, which are curricular modules designed to teach core mathematical topics from primary historical sources rather than from standard textbooks. Each PSP is designed to cover its topic in about the same number of course days as classes would otherwise. With PSPs, rather than learning a set of ideas, definitions, and theorems from a modern textbook, students learn directly from mathematicians such as Leonhard Euler, Augustin-Louis Cauchy, or Georg Cantor. This distinction is crucial to PSPs: they are not designed to teach history; rather, they use history as a tool to better teach mathematics.
PSPs employ a selection of excerpts from primary historical sources that follows the discovery and evolution of the topic in question. Each PSP contains commentary about the historical author, the problem the author wished to solve, and information about how the subject has evolved over time. Exercises are woven throughout the project, requiring that students actively engage with the mathematics as they read and work through each excerpt. At appropriate junctures, students are also introduced to present-day notations and terminology and are asked to reflect on how modern definitions have evolved to capture key properties of solutions to problems posed in the past. Learning from the PSP via in-class activities and discussions replaces standard lectures and template blackboard calculations.
As an example, the PSP Networks and Spanning Trees [19], opens with Arthur Cayley’s discovery of a pattern for the number of (labeled) trees on n vertices. Cayley used the term “tree,” without any definition, to describe the logical branching when iterating the basic process of (partial) differentiation. Students are asked to follow in Cayley’s footsteps and arrive at the same observations, hinting at an algebraic pattern for the number of such trees. Students are then asked to find the gaps in Cayley’s 1889 “proof” [7], and to reflect on what would constitute a valid argument. This is followed by Heinz Prüfer’s rigorous counting of (labeled) trees [21], motivated by the problem of enumerating all possible railway networks on n (fixed) towns so that (i) the least number of railway segments is used, yet (ii) travel remains possible between any two towns. The module continues with Otakar Borůvka’s 1926 solution to finding the most economical way to connect n towns in a rural region to an electrical network. He devised an ingenious algorithm to solve this problem by connecting each town to its nearest neighbor, and then iterating the algorithm on connected components until a connected graph (tree) resulted [4, 5, 10]. After working through these specific applied problems of Prüfer and Borůvka, students are asked to reflect on how the modern definition of a tree captures the proprieties sought by both authors, and what lemmas or theorems have evolved from these historical sources.
Classroom implementation of this and other PSPs is extremely flexible, thereby enhancing their adaptability to a wide variety of institutional settings. Although they work best by utilizing a combination of in-class activities and out-of-class homework, PSPs can be completed individually or in small groups, or assigned as a one-to-two week assignment requiring a written paper addressing all the exercises in the PSP. Other instructors assign a PSP in parts over a four-to-five week time period, and ask students to submit written answers to each exercise in installments. The written portion of a PSP allows students to react to the historical sources, organize their thoughts through mathematical exposition, and rediscover groundbreaking ideas for themselves.
Faculty interested in finding a wide collection of PSPs are encouraged to consult [1, 2, 3].
References
[1] J. Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D. Pengelley., I. Pivinka, and D. Ranjan, Learning Discrete Mathematics and Computer Science via Primary Historical Sources, http://www.cs.nmsu.edu/historical-projects, accessed on January 4, 2015.
[2] Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D Pengelley and D. Ranjan, Teaching discrete mathematics via primary historical sources, http://www.math.nmsu.edu/hist_projects, accessed on January 4, 2015.
[3] 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, http://www.maa.org/publications/periodicals/convergence/primary-historical-sources-in-the-classroom-discrete-mathematics-and-computer-science, accessed on January 18, 2015.
[4] O. Borůvka, O jistém problému minimálnim, (On a Certain Minimal Problem), Práce Moravské Přdovědecké Spolecnosti v Brně 3 (1926), 37–58.
[5] ______ , Přispěvek k řěsení oťazky ekonomické stavby elektrovodnich sítí (A Contribution to the Solution of a Problem on the Economical Construction of Power Networks), Elecktronický obzor 15 (1926), 153–154.
[6] R. Calinger, Classics of Mathematics, 2nd ed., Prentice-Hall, Engelwood Cliffs, New Jersey, 1995.
[7] A. Cayley, A Theorem on Trees, Quarterly Journal of Pure and Applied Mathematics 23 (1889), 376–378.
[8] L. Euler, The works of Leonhard Euler online, Available at http://eulerarchive.maa.org, accessed on January 4, 2015.
[9] M. Fried, Can Mathematics Education and History of Mathematics Coexist?, Science & Education 10 (2001), 391–408.
[10] R. L. Graham and P. Hell, On the History of the Minimum Spanning Tree Problem, Annals of the History of Computing 7 (1985), 43–57.
[11] M. de Guzmán, Enseñanza de las ciencias y la mathemática, Revista Iberoamericana de Education 043 (2007), 19–58.
[12] H. N. Jahnke, The use of original sources in the mathematics classroom, History in mathematics education: the ICMI study (Fauvel, J. and van Maanen, J., ed.), Kluwer Academic, Dordrecht, 2002, 291–328.
[13] U. T. Jankvist, The use of original sources and its possible relation to the recruitment problem, Proceedings of the eighth congress for the European society for research in mathematics education (B. Ubuz, Ç. Haser, and M. A. Mariotti, eds.), Middle East Technical University, Ankara, Turkey, 2013, 1900–1999.
[14] _______, On the Use of Primary Sources in the Teaching and Learning of Mathematics, International Handbook of Research in History, Philosophy and Science Teaching, Springer Verlag, New York, 2014, Matthews, M. (editor).
[15] V. Katz, A History of Mathematics: An Introduction, 2nd ed., Addison-Wesley, New York,1998.
[16] T. H. Kjeldsen and M. Blomhøj, Beyond motivation: history as a method for learning meta- discursive rules in mathematics, Educational Studies in Mathematics 10 (23 September 2011), DOI 10.1007/s 10649–011–9352–z.
[17] 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.
[18] R. Laubenbacher, D. Pengelley, and M. Siddoway, Recovering Motivation in Mathematics: Teaching with Original Sources, UME Trends 6, DOI 10.1007/s 11191–012–9470–8, Available at the website http://www.math.nmsu.edu/~history/ume.html, accessed January 4, 2015.
[19] J. Lodder, Networks and spanning trees, 32 page curricular module available at [1].
[20] J. Narum, Promising Practices in Undergraduate Science, Technology, Engineering, and Mathematics (STEM) Education, Evidence on Promising Practices in Undergraduate STEM Education – Commissioned Workshop Papers, National Academy of Sciences, June 30, 2008, Available at the website http://sites.nationalacademies.org/dbasse/bose/dbasse_080106, accessed January 4, 2015.
[21] H. Prüfer, Neuer Beweis eines Satzes über Permutationen, Archiv der Mathematik und Physik 3 (1918), no. 27, 142–144.
[22] N. Scoville, Georg Cantor at the Dawn of Point-Set Topology, Loci: Convergence (March 2012), DOI: 10.4169/loci003861.
[23] A. Sfard, Thinking as communicating: Human Development, the Growth of Discourse, and Mathematizing, Cambridge University Press, New York, 2008.