Primary Source Projects and Reading Apprenticeship in Mathematics History: Using Primary Sources to Teach Mathematics — Context, History, and Literature

Using Benjamin Wardhaugh’s terminology [2010], a primary source is created “at the time.” Primary sources are our only direct sources of information about the historical period. They may be artifacts other than written texts, such as photographs or sound recordings. Wardhaugh gives Isaac Newton’s Principia Mathematica as an example of a primary source. Although modern readers will typically read a translation of Newton’s Latin original, reading Principia in its original language gives the reader the opportunity to read Newton’s ideas as he presented them. Wardhaugh also describes secondary sources, which are based on primary sources and are one step away from the historical events they tell us about. Examples include biographies or writings that directly cite the original artifacts. Finally, there are tertiary sources, which include most encyclopedia articles, websites, and books aimed at a wide audience. Most modern textbooks and educational materials are secondary or tertiary sources.

Figure 2. Sample pages from the primary source Principia Mathematica. Convergence Mathematical Treasures.

Primary sources require the reader to slow down and carefully consider not only what writers knew, but how they knew it and why they understood what they knew the way that their presentations suggest they did. Wardhaugh writes

reading historical mathematics can also be hard. . . . The sources for historical mathematics were written in times and places that are mostly unfamiliar to us, by people whose ideas and values were very different from our own and whose mathematical culture, thoughts, and assumptions may have been very different from anything we are familiar with [Wardhaugh 2010, p. ix].

This inherent challenge in reading historical mathematics creates an opportunity to engage students as readers. Each discipline has its own conventions for reading—for example, most people read a mathematics text very differently than they do a magazine article. Yet, these techniques for disciplinary reading are often not explicitly taught to students. This can lead to frustration from both students and instructors. Utilizing primary sources in a classroom setting gives instructors the opportunity to include explicit instruction on reading practices and conventions in mathematics. This connects, in turn, to the utilization of RA routines (as described in detail in a later section), since these routines provide effective strategies for such instruction.

For centuries, students learned geometry via a primary source (often in translation), namely, Euclid’s Elements of Geometry. In his writings, Abraham Lincoln reflected: 

At last I said, Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies [quoted in Ketcham 1901, pp. 64–65].

Figure 3. Proposition from Book III of W. E. Dean’s American edition of John Playfair’s
Elements of Geometry, a widely-used adaptation of Euclid’s Elements. Internet Archive.

Lincoln studied Euclid’s Elements to learn how to construct a logical argument in his profession as a lawyer; indeed, he would have chosen it for self-study since he knew that Elements was commonly taught in British and American colleges in the 18th and 19th centuries precisely because it was seen as a model of clear reasoning that was useful for training ministers, politicians, and lawyers. (For more on this story, see [Roberts 2019, pp. 32–35] and [LaFantasie 2020].) However, Lincoln’s experience is not typical of 21st-century students. Although students in today's secondary school geometry classes are learning from a long history and tradition dating back to Euclid, they typically do not read the Elements itself as part of a standard course of study. Proponents of teaching mathematics with primary sources suggest an opportunity has been missed when students are not asked to engage with original texts. Reinhard Laubenbacher and David Pengelley write:

Stimulating problems are at the heart of many great advances in mathematics. In fact, whole subjects owe their existence to a single problem which resisted solution. Nevertheless, we tend to present only polished theories, devoid of both the motivating problems and the long road to their solution. As a consequence, we deprive our students of both an example of the process by which mathematics is created and of the central problems which fueled its development [Laubenbacher and Pengelley 1992, p. 313].

The inclusion of history of mathematics in a mathematics classroom allows students to engage with the process by which mathematics has been created and invites students to think about the intersection of mathematics with the cultures that have formed it [Bidwell 1993]. The use of the history of mathematics in teaching can also help increase motivation for learning, make mathematics less frightening, and change perceptions of mathematics [Fauvel 1991]. And, teaching mathematical content and history through the use of primary historical sources offers an opportunity for students to experience the “messy” processes by which original mathematics continues to be developed today.