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.

The reading of primary sources offers students and instructors a unique opportunity to grapple with original mathematical content. Student surveys in courses taught with primary sources have indicated that students engaged deeply with the content at hand and enjoyed the process. Instructors reflected that student behavior and writing on project work verified these comments [Barnett, Lodder, and Pengelley 2016]. Consistent implementation of teaching with primary sources over time allows researchers to further examine students’ experiences [Clark et al. 2022].

However, this approach can also be quite challenging for students and instructors. As Janet Heine Barnett, an author of many PSPs who has extensive experience with PSP implementation, explains, most undergraduate students will not have the mathematical experience and maturity necessary to interpret an original source without expert guidance [Barnett 2014]. Instructors therefore provide reading routines and instructional scaffolding to explicitly develop these skills. A typical PSP includes both of these components: carefully selected original excerpts and instructor-provided questions, as described in the introduction.

While these support materials are extremely helpful, further instruction and routines may be needed for students to fully engage with a PSP. Assigning a section of a PSP for pre-reading and in-class discussion may not yield the rich discussion an instructor envisions if the students have not carefully considered the original text excerpts and worked through the guided reading prompts. This, then, is the domain of Reading Apprenticeship—the use of routines and norms for participation to facilitate student engagement with primary historical materials.

Figure 4. Sample primary source excerpt and instructor-provided question from the PSP [Barnett 2017].

The Reading Apprenticeship (RA) framework is an excellent tool to further support and scaffold explicit instruction in reading historical mathematics. In Reading for Understanding [2012], authors Ruth Schoenbach, Cynthia Greenleaf, and Lynn Murphy describe four key dimensions of classroom life that are necessary to foster reading development: social, personal, cognitive, and knowledge-building. At the center of these four dimensions is metacognitive conversation or, to put it simply, “thinking about thinking.” The framework includes a variety of versatile techniques, or “routines,” intended to engage students as readers and to develop discipline-specific reading skills.

Figure 5. Reading Apprenticeship Framework, adapted from WestEd 2023.

Reading Apprenticeship was first developed in the 1990s, when a group of secondary teachers and researchers in San Francisco began working together to find new ways to deepen student learning. They found that there were many interventions to help elementary-aged students develop reading skills, but few methods or resources existed for older students or for content-specific areas. The San Francisco group thus developed the RA framework from objectives and approaches that were supported by educational research and that addressed both academic and social-emotional learning [WestEd 2023]. Although RA was originally developed for use in secondary schools, it has been adapted to all educational levels, including universities. Currently, WestEd, a nonprofit educational research agency, supports the RA community and has facilitated implementation at scale as well as evaluation projects [WestEd 2023]. A three-year evaluation for the California Community Colleges reported promising impacts of RA in STEM courses, including reductions in equity gaps [Edmunds 2017].

One of the core tenets of RA is its adaptability to different contexts. Common routines that can be used across disciplines and student levels include: discussion logs, think-alouds, personal reading histories, talking to the text, and metacognitive logs. Guides for these routines are provided in WestEd’s two RA texts: Reading for Understanding: How Reading Apprenticeship Improves Disciplinary Learning in Secondary and College Classrooms [Schoenbach et al. 2012] and Leading for Literacy: A Reading Apprenticeship Approach [Schoenbach et al. 2016]. A well-written PSP gives mathematical and historical content together in a guided reading approach. Overlaying RA routines on a PSP’s written materials can provide students with concrete mechanisms for discerning mathematical and historical meaning from the primary source excerpts in a PSP. In other words, the role of RA routines is to facilitate students’ engagement with the PSP materials and with one another during classroom discussion.

We implemented PSPs alongside RA routines during three Fall semesters (2021, 2022, 2023) in a junior-level college history of mathematics course at California State University, Monterey Bay, which is a public undergraduate institution with about 7,000 undergraduate students. The demographics of the university are such that 50% of the students are first-generation college students, 50% belong to an underrepresented racial or ethnic group, and 30% are considered low-income [Enrollment Fast Facts n.d.].

History of Mathematics is a required course for mathematics majors who plan to earn a single-subject teaching credential to teach mathematics at the high school level, and it is an elective course for mathematics and statistics majors in other concentrations. There are usually between 20 and 30 students in the course each semester. The course utilizes William Dunham’s text Journey Through Genius [1991] and surveys the development of mathematical topics throughout history. PSPs are used alongside Dunham’s text to include contributions from non-Western mathematicians and to explore additional ideas outside of the scope of the main text.

The next three sections of this paper present sample RA routines and the ways that they facilitated student reading of the PSP materials, as well as sample student work and survey responses from the Fall 2022 offering of the course.

In the mini-PSP Completing the Square: From the Roots of Algebra, by Daniel E. Otero [2019], students explore the rhetorical algebra approach to completing the square used by Muḥammad ibn Mūsā al-Khwārizmī (ca 780–850 CE). They are asked to make connections between modern approaches to solving quadratic equations algebraically (by using the quadratic formula) and al-Khwārizmī’s procedures for solving particular forms of quadratic equations, which he referred to as “compound species.” Figure 6 below shows a page from al-Khwārizmī’s algebra text, Kitāb al-jabr wa’l-muqābala. On this page is the proof of the rule that he gave for solving a quadratic equation of the form “squares and roots are equal to numbers.”

Figure 6. Excerpt from al-Khwārizmī’s Kitāb al-jabr wa’l-muqābala.
Convergence Mathematical Treasures.

In Tasks 3 and 4 in the PSP, students are given an excerpt from the original text and directed to formulate their problem-solving steps using symbolic algebra. We facilitated these tasks using an RA routine called “Talk to the Text.” In this routine, while they read students write comments about their reactions to the text, including predicting, questioning, clarifying, summarizing, and making connections (see downloadable resource for Talking to the Text Inquiry). This routine can also be done verbally as a “Think-Aloud.” The purpose of the routine is to write or say the thinking that we do, but typically do not record or verbalize, while reading. Figure 7 is a sample response for the Talk to the Text from Task 3.

Figure 7. Talk to the Text Sample.

A closer look at the content of Figure 7 illustrates this RA routine. The student first recognized that the phrase “Roots and Squares are equal to Numbers” is equivalent to the modern algebraic equation \(ax + bx^2 = c\). This is further reinforced by the student’s recognition of “one square, and ten roots of the same, amount to thirty-nine dirhams” as equivalent to the equation \[x^2 + 10x = 39,\] as seen in the student’s first note.

The student then traced the solution but not without asking the question, “Why do we call it the remainder?” In the student’s second note they asked if the number 5 (which appears as the result of half of 10) “is . . . related to \(\left(\frac{b}{2a}\right)^2\) from [the] algebraic approach.” Although the student did not pursue the answer to the question, student notes (1) and (2) both indicate a deeper level of thinking than might be accomplished with a purely symbolic approach. By asking the question, the student was demonstrating the ability to make connections, which is one of the aspects that the task was designed to support.

Overall, the student’s work showed evidence of engaging with the text in a way that was meaningful and informative. Following this, the student had no trouble completing Task 4 from the PSP, in which the student translated al-Khwārizmī’s steps into modern algebraic expressions.

In Janet Heine Barnett’s series of three mini-PSPs Gaussian Guesswork, students learn about results from Carl Friedrich Gauss’s (1777–1855) mathematical diary, including how infinite sequences are used to define the arithmetic-geometric mean. These PSPs are an excellent opportunity for students to read and make sense of Gauss’s original writing under the constraints of a short timeline and without study of advanced prerequisite material. We chose to use Gaussian Guesswork: The Arithmetic-Geometric Mean in this course.

We used a metacognitive reading log, which is an RA tool to help students make their own thinking explicit. Typically, a reading log prompts students to record important information and ideas in the first column and to write their thoughts, feelings, or questions in the second column (see downloadable resource for Metacognitive Reading Log). Because we anticipated students having prior knowledge of means, we used a three-column log in which students described their understanding prior to reading, their understanding after reading, and how they came to understanding. A sample student response is given in Table 1 below.

Table 1. Sample student response, metacognitive reading log for Gaussian Guesswork: The Arithmetic-Geometric Mean.

The student work sample presented in Table 1 gives insight as to what the student was thinking as she read. As expected, the student had a good understanding of the arithmetic mean. She noted that she has worked with it in other classes and accurately described how to calculate it. In the second row, she noted initially that she had encountered the geometric mean, but was not as clear on how it works (typically, about half of the students in this course will recognize the geometric mean and write a brief explanation in this first column). In the second column, the student accurately described how to calculate the geometric mean after reading. She noted in the final column that the reading helped her to remember more about it. This progression gives insight into the student’s prior knowledge and how well she has processed the information from the reading.

In the final row, the student noted that she had not previously encountered the arithmetic-geometric mean (AGM), which is typical for students in this mathematics history course. She gave a thoughtful (but incorrect) guess for how it is defined before reading. In the second column in this final row, she correctly described how to calculate the AGM after reading. She referenced the specific example from Gauss’s text of finding \(M(\sqrt{2},1)\). This is a particularly important example for this PSP, as the final section deals with Gauss’s conclusion:

Figure 8. Primary source excerpt from the PSP [Barnett 2017, p. 8].

Figure 9. Same excerpt in original Latin from facsimile copy of Gauss’ original diary
(from Volume X.1 of Gauss’ Werke).

Similar to the example given in Routine #1, the student’s work on this RA routine shows that she was prepared to tackle the more challenging tasks in the PSP. In general, this routine also draws on the students’ previous experiences as they acquire new knowledge through reading.

In Kenneth Monks’s Bhāskara’s Approximation to and Mādhava’s Series for Sine, students refine several approximations for the sine function to eventually obtain Bhāskara I’s (ca 600–680 CE) approximation formula for the sine function. They then translate Mādhava’s (1350–1425) sine series into modern notation and compare the accuracy of these methods. The tasks in this PSP require students to write solutions to mathematical problems and to explain their thinking in words and in symbols. Students also construct graphs and other diagrams as appropriate. This PSP was assigned as a course writing project at the end of the semester. Students wrote a full draft, engaged in peer review, and then wrote a final draft for submission.

One of the goals of consistently using RA routines is that students explain their thinking without being asked. In the sample response below, the student’s first draft included such an explanation of his reasoning, using a similar approach to a metacognitive reading log. Previously, students had been provided with the log and asked to respond in that format. In this instance, the student utilized the approach without being prompted.

Task 2 of the PSP prompts the student to translate the following excerpt into modern symbolic algebra.

Figure 10. Primary source excerpt giving Bhāskara's description of his sine approximation [Monks 2020, p. 3].

Figure 11. Same excerpt in original Sanskrit [Gupta 1967, p. 122].

The students’ response to this translation task was as follows.

Table 2. Sample first draft student response to translating Bhāskara’s approximation.

The student adapted the three-column log to draft his initial translation. The first column is his algebraic expression, which addresses the task prompt. The second column is Bhāskara’s text, “chunked” into one mathematical operation at a time. The third column is his explanation of “how I know,” similar to the final column in the three-column log from the Gaussian Guesswork project. His explanation of the final step suggests that he was still unsure about the meaning of the final sentence, but he gave his reasoning for translating it as he did.

The student was well prepared to complete the next task in the PSP because of his RA approach to the initial draft. In the next task, students need to algebraically manipulate their expression and convert to radians to obtain Bhāskara’s approximation that is used throughout the first half of the PSP. Students may find that their translation is incorrect and that they are thus unable to derive the given expression in radians. While this is part of the iterative process of problem solving in this PSP, it can lead to frustration if the student is not able to articulate how and why they have given their initial translation (and how it might need to be revised). The sample student’s response gives a clear insight into each step of his thought process and supports think-aloud problem solving with peers as they continue refining their translations.

At the end of the semester, students in the math history course that included the PSPs and RA routines described above were asked to reflect on their experiences in writing. Students were asked, “In your opinion, what are some benefits and challenges of learning about math history via primary sources?” Selected responses are given below with emphasis added.

I believe a challenge of learning math history via primary’s sources is how difficult it can sometimes be to translate and understand the wording of an older problem. As a benefit of using PSP’s would be gaining a better understanding and the origin of mathematical terms that one would use.

The biggest challenge was interpreting the language given in the primary source. Reading the source by itself you'd not know what some of the terms mean without outside explanation. The benefit would be learning the historical context in which the math concepts originated and puts the history in math history.

I really enjoyed reading the PSPs as it provided a lot of historical context and insight into some of the differences in the problem solving process and mathematical writing style employed in the past. Translating the writing into modern terms was a bit challenging at times, but I don't think this detracted from the process.

It’s challenging to get through the information since it’s written in a way that makes it difficult to process and understand but working with a group and along with the rest of the class makes it possible to complete.

These responses reflect the personal, cognitive, social, and knowledge-building dimensions that form the RA framework. However, the conclusions that can be drawn from these responses are limited. Students did not reflect explicitly on their experiences with making sense of a PSP using RA routines, nor on their use of RA routines otherwise in the course. Collecting these data would greatly strengthen the claim that RA routines are an effective strategy for facilitating PSPs in the classroom.

A set of quantitative survey items around these four dimensions was developed for the course involved in this study. Figure 5 describes specific skills associated with each of these four dimensions; the questionnaire items were developed using this language to specifically ask students about their experiences in using RA routines to facilitate learning with PSPs. While there were not a sufficient number of student responses to report aggregate findings, the survey is provided in Appendix I. Readers are encouraged to adapt the questionnaire for their own needs.

PSPs are a useful, classroom-ready resource for students and instructors to engage in learning mathematics via primary sources. Yet, students may still struggle to engage with the text, especially when they are still developing their own ability to read and comprehend written mathematics. Reading Apprenticeship routines can support students by providing the “missing piece” of explicitly teaching reading strategies, routines, and practices that are needed to make sense of a PSP. Used in concert, RA routines and PSPs can support student learning and engagement.

The RA routines presented in this paper are the “Talk to the Text” routine and metacognitive reading logs. These are introductory routines for instructors that can be used in a wide variety of settings. In the PSP tasks presented, the RA routines were mostly used to make sense of the original historical language in the PSPs before working through the guided reading tasks in the PSPs. These RA routines were particularly helpful for students when making sense of the transition from rhetorical to symbolic algebra, as in al-Khwārizmī’s and Bhāskara I’s work in the first and third examples, respectively. The second and third examples both utilize a metacognitive reading log approach, but with the difference that in the third example, the student is using the routine without being prompted by the instructor.

A brief description of additional RA routines that can be adapted to a variety of contexts, including PSPs, is given in Table 3 below. These routines are taken from Reading for Understanding: How Reading Apprenticeship Improves Disciplinary Learning in Secondary and College Classrooms [Schoenbach, Greenleaf, and Murphy 2016]. The sample student work and questionnaire responses presented in this article suggest that RA routines such as these and PSPs complement one another when used in the classroom.

Table 3. Additional sample RA routines.

The RA examples presented here may help students be more effective and open to exploring primary sources as well as perhaps to seeing the value in doing so. Using RA as a tool for students to access primary historical sources in mathematics gives students an explicit framework for making sense of mathematical texts that they can also apply to modern textbooks. Using PSPs and RAs together is a timely strategy as the move towards “alternative” assessment methods that promote equity and inclusion gains momentum and support in our learning institutions. Such assessment methods are more likely to be adapted by instructors if they are both effective for student learning and manageable for the instructor to learn about and implement. Readers are encouraged to download the PSPs and RA routines presented in this article and to adapt these tools for their own classrooms. The samples of student work may offer insight into how the methods could be applied in the classroom and what instructors might expect when planning their own courses. 

The first ten questions in this survey are Likert scale items, with the response options Strongly Disagree, Disagree, Agree, and Strongly Agree. The RA skills and dimensions are taken from the RA Framework depicted in Figure 5.

Barnett, Janet Heine. 2014. Learning Mathematics via Primary Historical Sources: Straight from the Source’s Mouth. PRIMUS 24(8): 722–36.

Barnett, Janet Heine. 2017. Gaussian Guesswork: Infinite Sequences and the Arithmetic Geometric Mean. Calculus, 2. https://digitalcommons.ursinus.edu/triumphs_calculus/2. Reprinted in Gaussian Guesswork: Three Mini-Primary Source Projects for Calculus 2 Students, Convergence 18(November 2021).

Barnett, Janet Heine, Jerry Lodder, and David Pengelley. 2016. Teaching and Learning Mathematics from Primary Historical Sources. PRIMUS 21(1): 1–18.

Bidwell, James K. 1993. Humanize Your Classroom with the History of Mathematics. The Mathematics Teacher 86(6): 461–64.

California State University Monterey Bay. n.d. Enrollment Fast Facts for 2023 Spring.

Clark, Kathleen Michelle, Cihan Can, Janet Heine Barnett, Mark Watford, and Oksana Melissa Rubis. 2022. Tales of Research Initiatitives on University-Level Mathematics and Primary Historical Sources. ZDM — Mathematics Education. 52: 1507–20.

Dunham, William. 1991. Journey Through Genius: The Great Theorems of Mathematics. London: Penguin Books.

Edmunds, Kimberly. 2017. Adopting Reading Apprenticeship at Pasadena City College. Equal Measure STEM Active Learning Vignette series.

Fauvel, John. 1991. Using History in Mathematics Education. For the Learning of Mathematics 11(2): 3–6. 

Gupta, Radha Charan. 1967. Bhāskara I’s Approximation to Sine. Indian Journal of History of Science 2(2): 121–136.

Ketcham, Henry. 1901. The Life of Abraham Lincoln. New York: J.A. Hill & Co.

LaFantasie, Glenn W. 2020. Lincoln, Euclid, and the Satisfaction of Success. Journal of the Abraham Lincoln Association 41(1): 24–46.

Laubenbacher, Reinhard C., and David J. Pengelley. 1992. Great Problems of Mathematics: A Course Based on Original Sources. American Mathematical Monthly 99(4): 313-17.

Monks, Kenneth M. 2020. Bhāskara's Approximation to and Mādhava's Series for Sine. Calculus, 15. https://digitalcommons.ursinus.edu/triumphs_calculus/15. Reprinted in Bhāskara’s Approximation to and Mādhava’s Series for Sine: A Mini-Primary Source Project for Calculus 2 Students, Convergence 18(May 2021).

Otero, Danny. 2019. Completing the Square: From the Roots of Algebra. Pre-calculus and Trigonometry, 4. https://digitalcommons.ursinus.edu/triumphs_precalc/4. Reprinted in Completing the Square: From the Roots of Algebra, A Mini-Primary Source Project for Students of Algebra and Their Teachers, Convergence 16(September 2019).

Roberts, David Lindsay. 2019. Republic of Numbers: Unexpected Stories of Mathematical Americans Through History. Baltimore: Johns Hopkins University Press.

Schoenbach, Ruth, Cynthia Greenleaf, and Lynn Murphy. 2012. Reading for Understanding: How Reading Apprenticeship Improves Disciplinary Learning in Secondary and College Classrooms. 2nd edition. San Francisco: Jossey-Bass.

Schoenbach, Ruth, Cynthia Greenleaf, and Lynn Murphy. 2016. Leading for Literacy: A Reading Apprenticeship Approach. San Francisco: Jossey-Bass.

Swetz, Frank J., and Victor J. Katz. 2011. Mathematical Treasures: al-Khwārizmī’s Algebra. Convergence 8(January).

TRIUMPHS. 2015. Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources. https://blogs.ursinus.edu/triumphs/.

Wardhaugh, Benjamin. 2010. How to Read Historical Mathematics. Princeton and Oxford: Princeton University Press.

WestEd Reading Apprenticeship. 2023. https://readingapprenticeship.org/about-us/.

Acknowledgements

The author wishes to thank the TRIUMPHS team for their support, including Janet Heine Barnett, Danny Otero, and Ken Monks, who authored the three PSPs described in this article. She also thanks Nelson Graff and Rebecca Kersnar in Teaching, Learning, and Assessment at California State University, Monterey Bay, for their mentoring and instruction in Reading Apprenticeship. Finally, she would like to acknowledge her father, Martin Bonsangue, who is the “primary source” of her interest in mathematics history.

About the Author

Jennifer Clinkenbeard is an associate professor in the Department of Mathematics and Statistics at California State University, Monterey Bay. For her, mathematics history is at the intersection of teaching, learning, research, and hobby. Her research includes work on teaching strategies that promote equity and inclusion as well as applying quantitative methods to evaluate their effectiveness. She is excited to work more with primary sources in the classroom and with the new TRIUMPHS Society.